Review of First- and Second-Order System Response 1 First ...

MASSACHUSETTS INSTITUTE OF TECHNOLOGYDEPARTMENT OF MECHANICAL Advanced System Dynamics and ControlReview of First - and Second-Order System Response11 First -Order Linear System Transient ResponseThe dynamics of many systems of interest to engineers may be represented by a simple modelcontaining 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 cupof coffee may all be approximated by a First -order differential equation, which may be written in astandard form as dydt+y(t) =f(t)(1)where the System is defined by the single parameter , the System time constant, andf(t) is aforcing function. For example, if the System is described by a linear First -order state equation andan associated output equation: x=ax+bu(2)y=cx+du.(3)and the selected output variable is the state-variable, that isy(t) =x(t), Eq.

Solution: The tank is represented as a °uid capacitance Cf with a value: Cf = A ‰g (i) where A is the area, g is the gravitational acceleration, and ‰ is the density of water. In this case Cf = 2=(1000£9:81) = 2:04£10¡4 m5/n and Rf = 1=10¡6 = 106 N-s/m5. The linear graph generates a state equation in terms of the pressure across the °uid

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