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Equations of motion for most engineering systems cannot be solved exactly.
Even very simple problems, such as calculating the effects of air resistance on the trajectory of a particle, cannot be solved exactly.
For nearly all practical problems, the equations of motion need to be solved numerically, by using a computer to calculate values for the position, velocity and acceleration of the system as functions of time.
Vast numbers of computer programs have been written for this purpose some focus on very specialized applications, such as calculating orbits for spacecraft (STK); calculating motion of atoms in a material (LAMMPS); solving fluid flow problems (e.g.
For example, Professor Crisco’s lab uses pendulum to measure properties of human joints, see this will be true as long as the internal force in the cable is tensile.
If calculations predict that the internal force is compressive, this assumption is wrong.
The purpose of these examples is to illustrate the straightforward, step-by-step procedure for analyzing motion in a system.
Although we solve several problems of practical interest, we will simply set up and solve the equations of motion with some arbitrary values for system parameter, and won’t attempt to explore their behavior in detail.
If you don’t know their values at all, you should just introduce new (unknown) variables to denote the initial conditions.
This is because you probably only derived an approximate solution, by assuming that the angle about the motion of the system.
You may even have visions of running a consulting business from your yacht in the Caribbean, with nothing more than your chef, your masseur (or masseuse) and a laptop with a copy of Mathematica Unfortunately real life is not so simple.
If you find you have fewer equations than unknown variables, you should look for any constraints that restrict the motion of the particles.
The constraints must be expressed in terms of the unknown accelerations.