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FirstSecondOrder

# FirstSecondOrder - MASSACHUSETTS INSTITUTE OF TECHNOLOGY...

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MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING 2.151 Advanced System Dynamics and Control Review of First- and Second-Order System Response 1 1 First-Order Linear System Transient Response The dynamics of many systems of interest to engineers may be represented by a simple model containing one independent energy storage element. For example, the braking of an automobile, the discharge of an electronic camera flash, the flow of fluid from a tank, and the cooling of a cup of coffee may all be approximated by a first-order differential equation, which may be written in a standard form as τ dy dt + y ( t ) = f ( t ) (1) where the system is defined by the single parameter τ , the system time constant, and f ( t ) is a forcing function. For example, if the system is described by a linear first-order state equation and an associated output equation: ˙ x = ax + bu (2) y = cx + du. (3) and the selected output variable is the state-variable, that is y ( t ) = x ( t ), Eq. (3) may be rearranged dy dt - ay = bu, (4) and rewritten in the standard form (in terms of a time constant τ = - 1 /a ), by dividing through by - a : - 1 a dy dt + y ( t ) = - b a u ( t ) (5) where the forcing function is f ( t ) = ( - b/a ) u ( t ). If the chosen output variable y ( t ) is not the state variable, Eqs. (2) and (3) may be combined to form an input/output differential equation in the variable y ( t ): dy dt - ay = d du dt + ( bc - ad ) u. (6) To obtain the standard form we again divide through by - a : - 1 a dy dt + y ( t ) = - d a du dt + ad - bc a u ( t ) . (7) Comparison with Eq. (1) shows the time constant is again τ = - 1 /a , but in this case the forcing function is a combination of the input and its derivative f ( t ) = - d a du dt + ad - bc a u ( t ) . (8) In both Eqs. (5) and (7) the left-hand side is a function of the time constant τ = - 1 /a only, and is independent of the particular output variable chosen. 1 D. Rowell 10/22/04 1

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Example 1 A sample of fluid, modeled as a thermal capacitance C t , is contained within an insulating vacuum flask. Find a pair of differential equations that describe 1) the temperature of the fluid, and 2) the heat flow through the walls of the flask as a function of the external ambient temperature. Identify the system time constant. T C T a m b q w a l l s R t f l u i d C t C t R t T C T a m b T r e f h e a t f l o w Figure 1: A first-order thermal model representing the heat exchange between a laboratory vacuum flask and the environment. Solution: The walls of the flask may be modeled as a single lumped thermal resistance R t and a linear graph for the system drawn as in Fig. 1. The environment is assumed to act as a temperature source T amb ( t ). The state equation for the system, in terms of the temperature T C of the fluid, is dT C dt = - 1 R t C t T C + 1 R t C t T amb ( t ) . (i) The output equation for the flow q R through the walls of the flask is q R = 1 R t T R = - 1 R t T C + 1 R t T amb ( t ) . (ii) The differential equation describing the dynamics of the fluid temperature T C is found directly by rearranging Eq. (i): R t C t dT C dt + T C = T amb ( t ) . (iii) from which the system time constant τ may be seen to be τ = R t C t .
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FirstSecondOrder - MASSACHUSETTS INSTITUTE OF TECHNOLOGY...

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