8/8/2019 A1-Introduction to Vibration[1]

1/32

0

A1- Introduction to Vibration

A1- Introduction to Vibration

http://www.linsly.org/tennis/physics.htm

8/8/2019 A1-Introduction to Vibration[1]

2/32

1

Chapter Outline

1. Oscillatory Motion

2. Degrees of Freedom

3. Classifications

4. Modeling of a Forging Hammer

5. Modeling of a Motorcycle with a Rider

6. Spring Elements7. Mass Elements

8. Damping Elements

9. Harmonic Motion

10. Terminology

11. Nomograph

8/8/2019 A1-Introduction to Vibration[1]

3/32

2

1- Oscillatory Motion

All bodies possessing mass and elasticity

A means for storing potential energy A means for storing kinetic energy

A means by which energy is gradually lost

Simple Pendulum

8/8/2019 A1-Introduction to Vibration[1]

4/32

3

2- Degrees of Freedom

The minimum number of independent coordinates to

determine completely the positions of all parts of a system atany instant of time

A rigid body has 6 DOF to describe its motion(3 translation, 3

rotation)

8/8/2019 A1-Introduction to Vibration[1]

5/32

4

Single DOF

8/8/2019 A1-Introduction to Vibration[1]

6/32

5

Two DOF

8/8/2019 A1-Introduction to Vibration[1]

7/32

6

Three DOF

8/8/2019 A1-Introduction to Vibration[1]

8/32

7

3- Classifications

Discrete (lumped) vs Cotinuous (distributed)

Free vs Forced vibration Undamped vs Damped

Linear vs Non-linear

Deterministic vs Random vibration

8/8/2019 A1-Introduction to Vibration[1]

9/32

8

Free Vibration System oscillates under inherent forces (not external

forces)

Vibration occurs at one or more natural frequencies which

are based on system mass and stiffness

Forced Vibration

System oscillates under external forces If external force is oscillatory, system response occurs at

excitation frequency

If excitation occurs at one of the natural frequencies, then

resonance occurs

8/8/2019 A1-Introduction to Vibration[1]

10/32

9

4- Modeling of a Forging Hammer

8/8/2019 A1-Introduction to Vibration[1]

11/32

10

8/8/2019 A1-Introduction to Vibration[1]

12/32

11

8/8/2019 A1-Introduction to Vibration[1]

13/32

12

5- Modeling a Motorcycle with a Rider

8/8/2019 A1-Introduction to Vibration[1]

14/32

8/8/2019 A1-Introduction to Vibration[1]

15/32

14

8/8/2019 A1-Introduction to Vibration[1]

16/32

15

6- Spring Elements

PE= 0.5 k (x)2

8/8/2019 A1-Introduction to Vibration[1]

17/32

16PE= 0.5 k (x)2

8/8/2019 A1-Introduction to Vibration[1]

18/32

17

8/8/2019 A1-Introduction to Vibration[1]

19/32

18

KE= 0.5 m (v)27- Mass Elements

8/8/2019 A1-Introduction to Vibration[1]

20/32

19

8/8/2019 A1-Introduction to Vibration[1]

21/32

20

8/8/2019 A1-Introduction to Vibration[1]

22/32

21

8- Damping Elements

Viscous Damping

Flat plates separated by a thin film of lubricant Piston-Cylinder Dashpot

Coulomb (dry friction) Damping

Material/Solid/Hysteretic Damping

- small, little effect on natural frequency

DAMPING: large effect on minimizing response, avoidsresonance

8/8/2019 A1-Introduction to Vibration[1]

23/32

22

8/8/2019 A1-Introduction to Vibration[1]

24/32

23

8/8/2019 A1-Introduction to Vibration[1]

25/32

8/8/2019 A1-Introduction to Vibration[1]

26/32

25

9- Harmonic Motion

Oscillatory motion may repeat itself regularly (wall clock withpendulum) or display irregularity (earthquakes). When themotion is repeated in equal intervals of time, it is calledperiodic motion. x(t) = x(t+)

The repetition time is called theperiod of oscillation (t) and itsreciprocal, thefrequency (f). The simplest form of periodicmotion is harmonic motion (demonstrated by a masssuspended from a light spring) given by x = A sin (2t/).

Writing the displacement as x= A sin (wt), the quantity iscircular frequency, measured in radians per second. Becausethe motion repeats itself in 2 radians, = 2/ = 2f

8/8/2019 A1-Introduction to Vibration[1]

27/32

26

Where and f are the period and frequency of the harmonic

motion, measured in seconds and hertz (cycles per second)respectively. If we write y = B sin (t+), then x and y are

synchronous and y leads x by radians (which is called a

phase angle).

The velocity and acceleration of harmonic motion can be

simply determined by differentiation. They lead the

displacement by /2 and . Verify this.

8/8/2019 A1-Introduction to Vibration[1]

28/32

27

10- Terminology

Average value (sort of like dc current versus ac)

(Avg. value for a complete cycle of sine wave, A sin t is zero,for a half-cycle is 0.637 A)

Mean Square Value of a function is

The root mean square value (rms) is the square root of theabove. (The rms value for the sine wave, A sin t is 0.707 A.Show this.)

=

0 )(

1

lim dttxx

=

0

22 )(1

lim dttxx

8/8/2019 A1-Introduction to Vibration[1]

29/32

28

The decibel (dB) is frequently used in vibration measurements.It is defined in terms of a power ratio. With power being

proportional to the square of amplitude or voltage,

dB = 10 log10 (p1/p2) = 20 log10 (x1/x2)

An amplifier with a voltage gain of 5 has a decibel gain of+14. Because the decibel is a log unit, it compresses orexpands the scale.

When the upper limit of the frequency range is twice its lowerlimit, the frequency span is called an octave. Octave bands,for example,

Band 1: 10-20 Hz (bw 10); Band 2: 20-40 Hz (bw 20); Band 3:

40-80 Hz (bw 40); Band 4: 200-400 Hz (bw 200)

8/8/2019 A1-Introduction to Vibration[1]

30/32

29

11- Nomograph

8/8/2019 A1-Introduction to Vibration[1]

31/32

30

Frequency Sensitivity

8/8/2019 A1-Introduction to Vibration[1]

32/32

31

Any Questions ?

Thank You