MathCAD/MatLab program for non-linear spring-mass-damper system

sailsinthesun

1. The problem statement, all variables and given/known data

The cutting force developed during a particular machining operation is shown in figure (a).
Model the system as a SDF with equivalent mass of the cutting head = 25 kg. The damping is linear viscous (ξ = 0.1), and the equivalent spring is nonlinear “hardening” spring of the form k = k1 + k2*x^2, where k1 = 400 kN/m, and k2 = 40 kN/m3. Assume the initial conditions x(0) = 0 and
x' (0) = 0.

I need to write a computer program in MathCAD, Matlab, or similar to solve this problem.(These two programs are available to me)

(a) Draw the analytical model and write the D.E. of the motion of the system.
(b) Computer printout of the program listing.
(c) Computer output of results (t, x, x' )
(d) Computer plot for x(t) vs. t
(e) Computer plot for x'(t) vs. t
(f) Compute the inaccuracies (in the vertical direction) in the surface finish due to the cutting force.

2. Relevant equations

m(d^2y/dt^2)+c(dy/dt)+ky=Fo*sin(wt)

3. The attempt at a solution

The first real problem I've run into is modeling the non-linear spring. In MathCAD, when I put k:= 400+40y^2 it's saying y is undefined, which is true, but how should I define it? I'm not given like 0<y<5 or anything, so again, how should it be defined?

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HDK

did you ever get this complete?

The Gringo

Not sure why you bumped a year old thread, but can't you just solve this by reducing it to a system of first order ODES and applying a runge kutta method or something similar?

NemoReally

Given that it's intended to be solved in Mathcad or Matlab (the latter with no presumption of the symbolic toolbox), it would seem that's the appropriate general method.

However, the first step is to express the actual system as a DE using the variables given. (Apart from not being in the correct form for an ODE, the Mathcad fragment doesn't even use the same variable).