Post on 07-Feb-2020
Chapter 6 Velocity in Mechanisms
(Instantaneous Centre
Method)
9282013
Dr Mohammad Abuhaiba PE 1
61 Introduction The combined motion of rotation and translation of link AB may be assumed to be a motion of pure rotation about some center I known as the instantaneous center of rotation (also called centro or virtual centre)
9282013
Dr Mohammad Abuhaiba PE
2
61 Introduction
Instantaneous center of a moving body
center which keeps changing from one instant to another
Centrode locus of all such ICs
instantaneous axis A line drawn through
an IC and perpendicular to plane of
motion Locus of this axis is known as
axode
9282013
Dr Mohammad Abuhaiba PE
3
62 Space and Body Centrodes
IC a point in the body which may be considered
fixed at any particular moment
Space centrode locus of IC in space during a
definite motion of the body
Body centrode locus of IC relative to body itself
The two centrodes have the IC as a common point
at any instant and during the motion of the body
Body centrode rolls without slipping over space
centrode
9282013
Dr Mohammad Abuhaiba PE
4
62 Space amp
Body Centrodes
I1 and I2 Ics for two different positions A1B1 and A2B2 of link A1B1
Space centrode number of positions of link A1B1 are considered and a curve is drawn passing through these ICs (I1 I2)
9282013
Dr Mohammad Abuhaiba PE
5
62 Space amp Body
Centrodes
C1 is attached to body or link A1B1 and moves with it in such a way that C1 coincides with I1 when the body is in position A1B1
C2 position of point C1 when link A1B1 occupies position A2B2
C2 will coincide with I2 (when link is in position A2B2) only if triangles A1B1C1 and A2B2C2 are identical
9282013
Dr Mohammad Abuhaiba PE
6
63 Methods for Determining the
Velocity of a Point on a Link
1 Instantaneous center method
2 Relative velocity method
9282013
Dr Mohammad Abuhaiba PE
7
64 Velocity of a Point on a Link
by Instantaneous Centre Method
vA is known in magnitude amp
direction amp vB in direction
9282013
Dr Mohammad Abuhaiba PE
8
66 Number of Instantaneous
Centers in a Mechanism
of ICs in a constrained kinematic chain
= of possible combinations of two links
of pairs of links or of ICs is the number
of combinations of n links taken two at a
time
Mathematically of ICs
9282013
Dr Mohammad Abuhaiba PE
9
67 Types of Instantaneous Centers
a primary instantaneous centers
1 Fixed instantaneous centers
2 Permanent instantaneous centers
b secondary instantaneous centers
3 Neither fixed nor permanent
instantaneous centers
9282013
Dr Mohammad Abuhaiba PE
10
67 Types of Instantaneous Centers
Fixed Instantaneous Centersgt I12 amp I14
remain in same place
for all configurations
of mechanism
9282013
Dr Mohammad Abuhaiba PE
11
67 Types of Instantaneous
Centers
I23 amp I34 permanent
instantaneous centers
they move when
mechanism moves
but the joints are of
permanent nature
I13 amp I24 neither fixed
nor permanent
instantaneous centers they vary with
configuration of
mechanism
9282013
Dr Mohammad Abuhaiba PE
12
68 Location of Instantaneous Centers
9282013
Dr Mohammad Abuhaiba PE
13
69 Aronhold Kennedy (Three
Centers in Line) Theorem if three bodies move
relatively to each other they have three ICs that lie on a straight line
Consider three kinematic links A B and C having relative plane motion
Number of ICs (N) is given by
9282013
Dr Mohammad Abuhaiba PE
14
69 Aronhold Kennedy (Three
Centers in Line) Theorem The two ICs at the pin joints of B with
A and C with A (Iab and Iac) are permanent instantaneous centers
According to Aronhold Kennedyrsquos theorem the third IC Ibc must lie on the line joining Iab and Iac
let us consider that instantaneous center Ibc lies outside the line joining Iab and Iac
point Ibc belongs to both links B and C Let us consider point Ibc on link B Its velocity vBC must be perpendicular to the line joining Iab and Ibc Now consider the point Ibc on link C Its velocity vBC must be perpendicular to the line joining Iac and Ibc
9282013
Dr Mohammad Abuhaiba PE
15
69 Aronhold Kennedy (or
Three Centers in Line) Theorem velocity of IC is same whether it is
regarded as a point on first link or as a point on second link
Therefore velocity of point Ibc cannot be perpendicular to both lines Iab Ibc and IacIbc unless the point Ibc lies on the line joining the points Iab and Iac
Thus the three instantaneous centers (Iab Iac and Ibc) must lie on the same straight line
The exact location of Ibc on line Iab Iac depends upon the directions and magnitudes of the angular velocities of B and C relative to A
9282013
Dr Mohammad Abuhaiba PE
16
610 Method of Locating Instantaneous
Centers in a Mechanism
Fig 68 (a)
1 Determine number of ICs (N)
2 Make a list of all instantaneous centers
9282013
Dr Mohammad Abuhaiba PE
17
Links 1 2 3 4
instantaneous
centers
12 23 34
13 24
14
610 Method of Locating Instantaneous
Centers in a Mechanism
3 Locate fixed amp permanent ICs by
inspection
I12 and I14 are fixed Ics
I23 and I34 are permanent IC
The four bar mechanism has
four turning pairs
four primary (fixed and permanent) ICs located at
the centers of the pin joints
9282013
Dr Mohammad Abuhaiba PE
18
610 Method of Locating Instantaneous
Centers in a Mechanism
4 Locate remaining neither fixed nor
permanent ICs (secondary centers) by
Kennedyrsquos theorem
Circle diagram Fig 68 (b)
Mark points on a circle equal to number
of links in a mechanism
5 Join points by solid lines to show that
these centers are already found
9282013
Dr Mohammad Abuhaiba PE
19
610 Method of Locating Instantaneous
Centers in a Mechanism
9282013
Dr Mohammad Abuhaiba PE
20
Example 61 In a pin jointed four bar mechanism Fig 69 AB = 300 mm BC = CD = 360 mm and AD = 600 mm The angle BAD = 60deg The crank AB rotates uniformly at 100 rpm Locate all instantaneous centers and find the angular velocity of link BC
9282013
Dr Mohammad Abuhaiba PE
21
Example 61
9282013
Dr Mohammad Abuhaiba PE
22
Example 62 Locate all ICs of the slider crank mechanism shown in Fig 612 The lengths of crank OB and connecting rod AB are 100 mm and 400 mm respectively If the crank rotates cw with an angular velocity of 10 rads find
1 Velocity of the slider A
2 Angular velocity of connecting rod AB
9282013
Dr Mohammad Abuhaiba PE
23
Example 62
9282013
Dr Mohammad Abuhaiba PE
24
Example 63
A mechanism Fig 615 has the following dimensions
OA = 200 mm AB = 1500 mm BC = 600 mm CD = 500
mm and BE = 400 mm Locate all the ICs If crank OA
rotates uniformly at 120 rpm cw find
1 velocity of B C and D
2 angular velocity of links AB BC and CD
9282013
Dr Mohammad Abuhaiba PE
25
Example 63 9282013
Dr Mohammad Abuhaiba PE
26
Example 64 The mechanism of a wrapping machine Fig 618 has the following dimensions O1A = 100 mm AC = 700 mm BC = 200 mm O3C = 200 mm O2E = 400 mm O2D = 200 mm and BD = 150 mm The crank O1A rotates at a uniform speed of 100 rads Find the velocity of point E of the bell crank lever by instantaneous center method
9282013
Dr Mohammad Abuhaiba PE
27
Example 64
9282013
Dr Mohammad Abuhaiba PE
28
Example 65 Fig 621 shows a sewing needle bar mechanism O1ABO2CD wherein the different dimensions are as follows Crank O1A = 16 mm b = 45deg Vertical distance between O1 and O2 = 40 mm Horizontal distance between O1 amp O2 = 13 mm O2B = 23 mm AB = 35 mm O2BC = 90deg BC = 16 mm CD = 40 mm D lies vertically below O1 Find the velocity of needle at D for the given configuration The crank O1A rotates at 400 rpm
9282013
Dr Mohammad Abuhaiba PE
29
Example 65
9282013
Dr Mohammad Abuhaiba PE
30
Example 66
Fig 624 shows a Whitworth
quick return motion
mechanism The various
dimensions in the mechanism
are as follows OQ = 100 mm
OA = 200 mm QC = 150 mm
and CD = 500 mm The crank
OA makes an angle of 60deg
with the vertical and rotates at
120 rpm cw Locate all ICs
and find the velocity of ram D
9282013
Dr Mohammad Abuhaiba PE
31
Example 66
9282013
Dr Mohammad Abuhaiba PE
32
61 Introduction The combined motion of rotation and translation of link AB may be assumed to be a motion of pure rotation about some center I known as the instantaneous center of rotation (also called centro or virtual centre)
9282013
Dr Mohammad Abuhaiba PE
2
61 Introduction
Instantaneous center of a moving body
center which keeps changing from one instant to another
Centrode locus of all such ICs
instantaneous axis A line drawn through
an IC and perpendicular to plane of
motion Locus of this axis is known as
axode
9282013
Dr Mohammad Abuhaiba PE
3
62 Space and Body Centrodes
IC a point in the body which may be considered
fixed at any particular moment
Space centrode locus of IC in space during a
definite motion of the body
Body centrode locus of IC relative to body itself
The two centrodes have the IC as a common point
at any instant and during the motion of the body
Body centrode rolls without slipping over space
centrode
9282013
Dr Mohammad Abuhaiba PE
4
62 Space amp
Body Centrodes
I1 and I2 Ics for two different positions A1B1 and A2B2 of link A1B1
Space centrode number of positions of link A1B1 are considered and a curve is drawn passing through these ICs (I1 I2)
9282013
Dr Mohammad Abuhaiba PE
5
62 Space amp Body
Centrodes
C1 is attached to body or link A1B1 and moves with it in such a way that C1 coincides with I1 when the body is in position A1B1
C2 position of point C1 when link A1B1 occupies position A2B2
C2 will coincide with I2 (when link is in position A2B2) only if triangles A1B1C1 and A2B2C2 are identical
9282013
Dr Mohammad Abuhaiba PE
6
63 Methods for Determining the
Velocity of a Point on a Link
1 Instantaneous center method
2 Relative velocity method
9282013
Dr Mohammad Abuhaiba PE
7
64 Velocity of a Point on a Link
by Instantaneous Centre Method
vA is known in magnitude amp
direction amp vB in direction
9282013
Dr Mohammad Abuhaiba PE
8
66 Number of Instantaneous
Centers in a Mechanism
of ICs in a constrained kinematic chain
= of possible combinations of two links
of pairs of links or of ICs is the number
of combinations of n links taken two at a
time
Mathematically of ICs
9282013
Dr Mohammad Abuhaiba PE
9
67 Types of Instantaneous Centers
a primary instantaneous centers
1 Fixed instantaneous centers
2 Permanent instantaneous centers
b secondary instantaneous centers
3 Neither fixed nor permanent
instantaneous centers
9282013
Dr Mohammad Abuhaiba PE
10
67 Types of Instantaneous Centers
Fixed Instantaneous Centersgt I12 amp I14
remain in same place
for all configurations
of mechanism
9282013
Dr Mohammad Abuhaiba PE
11
67 Types of Instantaneous
Centers
I23 amp I34 permanent
instantaneous centers
they move when
mechanism moves
but the joints are of
permanent nature
I13 amp I24 neither fixed
nor permanent
instantaneous centers they vary with
configuration of
mechanism
9282013
Dr Mohammad Abuhaiba PE
12
68 Location of Instantaneous Centers
9282013
Dr Mohammad Abuhaiba PE
13
69 Aronhold Kennedy (Three
Centers in Line) Theorem if three bodies move
relatively to each other they have three ICs that lie on a straight line
Consider three kinematic links A B and C having relative plane motion
Number of ICs (N) is given by
9282013
Dr Mohammad Abuhaiba PE
14
69 Aronhold Kennedy (Three
Centers in Line) Theorem The two ICs at the pin joints of B with
A and C with A (Iab and Iac) are permanent instantaneous centers
According to Aronhold Kennedyrsquos theorem the third IC Ibc must lie on the line joining Iab and Iac
let us consider that instantaneous center Ibc lies outside the line joining Iab and Iac
point Ibc belongs to both links B and C Let us consider point Ibc on link B Its velocity vBC must be perpendicular to the line joining Iab and Ibc Now consider the point Ibc on link C Its velocity vBC must be perpendicular to the line joining Iac and Ibc
9282013
Dr Mohammad Abuhaiba PE
15
69 Aronhold Kennedy (or
Three Centers in Line) Theorem velocity of IC is same whether it is
regarded as a point on first link or as a point on second link
Therefore velocity of point Ibc cannot be perpendicular to both lines Iab Ibc and IacIbc unless the point Ibc lies on the line joining the points Iab and Iac
Thus the three instantaneous centers (Iab Iac and Ibc) must lie on the same straight line
The exact location of Ibc on line Iab Iac depends upon the directions and magnitudes of the angular velocities of B and C relative to A
9282013
Dr Mohammad Abuhaiba PE
16
610 Method of Locating Instantaneous
Centers in a Mechanism
Fig 68 (a)
1 Determine number of ICs (N)
2 Make a list of all instantaneous centers
9282013
Dr Mohammad Abuhaiba PE
17
Links 1 2 3 4
instantaneous
centers
12 23 34
13 24
14
610 Method of Locating Instantaneous
Centers in a Mechanism
3 Locate fixed amp permanent ICs by
inspection
I12 and I14 are fixed Ics
I23 and I34 are permanent IC
The four bar mechanism has
four turning pairs
four primary (fixed and permanent) ICs located at
the centers of the pin joints
9282013
Dr Mohammad Abuhaiba PE
18
610 Method of Locating Instantaneous
Centers in a Mechanism
4 Locate remaining neither fixed nor
permanent ICs (secondary centers) by
Kennedyrsquos theorem
Circle diagram Fig 68 (b)
Mark points on a circle equal to number
of links in a mechanism
5 Join points by solid lines to show that
these centers are already found
9282013
Dr Mohammad Abuhaiba PE
19
610 Method of Locating Instantaneous
Centers in a Mechanism
9282013
Dr Mohammad Abuhaiba PE
20
Example 61 In a pin jointed four bar mechanism Fig 69 AB = 300 mm BC = CD = 360 mm and AD = 600 mm The angle BAD = 60deg The crank AB rotates uniformly at 100 rpm Locate all instantaneous centers and find the angular velocity of link BC
9282013
Dr Mohammad Abuhaiba PE
21
Example 61
9282013
Dr Mohammad Abuhaiba PE
22
Example 62 Locate all ICs of the slider crank mechanism shown in Fig 612 The lengths of crank OB and connecting rod AB are 100 mm and 400 mm respectively If the crank rotates cw with an angular velocity of 10 rads find
1 Velocity of the slider A
2 Angular velocity of connecting rod AB
9282013
Dr Mohammad Abuhaiba PE
23
Example 62
9282013
Dr Mohammad Abuhaiba PE
24
Example 63
A mechanism Fig 615 has the following dimensions
OA = 200 mm AB = 1500 mm BC = 600 mm CD = 500
mm and BE = 400 mm Locate all the ICs If crank OA
rotates uniformly at 120 rpm cw find
1 velocity of B C and D
2 angular velocity of links AB BC and CD
9282013
Dr Mohammad Abuhaiba PE
25
Example 63 9282013
Dr Mohammad Abuhaiba PE
26
Example 64 The mechanism of a wrapping machine Fig 618 has the following dimensions O1A = 100 mm AC = 700 mm BC = 200 mm O3C = 200 mm O2E = 400 mm O2D = 200 mm and BD = 150 mm The crank O1A rotates at a uniform speed of 100 rads Find the velocity of point E of the bell crank lever by instantaneous center method
9282013
Dr Mohammad Abuhaiba PE
27
Example 64
9282013
Dr Mohammad Abuhaiba PE
28
Example 65 Fig 621 shows a sewing needle bar mechanism O1ABO2CD wherein the different dimensions are as follows Crank O1A = 16 mm b = 45deg Vertical distance between O1 and O2 = 40 mm Horizontal distance between O1 amp O2 = 13 mm O2B = 23 mm AB = 35 mm O2BC = 90deg BC = 16 mm CD = 40 mm D lies vertically below O1 Find the velocity of needle at D for the given configuration The crank O1A rotates at 400 rpm
9282013
Dr Mohammad Abuhaiba PE
29
Example 65
9282013
Dr Mohammad Abuhaiba PE
30
Example 66
Fig 624 shows a Whitworth
quick return motion
mechanism The various
dimensions in the mechanism
are as follows OQ = 100 mm
OA = 200 mm QC = 150 mm
and CD = 500 mm The crank
OA makes an angle of 60deg
with the vertical and rotates at
