1. ME321 Kinematics and Dynamics of Machines
Steve Lambert
Mechanical Engineering,
U of Waterloo
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2. Rotating Unbalance m k c x
rt m0 e Fr xr m0
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3. Rotating Unbalance Equations of Motion: For the unbalanced mass:
For the net mass:
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4. Rotating Unbalance Assume a particular solution of the form:
By analogy to earlier solutions:
and
for r = r/n
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5. Rotating Unbalance
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6. Rotating Unbalance
Example 6.5: A machine has a rotating unbalance, which results in a maximum steady-state deflection of 1 cm at resonance. Based on a measurement of the free decay of the system, it is estimated that the damping ratio is  = The total mass of the system is 100 kg, and it is estimated, from manufacturing considerations, that the magnitude of the mass unbalance is 5 kg. Estimate the effective radius, e of the unbalance, and the amount of mass that would have to be added to the machine to reduce the vibration to 1 mm.
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7. Base Excitation m k c
x(t)
y(t)
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8. Base Excitation Assume a base excitation of the form:
This gives the following governing differential equation:
Or, in normalized form:
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9. Base Excitation There are two particular solutions.
One due to the force of the damper:
And one due to the force of the spring:
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10. Base Excitation
These two solutions have the same frequency, and can be combined as follows:
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11. Base Excitation
Displacement transmissibility:
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12. Base Excitation
The force acting on the mass through the damper and spring is:
or:
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13. Base Excitation The transmitted force can be rewritten as: with:
or, in normalized form:
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14. Base Excitation
Force transmissibility:
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15. Base Excitation
Force transmissibility (dashed line) and displacement transmissibility (solid line) for  = 0.05
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16. Base Excitation
Example 6.6: Consider a single degree-of-freedom model of an automobile suspension travelling over a rough road. The road is modeled as providing a base excitation, in m, of
The equivalent stiffness of the suspension is k = 4  105 N/m, a damping coefficient, c = 40  103 kg/s, and a mass of 1000 kg. Determine the steady-state amplitude and displacement of the automobile mass.
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