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 firstorder 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 StateSpace Expansion
One critical issue for the application of numerical integration to the differential equation of motion is that the equation of motion is secondorder, while the integration techniques require firstorder differential equations. This is not however a significant limitation. By a technique known as statespace expansion, a highorder differential equation can be converted into a set of simultaneous lowerorder differential equations. For the equation of motion, this means converting the single secondorder differential equation into a pair of simultaneous firstorder differential equations. S...
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 Fall '11
 Regalla
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