120 rpm cw Locate all ICs
and find the velocity of ram D
9282013
Dr Mohammad Abuhaiba PE
31
Example 66
9282013
Dr Mohammad Abuhaiba PE
32
61 Introduction
Instantaneous center of a moving body
center which keeps changing from one instant to another
Centrode locus of all such ICs
instantaneous axis A line drawn through
an IC and perpendicular to plane of
motion Locus of this axis is known as
axode
9282013
Dr Mohammad Abuhaiba PE
3
62 Space and Body Centrodes
IC a point in the body which may be considered
fixed at any particular moment
Space centrode locus of IC in space during a
definite motion of the body
Body centrode locus of IC relative to body itself
The two centrodes have the IC as a common point
at any instant and during the motion of the body
Body centrode rolls without slipping over space
centrode
9282013
Dr Mohammad Abuhaiba PE
4
62 Space amp
Body Centrodes
I1 and I2 Ics for two different positions A1B1 and A2B2 of link A1B1
Space centrode number of positions of link A1B1 are considered and a curve is drawn passing through these ICs (I1 I2)
9282013
Dr Mohammad Abuhaiba PE
5
62 Space amp Body
Centrodes
C1 is attached to body or link A1B1 and moves with it in such a way that C1 coincides with I1 when the body is in position A1B1
C2 position of point C1 when link A1B1 occupies position A2B2
C2 will coincide with I2 (when link is in position A2B2) only if triangles A1B1C1 and A2B2C2 are identical
9282013
Dr Mohammad Abuhaiba PE
6
63 Methods for Determining the
Velocity of a Point on a Link
1 Instantaneous center method
2 Relative velocity method
9282013
Dr Mohammad Abuhaiba PE
7
64 Velocity of a Point on a Link
by Instantaneous Centre Method
vA is known in magnitude amp
direction amp vB in direction
9282013
Dr Mohammad Abuhaiba PE
8
66 Number of Instantaneous
Centers in a Mechanism
of ICs in a constrained kinematic chain
= of possible combinations of two links
of pairs of links or of ICs is the number
of combinations of n links taken two at a
time
Mathematically of ICs
9282013
Dr Mohammad Abuhaiba PE
9
67 Types of Instantaneous Centers
a primary instantaneous centers
1 Fixed instantaneous centers
2 Permanent instantaneous centers
b secondary instantaneous centers
3 Neither fixed nor permanent
instantaneous centers
9282013
Dr Mohammad Abuhaiba PE
10
67 Types of Instantaneous Centers
Fixed Instantaneous Centersgt I12 amp I14
remain in same place
for all configurations
of mechanism
9282013
Dr Mohammad Abuhaiba PE
11
67 Types of Instantaneous
Centers
I23 amp I34 permanent
instantaneous centers
they move when
mechanism moves
but the joints are of
permanent nature
I13 amp I24 neither fixed
nor permanent
instantaneous centers they vary with
configuration of
mechanism
9282013
Dr Mohammad Abuhaiba PE
12
68 Location of Instantaneous Centers
9282013
Dr Mohammad Abuhaiba PE
13
69 Aronhold Kennedy (Three
Centers in Line) Theorem if three bodies move
relatively to each other they have three ICs that lie on a straight line
Consider three kinematic links A B and C having relative plane motion
Number of ICs (N) is given by
9282013
Dr Mohammad Abuhaiba PE
14
69 Aronhold Kennedy (Three
Centers in Line) Theorem The two ICs at the pin joints of B with
A and C with A (Iab and Iac) are permanent instantaneous centers
According to Aronhold Kennedyrsquos theorem the third IC Ibc must lie on the line joining Iab and Iac
let us consider that instantaneous center Ibc lies outside the line joining Iab and Iac
point Ibc belongs to both links B and C Let us consider point Ibc on link B Its velocity vBC must be perpendicular to the line joining Iab and Ibc Now consider the point Ibc on link C Its velocity vBC must be perpendicular to the line joining Iac and Ibc
9282013
Dr Mohammad Abuhaiba PE
15
69 Aronhold Kennedy (or
Three Centers in Line) Theorem velocity of IC is same whether it is
regarded as a point on first link or as a point on second link
Therefore velocity of point Ibc cannot be perpendicular to both lines Iab Ibc and IacIbc unless the point Ibc lies on the line joining the points Iab and Iac
Thus the three instantaneous centers (Iab Iac and Ibc) must lie on the same straight line
The exact location of Ibc on line Iab Iac depends upon the directions and magnitudes of the angular velocities of B and C relative to A
9282013
Dr Mohammad Abuhaiba PE
16
610 Method of Locating Instantaneous
Centers in a Mechanism
Fig 68 (a)
1 Determine number of ICs (N)
2 Make a list of all instantaneous centers
9282013
Dr Mohammad Abuhaiba PE
17
Links 1 2 3 4
instantaneous
centers
12 23 34
13 24
14
610 Method of Locating Instantaneous
Centers in a Mechanism
3 Locate fixed amp permanent ICs by
inspection
I12 and I14 are fixed Ics
I23 and I34 are permanent IC
The four bar mechanism has
four turning pairs
four primary (fixed and permanent) ICs located at
the centers of the pin joints
9282013
Dr Mohammad Abuhaiba PE
18
610 Method of Locating Instantaneous
Centers in a Mechanism
4 Locate remaining neither fixed nor
permanent ICs (secondary centers) by
Kennedyrsquos theorem
Circle diagram Fig 68 (b)
Mark points on a circle equal to number
of links in a mechanism
5 Join points by solid lines to show that
these centers are already found
9282013
Dr Mohammad Abuhaiba PE
19
610 Method of Locating Instantaneous
Centers in a Mechanism
9282013
Dr Mohammad Abuhaiba PE
20
Example 61 In a pin jointed four bar mechanism Fig 69 AB = 300 mm BC = CD = 360 mm and AD = 600 mm The angle BAD = 60deg The crank AB rotates uniformly at 100 rpm Locate all instantaneous centers and find the angular velocity of link BC
9282013
Dr Mohammad Abuhaiba PE
21
Example 61
9282013
Dr Mohammad Abuhaiba PE
22
Example 62 Locate all ICs of the slider crank mechanism shown in Fig 612 The lengths of crank OB and connecting rod AB are 100 mm and 400 mm respectively If the crank rotates cw with an angular velocity of 10 rads find
1 Velocity of the slider A
2 Angular velocity of connecting rod AB
9282013
Dr Mohammad Abuhaiba PE
23
Example 62
9282013
Dr Mohammad Abuhaiba PE
24
Example 63
A mechanism Fig 615 has the following dimensions
OA = 200 mm AB = 1500 mm BC = 600 mm CD = 500
mm and BE = 400 mm Locate all the ICs If crank OA
rotates uniformly at 120 rpm cw find
1 velocity of B C and D
2 angular velocity of links AB BC and CD
9282013
Dr Mohammad Abuhaiba PE
25
Example 63 9282013
Dr Mohammad Abuhaiba PE
26
Example 64 The mechanism of a wrapping machine Fig 618 has the following dimensions O1A = 100 mm AC = 700 mm BC = 200 mm O3C = 200 mm O2E = 400 mm O2D = 200 mm and BD = 150 mm The crank O1A rotates at a uniform speed of 100 rads Find the velocity of point E of the bell crank lever by instantaneous center method
9282013
Dr Mohammad Abuhaiba PE
27
Example 64
9282013
Dr Mohammad Abuhaiba PE
28
Example 65 Fig 621 shows a sewing needle bar mechanism O1ABO2CD wherein the different dimensions are as follows Crank O1A = 16 mm b = 45deg Vertical distance between O1 and O2 = 40 mm Horizontal distance between O1 amp O2 = 13 mm O2B = 23 mm AB = 35 mm O2BC = 90deg BC = 16 mm CD = 40 mm D lies vertically below O1 Find the velocity of needle at D for the given configuration The crank O1A rotates at 400 rpm
9282013
Dr Mohammad Abuhaiba PE
29
Example 65
9282013
Dr Mohammad Abuhaiba PE
30
Example 66
Fig 624 shows a Whitworth
quick return motion
mechanism The various
dimensions in the mechanism
are as follows OQ = 100 mm
OA = 200 mm QC = 150 mm
and CD = 500 mm The crank
OA makes an angle of 60deg
with the vertical and rotates at
120 rpm cw Locate all ICs
and find the velocity of ram D
9282013
Dr Mohammad Abuhaiba PE
31
Example 66
9282013
Dr Mohammad Abuhaiba PE
32
62 Space and Body Centrodes
IC a point in the body which may be considered
fixed at any particular moment
Space centrode locus of IC in space during a
definite motion of the body
Body centrode locus of IC relative to body itself
The two centrodes have the IC as a common point
at any instant and during the motion of the body
Body centrode rolls without slipping over space
centrode
9282013
Dr Mohammad Abuhaiba PE
4
62 Space amp
Body Centrodes
I1 and I2 Ics for two different positions A1B1 and A2B2 of link A1B1
Space centrode number of positions of link A1B1 are considered and a curve is drawn passing through these ICs (I1 I2)
9282013
Dr Mohammad Abuhaiba PE
5
62 Space amp Body
Centrodes
C1 is attached to body or link A1B1 and moves with it in such a way that C1 coincides with I1 when the body is in position A1B1
C2 position of point C1 when link A1B1 occupies position A2B2
C2 will coincide with I2 (when link is in position A2B2) only if triangles A1B1C1 and A2B2C2 are identical
9282013
Dr Mohammad Abuhaiba PE
6
63 Methods for Determining the
Velocity of a Point on a Link
1 Instantaneous center method
2 Relative velocity method
9282013
Dr Mohammad Abuhaiba PE
7
64 Velocity of a Point on a Link
by Instantaneous Centre Method
vA is known in magnitude amp
direction amp vB in direction
9282013
Dr Mohammad Abuhaiba PE
8
66 Number of Instantaneous
Centers in a Mechanism
of ICs in a constrained kinematic chain
= of possible combinations of two links
of pairs of links or of ICs is the number
of combinations of n links taken two at a
time
Mathematically of ICs
9282013
Dr Mohammad Abuhaiba PE
9
67 Types of Instantaneous Centers
a primary instantaneous centers
1 Fixed instantaneous centers
2 Permanent instantaneous centers
b secondary instantaneous centers
3 Neither fixed nor permanent
instantaneous centers
9282013
Dr Mohammad Abuhaiba PE
10
67 Types of Instantaneous Centers
Fixed Instantaneous Centersgt I12 amp I14
remain in same place
for all configurations
of mechanism
9282013
Dr Mohammad Abuhaiba PE
11
67 Types of Instantaneous
Centers
I23 amp I34 permanent
instantaneous centers
they move when
mechanism moves
but the joints are of
permanent nature
I13 amp I24 neither fixed
nor permanent
instantaneous centers they vary with
configuration of
mechanism
9282013
Dr Mohammad Abuhaiba PE
12
68 Location of Instantaneous Centers
9282013
Dr Mohammad Abuhaiba PE
13
69 Aronhold Kennedy (Three
Centers in Line) Theorem if three bodies move
relatively to each other they have three ICs that lie on a straight line
Consider three kinematic links A B and C having relative plane motion
Number of ICs (N) is given by
9282013
Dr Mohammad Abuhaiba PE
14
69 Aronhold Kennedy (Three
Centers in Line) Theorem The two ICs at the pin joints of B with
A and C with A (Iab and Iac) are permanent instantaneous centers
According to Aronhold Kennedyrsquos theorem the third IC Ibc must lie on the line joining Iab and Iac
let us consider that instantaneous center Ibc lies outside the line joining Iab and Iac
point Ibc belongs to both links B and C Let us consider point Ibc on link B Its velocity vBC must be perpendicular to the line joining Iab and Ibc Now consider the point Ibc on link C Its velocity vBC must be perpendicular to the line joining Iac and Ibc
9282013
Dr Mohammad Abuhaiba PE
15
69 Aronhold Kennedy (or
Three Centers in Line) Theorem velocity of IC is same whether it is
regarded as a point on first link or as a point on second link
Therefore velocity of point Ibc cannot be perpendicular to both lines Iab Ibc and IacIbc unless the point Ibc lies on the line joining the points Iab and Iac
Thus the three instantaneous centers (Iab Iac and Ibc) must lie on the same straight line
The exact location of Ibc on line Iab Iac depends upon the directions and magnitudes of the angular velocities of B and C relative to A
9282013
Dr Mohammad Abuhaiba PE
16
610 Method of Locating Instantaneous
Centers in a Mechanism
Fig 68 (a)
1 Determine number of ICs (N)
2 Make a list of all instantaneous centers
9282013
Dr Mohammad Abuhaiba PE
17
Links 1 2 3 4
instantaneous
centers
12 23 34
13 24
14
610 Method of Locating Instantaneous
Centers in a Mechanism
3 Locate fixed amp permanent ICs by
inspection
I12 and I14 are fixed Ics
I23 and I34 are permanent IC
The four bar mechanism has
four turning pairs
four primary (fixed and permanent) ICs located at
the centers of the pin joints
9282013
Dr Mohammad Abuhaiba PE
18
610 Method of Locating Instantaneous
Centers in a Mechanism
4 Locate remaining neither fixed nor
permanent ICs (secondary centers) by
Kennedyrsquos theorem
Circle diagram Fig 68 (b)
Mark points on a circle equal to number
of links in a mechanism
5 Join points by solid lines to show that
these centers are already found
9282013
Dr Mohammad Abuhaiba PE
19
610 Method of Locating Instantaneous
Centers in a Mechanism
9282013
Dr Mohammad Abuhaiba PE
20
Example 61 In a pin jointed four bar mechanism Fig 69 AB = 300 mm BC = CD = 360 mm and AD = 600 mm The angle BAD = 60deg The crank AB rotates uniformly at 100 rpm Locate all instantaneous centers and find the angular velocity of link BC
9282013
Dr Mohammad Abuhaiba PE
21
Example 61
9282013
Dr Mohammad Abuhaiba PE
22
Example 62 Locate all ICs of the slider crank mechanism shown in Fig 612 The lengths of crank OB and connecting rod AB are 100 mm and 400 mm respectively If the crank rotates cw with an angular velocity of 10 rads find
1 Velocity of the slider A
2 Angular velocity of connecting rod AB
9282013
Dr Mohammad Abuhaiba PE
23
Example 62
9282013
Dr Mohammad Abuhaiba PE
24
Example 63
A mechanism Fig 615 has the following dimensions
OA = 200 mm AB = 1500 mm BC = 600 mm CD = 500
mm and BE = 400 mm Locate all the ICs If crank OA
rotates uniformly at 120 rpm cw find
1 velocity of B C and D
2 angular velocity of links AB BC and CD
9282013
Dr Mohammad Abuhaiba PE
25
Example 63 9282013
Dr Mohammad Abuhaiba PE
26
Example 64 The mechanism of a wrapping machine Fig 618 has the following dimensions O1A = 100 mm AC = 700 mm BC = 200 mm O3C = 200 mm O2E = 400 mm O2D = 200 mm and BD = 150 mm The crank O1A rotates at a uniform speed of 100 rads Find the velocity of point E of the bell crank lever by instantaneous center method
9282013
Dr Mohammad Abuhaiba PE
27
Example 64
9282013
Dr Mohammad Abuhaiba PE
28
Example 65 Fig 621 shows a sewing needle bar mechanism O1ABO2CD wherein the different dimensions are as follows Crank O1A = 16 mm b = 45deg Vertical distance between O1 and O2 = 40 mm Horizontal distance between O1 amp O2 = 13 mm O2B = 23 mm AB = 35 mm O2BC = 90deg BC = 16 mm CD = 40 mm D lies vertically below O1 Find the velocity of needle at D for the given configuration The crank O1A rotates at 400 rpm
9282013
Dr Mohammad Abuhaiba PE
29
Example 65
9282013
Dr Mohammad Abuhaiba PE
30
Example 66
Fig 624 shows a Whitworth
quick return motion
mechanism The various
dimensions in the mechanism
are as follows OQ = 100 mm
OA = 200 mm QC = 150 mm
and CD = 500 mm The crank
OA makes an angle of 60deg
with the vertical and rotates at
120 rpm cw Locate all ICs
and find the velocity of ram D
9282013
Dr Mohammad Abuhaiba PE
31
Example 66
9282013
Dr Mohammad Abuhaiba PE
32
62 Space amp
Body Centrodes
I1 and I2 Ics for two different positions A1B1 and A2B2 of link A1B1
Space centrode number of positions of link A1B1 are considered and a curve is drawn passing through these ICs (I1 I2)
9282013
Dr Mohammad Abuhaiba PE
5
62 Space amp Body
Centrodes
C1 is attached to body or link A1B1 and moves with it in such a way that C1 coincides with I1 when the body is in position A1B1
C2 position of point C1 when link A1B1 occupies position A2B2
C2 will coincide with I2 (when link is in position A2B2) only if triangles A1B1C1 and A2B2C2 are identical
9282013
Dr Mohammad Abuhaiba PE
6
63 Methods for Determining the
