two lecture vibration

7/29/2019 Two Lecture Vibration

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Vibration in two lectures

Sudipto Mukherjee, IIT Delhi

This lecture is build upon a basic mechanics course. We assume

that you understand basic concepts of degree of freedom as in the

figures below having one, two, two and infinite degree of freedom.

Morever that if the mass at the end in the figure below is

significantly (factor of 10?) heavier than the beam, it can be

approximated by a single degree of freedom. Otherwise, design

using the lumped mass approach would be dicey.


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In mechanics, we usually restrict ourselves to studying position (x),

velocity (x ) and acceleration (x) dependent forces. While the

acceleration dependent term is always linear and proportional tothe mass, that is not necessarily true of the position and velocity

dependent terms, stiffness and damping. Luckily, for small

velocities and displacement, linearity is observed in most

mechanical systems.

a) Free vibrations: system disturbed from an equilibrium

position (pendulum)

b) Forced vibration: external disturbance independent of the

motion of the system (eccentric rotating mass)

c) Self excited vibration: External disturbance is dependent onthe motion of the system (singing of wires)

d) Parametrically excited vibration: system parameters changingwith time.

e) Random vibration: Excitation is not deterministic but only

probabilistically known.

Undamped free vibration


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0mx kx sin cos cosn n nx t A t B t X t

With nk

m being the natural frequency, which is a property of

the system

If the system is made vertical, how does gravity modify the

system? Clearly, the mass will descend by an amount = mg/k.Writing the force balance equation in a coordinate system whose

origin isshiftedto the new static equilibrium position: mx mg k x kx

which is in the same form as earlier!

Energy Conservation Approach

21

2T mx ,

21

2V mgx k x with the datum as the

equilibrium position. Then using 0d

T Vdt

we get:

0xmx kx and noting that is not equal to zero at alltimes,we get the same equation. This approach is sometime easier that

the force balance approach, but works only for conservative

systems.


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There are several derivations of free vibration equations in Ghosh

and Mallik that you can look into (especially the Trifilar

suspension which is perhaps the only experimental method tomeasure moment of inertia). Books on vibration will have many

more.

One of special interest is the rod on the two roller system shown

below:

1 2,2 2

a x a xN mg N mg

a a

and writing the equation of motion

in thex direction:

2 1( )mx N N or / 0x mg a x which is similar to that

of a SDOF system with /n

g a !

There is no restoring element, and this is an example of a self

excited system where the restoring force is generated by the

motion of the rod! This system is actually used to measure friction

coefficients, especially velocity dependent friction coefficients.

Damping

Energy of real life systems get dissipated and the rate of decay offree vibration is a measure of damping. This could be due to:

a) Dry friction between surfaces (Coulumb)

b) Drag between moving body and fluid environment (viscous)

c) Internal friction within a material (hysteretic)

d) Through radiation of energy into environment (wave)


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Consider free vibration with viscous damping.

0xmx c kx Assuming a solution of the form:x = e

st, we get:

2 0ms cs k which yields2

1.2 22 4c c ksm m m

a)2c km , over damped, in which case both roots are real

and negative. 1 2s t s tx Ae Be and does not have periodic

motion.

b) 2cc km , critically damped, in which case both roots are

identical and the solution is of the form 2ct

mA Bt e

(aperiodic) and the ratio, c/cc = , is called the damping

ratio.

c) 2c km , under-damped, in which case both roots are

imaginary and the solution is periodic and with the usual n,

1

2 2cos 1nt nx Xe t


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Rotating vector and phase-plane representation.

Harmonic quantities are conveniently represented in this form withspeed of rotation , and length of the vector as the amplitude. The

projection of the vector on the reference line represents the

instantaneous value of the harmonic quantity.

Differentiation multiplies the magnitude by and rotates thevector by 90

o(leading) in the direction of rotation.

The phase-plane has vertical axis asx and the horizontal axis is

/ nx . A point at the tip of the vector represents a particularphase

of the motion. Note that the arrowhead is such that for a +ve

velocity,x increases.


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Undamped Vibration with Harmonic ExcitationThis solution of a SDF system subjected to a harmonic force

Fcostis used to analyze general vibration problems by using

Fourier transforms.

cosmx kx F t And the solution is form:

0

2

/

sin cos cos1 /n n n

F k

x A t B t t

for

n and

0sin cos sin2

n n

n

F tx A t B t t

m

for=n

For zero initial conditions we see, beats as there are two

frequencies in the final solution for the first case. For the second

case, the magnitude of the frequency at grows linearly with time

and will finally go to infinity. This is called resonance.


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Please see http://www.vibrationdata.com/Tacoma.htm

http://www.youtube.com/watch?v=lXyG68_caV4&feature=related

One needs to remember that all systems have some damping. So

the initial free vibration will damp out and only the steady state

solution will remain in the long run.

0

2

/cos

1

F kx t

r

or

0

2

/

cos1

F k

x tr

forr>1. Thus the response is in

phase or out of phase in the

two zones.

Deflection is more than the

static deflection till r> 2

Viscously damped Vibration with Harmonic Excitation

0 cosmx cx kx F t or

2 02 / cosn nx x x F m t Which has a solution of the form:

cos tx X with


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0

12 2

22

/

1 2

F kX

r r

and

2

2tan

1

r

r

Define

1

2 20 2 2

1

1 2

XM

F kr r

as the magnification factorand the peak amplitude occurs at

1

2 21 2p n

At resonance, the phase is /2, with the inertia force being

neutralized by the spring force.

The half power band width is usedto measure the damping for small

damping : 2 1 / 2 n


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Systems with base excitation

0mx c x y k x y

1

22

2 22

1 2

1 2

r

X Yr r

and

3

22

2

tan 1 2

r

r r

Transmissibility

TR =X/Y in support motion

=FTo/F0 in force excitation

1

22

2 22

1 2

1 2

rTRr r

in force

transmission as well as base motion.


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Usually ris kept less than 5. Note that very low kvalues lead to

large static deflections and large vibrations even if the force

transmitted is reduced.

Given one more lecture, it would have been interesting to study

two body vibration problems, especially that of the vibration

absorber.

two lecture vibration

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