STRUCTURAL DYNAMICS, VOL. 9. Computational Dynamics

Transkript

1 STRUCTURAL DYNAMICS, VOL. 9 Computational Dynamics Søren R. K. Nielsen P (λ) y(λ) =P (µ k )+ ( P (µ k ) P (µ k ) ) λ µ k µ k µ k µ k µ k µ k+ λ λ λ 3 λ Aalborg tekniske Universitetsforlag June 005

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3 Contents INTRODUCTION 7. Fundamentals of Linear Structural Dynamics Solution of Initial Value Problem by Modal Decomposition Techniques Conclusions Exercises NUMERICAL INTEGRATION OF EQUATIONS OF MOTION 3. Newmark Algorithm Numerical accuracy Numerical stability Period Distortion Numerical Damping Generalized Alpha Algorithm Exercises LINEAR EIGENVALUE PROBLEMS Gauss Factorization of Characteristic Polynomials Eigenvalue Separation Principle Shift Transformation of GEVP to SEVP Exercises APPROXIMATE SOLUTION METHODS Static Condensation Rayleigh-Ritz Analysis Error Analysis Exercises VECTOR ITERATION METHODS Introduction Inverse and Forward Vector Iteration Shift in Vector Iteration Inverse Vector Iteration with Rayleigh Quotient Shift Vector Iteration with Gram-Schmidt Orthogonalization Exercises

4 4 Contents 6 SIMILARITY TRANSFORMATION METHODS 5 6. Introduction Special Jacobi Iteration General Jacobi Iteration Householder Reduction QR Iteration Exercises SOLUTION OF LARGE EIGENVALUE PROBLEMS Introduction Simultaneous Inverse Vector Iteration Subspace Iteration Characteristic Polynomial Iteration Exercises INDEX 7 A Solutions to Exercises 75 A. Exercise A. Exercise A.3 Exercise A.4 Exercise A.5 Exercise 3.4: Theory A.6 Exercise A.7 Exercise A.8 Exercise A.9 Exercise A.0 Exercise A. Exercise A. Exercise A.3 Exercise A.4 Exercise A.5 Exercise

5 Preface This book has been prepared for the course on Computational Mechanics given at the 8th semester at the structural engineering program in civil engineering at Aalborg University. The course presumes undergraduate knowledge of linear algebra and ordinary differential equations, as well as a basic graduate course in structural dynamics. Some of these prerequisites have been reviewed in an introductory chapter. The author wants to thank Jesper W. Larsen, Ph.D., and Ph.D. student Kristian Holm-Jørgensen for help with the preparation of figures and illustrations throughout the text. Answers to all exercises given at the end of each chapter can be downloaded from the home page of the course at the address: Aalborg University, June 005 Søren R.K. Nielsen 5

6 6 Contents

7 CHAPTER INTRODUCTION In this chapter the basic results in structural dynamics and linear algebra have been reviewed. In Section. the relevant initial and eigenvalue problems in structural dynamics are formulated. The initial value problems form the background for the numerical integration algorithms described in Chapter, whereas the related undamped generalized eigenvalue problem constitute the generic problem for the numerical eigenvalue solvers described in Chapters 3-7. Formal solutions to various formulations of the initial value problem are indicated, and their shortcomings in practical applications are emphasized. In Section. the semi-analytical solution approaches to the basic initial value problem of a multi degrees-of-freedom system in terms of expansion in various modal bases are presented. The application of these methods in relation to various reduction schemes, where typically merely the low-frequency modes are required, has been outlined.. Fundamentals of Linear Structural Dynamics The basic equation of motion for forced vibrations of a linear viscous damped n degree-offreedom system reads } Mẍ(t)+Cẋ(t)+Kx(t) =f(t), t > t 0 ( ) x(t 0 )=x 0, ẋ(t 0 )=ẋ 0 x(t) is the vector of displacements from the static equilibrium state, ẋ(t) is the velocity vector, ẍ(t) is the acceleration vector, and f(t) is the dynamic load vector. x 0 and x 0 denote the initial value vectors for the displacement and velocity, respectively. K, M and C indicate the stiffness matrix, mass matrix and damping matrices, all of the dimension n n. For any vector a 0 these fulfill the following positive definite and symmetry properties a T Ka > 0, K = K T a T Ma > 0, M = M T a T Ca > 0 ( ) S.R.K. Nielsen: Structural Dynamics, Vol.. Linear Structural Dynamics, 4th Ed.. Aalborg tekniske Universitetsforlag,