Velocity of a Point on a Link
1 Instantaneous center method
2 Relative velocity method
9282013
Dr Mohammad Abuhaiba PE
7
64 Velocity of a Point on a Link
by Instantaneous Centre Method
vA is known in magnitude amp
direction amp vB in direction
9282013
Dr Mohammad Abuhaiba PE
8
66 Number of Instantaneous
Centers in a Mechanism
of ICs in a constrained kinematic chain
= of possible combinations of two links
of pairs of links or of ICs is the number
of combinations of n links taken two at a
time
Mathematically of ICs
9282013
Dr Mohammad Abuhaiba PE
9
67 Types of Instantaneous Centers
a primary instantaneous centers
1 Fixed instantaneous centers
2 Permanent instantaneous centers
b secondary instantaneous centers
3 Neither fixed nor permanent
instantaneous centers
9282013
Dr Mohammad Abuhaiba PE
10
67 Types of Instantaneous Centers
Fixed Instantaneous Centersgt I12 amp I14
remain in same place
for all configurations
of mechanism
9282013
Dr Mohammad Abuhaiba PE
11
67 Types of Instantaneous
Centers
I23 amp I34 permanent
instantaneous centers
they move when
mechanism moves
but the joints are of
permanent nature
I13 amp I24 neither fixed
nor permanent
instantaneous centers they vary with
configuration of
mechanism
9282013
Dr Mohammad Abuhaiba PE
12
68 Location of Instantaneous Centers
9282013
Dr Mohammad Abuhaiba PE
13
69 Aronhold Kennedy (Three
Centers in Line) Theorem if three bodies move
relatively to each other they have three ICs that lie on a straight line
Consider three kinematic links A B and C having relative plane motion
Number of ICs (N) is given by
9282013
Dr Mohammad Abuhaiba PE
14
69 Aronhold Kennedy (Three
Centers in Line) Theorem The two ICs at the pin joints of B with
A and C with A (Iab and Iac) are permanent instantaneous centers
According to Aronhold Kennedyrsquos theorem the third IC Ibc must lie on the line joining Iab and Iac
let us consider that instantaneous center Ibc lies outside the line joining Iab and Iac
point Ibc belongs to both links B and C Let us consider point Ibc on link B Its velocity vBC must be perpendicular to the line joining Iab and Ibc Now consider the point Ibc on link C Its velocity vBC must be perpendicular to the line joining Iac and Ibc
9282013
Dr Mohammad Abuhaiba PE
15
69 Aronhold Kennedy (or
Three Centers in Line) Theorem velocity of IC is same whether it is
regarded as a point on first link or as a point on second link
Therefore velocity of point Ibc cannot be perpendicular to both lines Iab Ibc and IacIbc unless the point Ibc lies on the line joining the points Iab and Iac
Thus the three instantaneous centers (Iab Iac and Ibc) must lie on the same straight line
The exact location of Ibc on line Iab Iac depends upon the directions and magnitudes of the angular velocities of B and C relative to A
9282013
Dr Mohammad Abuhaiba PE
16
610 Method of Locating Instantaneous
Centers in a Mechanism
Fig 68 (a)
1 Determine number of ICs (N)
2 Make a list of all instantaneous centers
9282013
Dr Mohammad Abuhaiba PE
17
Links 1 2 3 4
instantaneous
centers
12 23 34
13 24
14
610 Method of Locating Instantaneous
Centers in a Mechanism
3 Locate fixed amp permanent ICs by
inspection
I12 and I14 are fixed Ics
I23 and I34 are permanent IC
The four bar mechanism has
four turning pairs
four primary (fixed and permanent) ICs located at
the centers of the pin joints
9282013
Dr Mohammad Abuhaiba PE
18
610 Method of Locating Instantaneous
Centers in a Mechanism
4 Locate remaining neither fixed nor
permanent ICs (secondary centers) by
Kennedyrsquos theorem
Circle diagram Fig 68 (b)
Mark points on a circle equal to number
of links in a mechanism
5 Join points by solid lines to show that
these centers are already found
9282013
Dr Mohammad Abuhaiba PE
19
610 Method of Locating Instantaneous
Centers in a Mechanism
9282013
Dr Mohammad Abuhaiba PE
20
Example 61 In a pin jointed four bar mechanism Fig 69 AB = 300 mm BC = CD = 360 mm and AD = 600 mm The angle BAD = 60deg The crank AB rotates uniformly at 100 rpm Locate all instantaneous centers and find the angular velocity of link BC
9282013
Dr Mohammad Abuhaiba PE
21
Example 61
9282013
Dr Mohammad Abuhaiba PE
22
Example 62 Locate all ICs of the slider crank mechanism shown in Fig 612 The lengths of crank OB and connecting rod AB are 100 mm and 400 mm respectively If the crank rotates cw with an angular velocity of 10 rads find
1 Velocity of the slider A
2 Angular velocity of connecting rod AB
9282013
Dr Mohammad Abuhaiba PE
23
Example 62
9282013
Dr Mohammad Abuhaiba PE
24
Example 63
A mechanism Fig 615 has the following dimensions
OA = 200 mm AB = 1500 mm BC = 600 mm CD = 500
mm and BE = 400 mm Locate all the ICs If crank OA
rotates uniformly at 120 rpm cw find
1 velocity of B C and D
2 angular velocity of links AB BC and CD
9282013
Dr Mohammad Abuhaiba PE
25
Example 63 9282013
Dr Mohammad Abuhaiba PE
26
Example 64 The mechanism of a wrapping machine Fig 618 has the following dimensions O1A = 100 mm AC = 700 mm BC = 200 mm O3C = 200 mm O2E = 400 mm O2D = 200 mm and BD = 150 mm The crank O1A rotates at a uniform speed of 100 rads Find the velocity of point E of the bell crank lever by instantaneous center method
9282013
Dr Mohammad Abuhaiba PE
27
Example 64
9282013
Dr Mohammad Abuhaiba PE
28
Example 65 Fig 621 shows a sewing needle bar mechanism O1ABO2CD wherein the different dimensions are as follows Crank O1A = 16 mm b = 45deg Vertical distance between O1 and O2 = 40 mm Horizontal distance between O1 amp O2 = 13 mm O2B = 23 mm AB = 35 mm O2BC = 90deg BC = 16 mm CD = 40 mm D lies vertically below O1 Find the velocity of needle at D for the given configuration The crank O1A rotates at 400 rpm
9282013
Dr Mohammad Abuhaiba PE
29
Example 65
9282013
Dr Mohammad Abuhaiba PE
30
Example 66
Fig 624 shows a Whitworth
quick return motion
mechanism The various
dimensions in the mechanism
are as follows OQ = 100 mm
OA = 200 mm QC = 150 mm
and CD = 500 mm The crank
OA makes an angle of 60deg
with the vertical and rotates at
120 rpm cw Locate all ICs
and find the velocity of ram D
9282013
Dr Mohammad Abuhaiba PE
31
Example 66
9282013
Dr Mohammad Abuhaiba PE
32
62 Space amp Body
Centrodes
C1 is attached to body or link A1B1 and moves with it in such a way that C1 coincides with I1 when the body is in position A1B1
C2 position of point C1 when link A1B1 occupies position A2B2
C2 will coincide with I2 (when link is in position A2B2) only if triangles A1B1C1 and A2B2C2 are identical
9282013
Dr Mohammad Abuhaiba PE
6
63 Methods for Determining the
Velocity of a Point on a Link
1 Instantaneous center method
2 Relative velocity method
9282013
Dr Mohammad Abuhaiba PE
7
64 Velocity of a Point on a Link
by Instantaneous Centre Method
vA is known in magnitude amp
direction amp vB in direction
9282013
Dr Mohammad Abuhaiba PE
8
66 Number of Instantaneous
Centers in a Mechanism
of ICs in a constrained kinematic chain
= of possible combinations of two links
of pairs of links or of ICs is the number
of combinations of n links taken two at a
time
Mathematically of ICs
9282013
Dr Mohammad Abuhaiba PE
9
67 Types of Instantaneous Centers
a primary instantaneous centers
1 Fixed instantaneous centers
2 Permanent instantaneous centers
b secondary instantaneous centers
3 Neither fixed nor permanent
instantaneous centers
9282013
Dr Mohammad Abuhaiba PE
10
67 Types of Instantaneous Centers
Fixed Instantaneous Centersgt I12 amp I14
remain in same place
for all configurations
of mechanism
9282013
Dr Mohammad Abuhaiba PE
11
67 Types of Instantaneous
Centers
I23 amp I34 permanent
instantaneous centers
they move when
mechanism moves
but the joints are of
permanent nature
I13 amp I24 neither fixed
nor permanent
instantaneous centers they vary with
configuration of
mechanism
9282013
Dr Mohammad Abuhaiba PE
12
68 Location of Instantaneous Centers
9282013
Dr Mohammad Abuhaiba PE
13
69 Aronhold Kennedy (Three
Centers in Line) Theorem if three bodies move
relatively to each other they have three ICs that lie on a straight line
Consider three kinematic links A B and C having relative plane motion
Number of ICs (N) is given by
9282013
Dr Mohammad Abuhaiba PE
14
69 Aronhold Kennedy (Three
Centers in Line) Theorem The two ICs at the pin joints of B with
A and C with A (Iab and Iac) are permanent instantaneous centers
According to Aronhold Kennedyrsquos theorem the third IC Ibc must lie on the line joining Iab and Iac
let us consider that instantaneous center Ibc lies outside the line joining Iab and Iac
point Ibc belongs to both links B and C Let us consider point Ibc on link B Its velocity vBC must be perpendicular to the line joining Iab and Ibc Now consider the point Ibc on link C Its velocity vBC must be perpendicular to the line joining Iac and Ibc
9282013
Dr Mohammad Abuhaiba PE
15
69 Aronhold Kennedy (or
Three Centers in Line) Theorem velocity of IC is same whether it is
regarded as a point on first link or as a point on second link
Therefore velocity of point Ibc cannot be perpendicular to both lines Iab Ibc and IacIbc unless the point Ibc lies on the line joining the points Iab and Iac
Thus the three instantaneous centers (Iab Iac and Ibc) must lie on the same straight line
The exact location of Ibc on line Iab Iac depends upon the directions and magnitudes of the angular velocities of B and C relative to A
9282013
Dr Mohammad Abuhaiba PE
16
610 Method of Locating Instantaneous
Centers in a Mechanism
Fig 68 (a)
1 Determine number of ICs (N)
2 Make a list of all instantaneous centers
9282013
Dr Mohammad Abuhaiba PE
17
Links 1 2 3 4
instantaneous
centers
12 23 34
13 24
14
610 Method of Locating Instantaneous
Centers in a Mechanism
3 Locate fixed amp permanent ICs by
inspection
I12 and I14 are fixed Ics
I23 and I34 are permanent IC
The four bar mechanism has
four turning pairs
four primary (fixed and permanent) ICs located at
the centers of the pin joints
9282013
Dr Mohammad Abuhaiba PE
18
610 Method of Locating Instantaneous
Centers in a Mechanism
4 Locate remaining neither fixed nor
permanent ICs (secondary centers) by
Kennedyrsquos theorem
Circle diagram Fig 68 (b)
Mark points on a circle equal to number
of links in a mechanism
5 Join points by solid lines to show that
these centers are already found
9282013
Dr Mohammad Abuhaiba PE
19
610 Method of Locating Instantaneous
Centers in a Mechanism
9282013
Dr Mohammad Abuhaiba PE
20
Example 61 In a pin jointed four bar mechanism Fig 69 AB = 300 mm BC = CD = 360 mm and AD = 600 mm The angle BAD = 60deg The crank AB rotates uniformly at 100 rpm Locate all instantaneous centers and find the angular velocity of link BC
9282013
Dr Mohammad Abuhaiba PE
21
Example 61
9282013
Dr Mohammad Abuhaiba PE
22
Example 62 Locate all ICs of the slider crank mechanism shown in Fig 612 The lengths of crank OB and connecting rod AB are 100 mm and 400 mm respectively If the crank rotates cw with an angular velocity of 10 rads find
1 Velocity of the slider A
2 Angular velocity of connecting rod AB
9282013
Dr Mohammad Abuhaiba PE
23
Example 62
9282013
Dr Mohammad Abuhaiba PE
24
Example 63
A mechanism Fig 615 has the following dimensions
OA = 200 mm AB = 1500 mm BC = 600 mm CD = 500
mm and BE = 400 mm Locate all the ICs If crank OA
rotates uniformly at 120 rpm cw find
1 velocity of B C and D
2 angular velocity of links AB BC and CD
9282013
Dr Mohammad Abuhaiba PE
25
Example 63 9282013
Dr Mohammad Abuhaiba PE
26
Example 64 The mechanism of a wrapping machine Fig 618 has the following dimensions O1A = 100 mm AC = 700 mm BC = 200 mm O3C = 200 mm O2E = 400 mm O2D = 200 mm and BD = 150 mm The crank O1A rotates at a uniform speed of 100 rads Find the velocity of point E of the bell crank lever by instantaneous center method
9282013
Dr Mohammad Abuhaiba PE
27
Example 64
9282013
Dr Mohammad Abuhaiba PE
28
Example 65 Fig 621 shows a sewing needle bar mechanism O1ABO2CD wherein the different dimensions are as follows Crank O1A = 16 mm b = 45deg Vertical distance between O1 and O2 = 40 mm Horizontal distance between O1 amp O2 = 13 mm O2B = 23 mm AB = 35 mm O2BC = 90deg BC = 16 mm CD = 40 mm D lies vertically below O1 Find the velocity of needle at D for the given configuration The crank O1A rotates at 400 rpm
9282013
Dr Mohammad Abuhaiba PE
29
Example 65
9282013
Dr Mohammad Abuhaiba PE
30
Example 66
Fig 624 shows a Whitworth
quick return motion
mechanism The various
dimensions in the mechanism
are as follows OQ = 100 mm
OA = 200 mm QC = 150 mm
and CD = 500 mm The crank
OA makes an angle of 60deg
with the vertical and rotates at
120 rpm cw Locate all ICs
and find the velocity of ram D
9282013
Dr Mohammad Abuhaiba PE
31
Example 66
9282013
Dr Mohammad Abuhaiba PE
32
63 Methods for Determining the
Velocity of a Point on a Link
1 Instantaneous center method
2 Relative velocity method
9282013
Dr Mohammad Abuhaiba PE
7
64 Velocity of a Point on a Link
by Instantaneous Centre Method
vA is known in magnitude amp
direction amp vB in direction
9282013
Dr Mohammad Abuhaiba PE
8
66 Number of Instantaneous
Centers in a Mechanism
of ICs in a constrained kinematic chain
= of possible combinations of two links
of pairs of links or of ICs is the number
of combinations of n links taken two at a
time
Mathematically of ICs
9282013
Dr Mohammad Abuhaiba PE
9
67 Types of Instantaneous Centers
a primary instantaneous centers
1 Fixed instantaneous centers
2 Permanent instantaneous centers
b secondary instantaneous centers
3 Neither fixed nor permanent
instantaneous centers
9282013
Dr Mohammad Abuhaiba PE
10
67 Types of Instantaneous Centers
Fixed Instantaneous Centersgt I12 amp I14
remain in same place
for all configurations
of mechanism
9282013
Dr Mohammad Abuhaiba PE
11
67 Types of Instantaneous
Centers
I23 amp I34 permanent
instantaneous centers
they move when
mechanism moves
but the joints are of
permanent nature
I13 amp I24 neither fixed
nor permanent
instantaneous centers they vary with
configuration of
mechanism
9282013
Dr Mohammad Abuhaiba PE
12
68 Location of Instantaneous Centers
9282013
Dr Mohammad Abuhaiba PE
13
69 Aronhold Kennedy (Three
Centers in Line) Theorem if three bodies move
relatively to each other they have three ICs that lie on a straight line
Consider three kinematic links A B and C having relative plane motion
Number of ICs (N) is given by
9282013
Dr Mohammad Abuhaiba PE
14
69 Aronhold Kennedy (Three
Centers in Line) Theorem The two ICs at the pin joints of B with
A and C with A (Iab and Iac) are permanent instantaneous centers
According to Aronhold Kennedyrsquos theorem the third IC Ibc must lie on the line joining Iab and Iac
let us consider that instantaneous center Ibc lies outside the line joining Iab and Iac
point Ibc belongs to both links B and C Let us consider point Ibc on link B Its velocity vBC must be perpendicular to the line joining Iab and Ibc Now consider the point Ibc on link C Its velocity vBC must be perpendicular to the line joining Iac and Ibc
9282013
Dr Mohammad Abuhaiba PE
15
69 Aronhold Kennedy (or
Three Centers in Line) Theorem velocity of IC is same whether it is
regarded as a point on first link or as a point on second link
Therefore velocity of point Ibc cannot be perpendicular to both lines Iab Ibc and IacIbc unless the point Ibc lies on the line joining the points Iab and Iac
Thus the three instantaneous centers (Iab Iac and Ibc) must lie on the same straight line
The exact location of Ibc on line Iab Iac depends upon the directions and magnitudes of the angular velocities of B and C relative to A
