ME 321 Kinematics and Dynamics of Machines Steve

Forced (Harmonic) Vibration F(t) x m or, in normalized form: with: 10/28/2020 c k

Summary of Undamped Response The solution is always the summation of the transient and

Steady-State Solution We assume a solution of the form: This has the same frequency,

Steady-State Solution Take the derivatives of the assumed solution with respect to time: And

Steady-State Solution Changing back to the amplitude and phase-angle form, the steady-state solution becomes:

Total Solution The total solution is the summation of this steady-state solution and the

Example 6. 4: Determine the steady-state response (amplitude and phase angle) for a mass-spring

Normalized Response (Steady) It is common to normalize the steady-state response as follows: This

Peak Response Notice that the maximum amplitude does not occur at r = 1.

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ME 321 Kinematics and Dynamics of Machines Steve Lambert Mechanical Engineering, U of Waterloo 10/28/2020

Forced (Harmonic) Vibration F(t) x m or, in normalized form: with: 10/28/2020 c k

Summary of Undamped Response The solution is always the summation of the transient and steady-state responses. When there is no damping, there is no phase shift, and the response is singular at the natural frequency. 10/28/2020

Steady-State Solution We assume a solution of the form: This has the same frequency, , as the excitation, and has a phase lag, , compared to the excitation. This can be rewritten as: with: This makes manipulations easier. 10/28/2020

Steady-State Solution Take the derivatives of the assumed solution with respect to time: And substitute into the governing differential equation: We can solve for As and Bs in terms of the system parameters: 10/28/2020

Steady-State Solution Changing back to the amplitude and phase-angle form, the steady-state solution becomes: with: 10/28/2020

Total Solution The total solution is the summation of this steady-state solution and the previous transient solution: The integration coefficients, A and , are again determined from the initial conditions, x 0 and v 0, but this time they also depend on the forcing function. 10/28/2020

Example 6. 4: Determine the steady-state response (amplitude and phase angle) for a mass-spring damper system that has the following properties: F 0 = 1000 N, m = 100 kg, =0. 1, n = 10 s-1, and = 5 s-1. What is the total response for an initial displacement of 0. 05 m and no initial velocity? 10/28/2020

Steady-State Response 10/28/2020

Total Response 10/28/2020

Normalized Response (Steady) It is common to normalize the steady-state response as follows: This can be expressed in terms of the frequency ratio: r = / n 10/28/2020

Steady-State Amplitude 10/28/2020

Steady-State Phase Angle 10/28/2020

Peak Response Notice that the maximum amplitude does not occur at r = 1. For < 0. 707, it occurs at: This maximum amplitude is: 10/28/2020