1. ME321 Kinematics and Dynamics of Machines
Steve Lambert
Mechanical Engineering,
U of Waterloo
11/30/2018

2. Forced (Harmonic) Vibrationk c x
F(t)
11/30/2018

3. Normalized Form of Equationsk c x
F(t)
where:
11/30/2018

4. Undamped Solution F(t) x m Assume the following form for the solution:k
C = 0 x
F(t)
Assume the following form for the solution:
Substitute into the governing differential equation to get:
So that ===>
11/30/2018

5. Steady-State Solution
The steady-state solution to:
is therefore:
Note that resonance occurs when  approachesn
11/30/2018

6. 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.
11/30/2018

7. 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:
11/30/2018

8. 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.
11/30/2018

9. Example Solution
11/30/2018

10. Beat Phenomenon
We get a beat frequency equal to the difference between the excitation frequency and the natural frequency when they are similar
11/30/2018