Lecture 2 example, mass-spring-damper (ENGR5001)

Lecture 2 example, mass-spring-damperTutorialCurrent spec2:28 min

ENGR5001 Lecture 2 worked example 3, mass-spring-damper (slides 18-20)

A mass $M$ hangs from a fixed support by a spring of stiffness $K'$ and a viscous damper of coefficient $C$ in parallel. An external force $F(t)$ acts on the mass, and $x(t)$ is the displacement of the mass from its static equilibrium position, measured in the direction of $F$ .

Formulas you may need

Newton's second law; spring force $K'x$ ; viscous damper force $C\dfrac{dx}{dt}$ (learn this)
Generalised second order form $\dfrac{d^2x}{dt^2} + 2\zeta\omega_n\dfrac{dx}{dt} + \omega_n^2 x = Ku(t)$ , steady state gain $K/\omega_n^2$ (learn this)

(a)
State the additional assumption needed compared with the undamped harmonic oscillator, and derive the model.
(b)
Put the model into generalised second order form, giving $K$ , $\omega_n$ , $\zeta$ and the steady state gain in terms of $M$ , $C$ and $K'$ .
(c)
How does the time response differ from that of the harmonic oscillator?