Hutton - Fundamentals of Finite Element Analysis (523155), страница 82
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Substituting Equation 10.182 into the upper partition of Equation 10.180, we obtain([K aa ] − [K ac ][K cc ]−1 [K ca ]){ A a } = 2 [M aa ]{ A a }(10.183)as the reduced eigenvalue problem. Note that all terms of the original stiffnessmatrix are retained but not those of the mass matrix. Another way of saying thisis that the stiffness matrix is exact but the mass matrix is approximate.The difficult part of this reduction procedure lies in selecting the degrees offreedom to be retained and associated with the lumped mass terms.
Fortunately,finite element software systems have such selection built into the software.The user generally need specify only the number of degrees of freedom to be retained, and the software selects those degrees of freedom based on the smallestratios of diagonal terms of the stiffness and mass matrices. Other algorithms areused if the user is interested in obtaining the dynamic modes within a specifiedfrequency. In any case, the retained degrees of freedom are most often calleddynamic degrees of freedom or master degrees of freedom.This discussion is meant to be for general information and does not representa hard and fast method for reducing and solving eigenvalue problems.
Indeed,reference to Equation 10.182 shows that the procedure requires finding the inverse of a huge matrix to accomplish the reduction. Nevertheless, several powerful techniques have been developed around the general reduction idea. Theseinclude subspace iteration [12] and the Lanczos method [13]. The user of a particular finite element analysis software system must become familiar with thevarious options presented for dynamic analysis, as multiple computationalschemes are available, depending on model size and user needs.10.13 SUMMARYThe application of the finite element method to structural dynamics is introduced in thegeneral context of linear systems. The basic ideas of natural frequency and mode shapes areintroduced using both discrete spring-mass systems and general structural elements.
Use ofthe natural modes of vibration to solve more-general problems of forced vibration is emphasized. In addition, the Newmark finite difference method for solving transient responseto general forcing functions is developed. The chapter is intended only as a general introduction to structural dynamics. Indeed, many fine texts are devoted completely to the topic.REFERENCES1.2.3.Inman, D. J. Engineering Vibration, 2nd ed. Upper Saddle River, NJ: PrenticeHall, 2001.Hutton, D. V. Applied Mechanical Vibrations.
New York: McGraw-Hill, 1980.Huebner, K. H., and E. A. Thornton. The Finite Element Method for Engineers,2nd ed. New York: John Wiley and Sons, 1982.443Hutton: Fundamentals ofFinite Element Analysis44410. Structural DynamicsC H A P T E R 104.5.6.7.8.9.10.11.12.13.Text© The McGraw−HillCompanies, 2004Structural DynamicsGinsberg, J. H. Advanced Engineering Dynamics, 2nd ed. New York: CambridgeUniversity Press, 1985.Goldstein, H. Classical Mechanics, 2nd ed. Reading, MA: Addison-Wesley, 1980.Bathe, K.-J.
Finite Element Procedures. Englewood Cliffs, NJ: Prentice-Hall, 1996.Newmark, N. M. “A Method of Computation for Structural Dynamics.” ASCEJournal of Engineering, Mechanics Division 85 (1959).Zienkiewicz, O. C. The Finite Method, 3rd ed. New York: McGraw-Hill, 1977.ANSYS User’s Reference Manual. Houghton, PA: Swanson Analysis Systems Inc.,2001.Zeid, I. CAD/CAM Theory and Practice. New York: McGraw-Hill, 1991.Guyan, R.
J. “Reduction of Stiffness and Mass Matrices.” AIAA Journal 3, no. 2(1965).Bathe, K.-J. “Convergence of Subspace Iteration.” In Formulations and NumericalAlgorithms in Finite Element Analysis. Cambridge, MA: MIT Press, 1977.Lanczos, C. “An Iteration Method for the Solution of the Eigenvalue Problemof Linear Differential and Integral Operators.” Journal of the Research of theNational Bureau of Standards 45 (1950).PROBLEMS10.110.210.310.4Verify by direct substitution that Equation 10.5 is the general solution ofEquation 10.4.A simple harmonic oscillator has m = 3 kg, k = 5 N/mm . The mass receivesan impact such that the initial velocity is 5 mm/sec and the initial displacementis zero.
