Moment and couple - Chulapioneer.chula.ac.th/~rchanat/2103213 MechI/ch2/presentation/statics...

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Moment and couple In 3-D, because the determination of the distance can be tedious, a vector approach becomes advantageous. z y x z y x o F F F r r r k j i F r M ˆ ˆ ˆ = × = r r r k F r F r j F r F r i F r F r M x y y x z x x z y z z y o ˆ ) ( ˆ ) ( ˆ ) ( + + = r F r M o r r r × = M x = F y r z + F z r y + x y + z + M y = F x r z F z r x M z = F x r y +F y r x x y z r r O A F r r z F x F y r x r y F z

Transcript of Moment and couple - Chulapioneer.chula.ac.th/~rchanat/2103213 MechI/ch2/presentation/statics...

Page 1: Moment and couple - Chulapioneer.chula.ac.th/~rchanat/2103213 MechI/ch2/presentation/statics ch2... · The resultant of the two forces and couple may be represented by a wrench. Determine

Moment and coupleIn 3-D, because the determination of the distance can be tedious, a vector approach becomes advantageous.

zyx

zyxo

FFFrrrkji

FrM

ˆˆˆ

=×=rrr

kFrFrjFrFriFrFrM xyyxzxxzyzzyoˆ)(ˆ)(ˆ)( −+−+−=

r

FrM o

rrr×=

Mx = − Fyrz + Fzry

+

x

y+

z+

My = Fxrz − Fzrx

Mz= −Fxry +Fyrxx

y

zrr

O

A

Fr

rzFx

Fy

rxry

Fz

Page 2: Moment and couple - Chulapioneer.chula.ac.th/~rchanat/2103213 MechI/ch2/presentation/statics ch2... · The resultant of the two forces and couple may be represented by a wrench. Determine

Moment about an arbitrary axis

O

OMr

rrFrn̂

λ

λMr 1. Calculate moment

FrM o

Find moment about λ axis

rrr×=

2. Calculate projection of moment on λ axis

nnFrnnMM O ˆ)ˆ(ˆ)ˆ( ⋅×=⋅=rrrr

λ

)ˆˆˆ.(

ˆˆˆ

)ˆ( kjiFFFrrrkji

nFr

zyx

zyx γβα ++=⋅×rr

γβα

γβα

zyx

zyx

zyx

zyx FFFrrr

FFFrrr ==

kji ˆˆˆ γβα ++

Page 3: Moment and couple - Chulapioneer.chula.ac.th/~rchanat/2103213 MechI/ch2/presentation/statics ch2... · The resultant of the two forces and couple may be represented by a wrench. Determine

Varignon’s Theorem

O

r

3Fr

A

1Fr

2Fr

...)(... 321321 +++×=+×+×+×= FFFrFrFrFrM o

rrrrrrrrrrr

)(∑×= Frrr

- Sum of the moments of a system of concurrent forces about a given point equals the moment of their sum about the same point

RrFrM o

rrrrr×=×=∑ )(

Page 4: Moment and couple - Chulapioneer.chula.ac.th/~rchanat/2103213 MechI/ch2/presentation/statics ch2... · The resultant of the two forces and couple may be represented by a wrench. Determine

Couples(1)

O

Fr

Fr

ArrBrr

rr

A

B

Mr

d

FrrFrFrM BABA

rrrrrrrr×−=−×+×= )()(

-Couple is a moment produced by two force of equal magnitude but opposite in direction.

FrMrrr

×=

- = vector from any point on the line of action of to any point on the line of action of

- Moment of a couple is the same about all point Couple may be represented as a free vector.

- Direction: normal to the plane of the two forces (right hand rule)- Recall: Moment of force about a point is a sliding vector.

Fr F

r−rr

Page 5: Moment and couple - Chulapioneer.chula.ac.th/~rchanat/2103213 MechI/ch2/presentation/statics ch2... · The resultant of the two forces and couple may be represented by a wrench. Determine

1Fr

2Fr

2Fr

1Fr

−2Mr

1Mr

1Mr

2Mr

Fr

Mr

Fr

Couples(2)

[Couple from F1]+[Couple from F2] = [Couple from F1+F2]

couples are free vector. the line of action or point of action are not needed!!!

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Force – couple systems

No changes in the net external effects

AB

Fr

A

B

Fr

AB

FrF

r

Fr

rr

FrMrrr

×=

- = Moment of about point B

- is a vector start from point B to any point on the line of action of

FrMrrr

×= Fr

rr

Fr

Couple

Page 7: Moment and couple - Chulapioneer.chula.ac.th/~rchanat/2103213 MechI/ch2/presentation/statics ch2... · The resultant of the two forces and couple may be represented by a wrench. Determine

Sample 1A Tension T of magnitude 10 kN is applied to the cable attached to the top A of the rigid mast and secured to the ground at B. Determine the moment Mz of T about the z-axis passing through the base O.

Page 8: Moment and couple - Chulapioneer.chula.ac.th/~rchanat/2103213 MechI/ch2/presentation/statics ch2... · The resultant of the two forces and couple may be represented by a wrench. Determine

Sample 2Determine the magnitude and direction of the couple M which will replace the two given couples and still produce the same external effect on the block. Specify the two force F and –F, applied in the two faces of the block parallel to the y-z plane, which may replace the four given forces. The 30-N forces act parallel to the y-z plane.

