Numerical Solutions to Harmonic Motion Problems
I wish to discuss the way to set up and solve harmonic motion - both simple harmonic and not - in this section. The process will look just like our previous numerical methods exercises. It will entail:
- Finding all the relevant forces and expressing them mathematically
- Setting the sum of the forces equal to mass times acceleration
- Solving for acceleration (v')
- Giving GeoGebra expressions for r', v' and having it solve them numerically
Simple Harmonic Motion Numerical Solution
There is only one force acting in a system that is simple harmonic. It is a linear restoring force. Therefore, our only force is Therefore we can write which gives us
Within GeoGebra, we need to enter three statements: 1)
r_x'(t,r_x,v_x)=v_x, 2) v_x'(t,r_x,v_x)=-k*r_x/m
, and 3) NSolveODE({r_x',v_x'},0,{1,0},100). In the last statement I used an initial position of 1 (assumed meter), zero initial velocity and run the calculation for 100 seconds into the future. Let's see what the solution looks like.Numerical Solution
Harmonic Motion (non-simple)
If you wish to calculate harmonic motion with a restoring force that is anything but linear, such as , the solution will still oscillate, but the position versus time will not lead to a true sinusoidal function. Adding another force that is velocity-dependent can make the motion damped. Try it out in GeoGebra.