9282013
Dr Mohammad Abuhaiba PE
16
610 Method of Locating Instantaneous
Centers in a Mechanism
Fig 68 (a)
1 Determine number of ICs (N)
2 Make a list of all instantaneous centers
9282013
Dr Mohammad Abuhaiba PE
17
Links 1 2 3 4
instantaneous
centers
12 23 34
13 24
14
610 Method of Locating Instantaneous
Centers in a Mechanism
3 Locate fixed amp permanent ICs by
inspection
I12 and I14 are fixed Ics
I23 and I34 are permanent IC
The four bar mechanism has
four turning pairs
four primary (fixed and permanent) ICs located at
the centers of the pin joints
9282013
Dr Mohammad Abuhaiba PE
18
610 Method of Locating Instantaneous
Centers in a Mechanism
4 Locate remaining neither fixed nor
permanent ICs (secondary centers) by
Kennedyrsquos theorem
Circle diagram Fig 68 (b)
Mark points on a circle equal to number
of links in a mechanism
5 Join points by solid lines to show that
these centers are already found
9282013
Dr Mohammad Abuhaiba PE
19
610 Method of Locating Instantaneous
Centers in a Mechanism
9282013
Dr Mohammad Abuhaiba PE
20
Example 61 In a pin jointed four bar mechanism Fig 69 AB = 300 mm BC = CD = 360 mm and AD = 600 mm The angle BAD = 60deg The crank AB rotates uniformly at 100 rpm Locate all instantaneous centers and find the angular velocity of link BC
9282013
Dr Mohammad Abuhaiba PE
21
Example 61
9282013
Dr Mohammad Abuhaiba PE
22
Example 62 Locate all ICs of the slider crank mechanism shown in Fig 612 The lengths of crank OB and connecting rod AB are 100 mm and 400 mm respectively If the crank rotates cw with an angular velocity of 10 rads find
1 Velocity of the slider A
2 Angular velocity of connecting rod AB
9282013
Dr Mohammad Abuhaiba PE
23
Example 62
9282013
Dr Mohammad Abuhaiba PE
24
Example 63
A mechanism Fig 615 has the following dimensions
OA = 200 mm AB = 1500 mm BC = 600 mm CD = 500
mm and BE = 400 mm Locate all the ICs If crank OA
rotates uniformly at 120 rpm cw find
1 velocity of B C and D
2 angular velocity of links AB BC and CD
9282013
Dr Mohammad Abuhaiba PE
25
Example 63 9282013
Dr Mohammad Abuhaiba PE
26
Example 64 The mechanism of a wrapping machine Fig 618 has the following dimensions O1A = 100 mm AC = 700 mm BC = 200 mm O3C = 200 mm O2E = 400 mm O2D = 200 mm and BD = 150 mm The crank O1A rotates at a uniform speed of 100 rads Find the velocity of point E of the bell crank lever by instantaneous center method
9282013
Dr Mohammad Abuhaiba PE
27
Example 64
9282013
Dr Mohammad Abuhaiba PE
28
Example 65 Fig 621 shows a sewing needle bar mechanism O1ABO2CD wherein the different dimensions are as follows Crank O1A = 16 mm b = 45deg Vertical distance between O1 and O2 = 40 mm Horizontal distance between O1 amp O2 = 13 mm O2B = 23 mm AB = 35 mm O2BC = 90deg BC = 16 mm CD = 40 mm D lies vertically below O1 Find the velocity of needle at D for the given configuration The crank O1A rotates at 400 rpm
9282013
Dr Mohammad Abuhaiba PE
29
Example 65
9282013
Dr Mohammad Abuhaiba PE
30
Example 66
Fig 624 shows a Whitworth
quick return motion
mechanism The various
dimensions in the mechanism
are as follows OQ = 100 mm
OA = 200 mm QC = 150 mm
and CD = 500 mm The crank
OA makes an angle of 60deg
with the vertical and rotates at
120 rpm cw Locate all ICs
and find the velocity of ram D
9282013
Dr Mohammad Abuhaiba PE
31
Example 66
9282013
Dr Mohammad Abuhaiba PE
32
64 Velocity of a Point on a Link
by Instantaneous Centre Method
vA is known in magnitude amp
direction amp vB in direction
9282013
Dr Mohammad Abuhaiba PE
8
66 Number of Instantaneous
Centers in a Mechanism
of ICs in a constrained kinematic chain
= of possible combinations of two links
of pairs of links or of ICs is the number
of combinations of n links taken two at a
time
Mathematically of ICs
9282013
Dr Mohammad Abuhaiba PE
9
67 Types of Instantaneous Centers
a primary instantaneous centers
1 Fixed instantaneous centers
2 Permanent instantaneous centers
b secondary instantaneous centers
3 Neither fixed nor permanent
instantaneous centers
9282013
Dr Mohammad Abuhaiba PE
10
67 Types of Instantaneous Centers
Fixed Instantaneous Centersgt I12 amp I14
remain in same place
for all configurations
of mechanism
9282013
Dr Mohammad Abuhaiba PE
11
67 Types of Instantaneous
Centers
I23 amp I34 permanent
instantaneous centers
they move when
mechanism moves
but the joints are of
permanent nature
I13 amp I24 neither fixed
nor permanent
instantaneous centers they vary with
configuration of
mechanism
9282013
Dr Mohammad Abuhaiba PE
12
68 Location of Instantaneous Centers
9282013
Dr Mohammad Abuhaiba PE
13
69 Aronhold Kennedy (Three
Centers in Line) Theorem if three bodies move
relatively to each other they have three ICs that lie on a straight line
Consider three kinematic links A B and C having relative plane motion
Number of ICs (N) is given by
9282013
Dr Mohammad Abuhaiba PE
14
69 Aronhold Kennedy (Three
Centers in Line) Theorem The two ICs at the pin joints of B with
A and C with A (Iab and Iac) are permanent instantaneous centers
According to Aronhold Kennedyrsquos theorem the third IC Ibc must lie on the line joining Iab and Iac
let us consider that instantaneous center Ibc lies outside the line joining Iab and Iac
point Ibc belongs to both links B and C Let us consider point Ibc on link B Its velocity vBC must be perpendicular to the line joining Iab and Ibc Now consider the point Ibc on link C Its velocity vBC must be perpendicular to the line joining Iac and Ibc
9282013
Dr Mohammad Abuhaiba PE
15
69 Aronhold Kennedy (or
Three Centers in Line) Theorem velocity of IC is same whether it is
regarded as a point on first link or as a point on second link
Therefore velocity of point Ibc cannot be perpendicular to both lines Iab Ibc and IacIbc unless the point Ibc lies on the line joining the points Iab and Iac
Thus the three instantaneous centers (Iab Iac and Ibc) must lie on the same straight line
The exact location of Ibc on line Iab Iac depends upon the directions and magnitudes of the angular velocities of B and C relative to A
9282013
Dr Mohammad Abuhaiba PE
16
610 Method of Locating Instantaneous
Centers in a Mechanism
Fig 68 (a)
1 Determine number of ICs (N)
2 Make a list of all instantaneous centers
9282013
Dr Mohammad Abuhaiba PE
17
Links 1 2 3 4
instantaneous
centers
12 23 34
13 24
14
610 Method of Locating Instantaneous
Centers in a Mechanism
3 Locate fixed amp permanent ICs by
inspection
I12 and I14 are fixed Ics
I23 and I34 are permanent IC
The four bar mechanism has
four turning pairs
four primary (fixed and permanent) ICs located at
the centers of the pin joints
9282013
Dr Mohammad Abuhaiba PE
18
610 Method of Locating Instantaneous
Centers in a Mechanism
4 Locate remaining neither fixed nor
permanent ICs (secondary centers) by
Kennedyrsquos theorem
Circle diagram Fig 68 (b)
Mark points on a circle equal to number
of links in a mechanism
5 Join points by solid lines to show that
these centers are already found
9282013
Dr Mohammad Abuhaiba PE
19
610 Method of Locating Instantaneous
Centers in a Mechanism
9282013
Dr Mohammad Abuhaiba PE
20
Example 61 In a pin jointed four bar mechanism Fig 69 AB = 300 mm BC = CD = 360 mm and AD = 600 mm The angle BAD = 60deg The crank AB rotates uniformly at 100 rpm Locate all instantaneous centers and find the angular velocity of link BC
9282013
Dr Mohammad Abuhaiba PE
21
Example 61
9282013
Dr Mohammad Abuhaiba PE
22
Example 62 Locate all ICs of the slider crank mechanism shown in Fig 612 The lengths of crank OB and connecting rod AB are 100 mm and 400 mm respectively If the crank rotates cw with an angular velocity of 10 rads find
1 Velocity of the slider A
2 Angular velocity of connecting rod AB
9282013
Dr Mohammad Abuhaiba PE
23
Example 62
9282013
Dr Mohammad Abuhaiba PE
24
Example 63
A mechanism Fig 615 has the following dimensions
OA = 200 mm AB = 1500 mm BC = 600 mm CD = 500
mm and BE = 400 mm Locate all the ICs If crank OA
rotates uniformly at 120 rpm cw find
1 velocity of B C and D
2 angular velocity of links AB BC and CD
9282013
Dr Mohammad Abuhaiba PE
25
Example 63 9282013
Dr Mohammad Abuhaiba PE
26
Example 64 The mechanism of a wrapping machine Fig 618 has the following dimensions O1A = 100 mm AC = 700 mm BC = 200 mm O3C = 200 mm O2E = 400 mm O2D = 200 mm and BD = 150 mm The crank O1A rotates at a uniform speed of 100 rads Find the velocity of point E of the bell crank lever by instantaneous center method
9282013
Dr Mohammad Abuhaiba PE
27
Example 64
9282013
Dr Mohammad Abuhaiba PE
28
Example 65 Fig 621 shows a sewing needle bar mechanism O1ABO2CD wherein the different dimensions are as follows Crank O1A = 16 mm b = 45deg Vertical distance between O1 and O2 = 40 mm Horizontal distance between O1 amp O2 = 13 mm O2B = 23 mm AB = 35 mm O2BC = 90deg BC = 16 mm CD = 40 mm D lies vertically below O1 Find the velocity of needle at D for the given configuration The crank O1A rotates at 400 rpm
9282013
Dr Mohammad Abuhaiba PE
29
Example 65
9282013
Dr Mohammad Abuhaiba PE
30
Example 66
Fig 624 shows a Whitworth
quick return motion
mechanism The various
dimensions in the mechanism
are as follows OQ = 100 mm
OA = 200 mm QC = 150 mm
and CD = 500 mm The crank
OA makes an angle of 60deg
with the vertical and rotates at
120 rpm cw Locate all ICs
and find the velocity of ram D
9282013
Dr Mohammad Abuhaiba PE
31
Example 66
9282013
Dr Mohammad Abuhaiba PE
32
66 Number of Instantaneous
Centers in a Mechanism
of ICs in a constrained kinematic chain
= of possible combinations of two links
of pairs of links or of ICs is the number
of combinations of n links taken two at a
time
Mathematically of ICs
9282013
Dr Mohammad Abuhaiba PE
9
67 Types of Instantaneous Centers
a primary instantaneous centers
1 Fixed instantaneous centers
2 Permanent instantaneous centers
b secondary instantaneous centers
3 Neither fixed nor permanent
instantaneous centers
9282013
Dr Mohammad Abuhaiba PE
10
67 Types of Instantaneous Centers
Fixed Instantaneous Centersgt I12 amp I14
remain in same place
for all configurations
of mechanism
9282013
Dr Mohammad Abuhaiba PE
11
67 Types of Instantaneous
Centers
I23 amp I34 permanent
instantaneous centers
they move when
mechanism moves
but the joints are of
permanent nature
I13 amp I24 neither fixed
nor permanent
instantaneous centers they vary with
configuration of
mechanism
9282013
Dr Mohammad Abuhaiba PE
12
68 Location of Instantaneous Centers
9282013
Dr Mohammad Abuhaiba PE
13
69 Aronhold Kennedy (Three
Centers in Line) Theorem if three bodies move
relatively to each other they have three ICs that lie on a straight line
Consider three kinematic links A B and C having relative plane motion
Number of ICs (N) is given by
9282013
Dr Mohammad Abuhaiba PE
14
69 Aronhold Kennedy (Three
Centers in Line) Theorem The two ICs at the pin joints of B with
A and C with A (Iab and Iac) are permanent instantaneous centers
According to Aronhold Kennedyrsquos theorem the third IC Ibc must lie on the line joining Iab and Iac
let us consider that instantaneous center Ibc lies outside the line joining Iab and Iac
point Ibc belongs to both links B and C Let us consider point Ibc on link B Its velocity vBC must be perpendicular to the line joining Iab and Ibc Now consider the point Ibc on link C Its velocity vBC must be perpendicular to the line joining Iac and Ibc
9282013
Dr Mohammad Abuhaiba PE
15
69 Aronhold Kennedy (or
Three Centers in Line) Theorem velocity of IC is same whether it is
regarded as a point on first link or as a point on second link
Therefore velocity of point Ibc cannot be perpendicular to both lines Iab Ibc and IacIbc unless the point Ibc lies on the line joining the points Iab and Iac
Thus the three instantaneous centers (Iab Iac and Ibc) must lie on the same straight line
The exact location of Ibc on line Iab Iac depends upon the directions and magnitudes of the angular velocities of B and C relative to A
9282013
Dr Mohammad Abuhaiba PE
16
610 Method of Locating Instantaneous
Centers in a Mechanism
Fig 68 (a)
1 Determine number of ICs (N)
2 Make a list of all instantaneous centers
9282013
Dr Mohammad Abuhaiba PE
17
Links 1 2 3 4
instantaneous
centers
12 23 34
13 24
14
610 Method of Locating Instantaneous
Centers in a Mechanism
3 Locate fixed amp permanent ICs by
inspection
I12 and I14 are fixed Ics
I23 and I34 are permanent IC
The four bar mechanism has
four turning pairs
four primary (fixed and permanent) ICs located at
the centers of the pin joints
9282013
Dr Mohammad Abuhaiba PE
18
610 Method of Locating Instantaneous
Centers in a Mechanism
4 Locate remaining neither fixed nor
permanent ICs (secondary centers) by
Kennedyrsquos theorem
Circle diagram Fig 68 (b)
Mark points on a circle equal to number
of links in a mechanism
5 Join points by solid lines to show that
these centers are already found
9282013
Dr Mohammad Abuhaiba PE
19
610 Method of Locating Instantaneous
Centers in a Mechanism
9282013
Dr Mohammad Abuhaiba PE
20
Example 61 In a pin jointed four bar mechanism Fig 69 AB = 300 mm BC = CD = 360 mm and AD = 600 mm The angle BAD = 60deg The crank AB rotates uniformly at 100 rpm Locate all instantaneous centers and find the angular velocity of link BC
9282013
Dr Mohammad Abuhaiba PE
21
Example 61
9282013
Dr Mohammad Abuhaiba PE
22
Example 62 Locate all ICs of the slider crank mechanism shown in Fig 612 The lengths of crank OB and connecting rod AB are 100 mm and 400 mm respectively If the crank rotates cw with an angular velocity of 10 rads find
1 Velocity of the slider A
2 Angular velocity of connecting rod AB
9282013
Dr Mohammad Abuhaiba PE
23
Example 62
9282013
Dr Mohammad Abuhaiba PE
24
Example 63
A mechanism Fig 615 has the following dimensions
OA = 200 mm AB = 1500 mm BC = 600 mm CD = 500
mm and BE = 400 mm Locate all the ICs If crank OA
rotates uniformly at 120 rpm cw find
1 velocity of B C and D
2 angular velocity of links AB BC and CD
9282013
Dr Mohammad Abuhaiba PE
25
Example 63 9282013
Dr Mohammad Abuhaiba PE
26
Example 64 The mechanism of a wrapping machine Fig 618 has the following dimensions O1A = 100 mm AC = 700 mm BC = 200 mm O3C = 200 mm O2E = 400 mm O2D = 200 mm and BD = 150 mm The crank O1A rotates at a uniform speed of 100 rads Find the velocity of point E of the bell crank lever by instantaneous center method
9282013
Dr Mohammad Abuhaiba PE
27
Example 64
9282013
Dr Mohammad Abuhaiba PE
28
Example 65 Fig 621 shows a sewing needle bar mechanism O1ABO2CD wherein the different dimensions are as follows Crank O1A = 16 mm b = 45deg Vertical distance between O1 and O2 = 40 mm Horizontal distance between O1 amp O2 = 13 mm O2B = 23 mm AB = 35 mm O2BC = 90deg BC = 16 mm CD = 40 mm D lies vertically below O1 Find the velocity of needle at D for the given configuration The crank O1A rotates at 400 rpm
9282013
Dr Mohammad Abuhaiba PE
29
Example 65
9282013
Dr Mohammad Abuhaiba PE
30
Example 66
Fig 624 shows a Whitworth
quick return motion
mechanism The various
dimensions in the mechanism
are as follows OQ = 100 mm
OA = 200 mm QC = 150 mm
and CD = 500 mm The crank
OA makes an angle of 60deg
with the vertical and rotates at
120 rpm cw Locate all ICs
and find the velocity of ram D
9282013
Dr Mohammad Abuhaiba PE
31
Example 66
9282013
Dr Mohammad Abuhaiba PE
32
67 Types of Instantaneous Centers
a primary instantaneous centers
1 Fixed instantaneous centers
2 Permanent instantaneous centers
b secondary instantaneous centers
3 Neither fixed nor permanent
instantaneous centers
9282013
Dr Mohammad Abuhaiba PE
10
