The mathematical model is obtained by applying laws of nature to the physical model. Our simplifying assumptions lead to a mathematical model consisting of a linear ordinary differential equation with constant coefficients. This model is less realistic, but it retains essential characteristics of the physical system and is much easier to solve and use.
Click here to see the assumptions for our model and the consequences which result from those assumptions.
To develop a mathematical model, first decide what the variables are and choose a coordinate system.
If the independent variable is time t, which of the following are the dependent variables?
What choices of the following should you make in order to specify the coordinate system?
The following clickable diagram represents our choices for the coordinate frame. As you proceed, keep in mind how the choice of the positive direction affects the derivation.
Note that x1 and x2 must be restricted so that the springs remain under tension.
The study of mechanical systems is based on the balance of forces. The applicable law of nature is Newton's Second Law F=ma (which states that the product of the mass m and its acceleration a is equal to the net force F acting on the mass.) To apply Newton's Second Law, you should identify the forces acting on each mass and show them in a Free Body Diagram (FBD).
Click on the question marks in the FBD below to label the forces when the masses have positive displacement and velocity.
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The forces acting on the left mass are:
The forces acting on the right mass are:
Each restoring force of a spring is given by Hooke's Law, which states that the force is proportional to the difference between the length of the unstretched spring and the length of the spring at time t. However, these quantities are simply the displacements of the masses at time t, x1(t) and x2(t).
What are the total restoring forces of the springs?
The damping forces are based on the assumption that the damping is proportional to the velocity of the mass. So, the damping forces can be expressed by:
Fd1 = -c1x1dot
Fd2 = -c2x2dot
Why are the damping forces negative?
The force of the motor, whose derivation can be explained in detail by clicking on DERIVE, is:
Applying Newton's Second Law, we get the following equation:
Why is left hand side of the above equation ?
Using and dividing by the effective mass
, we get the following differential equation:
where ,
and
(the natural frequency of the system).
To describe the problem completely, it is also necessary to specify the initial state of the system.