8 8 Chapter INTRODUCTION If the structural system is not supported against stiff-body motions, the stiffness matrix is merely positive semi-definite, soa T Ka 0. Correspondingly, if some degrees of freedom are not carrying kinetic energy (pseudo degrees of freedom with zero mass or zero mass moment of inertia), the mass matrix is merely positive semi-definite, so a T Ma 0. The positive definite property of the damping matrix is a formal statement of the physical property that any non-zero velocity of the system should be related with energy dissipation. However, C needs not fulfill any symmetry properties, although energy dissipation is confined to the symmetric part of the matrix. So-called aerodynamic damping loads are external dynamic loads proportional to the structural velocity, i.e. f(t) = C a ẋ(t). If the aerodynamic damping matrix C a is absorbed in the total damping matrix C, no definite property can be stated. The solution of the initial value problem (-) can be written in the following way x(t) = t h(t τ)f(τ)dτ + a 0 (t t 0 )x 0 + a (t t 0 )ẋ 0 t 0 a 0 (t) =h(t)c + ḣ(t)m a (t) =h(t)m ( 3) h(t) is the impulse response matrix. Formally, this matrix is obtained as a solution to the initial value problem Mḧ(t)+Cḣ(t)+Kh(t) =I δ(t) h(0 )=0, ḣ(0 )=0 } ( 4) I is the unit matrix of the dimension n n, and δ(t) is Dirac s delta function. The frequency response matrix H(iω) related to the system (-) is given as H(iω) =( (iω) M +(iω)c + K) ( 5) where i = is the complex unit. The impulse response matrix is related to the frequency response matrix in terms of the Fourier transform h(t) = H(iω)e iωt dt ( 6) π The convolution quadrature in (-3) is relative easily evaluated numerically. Hence, the solution of (-) is available, if the impulse response matrix h(t) is known. In turn, the n n components of this matrix can be calculated by the Fourier transforms (-6). Although these transforms may be evaluated numerically, the necessary calculation efforts become excessive even for a moderate number of degrees of freedom n. Hence, more direct analytical or numerical approaches are

9 . Fundamentals of Linear Structural Dynamics 9 mandatory. Undamped eigenvibrations ( C = 0, f(t) 0 ) are obtained as linear independent solutions to the homogeneous matrix differential equation Mẍ(t)+Kx(t) =0 ( 7) Solutions are searched for on the form x(t) =Φ (j) e iω jt ( 8) Insertion of (-8) into (-7) provides the following homogeneous system of linear equations for the determination of the amplitude Φ (j) and the unknown constant ω j ( ) K λ j M Φ (j) = 0, λ j = ωj ( 9) (-9) is a so-called generalized eigenvalue problem (GEVP). If M = I, the eigenvalue problem is referred to as a special eigenvalue problem (SEVP). The necessary condition for non-trivial solutions (i.e. Φ (j) 0) is that the determinant of the coefficient matrix is different from zero. This lead to the characteristic equation ( ) P (λ) =det K λm =0 ( 0) P (λ) indicates the characteristic polynomial. This may be expanded as P (λ) =a 0 λ n + a λ n + + a n λ + a n ( ) The constants a 0,a,...,a n are known as the invariants of the GEVP. This designation stems from the fact that the characteristic polynomial (-) is invariant under any rotation of the coordinate system. Obviously, a 0 =( ) n det(m), and a n =det(k). The nth order equation (-0) determines n solutions, λ,λ,...,λ n. Assume that either M or K are positive definite. Then, all eigenvalues λ j are non-negative real, which may be ordered in ascending magnitude as follows 0 λ λ λ n λ n ( ) λ n =, ifdet(m) =0. Similarly, λ =0,ifdet(K) =0. The eigenvalues are denotes as simple, ifλ <λ < <λ n <λ n. The undamped circular eigenfrequencies are related to the eigenvalues as follows

10 0 Chapter INTRODUCTION ω j = λ j ( 3) The corresponding solutions for the amplitude functions, Φ (),...,Φ (n), are denoted the undamped eigenmodes of the system, which are real as well. The eigenvalue problems (-9) can be assembled into following matrix formulation λ 0 0 K Φ () Φ () Φ (n) = M Φ () Φ () Φ (n) 0 λ λ n KΦ = MΦΛ where λ λ 0 Λ = ( 4) ( 5) 0 0 λ n and Φ is the so-called modal matrix of dimension n n, defined as Φ = Φ () Φ () Φ (n) ( 6) If the eigenvalues are simple, the eigenmodes fulfill the following orthogonality properties { Φ (i) T MΦ (j) 0, i j = M i, i = j { Φ (i) T KΦ (j) 0, i j = ωi M i, i = j where M i denotes the modal mass. The orthogonality properties (-7) can be assembled in the following matrix equation ( 7) ( 8) M 0 0 Φ () Φ () Φ (n) T M Φ () Φ () Φ (n) 0 M 0 = M n Φ T MΦ = m ( 9)

