Lecture 27

E12 Linear Physical Systems Analysis

April 28, 2026

In class
Groups of up to 3 allowed (but not required)
- Submission is still individual
Duration: Approx. 1 hour
Work on paper provided
Singer 034/035
Resources:
- Can access course website
- Cannot access any other website
- No calculators or calculator programs!!
Grade weightage TBD; minimum equal to 1 HW, maximum ?

A system is governed by the differential equation

\[6 \ddot{x} + 2 \dot{x} + 10 x = f(t), \tag{1}\]

where \(x(t)\) is its output and \(f(t)\) its input.

Make a sketch of the roots of the characteristic polynomial on the complex plane. Then, use your sketch to determine if this system is stable/unstable and overdamped/underdamped/critically damped.
Using the standard form \[T(s) = \frac{\omega_n^2}{s^2 + 2 \zeta \omega_n s + \omega_n^2}, \tag{2}\] find the transfer function \(T(s)\) for this system.
The free response of this system if \(x(0)=0\) and \(\dot{x}(0)=v_0\), a constant. Give your answer in the time domain as a mathematical expression in terms of \(t\). For this question, start by taking the Laplace Transform of Equation 1.
The step response of this system. Give your answer in the time domain as a mathematical expression in terms of \(t\). For this quesiton, use the transfer function \(T(s)\) from Equation 2.
Sketch the (magnitude) Bode plot for the system using Equation 2.
If \(T(s)\) is subjected to a sinusoidal input with a frequency of our choice and a magnitude of unity, determine the maximum possible magnitude and the coresponding phase shift in radians (relative to the input) of the steady-state output of this system. For full points, give your answers as exact numbers using square roots where necessary.