The book can be regarded as consisting of three parts: discrete systems, distributed systems and discretized systems. The contents of the book can be best reviewed through a brief discussion of the individual chapters. Any modern treatment of the vibration of discrete systems must rely heavily on liPrefaceThe last several decades have witnessed impressive progress jn structuraldynamics. This progress can be attributed to a happy confluence of newmethods of analysis such as the finite element method and substructuresynthesis, and the digital computer The newly developed methods ofanalysis permit the structural analyst to formulate increasingly complexproblems and the computer enables him to obtain numerical results tothese problems. As a result, the emphasis in the field of structuraldynamics has bcen shifting steadily toward computational methods. But,agood understanding of the techniques for deriving the cquations ofmotion for complex structures and of the computational algorithms fotsolving these equations demands an increasing mathematical sophistica-Lion onl the part of the structural dynamicist. This book presents the latestdevelopments in structural dynamics and it also provides the mathematical back ground necessary for the understanding of the various modellingtechniques and computational algorithms. The book is intended as a textfor a one-year graduate course in structural dynamics as well as arefcrence book for structural dynamicists. It assumes some prior know-ledge of the field of vibrations, such as that normally acquired in anelementary course in mechanical and structural vibrationsAn important aspect of structural dynamics is mathematical modelling. Mathematical models can be classified according to the spatialdistribution of the parameters describing the system properties. There aretwo major classes, namely, discrete and distributed-parameter modelsLess often one encounters hybrid systems, i. e, systems that are partdiscrete and part distributed Discrete systems are commonly describedby ordinary differential equations and distributed systems by partialdifferential equations most dynamical systems are actually distributedand, except for some simple classical examples, exact solutions for theresponse are difficult, if not impossible to produce. This points to approximate solutions as an alternative which almost invariably amounts to therepresentation of the distributed model by a discrete one in a procesVIPrefacePrefaceknown as spatial discretization. In approximating a distributed system bmethod occupies a special place in structural dynamics. In fact, as pointeda discrete one, the question of truncation of the order of the system playsout in Ch. 9, the finite element method itself can be regarded as aan important part. Of course, the subject is to produce a discrete modelRayleigh Ritz method. Note that the methods discussed in Chs. 8 and 9of relatively low order that is capable of simulating the behavior of thearc essentially discretization procedures. Complex structurcs require comdistributed system with the desired degree of accuracyplex mathematical models, in the sense that the mathematical simulationThe book can be regarded as consisting of thrcc parts: discreteis in terms of a large number of differential equations. Chapter 10 issystems, distributed systems and discretized systems. The contents of thedevoted to techniques for reducing the order of the system. A methodbook can be best reviewed through a brief discussion of the individualthat can be regarded as both a modelling and a reduction method ischapters. Any modern treatment of the vibration of discrete systems mustreferred to as substructure synthesis. A number of variants of the methodrely heavily on linear algebra. In Ch. 1, concepts from linear algebra ofare presented in Ch. 11particular interest in vibrations, such as vector spaces and matrices, areThe author wishes to acknowledge the help received from manyintroduced. Chapter 2 is devoted to the free vibration of discrete systemscolleagues and students during various stages of the manuscript. InIn this chapter, the