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Unformatted text preview: ize N ( N × N ). Therefore, this characteristic equation will yield 2 N modal frequencies. Expanding the matrix characteristic equation completely yields a high order ( 2N ) polynomial characteristic equation with scalar coefficients: Lecture Notes -74- 06/16/06 12:25 PM Mechanical Vibrations I α 2 N s 2 N + α 2 N −1s 2 N −1 + α 2 N −2 s 2 N −2 + + α 2 s 2 + α1s1 + α 0 s 0 = 0 The characteristic values (complex valued modal frequencies ( λr = σ r + jω r )) are found as the roots of this characteristic, high order ( 2 N ), scalar polynomial. Once the modal frequencies ( λr ) have been determined, the characteristic vectors (modal vectors {ψ r } ) can be found from the following relationship: s 2 [ M ] + s [C ] + [ K ] { X } = {0} Evaluating at s = λr : λr2 [ M ] + λr [C ] + [ K ] {ψ r } = {0} Note that this system of linear equations is always rank deficient by at least one since the equation system is being evaluated at one of the characteristic frequencies ( λr ). This process must be repeated for each modal frequency to determine each modal vector. Once the modal frequencies (complex valued, in general) and modal vectors (complex valued, in general) are determined, the final form of the complementary solution can be formulated as follows: { xc (t )} = ∑ (α r {ψ r } eλ t + α r* {ψ r*} eλ t ) N r * r r =1 Note that, in the above equation, the unknown coefficients α r appear in complex conjugate pairs. This will always be true in the underdamped case. It is not necessary to assume that the conjugate relationship exists; this will result when the solution method is followed. If there is no forcing function (the particular solution is zero), the unknown (complex valued) coefficients in the above equation ( α r ) can be determined by applying the initial conditions to the above equation and/or the derivative of the above equation. The derivative of the above equation is shown below. { xc (t )} = ∑ (α r λr {ψ r } eλ t + α r*λr* {ψ r*} eλ t ) N r * r r =1 If there is a forcing function, the solution for the unknown (complex-valued) coefficients must wait until the particular solution has been found. The initial conditions apply to the Lecture Notes -75- 06/16/06 12:25 PM Mechanical Vibrations I complete solution and the unknown coefficients must be found after the particular solution has been added to the complementary solution. Particular Solution Approach The particular solution is found by assuming a solution form for the response consistent with the forcing function characteristic. Since the forcing function (steady-state) is some form of harmonic (sine plus cosine terms), the forcing function and associated response can always be put into the following form (use the Euler identity for sine and cosine). This approach to solving for the particular solution is known as the method of undetermined coefficients. f (t ) = A cos(ω at ) + B sin(ω at ) e jωat + e − jωat e jωat + e − jωat f (t ) = A + B 2...
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