Applications of Second-Order Linear Equations with Constant Coefficients
Chapter 4 gave you the tools to solve . In this chapter we put those tools to work. A single equation of this form describes a car body bouncing on its suspension, a machine shaking on its mounts, a sensor needle settling on a reading and the charge sloshing back and forth in an electric circuit. Once you can solve it and, just as important, read its solution, you can predict how such systems behave and how to design them.
We build everything around one model: a mass on a spring, with a damper and an external force. The same mathematics then carries over, symbol for symbol, to the series RLC circuit.
- 5.1 The mass–spring equation: Hooke's law, the equilibrium position, damping and sign conventions lead to .
- 5.2 Free, undamped motion: simple harmonic motion, period, frequency, amplitude and phase angle.
- 5.3 Free, damped motion: underdamped, critically damped and overdamped motion; quasi-period and logarithmic decrement.
- 5.4 Forced motion: transient and steady-state response, steady-state amplitude and phase lag.
- 5.5 Resonance: unbounded growth without damping, beats, the resonance frequency and amplitude response curves.
- 5.6 Electric circuits: Kirchhoff's voltage law, the RLC equation, the electrical–mechanical analogy and steady-state current.
What you will be able to do
- Turn a physical description, with its units and directions, into an initial-value problem.
- Classify the motion from the coefficients alone and sketch it without solving.
- Extract the numbers engineers care about: amplitude, period, decay rate, overshoot, steady-state amplitude, phase lag and resonance frequency.
- Choose a damping constant or a component value to meet a design requirement.
Prerequisites
You need the characteristic equation and its three root cases (Section 4.2), the method of undetermined coefficients including the modification rule (Section 4.3), and the structure of the general solution (Section 4.1). From physics you need Newton's second law and the difference between mass (kg) and weight (N).