Kinematics, Dynamics and Vibrations

Dr. Mustafa ArafaMechanical Engineering Department

American University in Cairomharafa@aucegypt.edu

Kinematics, dynamics and vibration

• Kinematics: study of motion (displacement,

velocity, acceleration, time) without reference to

the cause of motion (i.e. regardless of forces).

• Dynamics: study of forces acting on a body, and

resulting motion.

• Vibration: Oscillatory motion of bodies &

associated forces.

Outline

A. Kinematics of mechanisms

B. Dynamics

C. Vibration: natural frequency and resonance

D. Balancing

Kinematics of mechanisms

Four-bar mechanism

Single Degree of Freedom

Slider-crank mechanism

Single Degree of Freedom

Position analysis

Given: a,b,c,d, the ground position, q2.

Find: q3 and q4

a

dq2

b

c

q3

q4

A

B

O2 O4

b

c

Graphical solution

• Draw an arc of radius b, centered at A

• Draw an arc of radius c, centered at O4

• The intersections are the two possible positions for the linkage, open and crossed

a

dq2

b

c

q3

q4

b

c

A

O2 O4

B1

B2

Analytical solution

2

2

sin

cos

q

q

aA

aA

y

x

Obtain coordinates of point A:

222

222

yx

yyxx

BdBc

ABABb

Obtain coordinates of point B:

These are 2 equations in 2 unknowns: Bx and By

See “position analysis” on page 242

Dynamics

Types of motion

Rectilinear Curvilinear

OverviewKinematics: equations for constant velocity and acceleration

Kinetics: Newton’s second law of motion: d

F mvdt

For constant mass: F ma

Kinetic energy: 212

T mv

Potential energy: U mgh

212

U kx

Gravity

Elastic

Friction

P

F

W

N

Basic equations

Projectile

Projectile

v

v

v

a

a

a

x

y

g

17

Plane Motion of a Rigid Body

x x y y G gF ma F ma M I

O OM I

For rotation about a fixed axis:

18

Example

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.

19

Solution

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:

VibrationsNatural frequency and resonance

Overview

mx cx kx f m

k c

x

fEquation of motion:

n k m Natural frequency:

2 f

rad/s Hz

/ 2 nc m

Damping

ratio

Damping

coeff.

critical

critical

critical

1 overdamping

1 critical damping

1 underdamping

c c

c c

c c

Free vibrations with no damping

0mx kx Equation of motion:

mx

1 2cos sinn nx C t C t

Solution:

C1 and C2 are constants to be determined from

initial conditions x0 and v0

0 0cos sinn n nx x t v t

k

Model of a vibrating system

Spring and mass

Spring, mass and damper

Forced vibration

Balancing of machinery

Static unbalance

Couple unbalance

Dynamic unbalance