Page 9: Moment and couple - Chulapioneer.chula.ac.th/~rchanat/2103213 MechI/ch2/presentation/statics ch2... · The resultant of the two forces and couple may be represented by a wrench. Determine

Sample 3A force of 400 N is applied at A to the handle of the control lever which is attached to the fixed shaft OB. In determining the effect of the force on the shaft at a cross section such as that at O, we may replace the force by an equivalent force at O and a couple. Describe this couple as a vector M.

Page 10: Moment and couple - Chulapioneer.chula.ac.th/~rchanat/2103213 MechI/ch2/presentation/statics ch2... · The resultant of the two forces and couple may be represented by a wrench. Determine

Sample 4If the magnitude of the moment of F about line CD is 50 Nm, determine the magnitude of F.

Page 11: Moment and couple - Chulapioneer.chula.ac.th/~rchanat/2103213 MechI/ch2/presentation/statics ch2... · The resultant of the two forces and couple may be represented by a wrench. Determine

Sample 5Tension in cable AB is 143.4 N. Determine the moment about the x-axis of this tension force acting on point A . Compare your result to the moment of the weight W of the 15-kg uniform plate about the x-axis. What is the moment of the tension force acting at A about line OB

Page 12: Moment and couple - Chulapioneer.chula.ac.th/~rchanat/2103213 MechI/ch2/presentation/statics ch2... · The resultant of the two forces and couple may be represented by a wrench. Determine

Summary (Force-Moment 3-D)Force1. Determine coordinate2. Determine unit vector3. Force can be calculate

Angle between force and x-,y-,z-axis1. Force = Fxi + Fyj + Fzk2. Determine amplitude of force F3. cosθx = Fx/F, cosθy = Fy/F, cosθz = Fz/F

Angle between force and arbitrary axis1. Determine unit vectors (nF, n)2. cosθ = nF・ n

Page 13: Moment and couple - Chulapioneer.chula.ac.th/~rchanat/2103213 MechI/ch2/presentation/statics ch2... · The resultant of the two forces and couple may be represented by a wrench. Determine

Summary (Force-Moment 3-D)

Vector method

Moment about an arbitrary point O1. Determine r and F2. Cross vectorMoment about an arbitrary axis1. Determine moment about any point on the axis MO2. Determine unit vector of the axis n3. Moment about the axis = MO・n

Angle between moment and axis

Same as angle between force and axis

Moment Consider to use vector method or scalar method

Page 14: Moment and couple - Chulapioneer.chula.ac.th/~rchanat/2103213 MechI/ch2/presentation/statics ch2... · The resultant of the two forces and couple may be represented by a wrench. Determine

Resultants(1)

Select a point to find moment

Replace forces with forces at point O + couples

Add forces and couples vectorially to get the resultant force and moment

Step1 Step2 Step3

∑=+++= FFFFRrrrrr

...321

∑ ×=+++= )(...321 FrMMMMrrrrrr

Page 15: Moment and couple - Chulapioneer.chula.ac.th/~rchanat/2103213 MechI/ch2/presentation/statics ch2... · The resultant of the two forces and couple may be represented by a wrench. Determine

Resultants(2)2-D

A Fr

B

A

BM=Fd

Fr

FMvv

⊥Force + couple can be replaced by a force F by changing the position of F.

3-D

Rr

Mr

O 1Mr

2Mr RM

vv⊥2

M2 and R can be replaced by one force R by changing the position of R.

RMvv

//1

M1 can not be replaced

Page 16: Moment and couple - Chulapioneer.chula.ac.th/~rchanat/2103213 MechI/ch2/presentation/statics ch2... · The resultant of the two forces and couple may be represented by a wrench. Determine

Wrench resultant(1)

M2=Rd

Page 17: Moment and couple - Chulapioneer.chula.ac.th/~rchanat/2103213 MechI/ch2/presentation/statics ch2... · The resultant of the two forces and couple may be represented by a wrench. Determine

Wrench resultant(2)

2-D: All force systems can be represented with only one resultant force or couple

3-D: All force systems can be represented with a wrench resultant

Wrench: resultant couple M parallel to the resultant force Rr r

Page 18: Moment and couple - Chulapioneer.chula.ac.th/~rchanat/2103213 MechI/ch2/presentation/statics ch2... · The resultant of the two forces and couple may be represented by a wrench. Determine

Sample 6Determine the resultant of the system of parallel forces which act on the plate. Solve with a vector approach.

Page 19: Moment and couple - Chulapioneer.chula.ac.th/~rchanat/2103213 MechI/ch2/presentation/statics ch2... · The resultant of the two forces and couple may be represented by a wrench. Determine

Sample 7Replace the two forces and the negative wrench by a single forceR applied at A and the corresponding couple M.

Page 20: Moment and couple - Chulapioneer.chula.ac.th/~rchanat/2103213 MechI/ch2/presentation/statics ch2... · The resultant of the two forces and couple may be represented by a wrench. Determine

Sample 8Determine the wrench resultant of the three forces acting on thebracket. Calculate the coordinates of the point P in the x-y plane through which the resultant force of the wrench acts. Also find the magnitude of the couple M of the wrench.

Page 21: Moment and couple - Chulapioneer.chula.ac.th/~rchanat/2103213 MechI/ch2/presentation/statics ch2... · The resultant of the two forces and couple may be represented by a wrench. Determine

Sample 9The resultant of the two forces and couple may be represented by a wrench. Determine the vector expression for the moment M of the wrench and find the coordinates of the point P in the x-z plane through which the resultant force of the wrench passes