Vibrations: Multiple Degrees of Freedom

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Vibrations Multiple Degrees of Freedom
So far one-dimensional motion (e.g. x, or y or
?) for a single body This is one degree of
freedom (d.o.f.) (1 body)x(1 dimension)1
Most systems consists of motion in multiple
dimensions (d2 or d3) and/or multiple
connected bodies (N)
2 bodies x, y motion 4 d.o.f.
N bodies in d dimensions N x d degrees of
freedom i.e. N x d different possible ways
the entire system can move.
This can mean that the system has N x d different
vibration frequencies
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Same Physics Fma Get Equations of Motion for
all d.o.f.s
Lets see how this works in detail for a simple
but instructive case
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2 masses, 3 springs, motion in lateral (x)
direction only
How many d.o.f.s?
Coordinates of motion x1 , x2
Draw F.B.D
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Determine Equations of Motion
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Lets guess solutions again look for both to
move with the same frequency
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Rewrite in Matrix Form (two simultaneous
equations)
This is an Eigenvalue Problem (another
Characteristic Equation)
What vector C1,C2, when acted upon by the
matrix M, gives the same vector just multiplied
by some constant (?2)? What is the constant ?2
? What is the vector C1,C2?
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Solution ? Determinant of the matrix M-?2I must
be zero
Can find determinants in Maple or compute
numerically in MATLAB. For simple 2x2 matrix
here, we can do the algebra
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Two natural frequencies of vibration for the
system (as advertised, since the system has 2
d.o.f.s)
Can we physically understand what motions
correspond to these two frequencies?
Yes, consider the two types of motion shown
earlier
Two masses move together Center spring doesnt
stretch System 2m mass connected to 2 k1
springs
Two masses move opposite Stretch of k2 spring is
double that of k1 springs
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Can we get those motions out of the math? Of
course..
Need to find the relationships between C1 and C2
Return to matrix equation, plug in value for ?,
and solve for C1, C2
The two masses move together, in sync, as argued
physically
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Similarly
The two masses move exactly opposite, as argued
physically
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Each natural frequency has a corresponding mode
of vibration, i.e. the masses move in a
particular way
General motion consists of a weighted sum of the
possible modes of vibration
where A, B, C, and D are determined by the
initial conditions for positions and velocities
of masses 1 and 2 (four conditions four
constants!)
Lets consider motion obtained by displacing mass
1 and releasing the system from rest
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Lets plot these solutions in MATLAB for various
k1, k2
Suppose that k2ltltk1, i.e. the two masses are
weakly coupled by the middle spring. Then..
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For times tltlt?/2??, we can approximate
cos(??t)1, sin(??t)0
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tltlt?/2??
Mass 1 oscillates like it is not connected to
mass 2 Mass 2 does not move
Over time t?/??, mass 2 motion builds up, mass
1 motion decreases to zero
Two frequencies appear to control motions
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Example
k20.05 k11 m1
Beating Phenomenon Interference of two close
frequencies to create oscillations at a much
lower frequency associated with the difference in
the two frequencies
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Another example of multiple degrees of freedom
One body but moving in multiple dimensions
x1,y1
k1
k3
x3,y3
k2
x,y
x2,y2
k4
x4,y4
As in prior HW, we could displace the mass to
position (x,y) and compute the free body diagram
to get the forces, and then the EOM
Here, take another approach springs are
conservative forces, so the forces can be
obtained from derivatives of the potential energy
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x/L, y/L ltlt1
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So, EOMs are
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Solution find eigenvalues again
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Greenhouse Gases
A case of forced vibrations but at the molecular
level
Light waves from the sun at high frequencies
(1015/s) visible and ultraviolet
Earth heats up
Greenhouse gas absorb this radiation and
ultimately re-radiate it, some of which heads
back toward the earth not escaping
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Why is Carbon Dioxide bad? Why not Oxygen or
Nitrogen? (certainly plenty of it in the
atomsphere)
A vibration resonance problem.
http//www.chemtube3d.com/vibrationsCO2.htm
Draw electromagnetic forces acting on each atom
in CO2
CO2 can very effectively absorb electromagnetic
radiation that is at a frequency equal to the
natural frequency of vibration for this
asymmetric mode of vibration
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We could work out the vibration frequencies if we
know the interatomic potential energy function
between the C and O atoms
CO2 has a vibration mode at a frequency that is
just in the range of the heat radiation
frequencies from the earth.
Wavelength 2.6x10-5 m