Kinematics and Dynamics of Machines: Steady-State and Transient Solutions

1. ME321 Kinematics and Dynamics of Machines Steve Lambert Mechanical Engineering, U of Waterloo

2. F(t) x m k c Forced (Harmonic) Vibration

3. F(t) x m k c Normalized Form of Equations where:

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

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

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.

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:

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.

9. Example Solution

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