4 | Forced Vibration
At the end of this section you should be able to
Identify a 1DOF, forced system
Find closed-form solution for damped or undamped 1DOF forced system under SHM motion
Model Coulomb friction in systems undergoing SHM
Account for distributed masses in modeling 1DOF systems
Introduction to Forced Vibration
Problem Solving Procedure
Model: Create a simplified schematic of the system (noting assumptions) and find equivalent values, may use Lagrange equations
EOM: Create EOM(s) with desired DOF(s) in standard form: dependent variables on LHS, independent variable on RHS
Identify: Based on EOM, identify the type of system (e.g. forced/free, damped/undamped) to find the form of solution
Parameters: Calculate natural frequency, etc. for system
ICs: Find initial conditions based on DOF(s) of EOM(s)
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