Question

a mass of 100g is hung from a spring of spring constant 100nm^-1 and is subject to a viscous force -bx' with b=0.1Nsm^-1. what is the steady state vibrational amplitude when subject to a driving force F0sin(wt) with f0=0.1N and w=10s^-1

Answer 1

This question is a little tricky.
Your force equation will be
\[F_{net} = m\ddot{x}= f(t) - b \dot{x} -kx \]
where f(t) is your driving force. So, you have to solve the non-homogenous second order differential equation
\[ f(t)= m\ddot{x} + b \dot{x} + kx \]
which is done in two parts. In the first part, you solve
\[ 0 = m\ddot{x} + b \dot{x} + kx \]
which will give you the transient part your solution because it drops fast. You won't have to do this. The steady state portion involves using some guess
\[x = A \sin{\omega t} + B \cos{\omega t}\]
and then plugging it in to get the answer. You'll get a system of equations with 2 variables which is easily solved. In most places, you'll see this answer with an amplitude
\[A = \frac{F_0}{m \sqrt{ (\omega_0^2 - \omega ^2 )^2- (2\gamma\omega})^2}\]
where
\[w_0 = \sqrt{\frac{k}{m}} \text{ and } \gamma = \frac{b}{2m}\]
To get this, you will have to use complex analysis and most textbooks and courses do not cover this. You can plug your numbers into this equation if you'd like the basic answer.

Answer 2

OOps, A is the amplitude in the second to last equation with F_0

Answer 3

Thank you. I hate waves so much far to abstract! give me electronics or thermal any day.