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The free vibrations of a linear n-degree-of-freedom system are governed by !! + Kx = 0 Mx where x is a nx1 vector of unknowns M is a nxn mass matrix and K is a nxn stiffness matrix

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The free vibrations of a linear n-degree-of-freedom system are governed by !! + Kx = 0 Mx where x is a nx1 vector of unknowns M is a nxn mass matrix and K is a nxn stiffness matrix. The normal mode solution x = Xeiwt is substituted into the differential equation to determine the natural frequencies, w and the mode shape vectors X . Substitution leads to the following eigenvalue problem M -1KX = l X where l = w 2 . In other words the natural frequencies are the square roots of the eigenvalues of M-1K . The mass and stiffness matrices are symmetric if formulated using energy methods but the product of M -1K is not necessarily symmetric. Assume that both matrices are positive definite. Define the energy inner products ( x, y )M = ( Mx, y ) and ( x, y )K = ( Kx, y ) . (a) Show that M -1K is self adjoint and positive definite with respect to the potential energy inner products. (b) For a four-degree-of-freedom system the mass and stiffness matrices are ? 0 ?=# 0 0 0 2? 0 0 0 0 3? 0 0 0 ( 0 ? 2? −? ?=# 0 0 −? 3? −2? 0 0 −2? 4? −2? 0 0 ( −2? 2? Let k = 1.2 ´106 N/m and m = 150 kg . Determine the natural frequencies for this system. (c) Determine the mode shape vectors for the system. Label these such that w1 < w2 < w3 < w4 (d) Determine a set of normalized mode shape vectors (normalize with respect to the kinetic energy inner product) (e) Demonstrate mode shape orthonormality with respect to the kinetic energy scalar product for X2 and X4 . (f) Demonstrate orthogonality with respect to the potential energy scalar product for X2 and X4 . (g) Determine the response of the system if the vibrations are initiated by giving the block of mass 3? a velocity ? while it is in equilibrium and the velocities of the other masses are zero. That is 0 ?(0) = 203, 0 0 0 03 ?? (0) = 2? 0 2. The equation governing free torsional oscillations of a tapered shaft is ¶ é ¶q ù ¶ 2q a ( x ) ú = b ( x) 2 ¶x êë ¶x û ¶t Note that a is a non-dimensional form of GJ where G is the shear modulus of the bar and J is the bar’s cross sectional polar moment of inertia and b is the non-dimensional form of r J q ( x, t ) = u ( x)eiwt . Substitution into the governing partial differential equation leads to Lu = lu where where r is the bar’s mass density. A normal mode solution is assumed as 1 d é du ù a ( x) ú and l = w 2 . A shaft that is fixed at x = 0 attached to a discrete ê b ( x) dx ë dx û du torsional spring at x = 1 has boundary conditions of u (0) = 0 and (1) + k u(1) = 0 dx (a) All eigenvalues of L are real and positive. How do we know this? (b) If the eigenvalues and normalized eigenvectors of L are designated by li and ui ( x) Lu = - respectively write out in integral form the orthonormality conditions satisfied by ui ( x) and u j ( x) . Write this out with respect to the kinetic energy inner product and the potential energy inner product. (c) Describe how you would approximate the eigenvalues and eigenvectors using the RayleighRitz method from the space of polynomials of degree 6 or less. That is (i) what basis functions would you use, (ii) write out the appropriate forms of the inner products and (iii) what are the equations for the coefficients in the linear combination? (d) If a bar is uniform and homogeneous (a ( x) = 1 and b ( x) = 1) and k = 1.5 determine the first five natural frequencies and mode shapes, ?! (?) for i=1,2,…5. HW 5 Consider a tapered bar that is attached to a spring at ? = 0 and fixed at ? = 1. The nondimensional partial differential equation governing the vibrations of the bar is ? ?? ?!? &?(?) + = ?(?) ! ?? ?? ?? The boundary conditions are ?? (0) − ??(0) = 0 ?? ?(1) = 0 "# where ? = $%(') is a nondimensional representation of the spring force. A normal mode solution is assumed as ?(?, ?) = ?(?)? )*+ where ?s are the natural frequencies and ?(?) is the spatial variation of the mode shape. Substitution into the above problem leads to ? ?? &?(?) + = −?! ?(?)? ?? ?? subject to ?? (0) − ??(0) = 0 ?? ?(1) = 0 The bar is rectangular in cross section and linearly tapered such that ?(?) = 1 − 0.1? ?(?) = 1 − 0.1? The spring force is such that ? = 2. (a) The differential equation has a Bessel function solution. Use this and approximate the first two natural frequencies. (b) The functions ?(?) and ?(?) can be written as 1 − ?? with ? = 0.1. Use a perturbation method and the Fredholm Alternative to approximate the first three natural frequencies of the system. Objectives: The objectives for this course are to provide the student with a mathematical foundation with which they can (1) solve problems in mechanical engineering and (2) read and understand the engineering literature. Syllabus 1. II. III. Mathematical modeling A. Vibrations problem B. Heat transfer problem C. Review of differential equations Non-dimensionalization A. Buckingham Pi Theorem and Pi groups B. When the mathematical model is known Linear Algebra A. Vector Spaces 1. R" 2. c?[a,b] 3. Subspaces B. Linear Independence C. Basis and Dimension D. Inner Products E. Norms F. Gram-Schmidt Orthogonalization G. Operators 1. Matrix operators 2. Differential operators H. Adjoint Operators 1. Self-adjoint operators 2. Stiffness matrix 3. Heat transfer differential operator with BC 4. Partial differential operator with BC defined in any region I. Positive Definite Operators J. Energy Inner Products 1. Kinetic energy inner products 2. Potential energy inner products IV. v. Variational Methods A. Least Squares B. Rayleigh-Ritz C. Finite Elements Eigenvalue Problems A. Definition B. Self-adjoint Operators C. Positive Definite Operators D. Finite Dimensional Vector Spaces (Matrix eigenvalue problems) E. Sturm-Liouville Problems F. Expansion Theorem G. Fourth-order Differential Equations H. Special Operators 1. Fredholm Alternative Special Functions A. Method of Frobenius B. Bessel Functions 1. Eigenvalue problems C. Legendre Functions Partial Differential Equations A. Laplace's Equation B. Wave Equation C. Three Dimensional Wave Equation D. Non-homogeneous Equations E. Green's Functions VI. VII.