ME321 Kinematics and Dynamics of Machines

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ME321 Kinematics and Dynamics of Machines

• Steve Lambert
• Mechanical Engineering,
• U of Waterloo

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Forced (Harmonic) Vibration
or, in normalized form
with
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Summary of Undamped Response
The solution is always the summation of the
transient and steady-state responses. When there
is no damping, there is no phase shift, and the
response is singular at the natural frequency.
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Steady-State Solution
We assume a solution of the form
This has the same frequency, ?, as the
excitation, and has a phase lag, ?, compared to
the excitation. This can be rewritten
as with This makes manipulations easier.
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Steady-State Solution
Take the derivatives of the assumed solution with
respect to time
And substitute into the governing differential
equation
We can solve for As and Bs in terms of the system
parameters
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Steady-State Solution
Changing back to the amplitude and phase-angle
form, the steady-state solution becomes
with
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Total Solution
The total solution is the summation of this
steady-state solution and the previous transient
solution
The integration coefficients, A and ?, are again
determined from the initial conditions, x0 and
v0, but this time they also depend on the forcing
function.
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Example
Example 6.4 Determine the steady-state response
(amplitude and phase angle) for a mass-spring
damper system that has the following properties
F0 1000 N, m 100 kg, ?0.1, ?n 10 s-1, and
? 5 s-1. What is the total response for an
initial displacement of 0.05 m and no initial
velocity?
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Steady-State Response
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Total Response
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Normalized Response (Steady)
It is common to normalize the steady-state
response as follows
This can be expressed in terms of the frequency
ratio r ?/?n
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Steady-State Amplitude
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Steady-State Phase Angle
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Peak Response
Notice that the maximum amplitude does not occur
at r 1. For ? lt 0.707, it occurs at
This maximum amplitude is