equations of motion for a large variety of dynamicalparticular, he wishes to express his appreciation to Professor robert Jsystems are derived The free vibration problem leads naturally to theMelosh for his many thoughtful and valuable comments. Many thanks arecharacteristic-value problem, or the eigenvalue problem, one of the mostalso due to Professor Earl H. Dowell, Mr. Arthur L. Hale, Dr. Dewey Hbasic problems in linear algebra. a widely used method for deriving theHodges and Professor Harold D. Nelson, as well as to Mr. Haim Baruh,response, known as modal analysis, is based on the solution of theDr. Hayrani Oz, Mr. Garnett Ryland and Mr. Lawrence M. Silverbergeigenvalue problem. Chapter 2 sets the stage for the following threeFinally, the author would like to thank mrs. Peggy Epperly, Miss Carolchapters. Indeed, Ch. 3 presents a general discussion of the algebraicAnn Sowder and Mrs. Marlene A. Taylor for their patience in typingeigenvalue problem and Ch 4 examines the behavior of the eigensolutionvarious portions of the manuscriptin a qualitative manner. In Ch. 4, some very powerful and far-reachingprinciples, such as Rayleighs principle, the maximum-minimum principleand the inclusion principle, are presented. In Ch. 5, the emphasis changesfrom qualitative considerations to quantitative methods. The chapter isdevoted almost exclusively to computational algorithms for the solutionof the algebraic eigenvalue problem including the most modern ones. Theresponse of discrete systems to arbitrary excitations is discussed in Ch 6In this chapter, various tcchniques from linear system thcory for solvinsets of simultaneous ordinary differential equations are presented. Consis-tent with the expectation that the response to arbitrary excitations mustbe ultimately simulated on a digital computer, the idea of discretization intime of the equations of motion is introduced. In ch. 7. the attention isshifted to the vibration of distributed-parameter systems. The chapter isdevoted to exact solutions to vibration problems and has a classicalappearance. One of its objectives is to highlight properties shared by allsolutions to vibration problems for a large class of distributed-parametersystems, namely, self-adjoint systems. These properties can be establishedregardless of whether solutions in terms of known functions can bederived or not. Knowledge of the solution properties plays an importantrolc in producing approximate solutions, as demonstrated in Ch. 8Among the approximate methods discussed in Ch. 8, the Rayleigh-RitzVIlIⅨXContentsChapter 1. Concepts from linear algebra1.1 Int1.2 Linear vector spaces1.3 Linear dependence1.4 Bases and dimension of a vector space1.5 Inner products and orthogonal vectors61.6 The Gram-Schmidt orthogonalization process81.7 Matrices1.8 Basic matrix operations131.9 Dete161.10 Inverse of a matrix1.11 Partitioned matrices2112 Systems of linear equations231.13 Matrix norms27Chapter 2. Free vibration of discrete systems2. 1 Introducti292.2 The system equations of motion292. 3 Small motions about equilibrium points312.4Ederations392.5 Free vibration and the eigenvalue problemChapter 3. The eigenvalue problem503.1 General discussion503.2 The gencral eigenvalue problem503.3 The eigenvalue problem for real symmetric matrices563.4 Geometric interpretation of the eigenvalue problem623.5 Hermitian matrices653.6 The eigenvalue problem for two nonpositive definite realsymmetric matrices683.7 The eigenvalue problem for real nonsymmetric matrices 70ContentsChapter 4. Qualitative behavior of the eigensolution736.8 Response of general dynamical systems2174.1 Introduction6.9 Discrete-time model for general dynamical systems221734.2 The Rayleigh principle6.10 Stability of motion in the neighborhood of equilibrium 2234.3 Rayleighs theorem for systems with constraints4.4 Maximum-minimum characterization of eigenvalues88Chapter 7. Vibration of continuous systems2297.1 Introduction2294.5 The inclusion principle4.6 A criterion for the positive definiteness of a Hermitian7.2granges equation for continuous systems. Boundary.matrixvalue problem2304.7 Eigenvalues of the sum of two Hermitian matrices7.3 The eigenvalue problem2392424.8 Gerschgorin's theorems997.4 Self-adjoint systems4. 