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
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Normalized Form of Equations
where
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Undamped Solution
Assume the following form for the solution
Substitute into the governing differential
equation to get
So that gt
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Steady-State Solution
The steady-state solution to
is therefore
Note that resonance occurs when ? approaches?n
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Transient Solution
Earlier, we obtained the following transient
solution for this problem
This can be rewritten as
Where the integration coefficients, A1 and A2,
can be determined from the initial conditions on
displacement and velocity.
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Total Solution
The total solution is the sum of our transient
and steady-state solutions
After substituting in our initial conditions
We get the following final equation
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Example
Example 6.3 Plot the full response for system
with a stiffness of 1000 N/m, a mass of 10 kg,
and an applied force magnitude of 25 N at twice
the natural frequency. The initial displacement,
x0, is 0 and the initial velocity, v0, is 0.2
m/s.
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Example Solution
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Beat Phenomenon
We get a beat frequency equal to the difference
between the excitation frequency and the natural
frequency when they are similar