12. Mathematical modelling in mechanics
From Mechanics
Mathematical modelling is used to describe the process of solving real world problems through the use of mathematics. The process begins when a problem statement passes through 3 main phases, formulation, solution and review, and culminates in a report. Mathematical models are often revised after the review stage and the formulation, solution, review process repeated. Each of these three phases is now considered in more detail.
Formulation
Before embarking on the formulation stage it is important to make sure that you understand the problem that has been posed. The first part of the formulation stage is to make a list of all the features of the problem that could influence the solution you will eventually give.
Some of the features identified may be very important while others are much more trivial. The second stage is to simplify the problem and to state a set of assumptions on which to base the problem.
The use of assumptions simplifies the problem considerably and allows a mathematical problem to be stated. The assumptions are important because they describe the conditions under which the mathematical solution will be obtained and they allow the problem to be simplified to an extent where it is possible to define a mathematical problem that can be solved.
Mathematical Solution
At this stage the mathematical problem defined in the formulation stage is solved to give a solution to the problem. This stage will involve many of the mathematical skills and techniques that you will have encountered in your mathematics and applied mathematics modules.
Review
This stage contains a number of elements which include interpretation, comparison with reality, criticism of results and reformulation. In some cases it will be important for interpretation of the solution to be given.
Comparison of results with real situations can be very useful for determining the validity of a solution.
Criticism of results is important to show any weaknesses of the solution, which may well be due to the initial assumptions.
In the light of the review of the solution it will very often be desirable to refine the solution to the problem and the first stage in this will be to reformulate the problem to take account of new factors or to revise the assumptions. The three stages of mathematical modelling are then often repeated until a satisfactory solution is obtained. The diagram shows how mathematical modelling has a cycle that can be repeated several times.
An Example of Mathematical Modelling in Mechanics
A car at an accident skids down a gentle slope leaving a skid mark of 20m before it collides with a stationary vehicle. How fast was the car travelling when it began to skid and was it breaking the 30mph speed limit?
Formulation
The first step is to draw up a feature list to include all those factors that might affect the solution of the problem. The list below includes a number of important factors.
• The road and tyre conditions.
• The gradient of the slope.
• The speed of impact between the two cars. The conditions of the road.
• Were all the wheels locked?
Are there any further factors that you think should be included?
The assumptions below allow a simple model to be formulated so that a mathematical problem can be defined.
• The road is horizontal.
• The speed of impact is zero.
• There is constant friction force and no air resistance.
• The coefficient of friction between the tyres and the road is 0.8. The car is to be modelled as a particle.
A car skids 20m to rest on a horizontal road. If the coefficient of friction is 0.8 find the initial speed of the car.
Mathematical Solution
As the car is skidding the friction force will take its limiting value of \displaystyle \mu R .
So,
\displaystyle F=-\mu mg=-0\textrm{.}8\times 10m
where \displaystyle m is the mass of the car and \displaystyle g is taken as
\displaystyle 10\ \text{m}{{\text{s}}^{-2}}.
The acceleration of the car is given by
\displaystyle a=\frac{F}{m}=-\frac{8m}{m}=-8\text{ m}{{\text{s}}^{\text{-2}}}.
Now the speed can be found using
\displaystyle {{v}^{2}}={{u}^{2}}+2as
\displaystyle \begin{align} & {{0}^{2}}={{u}^{2}}+2\times (-8)\times 20 \\ & {{u}^{2}}=320 \\ & u=17\textrm{.}9\text{ m}{{\text{s}}^{\text{-1}}} \\ \end{align}
Interpretation
The initial speed predicted can be converted to give 40mph. This figure clearly suggests that the car was breaking the speed limit.
Compare with Realitv
The Highway Code provides a useful source of data that can be used to compare the results with reality. The Highway Code quotes a distance of 80 feet or 24m as the braking distance for 40mph. This compares favourably with the prediction made above.
Criticism of the Model
The two major criticisms that can be made of this model are that the road is not horizontal and that the speed of impact is not taken into account.
Reformulation
The diagram shows the forces acting when the road is assumed to be at an angle of \displaystyle 5{}^\circ to the horizontal. The normal reaction now has the magnitude,
\displaystyle R=mg\cos 5{}^\circ
and the resultant force up the slope on the car is,
\displaystyle \mu mg\cos 5{}^\circ -mg\sin 5{}^\circ =mg(\mu \cos 5{}^\circ -\sin 5{}^\circ )
So the acceleration is
\displaystyle \begin{align} & a=g(\mu \cos 5{}^\circ -\sin 5{}^\circ ) \\ & =9\textrm{.}8(0\textrm{.}8\cos 5{}^\circ -\sin 5{}^\circ ) \\ & =-6\textrm{.}96\text{ m}{{\text{s}}^{\text{-2}}} \end{align}
Using this value for \displaystyle awith an impact speed of zero gives.
\displaystyle \begin{align} & {{0}^{2}}={{u}^{2}}+2\times (-6\textrm{.}96)\times 20 \\ & {{u}^{2}}=278\textrm{.}4 \\ & u=16\textrm{.}7\text{ m}{{\text{s}}^{\text{-1}}}\text{ or 37 mph} \\ \end{align}
Notice that taking account of the hill reduces the initial speed
As a final refinement to the model an impact speed
\displaystyle 9\ \text{m}{{\text{s}}^{-1}} or approximately 20 mph could be introduced. There was only minor damage to the vehicles which suggests that such a value is reasonable. Using this gives a revised initial speed
\displaystyle \begin{align} & {{9}^{2}}={{u}^{2}}+2\times (-6\textrm{.}96)\times 20 \\ & {{u}^{2}}=359\textrm{.}4 \\ & u=19\textrm{.}0\text{ m}{{\text{s}}^{\text{-1}}}\text{ or 42 mph} \\ \end{align}
It is interesting to note how close the two revised estimates are to the original prediction and that the conclusion that the car was breaking the speed limit was confirmed by both revisions.
