A scientific application

Physical problem and mathematical model

The task is to make a simulation program that can predict how a (simple) mechanical system oscillates in response to environmental forces. Introducing $ u(t) $ as some displacement of the system at time $ t $, application of Newton's second law of motion to such a mechanical system often results in the following type of equation for $ u $: $$ \begin{equation} mu'' + f(u') + s(u) = F(t), \tag{1} \end{equation} $$ The prime, as in $ u' $, denotes differentiation with respect to time ($ u'(t) $ or $ du/dt $). Furthermore, $ m $ is the mass of the system, $ f(u') $ is a friction force that gives rise to a damping of the motion, $ s(u) $ represents a restoring force, such as a spring, and $ F(t) $ models the external environmental forces on the system. Equation (1) must be accompanied by two initial conditions: $ u(0)=I $ and $ u'(0)=V $. The values of these have no effect on the steady state behavior of $ u(t) $ for large values of $ t $, since this behavior is determined by the force $ F(t) $ and the system parameters $ m $, $ f(u') $, and $ s(u) $.

There are two types of the friction force $ f $: linear damping $ f(u')=bu' $ and quadratic damping $ f(u')=bu'|u'| $. The input data consists of $ m $, $ b $, $ s(u) $, $ F(t) $, $ I $, $ V $, and specification of linear or quadratic damping. The unknown quantity to be computed is $ u(t) $ for $ t\in (0,T] $.

One example where the model above has relevance, is the vertical vibration of a vehicle in response to a bumpy road. Let $ h(x) $ be the height of the road at some coordinate $ x $ along the road. When driving along this road with constant velocity $ v $, the vehicle is moved up and down in time according to $ h(vt) $, resulting in an external vertical force $ F(t)=-mh''(vt)v^2 $. We assume that the vehicle has springs and dampers that here are modeled as $ bu' $ (damper) and $ s(s)=ku $ (spring), with given damping parameter $ b $ and spring constant $ k $. The unknown $ u(t) $ is the vertical displacement of the vehicle relative to the road. Figure 1 illustrates the situation. You may view an animation of such a motion on the web (by the way, this animation was created by coupling the Pysketcher tool for figure drawings with the differential equation solver presented later, see the script bumpy_road_fig.py for all details).

Figure 1: Vehicle on a bumpy road.

Another example regards the vertical shaking of a building due to earthquake-induced movement of the ground. If the vertical displacement of the ground is recorded as a function $ d(t) $, this results in a vertical force $ F(t)=-md''(t) $. The soil foundation acts as a spring and damper on the building, modeled through the damping parameter $ b $ and normally a linear spring term $ s(u)=ku $.

In both cases we drop the effect of gravity, which is just a constant compression of the spring.

Our task is to compute and analyze the vertical $ u(t) $ vibrations of a vehicle, given the shape $ h(x) $ of the road and some velocity $ v $.