ME 321 Kinematics and Dynamics of Machines Steve Lambert Mechanical Engineering, U of Waterloo 11/29/2020

Forced (Harmonic) Vibration F(t) x m c 11/29/2020 k

Normalized Form of Equations F(t) where: m c 11/29/2020 x k

Undamped Solution F(t) m Assume the following form for the solution: C=0 Substitute into the governing differential equation to get: So that ===> 11/29/2020 x k

Steady-State Solution The steady-state solution to: is therefore: Note that resonance occurs when approaches n 11/29/2020

Transient Solution Earlier, we obtained the following transient solution for this problem: This can be rewritten as: Where the integration coefficients, A 1 and A 2, can be determined from the initial conditions on displacement and velocity. 11/29/2020

Total Solution The total solution is the sum of our transient and steadystate solutions After substituting in our initial conditions: We get the following final equation: 11/29/2020

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, x 0, is 0 and the initial velocity, v 0, is 0. 2 m/s. 11/29/2020

Example Solution 11/29/2020

Beat Phenomenon We get a beat frequency equal to the difference between the excitation frequency and the natural frequency when they are similar 11/29/2020