67 Types of Instantaneous Centers
Fixed Instantaneous Centersgt I12 amp I14
remain in same place
for all configurations
of mechanism
9282013
Dr Mohammad Abuhaiba PE
11
67 Types of Instantaneous
Centers
I23 amp I34 permanent
instantaneous centers
they move when
mechanism moves
but the joints are of
permanent nature
I13 amp I24 neither fixed
nor permanent
instantaneous centers they vary with
configuration of
mechanism
9282013
Dr Mohammad Abuhaiba PE
12
68 Location of Instantaneous Centers
9282013
Dr Mohammad Abuhaiba PE
13
69 Aronhold Kennedy (Three
Centers in Line) Theorem if three bodies move
relatively to each other they have three ICs that lie on a straight line
Consider three kinematic links A B and C having relative plane motion
Number of ICs (N) is given by
9282013
Dr Mohammad Abuhaiba PE
14
69 Aronhold Kennedy (Three
Centers in Line) Theorem The two ICs at the pin joints of B with
A and C with A (Iab and Iac) are permanent instantaneous centers
According to Aronhold Kennedyrsquos theorem the third IC Ibc must lie on the line joining Iab and Iac
let us consider that instantaneous center Ibc lies outside the line joining Iab and Iac
point Ibc belongs to both links B and C Let us consider point Ibc on link B Its velocity vBC must be perpendicular to the line joining Iab and Ibc Now consider the point Ibc on link C Its velocity vBC must be perpendicular to the line joining Iac and Ibc
9282013
Dr Mohammad Abuhaiba PE
15
69 Aronhold Kennedy (or
Three Centers in Line) Theorem velocity of IC is same whether it is
regarded as a point on first link or as a point on second link
Therefore velocity of point Ibc cannot be perpendicular to both lines Iab Ibc and IacIbc unless the point Ibc lies on the line joining the points Iab and Iac
Thus the three instantaneous centers (Iab Iac and Ibc) must lie on the same straight line
The exact location of Ibc on line Iab Iac depends upon the directions and magnitudes of the angular velocities of B and C relative to A
9282013
Dr Mohammad Abuhaiba PE
16
610 Method of Locating Instantaneous
Centers in a Mechanism
Fig 68 (a)
1 Determine number of ICs (N)
2 Make a list of all instantaneous centers
9282013
Dr Mohammad Abuhaiba PE
17
Links 1 2 3 4
instantaneous
centers
12 23 34
13 24
14
610 Method of Locating Instantaneous
Centers in a Mechanism
3 Locate fixed amp permanent ICs by
inspection
I12 and I14 are fixed Ics
I23 and I34 are permanent IC
The four bar mechanism has
four turning pairs
four primary (fixed and permanent) ICs located at
the centers of the pin joints
9282013
Dr Mohammad Abuhaiba PE
18
610 Method of Locating Instantaneous
Centers in a Mechanism
4 Locate remaining neither fixed nor
permanent ICs (secondary centers) by
Kennedyrsquos theorem
Circle diagram Fig 68 (b)
Mark points on a circle equal to number
of links in a mechanism
5 Join points by solid lines to show that
these centers are already found
9282013
Dr Mohammad Abuhaiba PE
19
610 Method of Locating Instantaneous
Centers in a Mechanism
9282013
Dr Mohammad Abuhaiba PE
20
Example 61 In a pin jointed four bar mechanism Fig 69 AB = 300 mm BC = CD = 360 mm and AD = 600 mm The angle BAD = 60deg The crank AB rotates uniformly at 100 rpm Locate all instantaneous centers and find the angular velocity of link BC
9282013
Dr Mohammad Abuhaiba PE
21
Example 61
9282013
Dr Mohammad Abuhaiba PE
22
Example 62 Locate all ICs of the slider crank mechanism shown in Fig 612 The lengths of crank OB and connecting rod AB are 100 mm and 400 mm respectively If the crank rotates cw with an angular velocity of 10 rads find
1 Velocity of the slider A
2 Angular velocity of connecting rod AB
9282013
Dr Mohammad Abuhaiba PE
23
Example 62
9282013
Dr Mohammad Abuhaiba PE
24
Example 63
A mechanism Fig 615 has the following dimensions
OA = 200 mm AB = 1500 mm BC = 600 mm CD = 500
mm and BE = 400 mm Locate all the ICs If crank OA
rotates uniformly at 120 rpm cw find
1 velocity of B C and D
2 angular velocity of links AB BC and CD
9282013
Dr Mohammad Abuhaiba PE
25
Example 63 9282013
Dr Mohammad Abuhaiba PE
26
Example 64 The mechanism of a wrapping machine Fig 618 has the following dimensions O1A = 100 mm AC = 700 mm BC = 200 mm O3C = 200 mm O2E = 400 mm O2D = 200 mm and BD = 150 mm The crank O1A rotates at a uniform speed of 100 rads Find the velocity of point E of the bell crank lever by instantaneous center method
9282013
Dr Mohammad Abuhaiba PE
27
Example 64
9282013
Dr Mohammad Abuhaiba PE
28
Example 65 Fig 621 shows a sewing needle bar mechanism O1ABO2CD wherein the different dimensions are as follows Crank O1A = 16 mm b = 45deg Vertical distance between O1 and O2 = 40 mm Horizontal distance between O1 amp O2 = 13 mm O2B = 23 mm AB = 35 mm O2BC = 90deg BC = 16 mm CD = 40 mm D lies vertically below O1 Find the velocity of needle at D for the given configuration The crank O1A rotates at 400 rpm
9282013
Dr Mohammad Abuhaiba PE
29
Example 65
9282013
Dr Mohammad Abuhaiba PE
30
Example 66
Fig 624 shows a Whitworth
quick return motion
mechanism The various
dimensions in the mechanism
are as follows OQ = 100 mm
OA = 200 mm QC = 150 mm
and CD = 500 mm The crank
OA makes an angle of 60deg
with the vertical and rotates at
120 rpm cw Locate all ICs
and find the velocity of ram D
9282013
Dr Mohammad Abuhaiba PE
31
Example 66
9282013
Dr Mohammad Abuhaiba PE
32
67 Types of Instantaneous Centers
Fixed Instantaneous Centersgt I12 amp I14
remain in same place
for all configurations
of mechanism
9282013
Dr Mohammad Abuhaiba PE
11
67 Types of Instantaneous
Centers
I23 amp I34 permanent
instantaneous centers
they move when
mechanism moves
but the joints are of
permanent nature
I13 amp I24 neither fixed
nor permanent
instantaneous centers they vary with
configuration of
mechanism
9282013
Dr Mohammad Abuhaiba PE
12
68 Location of Instantaneous Centers
9282013
Dr Mohammad Abuhaiba PE
13
69 Aronhold Kennedy (Three
Centers in Line) Theorem if three bodies move
relatively to each other they have three ICs that lie on a straight line
Consider three kinematic links A B and C having relative plane motion
Number of ICs (N) is given by
9282013
Dr Mohammad Abuhaiba PE
14
69 Aronhold Kennedy (Three
Centers in Line) Theorem The two ICs at the pin joints of B with
A and C with A (Iab and Iac) are permanent instantaneous centers
According to Aronhold Kennedyrsquos theorem the third IC Ibc must lie on the line joining Iab and Iac
let us consider that instantaneous center Ibc lies outside the line joining Iab and Iac
point Ibc belongs to both links B and C Let us consider point Ibc on link B Its velocity vBC must be perpendicular to the line joining Iab and Ibc Now consider the point Ibc on link C Its velocity vBC must be perpendicular to the line joining Iac and Ibc
9282013
Dr Mohammad Abuhaiba PE
15
69 Aronhold Kennedy (or
Three Centers in Line) Theorem velocity of IC is same whether it is
regarded as a point on first link or as a point on second link
Therefore velocity of point Ibc cannot be perpendicular to both lines Iab Ibc and IacIbc unless the point Ibc lies on the line joining the points Iab and Iac
Thus the three instantaneous centers (Iab Iac and Ibc) must lie on the same straight line
The exact location of Ibc on line Iab Iac depends upon the directions and magnitudes of the angular velocities of B and C relative to A
9282013
Dr Mohammad Abuhaiba PE
16
610 Method of Locating Instantaneous
Centers in a Mechanism
Fig 68 (a)
1 Determine number of ICs (N)
2 Make a list of all instantaneous centers
9282013
Dr Mohammad Abuhaiba PE
17
Links 1 2 3 4
instantaneous
centers
12 23 34
13 24
14
610 Method of Locating Instantaneous
Centers in a Mechanism
3 Locate fixed amp permanent ICs by
inspection
I12 and I14 are fixed Ics
I23 and I34 are permanent IC
The four bar mechanism has
four turning pairs
four primary (fixed and permanent) ICs located at
the centers of the pin joints
9282013
Dr Mohammad Abuhaiba PE
18
610 Method of Locating Instantaneous
Centers in a Mechanism
4 Locate remaining neither fixed nor
permanent ICs (secondary centers) by
Kennedyrsquos theorem
Circle diagram Fig 68 (b)
Mark points on a circle equal to number
of links in a mechanism
5 Join points by solid lines to show that
these centers are already found
9282013
Dr Mohammad Abuhaiba PE
19
610 Method of Locating Instantaneous
Centers in a Mechanism
9282013
Dr Mohammad Abuhaiba PE
20
Example 61 In a pin jointed four bar mechanism Fig 69 AB = 300 mm BC = CD = 360 mm and AD = 600 mm The angle BAD = 60deg The crank AB rotates uniformly at 100 rpm Locate all instantaneous centers and find the angular velocity of link BC
9282013
Dr Mohammad Abuhaiba PE
21
Example 61
9282013
Dr Mohammad Abuhaiba PE
22
Example 62 Locate all ICs of the slider crank mechanism shown in Fig 612 The lengths of crank OB and connecting rod AB are 100 mm and 400 mm respectively If the crank rotates cw with an angular velocity of 10 rads find
1 Velocity of the slider A
2 Angular velocity of connecting rod AB
9282013
Dr Mohammad Abuhaiba PE
23
Example 62
9282013
Dr Mohammad Abuhaiba PE
24
Example 63
A mechanism Fig 615 has the following dimensions
OA = 200 mm AB = 1500 mm BC = 600 mm CD = 500
mm and BE = 400 mm Locate all the ICs If crank OA
rotates uniformly at 120 rpm cw find
1 velocity of B C and D
2 angular velocity of links AB BC and CD
9282013
Dr Mohammad Abuhaiba PE
25
Example 63 9282013
Dr Mohammad Abuhaiba PE
26
Example 64 The mechanism of a wrapping machine Fig 618 has the following dimensions O1A = 100 mm AC = 700 mm BC = 200 mm O3C = 200 mm O2E = 400 mm O2D = 200 mm and BD = 150 mm The crank O1A rotates at a uniform speed of 100 rads Find the velocity of point E of the bell crank lever by instantaneous center method
9282013
Dr Mohammad Abuhaiba PE
27
Example 64
9282013
Dr Mohammad Abuhaiba PE
28
Example 65 Fig 621 shows a sewing needle bar mechanism O1ABO2CD wherein the different dimensions are as follows Crank O1A = 16 mm b = 45deg Vertical distance between O1 and O2 = 40 mm Horizontal distance between O1 amp O2 = 13 mm O2B = 23 mm AB = 35 mm O2BC = 90deg BC = 16 mm CD = 40 mm D lies vertically below O1 Find the velocity of needle at D for the given configuration The crank O1A rotates at 400 rpm
9282013
Dr Mohammad Abuhaiba PE
29
Example 65
9282013
Dr Mohammad Abuhaiba PE
30
Example 66
Fig 624 shows a Whitworth
quick return motion
mechanism The various
dimensions in the mechanism
are as follows OQ = 100 mm
OA = 200 mm QC = 150 mm
and CD = 500 mm The crank
OA makes an angle of 60deg
with the vertical and rotates at
120 rpm cw Locate all ICs
and find the velocity of ram D
9282013
Dr Mohammad Abuhaiba PE
31
Example 66
9282013
Dr Mohammad Abuhaiba PE
32
67 Types of Instantaneous
Centers
I23 amp I34 permanent
instantaneous centers
they move when
mechanism moves
but the joints are of
permanent nature
I13 amp I24 neither fixed
nor permanent
instantaneous centers they vary with
configuration of
mechanism
9282013
Dr Mohammad Abuhaiba PE
12
68 Location of Instantaneous Centers
9282013
Dr Mohammad Abuhaiba PE
13
69 Aronhold Kennedy (Three
Centers in Line) Theorem if three bodies move
relatively to each other they have three ICs that lie on a straight line
Consider three kinematic links A B and C having relative plane motion
Number of ICs (N) is given by
9282013
Dr Mohammad Abuhaiba PE
14
69 Aronhold Kennedy (Three
Centers in Line) Theorem The two ICs at the pin joints of B with
A and C with A (Iab and Iac) are permanent instantaneous centers
According to Aronhold Kennedyrsquos theorem the third IC Ibc must lie on the line joining Iab and Iac
let us consider that instantaneous center Ibc lies outside the line joining Iab and Iac
point Ibc belongs to both links B and C Let us consider point Ibc on link B Its velocity vBC must be perpendicular to the line joining Iab and Ibc Now consider the point Ibc on link C Its velocity vBC must be perpendicular to the line joining Iac and Ibc
9282013
Dr Mohammad Abuhaiba PE
15
69 Aronhold Kennedy (or
Three Centers in Line) Theorem velocity of IC is same whether it is
regarded as a point on first link or as a point on second link
Therefore velocity of point Ibc cannot be perpendicular to both lines Iab Ibc and IacIbc unless the point Ibc lies on the line joining the points Iab and Iac
Thus the three instantaneous centers (Iab Iac and Ibc) must lie on the same straight line
The exact location of Ibc on line Iab Iac depends upon the directions and magnitudes of the angular velocities of B and C relative to A
9282013
Dr Mohammad Abuhaiba PE
16
610 Method of Locating Instantaneous
Centers in a Mechanism
Fig 68 (a)
1 Determine number of ICs (N)
2 Make a list of all instantaneous centers
9282013
Dr Mohammad Abuhaiba PE
17
Links 1 2 3 4
instantaneous
centers
12 23 34
13 24
14
610 Method of Locating Instantaneous
Centers in a Mechanism
3 Locate fixed amp permanent ICs by
inspection
I12 and I14 are fixed Ics
I23 and I34 are permanent IC
The four bar mechanism has
four turning pairs
four primary (fixed and permanent) ICs located at
the centers of the pin joints
9282013
Dr Mohammad Abuhaiba PE
18
610 Method of Locating Instantaneous
Centers in a Mechanism
4 Locate remaining neither fixed nor
permanent ICs (secondary centers) by
Kennedyrsquos theorem
Circle diagram Fig 68 (b)
Mark points on a circle equal to number
of links in a mechanism
5 Join points by solid lines to show that
these centers are already found
9282013
Dr Mohammad Abuhaiba PE
19
610 Method of Locating Instantaneous
Centers in a Mechanism
9282013
Dr Mohammad Abuhaiba PE
20
Example 61 In a pin jointed four bar mechanism Fig 69 AB = 300 mm BC = CD = 360 mm and AD = 600 mm The angle BAD = 60deg The crank AB rotates uniformly at 100 rpm Locate all instantaneous centers and find the angular velocity of link BC
9282013
Dr Mohammad Abuhaiba PE
21
Example 61
9282013
Dr Mohammad Abuhaiba PE
22
Example 62 Locate all ICs of the slider crank mechanism shown in Fig 612 The lengths of crank OB and connecting rod AB are 100 mm and 400 mm respectively If the crank rotates cw with an angular velocity of 10 rads find
1 Velocity of the slider A
2 Angular velocity of connecting rod AB
9282013
Dr Mohammad Abuhaiba PE
23
Example 62
9282013
Dr Mohammad Abuhaiba PE
24
Example 63
A mechanism Fig 615 has the following dimensions
OA = 200 mm AB = 1500 mm BC = 600 mm CD = 500
mm and BE = 400 mm Locate all the ICs If crank OA
rotates uniformly at 120 rpm cw find
1 velocity of B C and D
2 angular velocity of links AB BC and CD
9282013
Dr Mohammad Abuhaiba PE
25
Example 63 9282013
Dr Mohammad Abuhaiba PE
26
Example 64 The mechanism of a wrapping machine Fig 618 has the following dimensions O1A = 100 mm AC = 700 mm BC = 200 mm O3C = 200 mm O2E = 400 mm O2D = 200 mm and BD = 150 mm The crank O1A rotates at a uniform speed of 100 rads Find the velocity of point E of the bell crank lever by instantaneous center method
9282013
Dr Mohammad Abuhaiba PE
27
Example 64
9282013
Dr Mohammad Abuhaiba PE
28
Example 65 Fig 621 shows a sewing needle bar mechanism O1ABO2CD wherein the different dimensions are as follows Crank O1A = 16 mm b = 45deg Vertical distance between O1 and O2 = 40 mm Horizontal distance between O1 amp O2 = 13 mm O2B = 23 mm AB = 35 mm O2BC = 90deg BC = 16 mm CD = 40 mm D lies vertically below O1 Find the velocity of needle at D for the given configuration The crank O1A rotates at 400 rpm
9282013
Dr Mohammad Abuhaiba PE
29
Example 65
9282013
Dr Mohammad Abuhaiba PE
30
Example 66
Fig 624 shows a Whitworth
