ode

In Maple we can easily produce the explicit analytic solution of differential equations by invoking the Laplace Transform. As an illustration consider a Spring-Mass-Damper system. Use a 2nd Order Ordinary Differential Equation. The current 2nd ODE is expressed in terms of the damping coefficients and natural frequency. These two quantities is readily obtainable from the original ODE of the form,

M y'' + B y' + K y = U
where:
  M=mass, B=viscous damping, K=spring constant
  U=input
  y=mass displacement

#ax9.m  or ax9.txt
> ODE:=diff(y(t),t$2)+ 2* rho*omega*diff(y(t),t)+omega^2*y(t)=Heaviside(t);
> IC:=y(0)=0, D(y)(0)=0:
> solution:=dsolve( {ODE,IC}, y(t) , 'laplace');
# No damping coefficient
> rho:=0: simplify(solution);
# Fixed damping & varying frequencies
> rho:=1/6:simplify(solution);
> plot3d(rhs("),omega=2/3..4/3,t=0..20,style=hidden,orientation=[-30,45],axes=BOXED);
# Fixed frequency & varying damping coefficient  
> rho:='rho': omega:=1:
> plot3d( rhs(solution),rho=1/5..2,t=0..20,style=hidden,orientation=[-10,45],axes=BOXED);
>

Mountain Bike Suspension System.
(Uses Maple V): plots package, diff, dsolve, odeplot

This sample with photo & time responses is available in MATLAB section. Maple V code for Small Bump response is given below. Code uses system of three Ordinary Differential Equations with known initial conditions.

> #bike.txt
> #Moutain Bike Front Suspension System
> restart;
> with(plots):
> b:=2:
> sys := diff(x1(t),t)=x2(t),diff(x2(t),t)=-3*x1(t)+2*x3(t)+1,diff(x3(t),t)=2/b*x1(t)-2/b*x3(t): 
> fcns:={x1(t),x2(t),x3(t)}:
> p:= dsolve({sys,x1(0)=0,x2(0)=0,x3(0)=0},fcns,type=numeric):
> odeplot(p, [t,x1(t)], 0..16, numpoints=100,axes=BOXED,color=red);

Solution of Differential Equations