## Second Order Vibration Simulation in MATLAB

## Forced Sinusoidal Frequency Response and Free Response

This simulation tool in MATLAB displays a second order forced vibration system's response to sinusoidal input (the frequency response), and has the following features:

- Fully simulate the sinusoidal response of any spring-mass-damper or any series RLC circuit (when time and frequency units are normalized)
- View input and response together with gain and phase-lag on Bode plots
- Real-time slider control of parameters (input frequency, damping, etc)
- Option to display response split into transient and steady-state parts
- Simulate undamped response (beats, pure resonance) by setting \(\delta\) to \(0\) (where \(\delta\) is defined as \(\gamma/(2m)\) in the mechanical system and as \(R/(2L)\) in the electrical system)
- Simulate free response by setting the input amplitude \(A\) (and the input frequency \(\omega\)) to \(0\)
- For convenience, the natural undamped frequency \(\omega_0\) is fixed at 1, so that the input frequency \(\omega\) remains normalized

The Vibration Equation\[ \boxed{ \dfrac{d^2y}{dt^2} + 2\delta \dfrac{dy}{dt} + \omega_0^2 y = \omega_0^2 A \cos(\omega t - \theta)}, \] where the driving input (forcing term) \(A\cos(\omega t - \theta)\) produces the responce \(y = y(t)\). Thus \(\omega\) is the input frequency, \(A\) is the driving (input) amplitude, \(\theta\) is the input phase-lag, and:The \(Q\)-factor is defined as \(Q := \omega_0/(2\delta)\), so \(Q = \frac{1}{\gamma}\sqrt{km}\) in the mechanical system, and \(Q = \frac{1}{R}\sqrt{\frac{L}{C}}\) in the electrical system.

- In a mechanical spring-mass-damper system with mass \(m\), spring-constant \(k\), and damping ratio \(\gamma\), \[ \begin{aligned} \delta & := \frac{\gamma}{2m},\\ \omega_0 & := \sqrt{\frac{k}{m}},\\ A \cos(\omega t - \theta) & := \text{the driving displacement (input), and}\\ y & := \text{the displacement of the mass (response).} \end{aligned} \]
- In an electrical series RLC circuit with inductor \(L\), capacitor \(C\), and resistor \(R\), \[ \begin{aligned} \delta & := \frac{R}{2L},\\ \omega_0 & := \frac{1}{\sqrt{LC}},\\ A \cos(\omega t - \theta) & := \text{the driving (input) voltage (emf), and}\\ y & := \text{the capacitor voltage (response).} \end{aligned} \]
The MATLAB m-file is available.

Written by Abhijit Dasgupta for MTH-3720 (Differential Equations) at UDMercy

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