11 . Fundamentals of Linear Structural Dynamics where M M 0 m = M n ( 0) The corresponding grouping of the orthogonality properties (-8) reads Φ T KΦ = k ( ) where ωm ω k = M ωn M n ( ) If the eigenvalues are all simple, the eigenmodes become linear independent, which means that the inverse Φ exists. In the following it is generally assumed that the eigenmodes are normalized to unit modal mass, so m = I. For the special eigenvalue problem, where M = I, it then follows from (-9) that Φ = Φ T ( 3) A matrix fulfilling (-3) is known as orthonormal or unitary, and specifies a rotation of the coordinate system. All column and row vectors have the length, and are mutually orthogonal. It follows from (-9) and (-) that in case of simple eigenvalues a so-called similarity transformation exists, defined by the modal matrix Φ, that reduce the mass and stiffness matrices to a diagonal form. In case of multiple eigenvalues the problem becomes considerable more complicated. For the standard eigenvalue problem with multiple eigenvalues it can be shown that the stiffness matrix merely reduces to the so-called Jordan normal form under the considered similarity transformation, given as follows k k 0 k = k m ( 4) where m n denotes the number of different eigenvalues, and k i signifies the so-called Jordan boxes, which are block matrices of the form

12 Chapter INTRODUCTION ω i, ωi 0 ωi ωi ωi 0 0 0, 0 ωi 0 ω, i ωi 0 0 ωi ωi,... ( 5) Assume that the mass matrix is non-singular so M exists. Then, the equations of motion (-) may be reformulated in the following state vector form of coupled st order differential equations } ż(t) =Az(t)+F(t), t > t 0 z(t 0 )=z 0 ( 6) x(t) z(t) = ẋ(t), z 0 = x 0 ẋ 0 0 I, A = M K M C 0, F(t) = M f(t) ( 7) z(t) denotes the state vector. The corresponding homogeneous differential system reads ż(t) =Az(t) ( 8) The solution of (-6) becomes z(t) =e At (e At 0 z 0 + ) e Aτ F(τ)dτ t 0 t ( 9) The n n matrix e At is denoted the matrix exponential function. This forms a fundamental matrix to (-8), i.e. the columns of e At form n linearly independent solutions to (-8). Actually, e At is the fundamental matrix fulfilling the matrix initial value problem d dt eat = Ae At, t > 0 e A 0 = I ( 30) where I denotes a n n unit matrix. Now, ( e At) =e At as shown in Box.. Using this relation for t = t 0, (-9) is seen to fulfil the initial value of (-6). Since conventional differentiation rules also applies to matrix products, the fulfilment of the differential equation D.G. Zill and M.R. Cullen: Differential Equations with Boundary-Value Problems, 5th Ed. Brooks/Cole, 00.

13 . Fundamentals of Linear Structural Dynamics 3 in (-6) follows from differentiation of the right hand side of (-9), and application of (-30), i.e. d dt z(t) = d ( t ) ( ) dt eat e At 0 z 0 + e Aτ F(τ)dτ +e At 0 +e At F(t) = t 0 t ) Ae (e At At 0 z 0 + e Aτ F(τ)dτ + IF(t) =Az(t)+F(t) ( 3) t 0 The solution to (-30) can be represented by the following infinite series of matrix products e At = I + ta + t! A + t3 3! A3 + ( 3) where A = AA, A 3 = AAA etc. (-3) is seen to fulfil the initial value e A 0 = I. The fulfilment of the matrix differential equation (-30) follows from termwise differentiation of the right-hand side of (-3) d dt eat = 0 + A + t )! A + t! A3 + = 0 + A (I + ta + t! A + = Ae At ( 33) The right-hand side of (-3) converges for arbitrary values of t as the number of terms increases beyond limits on the right-hand side. Hence, e At can in principle be calculated using this representation. However, for large values of t the convergence is very slow. In (-9) the fundamental matrix e At is needed for arbitrary positive and negative values of t. Hence, the use of (-3) as an algorithm for e At in the solution (-9) becomes increasingly computational expensive as the integration time interval is increased. In Box. an analytical solution for e At has been indicated, which to some extent circumvents this problem. However, this approach requires that all eigenvectors and eigenvalues of A are available. Damped eigenvibrations are obtained as linear independent solutions to the homogeneous differential equation (-8). Analog to (-8) solutions are searched for on the form z(t) =Ψ (j) e λ jt ( 34) Insertion of (-34) into (-8) provides the following special eigenvalue problem of the dimension n for the determination of the damped eigenmodes Ψ (j) and the damped eigenvalues λ j ( ) A λ j I Ψ (j) = 0 ( 35) Since A is not symmetric, λ j and Ψ (j) are generally complex. Upon complex conjugation of (-35), it is seen that if (λ, Ψ) denotes an eigen-pair (solution) to (-35), then (λ, Ψ ) is also an eigen-pair, where * denotes complex conjugation. For lightly damped structures all eigenvalues are complex. In this case only n eigen-pairs (λ j, Ψ (j) ), j =,,...,nneed to be considered, where no eigen-pair is a complex conjugate of another in the set.

