1. ME321 Kinematics and Dynamics of Machines
Steve Lambert
Mechanical Engineering,
U of Waterloo
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2. Forced (Harmonic) Vibrationk c x
F(t)
or, in normalized form:
with:
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3. 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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4. 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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5. 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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6. Steady-State Solution
Changing back to the amplitude and phase-angle form, the steady-state solution becomes:
with:
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7. 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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8. 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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9. Steady-State Response
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10. Total Response
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11. 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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12. Steady-State Amplitude
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13. Steady-State Phase Angle
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14. Peak Response
Notice that the maximum amplitude does not occur at r = 1.
For  < 0.707, it occurs at:
This maximum amplitude is:
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