ETBX Linear System Time Response Module

Linear System Time Response

An EngineersToolbox Calculation Module

Background Information:

This module uses classical closed-form equations to calculate the time response of linear dynamic systems to user-specified initial conditions and forcing functions. The differential equations defining the linear system must be expressed in one of the forms listed in Table 1. The module will solve first- and second-order differential equations with positive constant coefficients and the input conditions shown in Table 2. For numerical solutions to other types of differential equations, please use the ETB Arbitrary System Dynamics module.

Table 1. Input forms for system differential equations

	Differential Equation	System Type
1)		First Order
2)		First Order
3)		Second Order
4)		Second Order

Table 2. Supported input conditions

Input Conditions
1)	Initial Conditions Only
2)	Unit Step Input
3)	Unit Ramp Input
4)	Harmonic Input

Input:

The Linear System Time Response module input form is shown in Figure 1. The module verifies that all coefficients in the differential equation are positive.

Figure 1. Linear System Time Response module input form

Output:

The module calculates the time response of the dynamic system due to the specified input. Results are tabulated and plotted using standard ETB output facilities as shown in Figures 2 and 3.

Figure 2. Linear System Time Response plotted results.

Figure 3. Linear System Time Response tabulated results.

Example Problems:

Example 1 - Harmonic Excitation of an Undamped SDOF System

References:

Craig, R. R. (1981), Structural Dynamics, John Wiley & Sons (New York).

Thomson W.T., and Dahleh, M. D. (1997) Theory of Vibrations with Applications (5th Edition), Prentice Hall (New Jersey).

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

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