ME 482

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Vibrations of Multi Degree of Freedom Systems
A Two Degree of Freedom System
Equation of Motion
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In Matrix Form
Or,
Solution in general,
where, X Mode shapes, ? Natural Frequencies
Substituting in the equation
Rearranging,
For the non-trivial solution, we set
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For the above example,
We find the eigenvalues (modes, natural
frequencies),
And the eigenvectors (mode shapes),
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Solution is then written in general as
With initial conditions specified as
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Unknowns Ai and ?i are found from
For the above problem
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Solution for the above example,
Or,
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Principal Coordinates
Find coordinates, called principal coordinates
p1, p2,,pn,
such that the equations are uncoupled, i.e.
let x Pp, where
Then the equations uncouple to become
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Viscous Damping (Proportional Damping)
The equations of motion for a multi-degree of
freedom system
with viscous damping
where, C ? K ? M Proportional Damping Matrix
? , ? constants
Equation with principal coordinates
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In standard form
Solution
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General Viscous Damping
The equations of motion for a multi-degree of
freedom system
with viscous damping
where, C General Damping Matrix
Solution is obtained from the equation
where,
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Solution is then
where, ? and ? are eigenvalues and eigenvectors
of matrix
Cj Constants (to be found from IC)
Review Example 6.17, p. 342
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Forced Vibrations of Multi-Degree-of-Freedom
Systems
Undamped Response for Harmonic Force (Excitation)
Solution is obtained from
where,
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Damped Response for General Excitation
Equation is converted to
where,
We do the transformation using the principal
coordinates
where,
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With the transformation the equations are
uncoupled as
where gi(t) is obtained by PT F
Solution is then obtained using the convolution
integral