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I will use the equation ms(Xs)''+C(Xs)'+kXs=C(Xu)'+kXu where Xs and Xu are the displacements of the sprung and unsprung mass, k-spring stiffness, C-stiffness of the shock absorber, ms-sprung mass. What methods are used to solve such equations? This is similar to an equation with two variables with respect to the parameter t?
If you factorize C and K your differential equation represents a single degree of freedom system.
There are quite a few ways of solving a SDOF mass spring damper system and a simple google search will produce many well presented methods, far better than what can be described on a forum.
What you need to understand is that a differential equation does not have a closed form solution. The solution is a function.
In this particular case I suppose Xu(t) is a known function and you are trying to determine Xs, Xs' and Xs''.
Xu' is simply found by integrating Xu(t). A simple and practical solution is to create an excel spreadsheet with a very fine timestep (say 1/1000 of the natural frequency), assume some initial conditions for Xs and its derivatives and calculate Xs'' from the force balance then integrate over the time step to find Xs' and Xs'' at each point in time. Having said that, care must be taken when integrating as small accuracy errors will accumulate and produce a drift in system response.
The advantage of this method is that it works for random excitations and non-linear systems (for example you can model variable spring stiffness, variable damping, limiting displacement conditions, etc). The downside is that it is very computationally expensive.
For linear systems where the excitation function is periodic and continuous (hence derivable) the solutions can be elegantly computed in frequency domain. For example in the study of engine driveline torsional vibration the torque pulses can be decomposed into a finite number of harmonic functions. The system can be solved separately for each harmonic excitation and the solutions assembled at the end using the principle of superposition.
Unfortunately for car suspensions I believe that the road excitation is mostly random and hence the system has to be solved in time domain.
