Lecture 2 example, harmonic oscillator with external force (ENGR5001)

Lecture 2 example, harmonic oscillator with external forceTutorialCurrent specThird8 min

ENGR5001 Lecture 2 worked example 1, harmonic oscillator with external force (slides 4-7 and 15)

A mass $M$ hangs from a spring of stiffness $K'$ attached to a fixed support. The displacement $x(t)$ of the mass is measured downwards from its static equilibrium position, and an external force $F(t)$ acts on the mass in the direction of $x$ . (The spring stiffness is written $K'$ so that it is not confused with the $K$ of the generalised form.)

Formulas you may need

Newton's second law $\sum F = M\ddot x$ ; linear spring force $K'x$ (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)
List the assumptions needed for a linear model.
(b)
Use Newton's second law to develop the model relating $F(t)$ to $x(t)$ .
(c)
Put the model into the generalised second order form and identify the output, the input (forcing function), $K$ , the natural frequency $\omega_n$ , the damping ratio $\zeta$ and the steady state gain.
(d)
What is the implication of the damping ratio you found for the free response?