Introduction to Forced Vibration

Problem Solving Procedure

1. Model: Create a simplified schematic of the system (noting assumptions) and find equivalent values, may use Lagrange equations

2. EOM: Create EOM(s) with desired DOF(s) in standard form: dependent variables on LHS, independent variable on RHS

3. Identify: Based on EOM, identify the type of system (e.g. forced/free, damped/undamped) to find the form of solution

4. Parameters: Calculate natural frequency, etc. for system

5. ICs: Find initial conditions based on DOF(s) of EOM(s)

6. Assemble Solution: Perform arithmetic to find the closed-form solution

Undamped (Simple Harmonic) Forced Vibration

The homogeneous solution is

and the particular (steady-state) solution is

where

The total solution is then

Beating

The beat frequency is calculated by

Examples

Underdamped (Simple Harmonic) Forced Vibration

Prescribed Force

The total solution is

where

Prescribed Displacement

The particular (steady-state) solution is

where

and

Coulomb (Dry) Friction

This approximation only holds up if

Note: the written steady-state solution at 20:39 should use “sin” instead of “cos”

State-Space Form

See http://sites.psu.edu/me357/lectures/unit-1/ss_model/ and https://sites.psu.edu/cmpsc200/content/unit-3-numerical-mathematics/topic-9/ for information on State Space form and using ODE45

Including Distributed Mass of Bodies

The kinetic energy in the distributed mass is found using

If the body is non-homogeneus, then the density may change along the length. The more general form is then

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Code