9 First-order perturbation of the eigenvalue problem1027.5 Non-self-adjoint systems57.6 Vibration of rods, shafts and strings255Chapter 5. Computational methods for the eigensolution1107.7 Bending vibration of bars2615.1 General discussion7.8 Two-dimensional problems2651105.2 Gaussian elimination7.9 Variational characterization of the eigenvalues2742785.3 Reduction to triangular form by elementary row opera-7.10 Integral formulation of the eigenvalue problemtions1157. 11 The response problem2815.4 Computation of cigenvcctors belonging to known eigen120Chapter 8. Discretization of continuous systems2855.5 Matrix iteration by the power method1238.1 Introduction2855.6 Hotelling's deflation1288.2 The Rayleigh-Ritz method2865.7 Wielandt's deflation8.3 The assumed-modes method2985. 8 The Cholesky decomposition138. 4 The method of weighted residuals3015.9 The Jacobi method8.5 Flutter of a cantilever aircraft wing3121385.10 Givens'method1468.6 Integral formulation of the method of weighted residuals 3 195.11 Householder's mcthod1518.7 Lumped-parameter method employing influence coffi5. 12 Eigenvalues of a tridiagonal symmetric matrix. Sturm'scients3211578.8 System response by approximate methods322theoreM5.13 The OR method1625. 14 The Cholesky algorithmChapter 9. The finite element method3285.15 Eigenvectors of a tridiagonal matrix3281769.1 Introduction5.16 Inverse iteration1779.2 Second-order problems. Linear elements3299.3 Higher-degree elements. Interpolation functions339Chapter 6. Response of discrete systems1839.4 Fourth-order problems3466.1 Introduction1839.5 Two-dimensional domains3496.2 Linear systems. The superposition principle1849.6 Errors in the eigenvalues and eigenfunctions3646. 3 Impulse response. The convolution integral9.7 Inconsistent mass matrices3656.4 Discrete-time systems6.5 Response of undamped nongyroscopic systems199Chapter 10. Systems with a large number of degrees of freedom 3686.6 Response of undamped gyroscopic systems20410.1 Introduction3686.7 Response of damped systems21010.2 Static condensation369XIIXIIIContents10.3 Mass condensation37010.4 Simultaneous iteration37310.5 Subspace iteration37710.6Thhe method of sectioning380Chapter 1l. Substructure svnthesis384Concepts from linear algebra11. 1 General discussion38411.2 Component-mode synthesis38411.3 Branch-mode analysis39111.4 Component-mode substitution3951.1 ntroduction11. 5 Substructure synthesis401BibliographyAs shown in subsequent chapters, the vibration of linear discrete systems410sd by a set of simultaneous lidinary diferentialSuggested problemsThe solution of such sets of equations can be obtained most conveniently415by means of a linear transformation rendering the set of equationsAuthor indexindependent. In seeking this linear transformation, the problem is con-432verted from that of a set of simultaneous differential equations to that ofa set of simultaneous algebraic equations. The latter problem is known asSubject index434the eigenvalue problem and it represents an important problem in linearalgebra. Hence, a brief discussion of certain pertinent concepts of linearalgebra is in order. In particular, a discussion of vector spaces, as well asof closely relatcd topics such as determinants and matrices, should provemost rewarding1.2 Linear vector spacesIn discussing vector spaces, it proves convenient to introduce the conceptof a field. a field is defined as a set of scalars possessing certain algebraicpropcrtics. The real numbers constitute a field, and so do the complexnumbersLet us consider a set of elements f such that for any two elenents aand B in F it is possible to define another two unique elements belongingto F, the first denoted by a+ B and called the sum of c and B, and thesecond denoted by as and called the product of a and B. The set Fiscalled a field if these two operations satisfy the five field postulates1. Commutative laws. For all o and B in F,(i)a+β=β+a,(i)aB=βa2. Associative laws. For all a, B and y in F.