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 $v$ .