14 4 Chapter INTRODUCTION Let the first n components of Ψ (j) be assembled in the n-dimensional sub-vector Φ (j). Then, from (-7) and (-34) it follows that x(t) =Φ (j) e λ jt ẋ(t) =λ j Φ (j) e λ jt ( 36) Consequently, the damped eigenmodes must have the structure Ψ (j) Φ = (j) λ j Φ (j) ( 37) Hence, merely the first n components of Ψ (j) need to be determined. The eigenvalue problems (-35) can be assembled into the following matrix formulation, cf. (-4)-(-6) λ 0 0 A Ψ () Ψ () Ψ (n) = Ψ () Ψ () Ψ (n) 0 λ λ n AΨ = ΨΛ A where λ λ 0 Λ A = λ n ( 38) ( 39) Ψ = Ψ () Ψ () Ψ (n) ( 40) The following representation of A in terms of the damped eigenmodes and eigenvalues follows from (-38) A = ΨΛ A Ψ ( 4) Assume that another n n matrix B has the same eigenvectors Ψ (j) as A, whereas the eigenvalues as stored in the diagonal matrix Λ B are different. Then, similar to (-4), B has the representation B = ΨΛ B Ψ ( 4) The matrix product of A and B becomes AB = ΨΛ A Ψ ΨΛ B Ψ = ΨΛ A Λ B Ψ ( 43)

15 . Fundamentals of Linear Structural Dynamics 5 Since Λ A and Λ B are diagonal matrices, matrix multiplication of these is commutative, i.e. Λ A Λ B = Λ B Λ A. Then, (-43) may be written AB = ΨΛ B Λ A Ψ = ΨΛ B Ψ ΨΛ A Ψ = BA ( 44) Consequently, if two matrices have the same eigenvectors, matrix multiplication of two matrices is commutative. Identical eigenvectors of two matrices can also be shown to constitute the necessary condition (the only if requirement) for commutative matrix multiplication. The so-called adjoint eigenvalue problem to (-35) reads ( ) A T ν i I Ψ (i) a = 0 ( 45) Hence, (ν i, Ψ (i) a ) denotes the eigenvalue and eigenvector to the transposed matrix A T. In Box. it is shown that the eigenvalues of the basic eigenvalue problem and the adjoint eigenvalue problem are identical, i.e. ν j = λ j. Further, it is shown that the eigenvectors Ψ (j) and Ψ (j) a fulfill the orthogonality properties Ψ (i) T a Ψ (j) = Ψ (i) T a AΨ (j) = { 0, i j m i, i = j { 0, i j λ i m i, i = j ( 46) ( 47) where m i is denoted the complex modal mass. Without any restriction this may be chosen as m j =. Then, the orthogonality conditions (-46) and (-47) may be assembled into the following matrix relation Ψ T a Ψ = I ( 48) Ψ T a AΨ = Λ A where Ψ a = Ψ () a Ψ () a Ψ (n) a ( 49) ( 50) From (-48) follows that Ψ a = ( Ψ ) T ( 5) Hence, the eigenvectors Ψ (i) a of the adjoint eigenvalue problem (the column vectors in Ψ a ) normalized to unit modal mass are determined as the row vectors of Ψ. The eigenvectors Ψ (i) of the direct eigenvalue problem may be arbitrarily normalized. Of course, if (λ i, Ψ (i) a ) is an eigen-solution to the adjoint eigenvalue problem, so is the complex conjugate (λ i, Ψ (i) a ).

