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Unformatted text preview: monic excitation, the forcing function can be represented as: x(t) f(t) f (t ) = Fe st k c For the assumed solution, the response will have the compatible form, x(t ) = Xe st , x(t ) = sXe st and x(t ) = s 2 Xe st where both F & X are complex scalars. Substituting, the above solution forms into the differential equation of motion produces a solution that must be valid for every value of s. This results in the following Laplace Domain solution. ms 2 Xe st + csXe st + kXe st = Fe st By collecting common terms, the expression reduces to: ( ms 2 + cs + k ) Xe st = Fe st Lecture Notes -34- 06/16/06 12:25 PM Mechanical Vibrations I At this point the response/excitation ratio can be calculated. This ratio is called the Transfer Function. It is normally written as H ( s ) . Multi-Degree of Freedom Transfer Function The same basic development is applicable to the multi-degree of freedom system, as well. H (s) X 1 (s) = 2 F ms + cs + k Starting from the equation While the Transfer Function is a [ M ] { x(t )} + [C ] { x(t )} + [ K ] { x(t )} = { f (t )} convenient mathematical model for and combining with defining the input-output relationship, { x(t )} = { X } e st & { f (t )} = {F } e st well suited to controls applications, it is yields the matrix equation less appealing for vibrations purposes primarily because it a continuous [ M ] s 2 + [C ] s + [ K ] { X } e st = { F } e st mathematical expression. However, By recognizing the basic definition of the because the complex surface defined by Transfer Function, the expression the H ( s ) expression is analytic, any slice becomes, through the surface contains all the −1 H ( s ) = [ M ] s 2 + [C ] s + [ K ] information necessary to reconstruct the entire surface. This recognition leads to the use of the Fourier Transform, that while its mathematical background is different, it is effectively the Laplace Surface evaluated at s = jω . (Another argument for the equivalence of the Laplace Transform and the Fourier Transform lies in the fact that both transforms start with the same Time Domain function, i.e. g (t ) = L−1 (G ( s )) = F −1 (G (ω )) .) Evaluating the Transfer Function at s = jω yields another input-output form known as the Frequency Response Function (FRF). H (ω ) X 1 (ω ) = 2 F −ω m + jω c + k Multi-Degree of Freedom Frequency Response Function Just as the Transfer Function has a multidegree of freedom form, so does the Frequency Response Function. By evaluating the multi-degree of freedom (matrix) Transfer Function at s = jω , the multi-degree of freedom Frequency Response Function results. By contrast with the Transfer Function, the Frequency Response Function is well suited for vibration applications, particularly experimental applications. While there is no discrete equivalent to the Laplace Transform, there is a discrete Fourier Transform that yields essentially the same information as the continuous integral transform. This is particularly important when working with d...
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