Calculate the ensuing free vibration.The equilibrium deflection of a spring-mass system as in Figure 10.1 ismeasured to be 1.4 in. Calculate the natural circular frequency, the cyclicfrequency, and period of free vibrations.Show that the forced amplitude given by Equation 10.28 can be expressed asU =X01 − r2r = 1with X 0 = F0 / k equivalent static deflection and r = f / ≡ frequency ratio.Determine the solution to Equation 10.26 for the case f = . Note that, forthis condition, Equation 10.29 is not the correct solution.10.6 Combine Equations 10.5 and 10.29 to obtain the complete response of a simpleharmonic oscillator, including both free and forced vibration terms. Show that,for initial conditions given by x (t = 0) = x 0 and ẋ (t = 0) = v0 , the completeresponse becomes10.5x (t ) =10.7v0X0sin t + x 0 cos t +(sin f t − r sin t )1 − r2with X 0 and r as defined in Problem 10.4.Use the result of Problem 10.6 with x 0 = v0 = 0, r = 0.95, X 0 = 2, f = 10 rad/sec and plot the complete response x(t) for several motion cycles.Hutton: Fundamentals ofFinite Element Analysis10.
Structural DynamicsText© The McGraw−HillCompanies, 2004Problems10.810.910.1010.1110.12For the problem in Example 10.2, what initial conditions would be required sothat the system moved (a) in the fundamental mode only or (b) in the secondmode only?Using the data and solution of Example 10.2, normalize the modal matrix perthe procedure of Section 10.7 and verify that the differential equations areuncoupled by the procedure.Using the two-element solution given in Example 10.4, determine the modalamplitude vectors. Normalize the modal amplitude vectors and show that matrixproduct [ A] T [M ][ A] is the identity matrix.The 2 degrees-of-freedom system in Figure 10.4 is subjected to an externalforce F2 = 10 sin 8t lb applied to node 2 and external force F3 = 6 sin 4t lbapplied to node 3. Use the normalized modal matrix to uncouple the differentialequations and solve for the forced response of the nodal displacements.
Use thenumerical data of Example 10.2.Solve the problem of Example 10.4 using two equal-length bar elements exceptthat the mass matrices are lumped; that is, take the element mass matrices as10.1310.1410.1510.1610.1710.1810.1910.20 AL 1m (1) = m (2) =0401How do the computed natural frequencies compare with those obtained usingconsistent mass matrices?Obtain a refined solution for Example 10.4 using three equal-length elementsand lumped mass matrices. How do the frequencies compare to the two-elementsolution?Considering the rotational degrees of freedom involved in a beam element, howwould one define a lumped mass matrix for a beam element?Verify the consistent mass matrix for the beam element given by Equation 10.78by direct integration.Verify the mass matrix result of Example 10.6 using Gaussian quadraturenumerical integration.Show that, within the accuracy of the calculations as given, the sum of all termsin the rectangular element mass matrix in Example 10.6 is twice the total massof the element.
Why?What are the values of the terms of a lumped mass matrix for the element inExample 10.6?Assume that the dynamic response equations for a finite element have beenuncoupled and are given by Equation 10.120 but the external forces are notsinusoidal. How would you solve the differential equations for a general forcingfunction or functions?Given the solution data of Example 10.7, assume that the system is changed toinclude damping such that the system damping matrix (after setting u 1 = 0 ) isgiven by2c[C] = −c0−c2c−c0−c cShow that the matrix product [ A] T [C ][ A] does not result in a diagonal matrix.445Hutton: Fundamentals ofFinite Element Analysis44610.
Structural DynamicsC H A P T E R 1010.2110.2210.23Text© The McGraw−HillCompanies, 2004Structural DynamicsPerform the matrix multiplications indicated in Equation 10.177 to verify theresult given in Equation 10.178.For the truss in Example 10.10, reformulate the system mass matrix usinglumped element mass matrices. Resolve for the frequencies and mode shapesusing the finite element software available to you, if it has the lumped matrixavailable as an option (most finite element software includes this option).If you formally apply a reduction procedure such as outlined in Section 10.12,which degrees of freedom would be important to retain if, say, we wish tocompute only four of the eight frequencies?Hutton: Fundamentals ofFinite Element AnalysisBack MatterAppendix A: MatrixMathematicsA P P E N D I X© The McGraw−HillCompanies, 2004AMatrix MathematicsA.1 DEFINITIONSThe mathematical description of many physical problems is often simplified bythe use of rectangular arrays of scalar quantities of the forma11 a12 · · · a1n a21 a22 · · · a2n [A] = ..(A.1)......
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