Kinematics, Dynamics and Vibrations

Dr. Mustafa ArafaMechanical Engineering Department

American University in Cairomharafa@aucegypt.edu

Outline

A. Kinematics of mechanisms

B. Dynamics of mechanisms

C. Rigid body dynamics

D. Natural frequency and resonance

E. Balancing of rotating &reciprocating equipment

F. Forced vibrations (e.g., isolation, force transmission, support motion)

References

• Shigley & Uicker: Theory of Machines and Mechanisms

• Norton: Design of Machinery

• Rao: Mechanical Vibrations

• Beer & Johnston: Vector Mechanics for Engineers

A.Kinematics of mechanisms

Four-bar linkage

Slider-crank mechanism

Scotch yoke mechanism

Four-bar mechanism

2

2

sin

cos

aA

aA

y

x

Obtain coordinates of point A:

222

222

yx

yyxx

BdBc

ABABb

Obtain coordinates of point B:

2 equations in 2 unknowns: Bx and By

Analytical position analysis

Write vector loop equation:

Using complex vectors:

01432 iiii

decebeae

01432 RRRR

Solve for 3 and 4

Velocity analysis

10

Vector loop equation

After solving for position, take the derivative:

01432 iiii

decebeae

0432432 iceibeiae

iii

0432432

iiiceibeiaei

Solve for 3 and 4

Acceleration analysis

01432 iiii

decebeae

0432432

iiiceibeiaei

04433224

243

232

22

iiiiiiceicebeibeaeiae

Write vector loop equation:

Take two derivatives:

B. Dynamics of mechanisms

• Free Body Diagrams

Force analysis

Links 2 and 3Link 2

232323232

1212121212

23212

23212

2

2

2

G

G

G

IFRFR

FRFRT

amFF

amFF

xyyx

xyyx

yyy

xxx

3

32233223

43434343

33243

33243

3

3

3

GPPPP

GP

GP

IFRFR

FRFR

FRFR

amFFF

amFFF

xyyx

xyyx

xyyx

yyyy

xxxx

Link 3 (F23=-F32)

Link 4

• F34=-F43

443434343

141414144

44314

44314

4

4

4

G

G

G

IFRFR

FRFRT

amFF

amFF

xyyx

xyyx

yyy

xxx

In one matrix equation

• We have 9 equations and 9 unknowns

44

4

4

3

3

3

2

2

2

12

14

14

43

43

32

32

12

12

14144334

43432323

32321212

4

4

4

3

3

3

2

2

2

00000

010100000

001010000

00000

000101000

000010100

10000

000001010

000000101

TI

am

am

FRFRI

Fam

Fam

I

am

am

T

F

F

F

F

F

F

F

F

RRRR

RRRR

RRRR

G

G

G

PPPPG

PG

PG

G

G

G

y

x

xyyx

yy

xx

y

x

y

x

y

x

y

x

y

x

xyxy

xyxy

xyxy

C. Rigid body dynamics

Basic equations

Particle dynamics

A thin circular rod is supported in a vertical plane by a bracket at A. A spring of stiffness k = 40 N/m is attached at A and fits loosely on the rod. The spring has an undeformed length equal to the arc of the circle AB. A 200-g collar C (not attached to the spring) can slide without friction. Knowing that the collar is released from rest when = 30⁰, determine the maximum height above point B reached by the collar. Determine also the maximum velocity.

20

Rigid body dynamics

At a forward speed of 30 ft/s, the truck brakes were applied, causing the wheels to stop rotating. It was observed that the truck skidded to a stop in 20 ft.

Determine the magnitude of the normal reaction and the friction force at each wheel as the truck skidded to a stop.

21

Rigid body dynamics

2 2

0 0

2

2

0 30 2 20

v v a x x

a

22.5 ft sa

x GxF ma

Free-body diagram:

22.5A BF F m

y GyF maA BW mg N N

G GM I 7 4 4 5B B A AN F F N

But ,A A B BF N F N

22.5A BN N m

A BN N mg 0.699

0.35 , 0.65A BN mg N mg

, ,A BN N Unknowns:

D. Natural frequency and resonance

Model of a vibrating system

Mass and spring

Mass, spring & damper

Forced vibration

2 DOF systems

Modal Analysis

E. Balancing of rotating & reciprocating equipment

Static Unbalance

• Acting through the center of mass of the rotor

• Can be corrected at a single location (plane)

• Can be detected without spinning the rotor

Couple Unbalance

• Rotor is statically balanced, but dynamically unbalanced

Dynamic Unbalance

• Combination of static & couple unbalance

F. Forced vibrations (e.g., isolation, force transmission, support motion)

Forced vibration

Equation of motion:

Steady-state solution:

Where:

Or:

Frequency response

Support motion

Relative motion:

Support motion

Absolute motion:

Vibration isolation

Force transmitted