quick return motion
mechanism The various
dimensions in the mechanism
are as follows OQ = 100 mm
OA = 200 mm QC = 150 mm
and CD = 500 mm The crank
OA makes an angle of 60deg
with the vertical and rotates at
120 rpm cw Locate all ICs
and find the velocity of ram D
9282013
Dr Mohammad Abuhaiba PE
31
Example 66
9282013
Dr Mohammad Abuhaiba PE
32
68 Location of Instantaneous Centers
9282013
Dr Mohammad Abuhaiba PE
13
69 Aronhold Kennedy (Three
Centers in Line) Theorem if three bodies move
relatively to each other they have three ICs that lie on a straight line
Consider three kinematic links A B and C having relative plane motion
Number of ICs (N) is given by
9282013
Dr Mohammad Abuhaiba PE
14
69 Aronhold Kennedy (Three
Centers in Line) Theorem The two ICs at the pin joints of B with
A and C with A (Iab and Iac) are permanent instantaneous centers
According to Aronhold Kennedyrsquos theorem the third IC Ibc must lie on the line joining Iab and Iac
let us consider that instantaneous center Ibc lies outside the line joining Iab and Iac
point Ibc belongs to both links B and C Let us consider point Ibc on link B Its velocity vBC must be perpendicular to the line joining Iab and Ibc Now consider the point Ibc on link C Its velocity vBC must be perpendicular to the line joining Iac and Ibc
9282013
Dr Mohammad Abuhaiba PE
15
69 Aronhold Kennedy (or
Three Centers in Line) Theorem velocity of IC is same whether it is
regarded as a point on first link or as a point on second link
Therefore velocity of point Ibc cannot be perpendicular to both lines Iab Ibc and IacIbc unless the point Ibc lies on the line joining the points Iab and Iac
Thus the three instantaneous centers (Iab Iac and Ibc) must lie on the same straight line
The exact location of Ibc on line Iab Iac depends upon the directions and magnitudes of the angular velocities of B and C relative to A
9282013
Dr Mohammad Abuhaiba PE
16
610 Method of Locating Instantaneous
Centers in a Mechanism
Fig 68 (a)
1 Determine number of ICs (N)
2 Make a list of all instantaneous centers
9282013
Dr Mohammad Abuhaiba PE
17
Links 1 2 3 4
instantaneous
centers
12 23 34
13 24
14
610 Method of Locating Instantaneous
Centers in a Mechanism
3 Locate fixed amp permanent ICs by
inspection
I12 and I14 are fixed Ics
I23 and I34 are permanent IC
The four bar mechanism has
four turning pairs
four primary (fixed and permanent) ICs located at
the centers of the pin joints
9282013
Dr Mohammad Abuhaiba PE
18
610 Method of Locating Instantaneous
Centers in a Mechanism
4 Locate remaining neither fixed nor
permanent ICs (secondary centers) by
Kennedyrsquos theorem
Circle diagram Fig 68 (b)
Mark points on a circle equal to number
of links in a mechanism
5 Join points by solid lines to show that
these centers are already found
9282013
Dr Mohammad Abuhaiba PE
19
610 Method of Locating Instantaneous
Centers in a Mechanism
9282013
Dr Mohammad Abuhaiba PE
20
Example 61 In a pin jointed four bar mechanism Fig 69 AB = 300 mm BC = CD = 360 mm and AD = 600 mm The angle BAD = 60deg The crank AB rotates uniformly at 100 rpm Locate all instantaneous centers and find the angular velocity of link BC
9282013
Dr Mohammad Abuhaiba PE
21
Example 61
9282013
Dr Mohammad Abuhaiba PE
22
Example 62 Locate all ICs of the slider crank mechanism shown in Fig 612 The lengths of crank OB and connecting rod AB are 100 mm and 400 mm respectively If the crank rotates cw with an angular velocity of 10 rads find
1 Velocity of the slider A
2 Angular velocity of connecting rod AB
9282013
Dr Mohammad Abuhaiba PE
23
Example 62
9282013
Dr Mohammad Abuhaiba PE
24
Example 63
A mechanism Fig 615 has the following dimensions
OA = 200 mm AB = 1500 mm BC = 600 mm CD = 500
mm and BE = 400 mm Locate all the ICs If crank OA
rotates uniformly at 120 rpm cw find
1 velocity of B C and D
2 angular velocity of links AB BC and CD
9282013
Dr Mohammad Abuhaiba PE
25
Example 63 9282013
Dr Mohammad Abuhaiba PE
26
Example 64 The mechanism of a wrapping machine Fig 618 has the following dimensions O1A = 100 mm AC = 700 mm BC = 200 mm O3C = 200 mm O2E = 400 mm O2D = 200 mm and BD = 150 mm The crank O1A rotates at a uniform speed of 100 rads Find the velocity of point E of the bell crank lever by instantaneous center method
9282013
Dr Mohammad Abuhaiba PE
27
Example 64
9282013
Dr Mohammad Abuhaiba PE
28
Example 65 Fig 621 shows a sewing needle bar mechanism O1ABO2CD wherein the different dimensions are as follows Crank O1A = 16 mm b = 45deg Vertical distance between O1 and O2 = 40 mm Horizontal distance between O1 amp O2 = 13 mm O2B = 23 mm AB = 35 mm O2BC = 90deg BC = 16 mm CD = 40 mm D lies vertically below O1 Find the velocity of needle at D for the given configuration The crank O1A rotates at 400 rpm
9282013
Dr Mohammad Abuhaiba PE
29
Example 65
9282013
Dr Mohammad Abuhaiba PE
30
Example 66
Fig 624 shows a Whitworth
quick return motion
mechanism The various
dimensions in the mechanism
are as follows OQ = 100 mm
OA = 200 mm QC = 150 mm
and CD = 500 mm The crank
OA makes an angle of 60deg
with the vertical and rotates at
120 rpm cw Locate all ICs
and find the velocity of ram D
9282013
Dr Mohammad Abuhaiba PE
31
Example 66
9282013
Dr Mohammad Abuhaiba PE
32
69 Aronhold Kennedy (Three
Centers in Line) Theorem if three bodies move
relatively to each other they have three ICs that lie on a straight line
Consider three kinematic links A B and C having relative plane motion
Number of ICs (N) is given by
9282013
Dr Mohammad Abuhaiba PE
14
69 Aronhold Kennedy (Three
Centers in Line) Theorem The two ICs at the pin joints of B with
A and C with A (Iab and Iac) are permanent instantaneous centers
According to Aronhold Kennedyrsquos theorem the third IC Ibc must lie on the line joining Iab and Iac
let us consider that instantaneous center Ibc lies outside the line joining Iab and Iac
point Ibc belongs to both links B and C Let us consider point Ibc on link B Its velocity vBC must be perpendicular to the line joining Iab and Ibc Now consider the point Ibc on link C Its velocity vBC must be perpendicular to the line joining Iac and Ibc
9282013
Dr Mohammad Abuhaiba PE
15
69 Aronhold Kennedy (or
Three Centers in Line) Theorem velocity of IC is same whether it is
regarded as a point on first link or as a point on second link
Therefore velocity of point Ibc cannot be perpendicular to both lines Iab Ibc and IacIbc unless the point Ibc lies on the line joining the points Iab and Iac
Thus the three instantaneous centers (Iab Iac and Ibc) must lie on the same straight line
The exact location of Ibc on line Iab Iac depends upon the directions and magnitudes of the angular velocities of B and C relative to A
9282013
Dr Mohammad Abuhaiba PE
16
610 Method of Locating Instantaneous
Centers in a Mechanism
Fig 68 (a)
1 Determine number of ICs (N)
2 Make a list of all instantaneous centers
9282013
Dr Mohammad Abuhaiba PE
17
Links 1 2 3 4
instantaneous
centers
12 23 34
13 24
14
610 Method of Locating Instantaneous
Centers in a Mechanism
3 Locate fixed amp permanent ICs by
inspection
I12 and I14 are fixed Ics
I23 and I34 are permanent IC
The four bar mechanism has
four turning pairs
four primary (fixed and permanent) ICs located at
the centers of the pin joints
9282013
Dr Mohammad Abuhaiba PE
18
610 Method of Locating Instantaneous
Centers in a Mechanism
4 Locate remaining neither fixed nor
permanent ICs (secondary centers) by
Kennedyrsquos theorem
Circle diagram Fig 68 (b)
Mark points on a circle equal to number
of links in a mechanism
5 Join points by solid lines to show that
these centers are already found
9282013
Dr Mohammad Abuhaiba PE
19
610 Method of Locating Instantaneous
Centers in a Mechanism
9282013
Dr Mohammad Abuhaiba PE
20
Example 61 In a pin jointed four bar mechanism Fig 69 AB = 300 mm BC = CD = 360 mm and AD = 600 mm The angle BAD = 60deg The crank AB rotates uniformly at 100 rpm Locate all instantaneous centers and find the angular velocity of link BC
9282013
Dr Mohammad Abuhaiba PE
21
Example 61
9282013
Dr Mohammad Abuhaiba PE
22
Example 62 Locate all ICs of the slider crank mechanism shown in Fig 612 The lengths of crank OB and connecting rod AB are 100 mm and 400 mm respectively If the crank rotates cw with an angular velocity of 10 rads find
1 Velocity of the slider A
2 Angular velocity of connecting rod AB
9282013
Dr Mohammad Abuhaiba PE
23
Example 62
9282013
Dr Mohammad Abuhaiba PE
24
Example 63
A mechanism Fig 615 has the following dimensions
OA = 200 mm AB = 1500 mm BC = 600 mm CD = 500
mm and BE = 400 mm Locate all the ICs If crank OA
rotates uniformly at 120 rpm cw find
1 velocity of B C and D
2 angular velocity of links AB BC and CD
9282013
Dr Mohammad Abuhaiba PE
25
Example 63 9282013
Dr Mohammad Abuhaiba PE
26
Example 64 The mechanism of a wrapping machine Fig 618 has the following dimensions O1A = 100 mm AC = 700 mm BC = 200 mm O3C = 200 mm O2E = 400 mm O2D = 200 mm and BD = 150 mm The crank O1A rotates at a uniform speed of 100 rads Find the velocity of point E of the bell crank lever by instantaneous center method
9282013
Dr Mohammad Abuhaiba PE
27
Example 64
9282013
Dr Mohammad Abuhaiba PE
28
Example 65 Fig 621 shows a sewing needle bar mechanism O1ABO2CD wherein the different dimensions are as follows Crank O1A = 16 mm b = 45deg Vertical distance between O1 and O2 = 40 mm Horizontal distance between O1 amp O2 = 13 mm O2B = 23 mm AB = 35 mm O2BC = 90deg BC = 16 mm CD = 40 mm D lies vertically below O1 Find the velocity of needle at D for the given configuration The crank O1A rotates at 400 rpm
9282013
Dr Mohammad Abuhaiba PE
29
Example 65
9282013
Dr Mohammad Abuhaiba PE
30
Example 66
Fig 624 shows a Whitworth
quick return motion
mechanism The various
dimensions in the mechanism
are as follows OQ = 100 mm
OA = 200 mm QC = 150 mm
and CD = 500 mm The crank
OA makes an angle of 60deg
with the vertical and rotates at
120 rpm cw Locate all ICs
and find the velocity of ram D
9282013
Dr Mohammad Abuhaiba PE
31
Example 66
9282013
Dr Mohammad Abuhaiba PE
32
69 Aronhold Kennedy (Three
Centers in Line) Theorem The two ICs at the pin joints of B with
A and C with A (Iab and Iac) are permanent instantaneous centers
According to Aronhold Kennedyrsquos theorem the third IC Ibc must lie on the line joining Iab and Iac
let us consider that instantaneous center Ibc lies outside the line joining Iab and Iac
point Ibc belongs to both links B and C Let us consider point Ibc on link B Its velocity vBC must be perpendicular to the line joining Iab and Ibc Now consider the point Ibc on link C Its velocity vBC must be perpendicular to the line joining Iac and Ibc
9282013
Dr Mohammad Abuhaiba PE
15
69 Aronhold Kennedy (or
Three Centers in Line) Theorem velocity of IC is same whether it is
regarded as a point on first link or as a point on second link
Therefore velocity of point Ibc cannot be perpendicular to both lines Iab Ibc and IacIbc unless the point Ibc lies on the line joining the points Iab and Iac
Thus the three instantaneous centers (Iab Iac and Ibc) must lie on the same straight line
The exact location of Ibc on line Iab Iac depends upon the directions and magnitudes of the angular velocities of B and C relative to A
9282013
Dr Mohammad Abuhaiba PE
16
610 Method of Locating Instantaneous
Centers in a Mechanism
Fig 68 (a)
1 Determine number of ICs (N)
2 Make a list of all instantaneous centers
9282013
Dr Mohammad Abuhaiba PE
17
Links 1 2 3 4
instantaneous
centers
12 23 34
13 24
14
610 Method of Locating Instantaneous
Centers in a Mechanism
3 Locate fixed amp permanent ICs by
inspection
I12 and I14 are fixed Ics
I23 and I34 are permanent IC
The four bar mechanism has
four turning pairs
four primary (fixed and permanent) ICs located at
the centers of the pin joints
9282013
Dr Mohammad Abuhaiba PE
18
610 Method of Locating Instantaneous
Centers in a Mechanism
4 Locate remaining neither fixed nor
permanent ICs (secondary centers) by
Kennedyrsquos theorem
Circle diagram Fig 68 (b)
Mark points on a circle equal to number
of links in a mechanism
5 Join points by solid lines to show that
these centers are already found
9282013
Dr Mohammad Abuhaiba PE
19
610 Method of Locating Instantaneous
Centers in a Mechanism
9282013
Dr Mohammad Abuhaiba PE
20
Example 61 In a pin jointed four bar mechanism Fig 69 AB = 300 mm BC = CD = 360 mm and AD = 600 mm The angle BAD = 60deg The crank AB rotates uniformly at 100 rpm Locate all instantaneous centers and find the angular velocity of link BC
9282013
Dr Mohammad Abuhaiba PE
21
Example 61
9282013
Dr Mohammad Abuhaiba PE
22
Example 62 Locate all ICs of the slider crank mechanism shown in Fig 612 The lengths of crank OB and connecting rod AB are 100 mm and 400 mm respectively If the crank rotates cw with an angular velocity of 10 rads find
1 Velocity of the slider A
2 Angular velocity of connecting rod AB
9282013
Dr Mohammad Abuhaiba PE
23
Example 62
9282013
Dr Mohammad Abuhaiba PE
24
Example 63
A mechanism Fig 615 has the following dimensions
OA = 200 mm AB = 1500 mm BC = 600 mm CD = 500
mm and BE = 400 mm Locate all the ICs If crank OA
rotates uniformly at 120 rpm cw find
1 velocity of B C and D
2 angular velocity of links AB BC and CD
9282013
Dr Mohammad Abuhaiba PE
25
Example 63 9282013
Dr Mohammad Abuhaiba PE
26
Example 64 The mechanism of a wrapping machine Fig 618 has the following dimensions O1A = 100 mm AC = 700 mm BC = 200 mm O3C = 200 mm O2E = 400 mm O2D = 200 mm and BD = 150 mm The crank O1A rotates at a uniform speed of 100 rads Find the velocity of point E of the bell crank lever by instantaneous center method
9282013
Dr Mohammad Abuhaiba PE
27
Example 64
9282013
Dr Mohammad Abuhaiba PE
28
Example 65 Fig 621 shows a sewing needle bar mechanism O1ABO2CD wherein the different dimensions are as follows Crank O1A = 16 mm b = 45deg Vertical distance between O1 and O2 = 40 mm Horizontal distance between O1 amp O2 = 13 mm O2B = 23 mm AB = 35 mm O2BC = 90deg BC = 16 mm CD = 40 mm D lies vertically below O1 Find the velocity of needle at D for the given configuration The crank O1A rotates at 400 rpm
9282013
Dr Mohammad Abuhaiba PE
29
Example 65
9282013
Dr Mohammad Abuhaiba PE
30
Example 66
Fig 624 shows a Whitworth
quick return motion
mechanism The various
dimensions in the mechanism
are as follows OQ = 100 mm
OA = 200 mm QC = 150 mm
and CD = 500 mm The crank
OA makes an angle of 60deg
with the vertical and rotates at
120 rpm cw Locate all ICs
and find the velocity of ram D
9282013
Dr Mohammad Abuhaiba PE
31
Example 66
9282013
Dr Mohammad Abuhaiba PE
32
69 Aronhold Kennedy (or
Three Centers in Line) Theorem velocity of IC is same whether it is
regarded as a point on first link or as a point on second link
Therefore velocity of point Ibc cannot be perpendicular to both lines Iab Ibc and IacIbc unless the point Ibc lies on the line joining the points Iab and Iac
Thus the three instantaneous centers (Iab Iac and Ibc) must lie on the same straight line
The exact location of Ibc on line Iab Iac depends upon the directions and magnitudes of the angular velocities of B and C relative to A
9282013
Dr Mohammad Abuhaiba PE
16
610 Method of Locating Instantaneous
Centers in a Mechanism
Fig 68 (a)
1 Determine number of ICs (N)
2 Make a list of all instantaneous centers
9282013
Dr Mohammad Abuhaiba PE
17
Links 1 2 3 4
instantaneous
centers
12 23 34
13 24
14
610 Method of Locating Instantaneous
Centers in a Mechanism
3 Locate fixed amp permanent ICs by
inspection
I12 and I14 are fixed Ics
