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.
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