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Unformatted text preview: rt ˆ e sin(ω r t ) ≡ Fh(t ) mω r where h(t ) = 1 σ rt e sin(ω r t ) mω r The unit impulse response function ( h(t ) ) is important for the analysis of transients! Arbitrary Excitation Having developed the unit impulse response ( h(t ) ), it is now possible to establish the response ( x(t ) ) of a system to an arbitrary excitation ( f (t ) ). First identify the strength of the ˆ impulse at time ξ ( F = f (ξ )∆ξ ) and its contribution to the response ( x(t ) )) at time t ( x(t ) = f (ξ )∆ξ h(t − ξ ) ). Recognized that t is constant for the above expressions, it is f (t ) ξ that is variable. f (t ) ξ ∆ξ t f (ξ )∆ξ f (ξ )∆ξ h(t − ξ ) Because the system is linear, the principal of superposition ξ t −ξ applies and it is possible to ξ =t add up all the impulse contributions. Adding up all the contributions of the individual impulses yields the expression for the response of a system to an arbitrary excitation, (assuming zero initial conditions.) Lecture Notes -68- 06/16/06 12:25 PM Mechanical Vibrations I x(t ) = ∫ f (ξ )h(t − ξ )dξ 0 t This expression is called the convolution integral. As can be seen from the form, the forcing function ( f (ξ ) ) could be any arbitrary function and a general closed form solution is impractical. Lecture Notes -69- 06/16/06 12:25 PM Mechanical Vibrations I #14 – Numerical Solution of Equation of Motion The general solution of the differential equation of motion with initial conditions and an arbitrary forcing function is most easily done numerically. While, there are numerous techniques for solving linear, simultaneous, ordinary differential equations, this section will focus upon only one method, Euler’s Method, both for its simplicity and its representative character. Euler’s Method Euler’s Method is the simplest first-order method. It can be used to solve differential equations of the form, y′ = f ( x, y ) . It is an explicit method, which means that the derivative ( y′ ) is an explicit function of the variable x and all previous values of both x and y . Euler’s Method is very simple. Given the current value of the function ( yn ), the current derivative of the function ( y′ ) and the step size ( ∆x ), simply n evaluate: ′ yn+1 ≈ yn + yn ∆x Clearly the accuracy of this method is determined by the step size ∆x . y ( x) yn yn+1 ′ yn xn ∆x x State-Space Expansion One critical issue for the application of numerical integration to the differential equation of motion is that the equation of motion is second-order, while the integration techniques require first-order differential equations. This is not however a significant limitation. By a technique known as state-space expansion, a high-order differential equation can be converted into a set of simultaneous lower-order differential equations. For the equation of motion, this means converting the single second-order differential equation into a pair of simultaneous first-order differential equations. S...
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