A block of mass is attached to a spring of stiffness and a damper of coefficient

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Oscillations, Waves & Acoustics · Simple, damped and driven oscillations

Conditions of Validity, Ch Linear ordinary differential equations: existence, the Wronskian, and a kernel in disguise

Problem

A block of mass

m=0.20\,\mathrm{kg}

is attached to a spring of stiffness

k=500\,\mathrm{N\,m^{-1}}

and a damper of coefficient

b=4.0\,\mathrm{kg\,s^{-1}}

, obeying

m\ddot{x}+b\dot{x}+kx=F(t)

with

x(0)=0

\dot{x}(0)=0

. Air resistance beyond the modelled damper is ignored. At

t=0

a constant force

F_0=100\,\mathrm{N}

is switched on and held on for

\tau=20\,\mathrm{ms}

, then switched off. ((a)) Write down the initial-value Green function

G(t,\xi)

of the operator

m\,\mathrm{d}^2/\mathrm{d}t^2+b\,\mathrm{d}/\mathrm{d}t+k

in terms of

\gamma=b/m

and

\omega_d=\sqrt{k/m-(\gamma/2)^2}

, and verify that its jump

\left[\partial G/\partial t\right]_{t=\xi^-}^{t=\xi^+}

equals

1/m

. ((b)) Using

x(t)=\int_0^tG(t,\xi)F(\xi)\,\mathrm{d}\xi

, derive a closed form for

x(t)

valid while the force is still on,

0\le t\le\tau

. ((c)) Evaluate the displacement at the instant the force is switched off,

x(\tau)

Related theory

Ordinary differential equations

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