()(a+β)+y=a+(β+y),(i)(cB)y=a(β1 Concepts from linear algebra1.3 Linear dependence3. Distributive laws. For all a, B and y in F,such vectors in L asa(B+y)=aB+ayy14. Identity elements. There exist in F elements 0 and 1 called the zero(11)and the unity elements, respectively, such that 0* 1, and for all a in F(i)a+0-a,(i)1a=ay5. nverse elementsand refer to them as n-vectors. The set of all n-vectors is called the vectorspace Ln. Then, the addition of these two vectors is defined as(i) For every element a in F there exists a unique element-a, calledthe additive inverse of a, such that a+(a=0x1+y1(ii) For element c*0 in F there exists a unique element a,calledx+y=|x2+y2(1.2)the multiplicative inverse of a, such that aa -1=1It is clear that the set of all real numbers satisfy all five postulates.Moreover, if a is a scalar in f, then the product of a scalar and a vector isNext, we wish to define the concept of linear vector space. alsodefined asreferred to as linear space and vector space. Let L be a set of elementsxalled vectors and F a field of scalars. Then if l and f are such that twooperations, namely vector addition and scalar multiplication, are definedfor L and f, the set of vectors together with the two operations are calleda linear vector space L over a field F. For every two elements x and y in L,Let s be a subset of the vector space L. Then, s is a subspace of L if theit satisfies the postulates:following statements are truc1. Commutativity. x+y-y+x1. If x and y are in S, then x+y is in s2. Associativity. (x+y)+z=x+(y+z2. If x is in s and a is in f, then ax is in S.3. There exists a unique vector 0 in L such that x+0=0+x=X4. For every vector x in L there exists a unique vector -x such that1.3 Linear dependencex+(-x)=(x)+x=0.Hence, the rules of vector addition are similar to those of ordinaryLet us consider a set of vectors x1, x2,..., xn in a linear space L and a setalgebra. Moreover, for any vector x in L and any scalar a in f, there isof scalars a1, a2,..., an in F. Then the vector x given bydefined a unique scalar product ax which is also an element of L.Thex=a1X1+a2x2+……+anXnscalar multiplication must be such that, for all a and B in Fand all x and yin L, it satisfies the postulatesis said to be a linear combination of x1, x2,..., xn with coefficientsa1,a2,,.,an. The vectors x1, x2,..., Xn are said to be linearly indepen5. Associativity.c(βx)=(aβ)xdent if thc relation6. Distributivity.()a(x+y)=αx+ay,(i)(a+β)x=αx+Bx7. lx=x, where 1 is the unit scalar, and Ox=0α1x1+c2X2+…+anxn=0(15)can be satisfied only for the trivial case, i.e., only when all the coeficientsWe have considerable interest in a vector space L possessing n elementsa1, a2, .. an are identically zero. If the relation (1.5)is satisfied and atof the field F, i.e, in a vector space of n-tuples. We shall write any twoleast one of the coefficients al, a2,..., an is different from zero, then the1 Concepts from linear algebra1. 4 Bases and dimension of a vector spacevectors x1, x2,..., xn are said to be linearly dependent, with the implica-On the other hand. the three vectorstion that one vector is a linear combination of the remaining n-1vectorsThe subspace S of L consisting of all the linear combinations of the目·{vectors xi, X2,..., Xn is called a subspace spanned by the vectorsX1, X2,..., xn. If S-L, then x1, x2,..., xn are said to span Lare linearly indcpcndent becauseExample 1.1c1x1+a2x2+a4X4≠0Consider the two independent vectorsfor all cases other than the trivial one. The three vectors x1, x2, and gspan a three-dimensional space1.4 Bases and dimension of a vector spaceA vector space L over F is said to be finite dimensional if there exists ain a three-dimensional space. The set of all linear combinations of x, andfinite set of vectors x,, x,,..., Xn which span L, i.e., such that everyx2 span a plane passing through the origin and the tips of xi and x2. Thevector in L is a linear combination of x,, x2sthree vectorsLet L be a vcctor space over F. A set of vectors x1, x2,..., xn whichspan L is called a generating system of L. If x1, x2,..., Xn are linearlyindependent and span L, then the generating system is