ETBX Arbitrary Dynamic System Response

Arbitrary Dynamic System Response

An EngineersToolbox Calculation Module

Example 1 - Harmonic Excitation of an Undamped SDOF System

The system shown in Figure 1 has a spring with a stiffness of 40 lb/in, and a mass of weight 38.6 lb. Calculate the dynamic response of the mass to an excitation u(t) = 10cos(wt) assuming the system is initially at rest; that is, x(0) = 0 and dx(0) = 0.

Figure 1. Harmonic excitation of an undamped SDOF system

The equation of motion for this single degree-of-freedom system is:

To solve this differential equation using the Arbitrary System Dynamic Response module, it must first be expressed in a state-space convention as follows:


where:

This second-order system of differential equations can be expressed by the following code, assuming a system initially at rest and an input u(t) = G*cos(wt):

x0[1] = 0;
x0[2] = 0;

dx[1]=x[2];
dx[2]=1/m*(G*cos(w*t)-c*x[2]-k*x[1]);

The module main input form used to solve the given example problem is shown in Figure 2. A screen shot of the input code required to solve the given example problem is shown in Figure 3. The calculated results are plotted in Figure 4 for t = 0 to t = p/4 for comparison to Example 4.1 of Craig (Reference 1 below). Tabulated output of the calculated results is shown in Figure 5 for t = 0 to t = 0.2 for comparison to Example 7.2 of Craig (Reference 1 below).

Figure 2. Arbitrary System Dynamic Response module input form

Figure 3. System input form for SDOF system with harmonic input

Figure 4. Plot of calculated time response

Figure 5. Tabulated results

References

Craig Jr., R. R., (1981) Structural Dynamics - An Introduction to Computer Methods, John Wiley and Sons (New York).

Forsythe, G.E., Malcom M.A., Moler, C.B. (1977) Computer Methods for Mathematical Computations, Prentice-Hall Inc. (New Jersey).

Shampine L. (1994) Numerical Solution of Ordinary Differential Equations, Capman and Hall (London).

Woods, R. L., Lawrence, K. L. (1997), Modeling and Simulation of Dynamic Systems, Prentice-Hall Inc. (New Jersey).

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