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cxs k xs f mg but recall that k

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Unformatted text preview: with the applied external forces. Lecture Notes -12- 06/16/06 12:25 PM Mechanical Vibrations I Newton Example B Again, in order to solve Newton’s Equation, ∑M 0 = J cgθ + L ⋅ mLθ , L m, J cg for the given single degree-of-freedom system, it is first necessary to draw the θ complete free-body diagram. This involves identifying all internal and external forces acting upon the degree-of-freedom. In this case, there is no specific external moment applied at point ‘o’. Also, recall the parallel axis theorem, J 0 = J cg + mL2 . (See above for the general expression.) All the internal and external moments are combined to form the actual equation of motion. Notice that neither the support force ‘T’ nor the centripetal acceleration force ‘ mLθ 2 ’ contribute to the moment about point ‘o’. (However ‘T’ would contribute to the moments about the ‘cg’.) − mgL sin θ = ( J cg + mL2 ) θ T = mLθ mLθ 2 θ mg J cgθ collecting all the coordinate based terms to one side of the equation and all the externally applied moments to the other yields (J cg + mL2 ) θ + mgL sin θ = 0 This time, since the mg term ends up on the left hand side (multiplied by the coordinate), it is not possible to eliminate the mg term from the solution. So the exact equation of motion remains, (J cg + mL2 )θ + mgL sin θ = 0 However, because the mgL sin θ term is non-linear in the desired coordinate, it is possible to reduce the solution to a linear approximation by using the first order McClaurin Series expansion, sin θ ≈ θ for θ 1 , as follows. (J cg + mL2 )θ + mgLθ = 0 Lecture Notes -13- 06/16/06 12:25 PM Mechanical Vibrations I #3 - The Eigenvalue Problem In many areas of engineering, a mathematical concept referred to as the eigenvalue problem arises. While the general application of eigenvalue methods has been used to solve a wide variety of mathematical, scientific and engineering problems, in this course, we will focus upon its application to the area of mechanical vibrations. Because of this, we will not examine the eigenvalue/eigenvector problem in detail, but only its most basic characteristics. Mathematically, the eigenvalue problem can be expressed as: [ A]{ x} = λ { x} In other words, given a matrix [ A] , the object is to find a vector { x} that when multiplied times the matrix [ A] does not change direction, but only magnitude (and possibly sense.) If this vector can be found, it is referred to as an eigenvector of [ A] , and the scalar λ , which describes the change in magnitude and sense, is referred to as an eigenvalue of [ A] . (Note that in general, there will be as many eigenvalue/eigenvector pairs as the size of the matrix. Also, it is only defined for square matrices.) The solution is essentially a two-step process. First identify the eigenvalues of the matrix, and then solve for their associated eigenvectors. Solving for the eigenvalues involves manipulating the equation form as follows. [ A]{ x} − λ { x} = {0} [...
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