called a basis forIf L is a finite-dimensional vector space, any two bases for L containthe same number of vectors. The basis can be regarded as the generalizaspan the same plane, because x3 lies in the plane spanned by x1 and x2tion of the concept of a coordinate system.Hence, the three vectors are linearly dependent. Indeed, it can be easilyLet L be a finite dimcnsional vector space over F. the dimension ofverified thatL is defined as the number of vectors in any basis for L. This integer isdenoted by dim L The vector space Ln is spanned by n linearly indepenx1+2x2x3=0dent vectors so that dim L=no that x3 is really a linear combination of x, and x2(see Fig. 1.1)Consider an arbitrary n-vector x in L with componentsx1,x2,..., x and introduce a set of n-vcctors given by060The vector x can be written in terms of the vectors e(i=1, 2, .,, n)as2x2followTHE PLANESPANNED BY EI(AND X3x-x1e1+x2e24……+xnen=2xe(1.7)Figure 1.1It follows that Ln is spanned by the set of vectors e; (i=1,2,.,n),so1 Concepts from linear algebra1.5 Inner products and orthogonal vectorsthat the vectors e i constitute a generating system of L The set of vectorswhich represents a complex number. An inner product space defined overei can be verified as being linearly independent and they are generalthe field of complex numbers is called a unitary spacereferred to as the standard basis for LWhen x and y are real vectors, Eq .(1. 8)reduces to(x,y)=x1y1+x2y2+…+xy(19)Example 1.2which defines the real inner product, a real number. A finite-dimensionalThe vectors x,, x2, and x4 of Example 1.1 form a basis for a threeinner product space defined over the real scalar field is called a euclideandimensional vector space. Any vector x in L can be written as a uniquespace.linear combination of x1, x2, and x4. For example, it can be verified thatIt is often desirable to have a measure of the size of a vector Such athe vectormeasure is called the norm. It is designated by the symbol xll and isrequired to possess the following properties1.x=0 and x 0 if and only if x=02. Ax=al x for any scalar Acan be represented in the fcorn3.|x+ylsk‖+|yx=2x1+3x2-x4where property 3 is known as the triangle inequality. Note that aThe same vector x can be also represented in the terms of the standarddenotes the absolute value, or modulus of aA commonly used norm is the quadratic normbasis ei,e,, e, for L3. Indecd, it is easy to see that(1.10)3e1+0e,+4which defines the length of the vector x In the case of real vector spaces1.5 Inner products and orthogonal vectorsEq.(1.10)reduces toVarious concepts encountered in two-and three-dimensional spaces, such(111)as the length of a vector and orthogonality, can be generalized ton-dimensional spaces. This requires the introduction of additional definiwhich defines the Euclidean norm. Equation (1. 11)can be recognized astionsthc extension to n dimensions of the ordinary concept of length of aLet L be an n-dimensional vector space defined over the field F ofvector in two and three dimensionsscalars. If to each pair of vectors x and y in L is assigned a unique scalara vector whose norm is equal to unity, x=(x, x)=1, is called ain F, called the inner product of x and y, then Ln is said to be an innerunit vector. Any nonzero vector can be normalized so as to form a unitproduct space. The vectors x and y can be complex, in which case x andvector by simply dividing the vector by its normdenote their complex conjugates. The inner product is denoted by (x, y)and must satisfy the following postulates(1.12)1.(x,x20 for all x in L" and(x, x)=0 if and only if x=02.(X,y)=(y,x)It is easy to verify that the vectors e i defined by Eqs.(1.6)are unit3.(Ax, y)=A(x, y) and(x, Ay)=A(x,y)for all a in 1vectors4.(x,y+z)=(x, y)+(x, z) for all x, y, and z in LnWhen the vectors x and y are real, the inner product is sometimesThe most common definition of the complex inner product isreferred to as the dot product We recall from ordinary vector analysis thatthe dot product of two vectors in the two-and three-dimensional space(x,y)=x1y1+x2y2x,8)can be used to define the cosine of the angle between the two vectors6