I23 and I34 are permanent IC
The four bar mechanism has
four turning pairs
four primary (fixed and permanent) ICs located at
the centers of the pin joints
9282013
Dr Mohammad Abuhaiba PE
18
610 Method of Locating Instantaneous
Centers in a Mechanism
4 Locate remaining neither fixed nor
permanent ICs (secondary centers) by
Kennedyrsquos theorem
Circle diagram Fig 68 (b)
Mark points on a circle equal to number
of links in a mechanism
5 Join points by solid lines to show that
these centers are already found
9282013
Dr Mohammad Abuhaiba PE
19
610 Method of Locating Instantaneous
Centers in a Mechanism
9282013
Dr Mohammad Abuhaiba PE
20
Example 61 In a pin jointed four bar mechanism Fig 69 AB = 300 mm BC = CD = 360 mm and AD = 600 mm The angle BAD = 60deg The crank AB rotates uniformly at 100 rpm Locate all instantaneous centers and find the angular velocity of link BC
9282013
Dr Mohammad Abuhaiba PE
21
Example 61
9282013
Dr Mohammad Abuhaiba PE
22
Example 62 Locate all ICs of the slider crank mechanism shown in Fig 612 The lengths of crank OB and connecting rod AB are 100 mm and 400 mm respectively If the crank rotates cw with an angular velocity of 10 rads find
1 Velocity of the slider A
2 Angular velocity of connecting rod AB
9282013
Dr Mohammad Abuhaiba PE
23
Example 62
9282013
Dr Mohammad Abuhaiba PE
24
Example 63
A mechanism Fig 615 has the following dimensions
OA = 200 mm AB = 1500 mm BC = 600 mm CD = 500
mm and BE = 400 mm Locate all the ICs If crank OA
rotates uniformly at 120 rpm cw find
1 velocity of B C and D
2 angular velocity of links AB BC and CD
9282013
Dr Mohammad Abuhaiba PE
25
Example 63 9282013
Dr Mohammad Abuhaiba PE
26
Example 64 The mechanism of a wrapping machine Fig 618 has the following dimensions O1A = 100 mm AC = 700 mm BC = 200 mm O3C = 200 mm O2E = 400 mm O2D = 200 mm and BD = 150 mm The crank O1A rotates at a uniform speed of 100 rads Find the velocity of point E of the bell crank lever by instantaneous center method
9282013
Dr Mohammad Abuhaiba PE
27
Example 64
9282013
Dr Mohammad Abuhaiba PE
28
Example 65 Fig 621 shows a sewing needle bar mechanism O1ABO2CD wherein the different dimensions are as follows Crank O1A = 16 mm b = 45deg Vertical distance between O1 and O2 = 40 mm Horizontal distance between O1 amp O2 = 13 mm O2B = 23 mm AB = 35 mm O2BC = 90deg BC = 16 mm CD = 40 mm D lies vertically below O1 Find the velocity of needle at D for the given configuration The crank O1A rotates at 400 rpm
9282013
Dr Mohammad Abuhaiba PE
29
Example 65
9282013
Dr Mohammad Abuhaiba PE
30
Example 66
Fig 624 shows a Whitworth
quick return motion
mechanism The various
dimensions in the mechanism
are as follows OQ = 100 mm
OA = 200 mm QC = 150 mm
and CD = 500 mm The crank
OA makes an angle of 60deg
with the vertical and rotates at
120 rpm cw Locate all ICs
and find the velocity of ram D
9282013
Dr Mohammad Abuhaiba PE
31
Example 66
9282013
Dr Mohammad Abuhaiba PE
32
610 Method of Locating Instantaneous
Centers in a Mechanism
Fig 68 (a)
1 Determine number of ICs (N)
2 Make a list of all instantaneous centers
9282013
Dr Mohammad Abuhaiba PE
17
Links 1 2 3 4
instantaneous
centers
12 23 34
13 24
14
610 Method of Locating Instantaneous
Centers in a Mechanism
3 Locate fixed amp permanent ICs by
inspection
I12 and I14 are fixed Ics
I23 and I34 are permanent IC
The four bar mechanism has
four turning pairs
four primary (fixed and permanent) ICs located at
the centers of the pin joints
9282013
Dr Mohammad Abuhaiba PE
18
610 Method of Locating Instantaneous
Centers in a Mechanism
4 Locate remaining neither fixed nor
permanent ICs (secondary centers) by
Kennedyrsquos theorem
Circle diagram Fig 68 (b)
Mark points on a circle equal to number
of links in a mechanism
5 Join points by solid lines to show that
these centers are already found
9282013
Dr Mohammad Abuhaiba PE
19
610 Method of Locating Instantaneous
Centers in a Mechanism
9282013
Dr Mohammad Abuhaiba PE
20
Example 61 In a pin jointed four bar mechanism Fig 69 AB = 300 mm BC = CD = 360 mm and AD = 600 mm The angle BAD = 60deg The crank AB rotates uniformly at 100 rpm Locate all instantaneous centers and find the angular velocity of link BC
9282013
Dr Mohammad Abuhaiba PE
21
Example 61
9282013
Dr Mohammad Abuhaiba PE
22
Example 62 Locate all ICs of the slider crank mechanism shown in Fig 612 The lengths of crank OB and connecting rod AB are 100 mm and 400 mm respectively If the crank rotates cw with an angular velocity of 10 rads find
1 Velocity of the slider A
2 Angular velocity of connecting rod AB
9282013
Dr Mohammad Abuhaiba PE
23
Example 62
9282013
Dr Mohammad Abuhaiba PE
24
Example 63
A mechanism Fig 615 has the following dimensions
OA = 200 mm AB = 1500 mm BC = 600 mm CD = 500
mm and BE = 400 mm Locate all the ICs If crank OA
rotates uniformly at 120 rpm cw find
1 velocity of B C and D
2 angular velocity of links AB BC and CD
9282013
Dr Mohammad Abuhaiba PE
25
Example 63 9282013
Dr Mohammad Abuhaiba PE
26
Example 64 The mechanism of a wrapping machine Fig 618 has the following dimensions O1A = 100 mm AC = 700 mm BC = 200 mm O3C = 200 mm O2E = 400 mm O2D = 200 mm and BD = 150 mm The crank O1A rotates at a uniform speed of 100 rads Find the velocity of point E of the bell crank lever by instantaneous center method
9282013
Dr Mohammad Abuhaiba PE
27
Example 64
9282013
Dr Mohammad Abuhaiba PE
28
Example 65 Fig 621 shows a sewing needle bar mechanism O1ABO2CD wherein the different dimensions are as follows Crank O1A = 16 mm b = 45deg Vertical distance between O1 and O2 = 40 mm Horizontal distance between O1 amp O2 = 13 mm O2B = 23 mm AB = 35 mm O2BC = 90deg BC = 16 mm CD = 40 mm D lies vertically below O1 Find the velocity of needle at D for the given configuration The crank O1A rotates at 400 rpm
9282013
Dr Mohammad Abuhaiba PE
29
Example 65
9282013
Dr Mohammad Abuhaiba PE
30
Example 66
Fig 624 shows a Whitworth
quick return motion
mechanism The various
dimensions in the mechanism
are as follows OQ = 100 mm
OA = 200 mm QC = 150 mm
and CD = 500 mm The crank
OA makes an angle of 60deg
with the vertical and rotates at
120 rpm cw Locate all ICs
and find the velocity of ram D
9282013
Dr Mohammad Abuhaiba PE
31
Example 66
9282013
Dr Mohammad Abuhaiba PE
32
610 Method of Locating Instantaneous
Centers in a Mechanism
3 Locate fixed amp permanent ICs by
inspection
I12 and I14 are fixed Ics
I23 and I34 are permanent IC
The four bar mechanism has
four turning pairs
four primary (fixed and permanent) ICs located at
the centers of the pin joints
9282013
Dr Mohammad Abuhaiba PE
18
610 Method of Locating Instantaneous
Centers in a Mechanism
4 Locate remaining neither fixed nor
permanent ICs (secondary centers) by
Kennedyrsquos theorem
Circle diagram Fig 68 (b)
Mark points on a circle equal to number
of links in a mechanism
5 Join points by solid lines to show that
these centers are already found
9282013
Dr Mohammad Abuhaiba PE
19
610 Method of Locating Instantaneous
Centers in a Mechanism
9282013
Dr Mohammad Abuhaiba PE
20
Example 61 In a pin jointed four bar mechanism Fig 69 AB = 300 mm BC = CD = 360 mm and AD = 600 mm The angle BAD = 60deg The crank AB rotates uniformly at 100 rpm Locate all instantaneous centers and find the angular velocity of link BC
9282013
Dr Mohammad Abuhaiba PE
21
Example 61
9282013
Dr Mohammad Abuhaiba PE
22
Example 62 Locate all ICs of the slider crank mechanism shown in Fig 612 The lengths of crank OB and connecting rod AB are 100 mm and 400 mm respectively If the crank rotates cw with an angular velocity of 10 rads find
1 Velocity of the slider A
2 Angular velocity of connecting rod AB
9282013
Dr Mohammad Abuhaiba PE
23
Example 62
9282013
Dr Mohammad Abuhaiba PE
24
Example 63
A mechanism Fig 615 has the following dimensions
OA = 200 mm AB = 1500 mm BC = 600 mm CD = 500
mm and BE = 400 mm Locate all the ICs If crank OA
rotates uniformly at 120 rpm cw find
1 velocity of B C and D
2 angular velocity of links AB BC and CD
9282013
Dr Mohammad Abuhaiba PE
25
Example 63 9282013
Dr Mohammad Abuhaiba PE
26
Example 64 The mechanism of a wrapping machine Fig 618 has the following dimensions O1A = 100 mm AC = 700 mm BC = 200 mm O3C = 200 mm O2E = 400 mm O2D = 200 mm and BD = 150 mm The crank O1A rotates at a uniform speed of 100 rads Find the velocity of point E of the bell crank lever by instantaneous center method
9282013
Dr Mohammad Abuhaiba PE
27
Example 64
9282013
Dr Mohammad Abuhaiba PE
28
Example 65 Fig 621 shows a sewing needle bar mechanism O1ABO2CD wherein the different dimensions are as follows Crank O1A = 16 mm b = 45deg Vertical distance between O1 and O2 = 40 mm Horizontal distance between O1 amp O2 = 13 mm O2B = 23 mm AB = 35 mm O2BC = 90deg BC = 16 mm CD = 40 mm D lies vertically below O1 Find the velocity of needle at D for the given configuration The crank O1A rotates at 400 rpm
9282013
Dr Mohammad Abuhaiba PE
29
Example 65
9282013
Dr Mohammad Abuhaiba PE
30
Example 66
Fig 624 shows a Whitworth
quick return motion
mechanism The various
dimensions in the mechanism
are as follows OQ = 100 mm
OA = 200 mm QC = 150 mm
and CD = 500 mm The crank
OA makes an angle of 60deg
with the vertical and rotates at
120 rpm cw Locate all ICs
and find the velocity of ram D
9282013
Dr Mohammad Abuhaiba PE
31
Example 66
9282013
Dr Mohammad Abuhaiba PE
32
610 Method of Locating Instantaneous
Centers in a Mechanism
4 Locate remaining neither fixed nor
permanent ICs (secondary centers) by
Kennedyrsquos theorem
Circle diagram Fig 68 (b)
Mark points on a circle equal to number
of links in a mechanism
5 Join points by solid lines to show that
these centers are already found
9282013
Dr Mohammad Abuhaiba PE
19
610 Method of Locating Instantaneous
Centers in a Mechanism
9282013
Dr Mohammad Abuhaiba PE
20
Example 61 In a pin jointed four bar mechanism Fig 69 AB = 300 mm BC = CD = 360 mm and AD = 600 mm The angle BAD = 60deg The crank AB rotates uniformly at 100 rpm Locate all instantaneous centers and find the angular velocity of link BC
9282013
Dr Mohammad Abuhaiba PE
21
Example 61
9282013
Dr Mohammad Abuhaiba PE
22
Example 62 Locate all ICs of the slider crank mechanism shown in Fig 612 The lengths of crank OB and connecting rod AB are 100 mm and 400 mm respectively If the crank rotates cw with an angular velocity of 10 rads find
1 Velocity of the slider A
2 Angular velocity of connecting rod AB
9282013
Dr Mohammad Abuhaiba PE
23
Example 62
9282013
Dr Mohammad Abuhaiba PE
24
Example 63
A mechanism Fig 615 has the following dimensions
OA = 200 mm AB = 1500 mm BC = 600 mm CD = 500
mm and BE = 400 mm Locate all the ICs If crank OA
rotates uniformly at 120 rpm cw find
1 velocity of B C and D
2 angular velocity of links AB BC and CD
9282013
Dr Mohammad Abuhaiba PE
25
Example 63 9282013
Dr Mohammad Abuhaiba PE
26
Example 64 The mechanism of a wrapping machine Fig 618 has the following dimensions O1A = 100 mm AC = 700 mm BC = 200 mm O3C = 200 mm O2E = 400 mm O2D = 200 mm and BD = 150 mm The crank O1A rotates at a uniform speed of 100 rads Find the velocity of point E of the bell crank lever by instantaneous center method
9282013
Dr Mohammad Abuhaiba PE
27
Example 64
9282013
Dr Mohammad Abuhaiba PE
28
Example 65 Fig 621 shows a sewing needle bar mechanism O1ABO2CD wherein the different dimensions are as follows Crank O1A = 16 mm b = 45deg Vertical distance between O1 and O2 = 40 mm Horizontal distance between O1 amp O2 = 13 mm O2B = 23 mm AB = 35 mm O2BC = 90deg BC = 16 mm CD = 40 mm D lies vertically below O1 Find the velocity of needle at D for the given configuration The crank O1A rotates at 400 rpm
9282013
Dr Mohammad Abuhaiba PE
29
Example 65
9282013
Dr Mohammad Abuhaiba PE
30
Example 66
Fig 624 shows a Whitworth
quick return motion
mechanism The various
dimensions in the mechanism
are as follows OQ = 100 mm
OA = 200 mm QC = 150 mm
and CD = 500 mm The crank
OA makes an angle of 60deg
with the vertical and rotates at
120 rpm cw Locate all ICs
and find the velocity of ram D
9282013
Dr Mohammad Abuhaiba PE
31
Example 66
9282013
Dr Mohammad Abuhaiba PE
32
610 Method of Locating Instantaneous
Centers in a Mechanism
9282013
Dr Mohammad Abuhaiba PE
20
Example 61 In a pin jointed four bar mechanism Fig 69 AB = 300 mm BC = CD = 360 mm and AD = 600 mm The angle BAD = 60deg The crank AB rotates uniformly at 100 rpm Locate all instantaneous centers and find the angular velocity of link BC
9282013
Dr Mohammad Abuhaiba PE
21
Example 61
9282013
Dr Mohammad Abuhaiba PE
22
Example 62 Locate all ICs of the slider crank mechanism shown in Fig 612 The lengths of crank OB and connecting rod AB are 100 mm and 400 mm respectively If the crank rotates cw with an angular velocity of 10 rads find
1 Velocity of the slider A
2 Angular velocity of connecting rod AB
9282013
Dr Mohammad Abuhaiba PE
23
Example 62
9282013
Dr Mohammad Abuhaiba PE
24
Example 63
A mechanism Fig 615 has the following dimensions
OA = 200 mm AB = 1500 mm BC = 600 mm CD = 500
mm and BE = 400 mm Locate all the ICs If crank OA
rotates uniformly at 120 rpm cw find
1 velocity of B C and D
2 angular velocity of links AB BC and CD
9282013
Dr Mohammad Abuhaiba PE
25
Example 63 9282013
Dr Mohammad Abuhaiba PE
26
Example 64 The mechanism of a wrapping machine Fig 618 has the following dimensions O1A = 100 mm AC = 700 mm BC = 200 mm O3C = 200 mm O2E = 400 mm O2D = 200 mm and BD = 150 mm The crank O1A rotates at a uniform speed of 100 rads Find the velocity of point E of the bell crank lever by instantaneous center method
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Dr Mohammad Abuhaiba PE
27
Example 64
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Dr Mohammad Abuhaiba PE
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Example 65 Fig 621 shows a sewing needle bar mechanism O1ABO2CD wherein the different dimensions are as follows Crank O1A = 16 mm b = 45deg Vertical distance between O1 and O2 = 40 mm Horizontal distance between O1 amp O2 = 13 mm O2B = 23 mm AB = 35 mm O2BC = 90deg BC = 16 mm CD = 40 mm D lies vertically below O1 Find the velocity of needle at D for the given configuration The crank O1A rotates at 400 rpm
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Dr Mohammad Abuhaiba PE
29
Example 65
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Dr Mohammad Abuhaiba PE
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Example 66
Fig 624 shows a Whitworth
quick return motion
mechanism The various
dimensions in the mechanism
are as follows OQ = 100 mm
OA = 200 mm QC = 150 mm
and CD = 500 mm The crank
OA makes an angle of 60deg
with the vertical and rotates at
120 rpm cw Locate all ICs
and find the velocity of ram D
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Dr Mohammad Abuhaiba PE
31
Example 66
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Dr Mohammad Abuhaiba PE
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Example 61 In a pin jointed four bar mechanism Fig 69 AB = 300 mm BC = CD = 360 mm and AD = 600 mm The angle BAD = 60deg The crank AB rotates uniformly at 100 rpm Locate all instantaneous centers and find the angular velocity of link BC
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Dr Mohammad Abuhaiba PE
21
Example 61
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Dr Mohammad Abuhaiba PE
22
Example 62 Locate all ICs of the slider crank mechanism shown in Fig 612 The lengths of crank OB and connecting rod AB are 100 mm and 400 mm respectively If the crank rotates cw with an angular velocity of 10 rads find