16 6 Chapter INTRODUCTION Box.: Matrix exponential function Multiple application of (-4) provides for k =,,... A = AA A 3 = AA = ΨΛ A Ψ ΨΛ A Ψ = ΨΛ A Ψ = ΨΛ A Ψ ΨΛ A Ψ = ΨΛ 3 A Ψ. A j+ = AA j = ΨΛ A Ψ ΨΛ j A Ψ = ΨΛ j+ A Ψ ( 5) Λ j+ A is a product of diagonal matrices, and then becomes a diagonal matrix itself. The diagonal elements become λ j+ k, where λ k is the corresponding diagonal element in Λ A. Consider the matrix exponential function, cf. (-3) e ΛAt = I + tλ A + t! Λ A + t3 3! Λ3 A + ( 53) Since all addends on the right-hand side of (-53) are diagonal matrices, it follows that also e ΛAt becomes diagonal with the diagonal elements +tλ k + t! λ k + t3 3! λ3 k + =eλ k t ( 54) where the Maclaurin series for the exponential function has been used in the last statement. Then, from (-3), (-5) and (-53) follows e At = Ψ (I ) + tλ A + t! Λ A + t3 3! Λ3 A + Ψ = Ψe ΛAt Ψ ( 55) For arbitrary positive or negative t and t it then follows that e At e At = Ψe Λ At Ψ Ψe Λ At Ψ = Ψe Λ At e Λ At Ψ = Ψe Λ A(t +t ) Ψ = e A(t +t ) ( 56) (-56) represents the fundamental multiplication rule of matrix exponential functions. Especially for t = t and t = t we have e At e At =e A 0 = I e At = ( e At) ( 57) Further, A n = A A = ΨΛ n A Ψ, n =,,... ( 58) (-58) is proved by insertion of (-5) and (-58) into the identity A n A n = I. As seen, e At and A n have identical eigenvectors. Then, from (-44) it follows that A n e At =e At A n, n =,,... ( 59)

17 . Fundamentals of Linear Structural Dynamics 7 Box.: Proof of orthogonality properties of eigenvectors and adjoint eigenvectors (-35) is pre-multiplied with Ψ (i) a T, and (-45) is pre-multiplied with Ψ (j) T, leading to the identities Ψ (i) a T AΨ (j) = λ j Ψ (i) a T Ψ (j) ( 60) Ψ (j) T A T Ψ (i) a = ν i Ψ (j) T Ψ (i) a Ψ (i) T a AΨ (j) = ν i Ψ (i) T a Ψ (j) ( 6) The last statement follows from transposing the previous one. Withdrawal of (-6) from (-60) provides ( λj ν i ) Ψ (i) T a Ψ (j) =0 ( 6) For i = j, (-6) can only be fulfilled for ν i = λ i, since Ψ (i) T a Ψ (i) 0. Next, presume simple eigenvalues, so λ i λ j. Then, for i j, (-6) can only be fulfilled, if Ψ (i) a T Ψ (j) =0, corresponding to (-46). Since the right-hand side of (-60) is zero for i j, this must also hold true for the lefthand side, i.e. Ψ (i) a T AΨ (j) =0for i j. Then for i = j, (-60) provides the result Ψ (i) a T AΨ (i) = λ i m i, which completes the proof of (-47). Example.: Equations of motion of linear viscous damped DOF system f k k k 3 f m m c c c 3 x x k x c ẋ m f k (x x ) c (ẋ ẋ ) f m k 3 x c 3 ẋ Fig. Equation of motion of linear viscous damped DOF system. The two-degree-of-freedom system shown on Fig. - consists of the masses m and m connected with linear elastic springs with the spring constants k, k, k 3, and linear viscous damper elements with the damper constants

18 8 Chapter INTRODUCTION c, c, c 3. The displacement of the masses from the static equilibrium state are denoted as x (t) and x (t). The velocities ẋ i (t) and accelerations ẍ i (t) are considered positive in the same direction as the displacements x i (t) and the external forces f i (t). The masses are cut free from the springs and dampers in the deformed state, and the damper- and spring forces are applied as equivalent external forces. Next, Newton s nd law of motion is formulated for each of the masses leading to } m ẍ = k x + k (x x ) c ẋ + c (ẋ ẋ )+f (t) m ẍ = k 3 x k (x x ) c 3 ẋ c (ẋ ẋ )+f (t) ( 63) (-63) may be formulated as the following matrix differential equations Mẍ(t)+Cẋ(t)+Kx(t) =f(t), t > t 0 f, f(t) = (t) f (t) x x(t) = (t) x (t) m M = 0 0 m, C = c + c c c c + c 3, K = k + k k k k + k 3 ( 64) For each of the masses an initial displacement x i (t 0 )=x i,0 from the static equilibrium state and an initial velocity ẋ i (t 0 )=ẋ i,0 are specified. These are assembled into the following initial value vectors x x 0 = x(t 0 )=,0 x,0 ẋ, ẋ 0 = ẋ(t 0 )=,0 ẋ,0 ( 65) The presented system will be further analyzed in various numerical examples throughout the book. Example.: Discretized equations of motion of a vibrating string l F j n n u u u(x, t) u j u j+ u n u n F x l Fig. Discretization of vibrating string. Fig. - shows a vibrating string with the pre-stress force F, and the mass per unit length µ. The string has been divided into n identical elements, each of the length l. Hence, the total length of the string is l = n l. The displacement u(x, t) of the string at the position x and time t in the transverse direction is given by the wave equation with homogeneous boundary conditions µ u t F u =0, x 0,l x u(0,t)=u(l, t) =0 ( 66)

