Example 1: Linear analysis with a simple mass-spring system

In this example we will consider the following mass-spring system:

Figure 60: Simple mass-spring system

Objectives:

• Performing Linear analysis,

• Introducing Eigenvalue analysis,

• Getting an Equilibrium position.

To demonstrate linearization we will start with the mass-spring system shown in figure Figure 60. This example has the advantage of being simple enough to check many of the results analytically.

Procedure

1. Construct the system.

   Note Simcenter Amesim Run users can get this system through Help > Get demo, from the AMERunTutorials directory. The file is called SimpleMassSpring.ame.

2. Use Premier Submodel.

3. Name it SimpleMassSpring.

4. Leave the parameters at default values.

5. Run a simulation.

Next you will use Simcenter Amesim to linearize the system about an operating point producing the standard A, B, C and D matrices for the state space equations with:

where x, u and y are respectively the state, control and observer variables. Strictly speaking this is true only for classical ordinary differential equations (O.D.E.s). Simcenter Amesim can also solve differential algebraic equations (D.A.E.s). The following indented paragraph is intended for the more mathematically inclined and can be skipped if you prefer.

For D.A.E.s, Simcenter Amesim will linearize the system to produce

where E is in general a singular matrix. Simcenter Amesim then processes these equations to give:

where s = d/dt and p is called the index of nil-potency index of nil-potency of the D.A.E.s.

We stress that you do not have to do the processing of the equations. This index is a measure of how difficult the system is to solve.

If p=0, the system of equations is actually an ODE. Normally p=1 and p=2 systems can be solved without trouble but p=3 is very difficult and p=4 is almost guaranteed to fail. The problem is that the equations become progressively more badly conditioned as the index increases.