1 Velocity of the slider A
2 Angular velocity of connecting rod AB
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Dr Mohammad Abuhaiba PE
23
Example 62
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Dr Mohammad Abuhaiba PE
24
Example 63
A mechanism Fig 615 has the following dimensions
OA = 200 mm AB = 1500 mm BC = 600 mm CD = 500
mm and BE = 400 mm Locate all the ICs If crank OA
rotates uniformly at 120 rpm cw find
1 velocity of B C and D
2 angular velocity of links AB BC and CD
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Dr Mohammad Abuhaiba PE
25
Example 63 9282013
Dr Mohammad Abuhaiba PE
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Example 64 The mechanism of a wrapping machine Fig 618 has the following dimensions O1A = 100 mm AC = 700 mm BC = 200 mm O3C = 200 mm O2E = 400 mm O2D = 200 mm and BD = 150 mm The crank O1A rotates at a uniform speed of 100 rads Find the velocity of point E of the bell crank lever by instantaneous center method
9282013
Dr Mohammad Abuhaiba PE
27
Example 64
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Dr Mohammad Abuhaiba PE
28
Example 65 Fig 621 shows a sewing needle bar mechanism O1ABO2CD wherein the different dimensions are as follows Crank O1A = 16 mm b = 45deg Vertical distance between O1 and O2 = 40 mm Horizontal distance between O1 amp O2 = 13 mm O2B = 23 mm AB = 35 mm O2BC = 90deg BC = 16 mm CD = 40 mm D lies vertically below O1 Find the velocity of needle at D for the given configuration The crank O1A rotates at 400 rpm
9282013
Dr Mohammad Abuhaiba PE
29
Example 65
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Dr Mohammad Abuhaiba PE
30
Example 66
Fig 624 shows a Whitworth
quick return motion
mechanism The various
dimensions in the mechanism
are as follows OQ = 100 mm
OA = 200 mm QC = 150 mm
and CD = 500 mm The crank
OA makes an angle of 60deg
with the vertical and rotates at
120 rpm cw Locate all ICs
and find the velocity of ram D
9282013
Dr Mohammad Abuhaiba PE
31
Example 66
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Dr Mohammad Abuhaiba PE
32
Example 61
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Dr Mohammad Abuhaiba PE
22
Example 62 Locate all ICs of the slider crank mechanism shown in Fig 612 The lengths of crank OB and connecting rod AB are 100 mm and 400 mm respectively If the crank rotates cw with an angular velocity of 10 rads find
1 Velocity of the slider A
2 Angular velocity of connecting rod AB
9282013
Dr Mohammad Abuhaiba PE
23
Example 62
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Dr Mohammad Abuhaiba PE
24
Example 63
A mechanism Fig 615 has the following dimensions
OA = 200 mm AB = 1500 mm BC = 600 mm CD = 500
mm and BE = 400 mm Locate all the ICs If crank OA
rotates uniformly at 120 rpm cw find
1 velocity of B C and D
2 angular velocity of links AB BC and CD
9282013
Dr Mohammad Abuhaiba PE
25
Example 63 9282013
Dr Mohammad Abuhaiba PE
26
Example 64 The mechanism of a wrapping machine Fig 618 has the following dimensions O1A = 100 mm AC = 700 mm BC = 200 mm O3C = 200 mm O2E = 400 mm O2D = 200 mm and BD = 150 mm The crank O1A rotates at a uniform speed of 100 rads Find the velocity of point E of the bell crank lever by instantaneous center method
9282013
Dr Mohammad Abuhaiba PE
27
Example 64
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Dr Mohammad Abuhaiba PE
28
Example 65 Fig 621 shows a sewing needle bar mechanism O1ABO2CD wherein the different dimensions are as follows Crank O1A = 16 mm b = 45deg Vertical distance between O1 and O2 = 40 mm Horizontal distance between O1 amp O2 = 13 mm O2B = 23 mm AB = 35 mm O2BC = 90deg BC = 16 mm CD = 40 mm D lies vertically below O1 Find the velocity of needle at D for the given configuration The crank O1A rotates at 400 rpm
9282013
Dr Mohammad Abuhaiba PE
29
Example 65
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Dr Mohammad Abuhaiba PE
30
Example 66
Fig 624 shows a Whitworth
quick return motion
mechanism The various
dimensions in the mechanism
are as follows OQ = 100 mm
OA = 200 mm QC = 150 mm
and CD = 500 mm The crank
OA makes an angle of 60deg
with the vertical and rotates at
120 rpm cw Locate all ICs
and find the velocity of ram D
9282013
Dr Mohammad Abuhaiba PE
31
Example 66
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Dr Mohammad Abuhaiba PE
32
Example 62 Locate all ICs of the slider crank mechanism shown in Fig 612 The lengths of crank OB and connecting rod AB are 100 mm and 400 mm respectively If the crank rotates cw with an angular velocity of 10 rads find
1 Velocity of the slider A
2 Angular velocity of connecting rod AB
9282013
Dr Mohammad Abuhaiba PE
23
Example 62
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Dr Mohammad Abuhaiba PE
24
Example 63
A mechanism Fig 615 has the following dimensions
OA = 200 mm AB = 1500 mm BC = 600 mm CD = 500
mm and BE = 400 mm Locate all the ICs If crank OA
rotates uniformly at 120 rpm cw find
1 velocity of B C and D
2 angular velocity of links AB BC and CD
9282013
Dr Mohammad Abuhaiba PE
25
Example 63 9282013
Dr Mohammad Abuhaiba PE
26
Example 64 The mechanism of a wrapping machine Fig 618 has the following dimensions O1A = 100 mm AC = 700 mm BC = 200 mm O3C = 200 mm O2E = 400 mm O2D = 200 mm and BD = 150 mm The crank O1A rotates at a uniform speed of 100 rads Find the velocity of point E of the bell crank lever by instantaneous center method
9282013
Dr Mohammad Abuhaiba PE
27
Example 64
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Dr Mohammad Abuhaiba PE
28
Example 65 Fig 621 shows a sewing needle bar mechanism O1ABO2CD wherein the different dimensions are as follows Crank O1A = 16 mm b = 45deg Vertical distance between O1 and O2 = 40 mm Horizontal distance between O1 amp O2 = 13 mm O2B = 23 mm AB = 35 mm O2BC = 90deg BC = 16 mm CD = 40 mm D lies vertically below O1 Find the velocity of needle at D for the given configuration The crank O1A rotates at 400 rpm
9282013
Dr Mohammad Abuhaiba PE
29
Example 65
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Dr Mohammad Abuhaiba PE
30
Example 66
Fig 624 shows a Whitworth
quick return motion
mechanism The various
dimensions in the mechanism
are as follows OQ = 100 mm
OA = 200 mm QC = 150 mm
and CD = 500 mm The crank
OA makes an angle of 60deg
with the vertical and rotates at
120 rpm cw Locate all ICs
and find the velocity of ram D
9282013
Dr Mohammad Abuhaiba PE
31
Example 66
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Dr Mohammad Abuhaiba PE
32
Example 62
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Dr Mohammad Abuhaiba PE
24
Example 63
A mechanism Fig 615 has the following dimensions
OA = 200 mm AB = 1500 mm BC = 600 mm CD = 500
mm and BE = 400 mm Locate all the ICs If crank OA
rotates uniformly at 120 rpm cw find
1 velocity of B C and D
2 angular velocity of links AB BC and CD
9282013
Dr Mohammad Abuhaiba PE
25
Example 63 9282013
Dr Mohammad Abuhaiba PE
26
Example 64 The mechanism of a wrapping machine Fig 618 has the following dimensions O1A = 100 mm AC = 700 mm BC = 200 mm O3C = 200 mm O2E = 400 mm O2D = 200 mm and BD = 150 mm The crank O1A rotates at a uniform speed of 100 rads Find the velocity of point E of the bell crank lever by instantaneous center method
9282013
Dr Mohammad Abuhaiba PE
27
Example 64
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Dr Mohammad Abuhaiba PE
28
Example 65 Fig 621 shows a sewing needle bar mechanism O1ABO2CD wherein the different dimensions are as follows Crank O1A = 16 mm b = 45deg Vertical distance between O1 and O2 = 40 mm Horizontal distance between O1 amp O2 = 13 mm O2B = 23 mm AB = 35 mm O2BC = 90deg BC = 16 mm CD = 40 mm D lies vertically below O1 Find the velocity of needle at D for the given configuration The crank O1A rotates at 400 rpm
9282013
Dr Mohammad Abuhaiba PE
29
Example 65
9282013
Dr Mohammad Abuhaiba PE
30
Example 66
Fig 624 shows a Whitworth
quick return motion
mechanism The various
dimensions in the mechanism
are as follows OQ = 100 mm
OA = 200 mm QC = 150 mm
and CD = 500 mm The crank
OA makes an angle of 60deg
with the vertical and rotates at
120 rpm cw Locate all ICs
and find the velocity of ram D
9282013
Dr Mohammad Abuhaiba PE
31
Example 66
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Dr Mohammad Abuhaiba PE
32
Example 63
A mechanism Fig 615 has the following dimensions
OA = 200 mm AB = 1500 mm BC = 600 mm CD = 500
mm and BE = 400 mm Locate all the ICs If crank OA
rotates uniformly at 120 rpm cw find
1 velocity of B C and D
2 angular velocity of links AB BC and CD
9282013
Dr Mohammad Abuhaiba PE
25
Example 63 9282013
Dr Mohammad Abuhaiba PE
26
Example 64 The mechanism of a wrapping machine Fig 618 has the following dimensions O1A = 100 mm AC = 700 mm BC = 200 mm O3C = 200 mm O2E = 400 mm O2D = 200 mm and BD = 150 mm The crank O1A rotates at a uniform speed of 100 rads Find the velocity of point E of the bell crank lever by instantaneous center method
9282013
Dr Mohammad Abuhaiba PE
27
Example 64
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Dr Mohammad Abuhaiba PE
28
Example 65 Fig 621 shows a sewing needle bar mechanism O1ABO2CD wherein the different dimensions are as follows Crank O1A = 16 mm b = 45deg Vertical distance between O1 and O2 = 40 mm Horizontal distance between O1 amp O2 = 13 mm O2B = 23 mm AB = 35 mm O2BC = 90deg BC = 16 mm CD = 40 mm D lies vertically below O1 Find the velocity of needle at D for the given configuration The crank O1A rotates at 400 rpm
9282013
Dr Mohammad Abuhaiba PE
29
Example 65
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Dr Mohammad Abuhaiba PE
30
Example 66
Fig 624 shows a Whitworth
quick return motion
mechanism The various
dimensions in the mechanism
are as follows OQ = 100 mm
OA = 200 mm QC = 150 mm
and CD = 500 mm The crank
OA makes an angle of 60deg
with the vertical and rotates at
120 rpm cw Locate all ICs
and find the velocity of ram D
9282013
Dr Mohammad Abuhaiba PE
31
Example 66
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Dr Mohammad Abuhaiba PE
32
Example 63 9282013
Dr Mohammad Abuhaiba PE
26
Example 64 The mechanism of a wrapping machine Fig 618 has the following dimensions O1A = 100 mm AC = 700 mm BC = 200 mm O3C = 200 mm O2E = 400 mm O2D = 200 mm and BD = 150 mm The crank O1A rotates at a uniform speed of 100 rads Find the velocity of point E of the bell crank lever by instantaneous center method
9282013
Dr Mohammad Abuhaiba PE
27
Example 64
9282013
Dr Mohammad Abuhaiba PE
28
Example 65 Fig 621 shows a sewing needle bar mechanism O1ABO2CD wherein the different dimensions are as follows Crank O1A = 16 mm b = 45deg Vertical distance between O1 and O2 = 40 mm Horizontal distance between O1 amp O2 = 13 mm O2B = 23 mm AB = 35 mm O2BC = 90deg BC = 16 mm CD = 40 mm D lies vertically below O1 Find the velocity of needle at D for the given configuration The crank O1A rotates at 400 rpm
9282013
Dr Mohammad Abuhaiba PE
29
Example 65
9282013
Dr Mohammad Abuhaiba PE
30
Example 66
Fig 624 shows a Whitworth
quick return motion
mechanism The various
dimensions in the mechanism
are as follows OQ = 100 mm
OA = 200 mm QC = 150 mm
and CD = 500 mm The crank
OA makes an angle of 60deg
with the vertical and rotates at
120 rpm cw Locate all ICs
and find the velocity of ram D
9282013
Dr Mohammad Abuhaiba PE
31
Example 66
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Dr Mohammad Abuhaiba PE
32
Example 64 The mechanism of a wrapping machine Fig 618 has the following dimensions O1A = 100 mm AC = 700 mm BC = 200 mm O3C = 200 mm O2E = 400 mm O2D = 200 mm and BD = 150 mm The crank O1A rotates at a uniform speed of 100 rads Find the velocity of point E of the bell crank lever by instantaneous center method
9282013
Dr Mohammad Abuhaiba PE
27
Example 64
9282013
Dr Mohammad Abuhaiba PE
28
Example 65 Fig 621 shows a sewing needle bar mechanism O1ABO2CD wherein the different dimensions are as follows Crank O1A = 16 mm b = 45deg Vertical distance between O1 and O2 = 40 mm Horizontal distance between O1 amp O2 = 13 mm O2B = 23 mm AB = 35 mm O2BC = 90deg BC = 16 mm CD = 40 mm D lies vertically below O1 Find the velocity of needle at D for the given configuration The crank O1A rotates at 400 rpm
9282013
Dr Mohammad Abuhaiba PE
29
Example 65
9282013
Dr Mohammad Abuhaiba PE
30
Example 66
Fig 624 shows a Whitworth
quick return motion
mechanism The various
dimensions in the mechanism
are as follows OQ = 100 mm
OA = 200 mm QC = 150 mm
and CD = 500 mm The crank
OA makes an angle of 60deg
with the vertical and rotates at
120 rpm cw Locate all ICs
and find the velocity of ram D
9282013
Dr Mohammad Abuhaiba PE
31
Example 66
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Dr Mohammad Abuhaiba PE
32
Example 64
9282013
Dr Mohammad Abuhaiba PE
28
Example 65 Fig 621 shows a sewing needle bar mechanism O1ABO2CD wherein the different dimensions are as follows Crank O1A = 16 mm b = 45deg Vertical distance between O1 and O2 = 40 mm Horizontal distance between O1 amp O2 = 13 mm O2B = 23 mm AB = 35 mm O2BC = 90deg BC = 16 mm CD = 40 mm D lies vertically below O1 Find the velocity of needle at D for the given configuration The crank O1A rotates at 400 rpm
9282013
Dr Mohammad Abuhaiba PE
29
Example 65
9282013
Dr Mohammad Abuhaiba PE
30
Example 66
Fig 624 shows a Whitworth
quick return motion
mechanism The various
dimensions in the mechanism
are as follows OQ = 100 mm
OA = 200 mm QC = 150 mm
and CD = 500 mm The crank
OA makes an angle of 60deg
with the vertical and rotates at
120 rpm cw Locate all ICs
and find the velocity of ram D
9282013
Dr Mohammad Abuhaiba PE
31
Example 66
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Dr Mohammad Abuhaiba PE
32
Example 65 Fig 621 shows a sewing needle bar mechanism O1ABO2CD wherein the different dimensions are as follows Crank O1A = 16 mm b = 45deg Vertical distance between O1 and O2 = 40 mm Horizontal distance between O1 amp O2 = 13 mm O2B = 23 mm AB = 35 mm O2BC = 90deg BC = 16 mm CD = 40 mm D lies vertically below O1 Find the velocity of needle at D for the given configuration The crank O1A rotates at 400 rpm
9282013
Dr Mohammad Abuhaiba PE
29
Example 65
9282013
Dr Mohammad Abuhaiba PE
30
Example 66
Fig 624 shows a Whitworth
quick return motion
mechanism The various
dimensions in the mechanism
are as follows OQ = 100 mm
OA = 200 mm QC = 150 mm
and CD = 500 mm The crank
OA makes an angle of 60deg
with the vertical and rotates at
120 rpm cw Locate all ICs
and find the velocity of ram D
9282013
Dr Mohammad Abuhaiba PE
31
Example 66
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Dr Mohammad Abuhaiba PE
32
Example 65
9282013
Dr Mohammad Abuhaiba PE
30
Example 66
Fig 624 shows a Whitworth
quick return motion
mechanism The various
dimensions in the mechanism
are as follows OQ = 100 mm
OA = 200 mm QC = 150 mm
and CD = 500 mm The crank
OA makes an angle of 60deg
with the vertical and rotates at
120 rpm cw Locate all ICs
and find the velocity of ram D
9282013
Dr Mohammad Abuhaiba PE
31
Example 66
9282013
Dr Mohammad Abuhaiba PE
32
Example 66
Fig 624 shows a Whitworth
quick return motion
mechanism The various
dimensions in the mechanism
are as follows OQ = 100 mm
OA = 200 mm QC = 150 mm
and CD = 500 mm The crank
OA makes an angle of 60deg
with the vertical and rotates at
120 rpm cw Locate all ICs
and find the velocity of ram D
9282013
Dr Mohammad Abuhaiba PE
31
Example 66
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Dr Mohammad Abuhaiba PE
32