19 . Fundamentals of Linear Structural Dynamics 9 where x is measured from the left support point. The spatial differential operator in (-66) is discretized by means of a central difference operator, i.e. F u(x i,t) x F l ( ui+ u i + u i ), i =,...,n ( 67) where u i (t) =u(x i,t), x i = i l. Further, let ü i (t) = t u(x i,t). The boundary conditions imply that u 0 (t) = u n (t) =0. Then, the discretized wave equation may be represented by the matrix differential equation Mẍ(t)+Kx(t) =0 ( 68) x(t) = u (t) u (t) u 3 (t). u n (t) u n (t) , M = µ , K = F l ( 69) Alternatively, the wave equation may be discretized by means of a finite element approach. Assuming linear interpolation between the nodal values stored in the vector x(t), and using the same interpolation for the displacement field and the variational field (Galerkin variation), the following mass- and stiffness matrices are obtained M = µ l , K = F l ( 70) (-70) represents the so-called consistent mass matrix, for which the same interpolation algorithm is used for discretizing the kinetic and the potential energy. By contrast the diagonal mass matrix in (-69) is referred to as a lumped mass matrix. As seen the central difference operator and Galerkin variation with piecewise linear interpolation leads to the same stiffness matrix. The presented system will be further analyzed in various numerical examples in what follows. The calculated eigenvalues based on the system matrices (-69) and (-70) are shown in Fig. -3 as a function of the number of elements n. The solutions based on the lumped mass matrix (-69) and the consistent mass (-70) are shown with dotted and dashed signature, respectively. The numerical solutions have been given relative to the analytical solutions F ω j,a = jπ µl, j =,...,4 ( 7) As seen, the consistent mass matrix provides upper-bounds in accordance with the Rayleigh-Ritz principle described in Section 4.. By contrast the lumped mass matrix provides lower bounds, when used in combination with the consistent stiffness matrix. There is no formal proof of this property, which merely is an empirical observation fulfilled in many dynamical problems. The indicated observation immediately suggest that an improvement

20 0 Chapter INTRODUCTION of the numerical solutions may be obtained by using a linear combination of the consistent and the lumped mass matrix. Typically, the mean value is used leading to the mass matrix M = µ l µ l = µ l ( 7) (-7) is solved with the consistent stiffness matrix (-70). The results are showed with a dashed-dotted signature on Fig. -3. As expected the results show a significant improvement. A theoretical argument for using the mean value of the consistent and lumped mass matrices for the combined mass matrix has been given by Krenk. 3 ω/ω,a n ω/ω,a n ω3/ω3,a ω4/ω4,a n Fig. 3 Undamped eigenvibrations of string. : Analytical solution : Consistent mass matrix.... : Lumped mass matrix : Combined mass matrix. n 3 S. Krenk: Dispersion-corrected explicit integration of the wave equation. Computer Methods in Applied Mechanics and Engineering, 9, pp , 00.

21 . Fundamentals of Linear Structural Dynamics Example.3: Verification of eigensolutions Given the following mass- and stiffness matrices 5 M = 4 0 5, K = 0 5 Verify that the eigensolutions with modal masses normalized to are given by ( 73) Λ = ω 0 0 = 0 ω 0, Φ = Φ () Φ () 4 = 5 5 ( 74) Based on the proposed eigensolutions the following calculations are performed, cf. (-4) 5 4 KΦ = = MΦΛ = = ( 75) This proofs the validity of the proposed eigensolutions. The orthonormality follows from the following calculations, cf. (-9) and (-) Φ T MΦ = Φ T KΦ = T = 0 0 T = 0 0 ( 76) Example.4: M- and K-orthogonal vectors Given the following mass- and stiffness matrices M = 0 0, K = 4 ( 77) Additionally, the following vectors are considered v =, v = 0 0 From (-78) the following matrix is formed ( 78) V =v v = ( 79) 0 0

22 Chapter INTRODUCTION We may then perform the following calculations, cf. (-9) and (-) V T MV = 0 0 V T KV = T T = = ( 80) (-80) shows that the vectors v and v are mutual orthogonal with weights M and K, and that both have been normalized to unit modal mass. As will be shown in Example.5 neither v nor v are eigenmodes, and the eigenvalues are different from.5858 and However, if three linear independent vectors are mutual orthogonal weighted with the three-dimensional matrices M and K, they will be eigenmodes to the system. Example.5: Analytical calculation of eigensolutions The mass- and stiffness matrices defined in Example.4 are considered again. Now, an analytical solution of the eigenmodes and eigenvalues is wanted. The generalized eigenvalue problem (-9) becomes λ j 0 4 λ j 0 λ j Φ (j) Φ (j) Φ (j) 3 The characteristic equation (-0) becomes λ j 0 P (λ) =det 4 λ j = 0 λ j ( (4 j)( λ ) ( λj ) ) ( ( λ )) j + λ j, j = λ j = 4, j = 6, j =3 0 = 0 ( 8) 0 = ( )( λ j 6 4λ j + ) λ j =0 ( 8) Initially, the eigenmodes are normalized by setting an arbitrary component to. Here we shall choose Φ (j) 3 =. The remaining components Φ (j) and Φ (j) are then determined from any two of the three equations (-8). The first and the second equations are chosen, corresponding to λ j 4 λ j (j) Φ Φ (j) = 0 Φ (j) Φ (j) Φ (j) 3 = 4 8λ j+λ j 4 λ j 4 8λ j+λ j ( 83) The modal matrix with eigenmodes normalized as indicated in (-83) is denoted as Φ. This becomes

23 . Fundamentals of Linear Structural Dynamics 3 Φ = 0 ( 84) The modal masses become, cf. (-9) 0 0 m = Φ T M Φ = 0 0 ( 85) 0 0 Φ () denotes the st eigenmode normalized to unit modal mass. This is related to Φ () in the following way Φ () = M Φ() = ( 86) The other modes are treated in the same manner, which results in the following eigensolutions ω Λ = 0 ω 0 = 0 4 0, Φ = 0 0 ω Φ () Φ () Φ (3) = 0 ( 87) Example.6: Undamped and damped eigenvibrations of DOF system 00 N/m 00 N/m 300 N/m kg kg 3 kg/s kg/s kg/s x x Fig. 4 Eigenvibrations of DOF system. The system in Example. is considered again with the structural parameters defined in Fig. -3. The massdamping and stiffness matrices become, cf. (-64) 0 5 M = kg, C = 0 3 kg s N, K = m ( 88) The eigensolutions with modal masses normalized to become Λ = ω = s, Φ = Φ () Φ () = 0 ω ( 89)

24 4 Chapter INTRODUCTION The matrix A defined by (-7) becomes I A = = M K M C ( 90) The eigenvalues and eigenfunctions become λ λ Λ A = λ 3 0 = λ i i i i ( 9) Ψ = Ψ () Ψ () Ψ (3) Ψ (4) = Φ() Φ () Φ () Φ () = λ Φ () λ Φ () λ Φ() λ Φ() i i i i i i i i i i i i ( 9) As seen from (-9) the second component of the sub-vectors Φ () and Φ () has been normalized to one. Hence, the entire modal matrix with 6 components is defined by merely 4 entities, namely the the first component of the sub-vectors Φ () and Φ () and the eigenvalues λ and λ. The eigenvectors of the adjoint eigenvalue problem follows from (-5) and (-9) Ψ a = ( Ψ ) T = Ψ () a Ψ () a Ψ (3) a Ψ (4) a = Ψ () a Ψ () a Ψ () a Ψ () a = i i i i i i i i i i i i i i i i ( 93) As seen Ψ (3) a and Ψ (3) a become the complex conjugates of Ψ () a and Ψ () a, cf. remarks subsequent to (-5).

25 . Solution of Initial Value Problem by Modal Decomposition Techniques 5. Solution of Initial Value Problem by Modal Decomposition Techniques Assume that undamped eigenmodes Φ (i) in addition to the orthogonality properties (-7) and (-8) also are orthogonal weighted with the damping matrix, i.e. { Φ (i) T CΦ (j) 0, i j = ζ i ω i M i, i = j ( 94) ζ i denotes the modal damping ratio. In practice (-94) is fulfilled, if the structure is lightly damped and the eigenfrequencies are well separated. The orthogonality properties may be assembled into the following matrix relation similar to (-9) and (-) Φ T CΦ = c ( 95) where ω ζ M ω ζ M 0 c = ω n ζ n M n ( 96) The undamped eigenmodes are linear independent and may be used as a basis in the n-dimensional vector space. Hence, the displacement vector x(t) may be written as x(t) = q (t) n Φ (j) q (t) q j (t) =Φq(t), q(t) =. q n (t) j= ( 97) where q (t),..., q n (t) represent the undamped modal coordinates, i.e. the coordinates in the vector basis formed by the undamped eigenmodes Φ (),..., Φ (n). Insertion of (-95) into (- ), followed by a pre-multiplication with Φ T and use of (-9), (-), (-94), provides the following matrix differential equation for the modal coordinates } m q(t)+c q(t)+kq(t) =F(t), t > t 0 q(t 0 )=Φ x 0, q(t 0 )=Φ ẋ 0 ( 98) where

26 6 Chapter INTRODUCTION F (t) F(t) =Φ T F (t) f(t) =. F n (t) ( 99) F (t),..., F n (t) are denoted the modal load. Since m, c and k are diagonal matrices the component differential equations related to (-98) decouple completely. This is caused by the orthogonality condition (-94) for which reason this relation is referred to as the decoupling condition. The differential equation for the kth modal coordinate reads ) M k ( q k (t)+ζ k ω k q k (t)+ωkq k (t) = F k (t), k =,..., n ( 00) Hence, the decoupling condition reduces the integration of a linear n degrees-of-freedom system to the integration of n single-degree-of-freedom oscillators. Typically, the dynamic response is carried by lowest modes in the expansion (-97). Assume that the modal response above the first n n may be disregarded. Then (-97) reduces to n x(t) Φ (j) q j (t) =Φ q (t) ( 0) j= where Φ is a reduced modal matrix of dimension n n, and q (t) is a sub-vector of modal coordinates defined as Φ = Φ () Φ () Φ (n ), q (t) = q (t) q (t). q n (t) ( 0) (-0) completely ignores the influence of the higher modes. Although the dynamic response of these modes are ignorable, they may still influence the low-frequency components via a quasi-static response component. A consistent correction taken this effect into consideration reads ( n x(t) Φ (j) q j (t)+ K j= n j= ) ωj M Φ (j) Φ (j) T f(t) ( 03) j (-03) may be represented in terms of the following equivalent matrix formulation x(t) Φ q (t)+ ( ) K Φ k Φ T f(t) ( 04)

27 . Solution of Initial Value Problem by Modal Decomposition Techniques 7 where ω M ω k = M ωn M n ( 05) Both (-0) and (-03) requires knowledge of the first n eigen-pairs (ω j, Φ (j) ). The corresponding modal coordinates are determined from the first n equations in (-00). Correspondingly, the n eigenvectors Ψ (j), j =,...,n of the matrix A form a vector basis in the n-dimensional vector space. Then, the state vector z(t) admits the representation z(t) = n j= Ψ (j) q j (t) =Ψq(t) ( 06) where q (t) q (t) q(t) =. q n (t) ( 07) q (t),...,q n (t) represent the damped modal coordinates, i.e. the coordinates in the vector basis made up of the damped eigenmodes Ψ (),... Ψ (n). Insertion of (-06) into (-6), followed by pre-multiplication of Ψ T a and use of (-48), (-49), the following matrix differential differential equations for the damped modal coordinates is obtained } q(t) =Λ A q(t)+g(t), t > t 0 q(t 0 )=Ψ z 0 = Ψ T a z 0 ( 08) where G (t) G(t) =Ψ T a F(t) = G (t). G n (t) ( 09) In the initial value statement of (-06) the relation (-5) between the adjoint and direct modal matrices has been used. G j (t) =Ψ a (j)t F(t) denotes the jth damped modal load.

28 8 Chapter INTRODUCTION (-08) indicates n decoupled complex st order differential equations. The differential equation for the jth modal coordinate reads q j (t) =λ j q j (t)+g j (t), j =,..., n ( 0) Since, ( λ j+n, Ψ (j+n) ) ( a = λ j, Ψ a (j) ) for n =,..., n, it follows that Gj+n (t) =G j (t), and in turn that q j+n (t) =qj (t). Hence, merely the first n differential equations (-0) need to be integrated. Then, (-04) may be written ( n ) z(t) =Re Ψ (j) q j (t) j= ( ) As is the case for expansion in undamped modal coordinates the response is primarily carried by the lowest n modes leading to the following reduced form of (-) z(t) Re ) Ψ q (t) ( n j= ( ) where Ψ = Ψ () Ψ () Ψ (n ), q (t) = q (t) q (t). q n (t) ( 3) (-0), (-03) and (-) describes the dynamic system with less coordinates than the original formulation (-). For this reason such formulations are referred to as system reduction schemes. A system reduction scheme with due consideration to the quasi-static response may also be formulated as a correction to (-)..3 Conclusions On condition that the convolution integral is evaluated numerically an analytical solution to the initial value problem (-) is provided by the result (-3). Since this solution relies on the Fourier transform of the frequency matrix (-6) for the impulse response matrix, the approach becomes computational prohibitive for a large number of degrees of freedom. Alternatively, if the initial value problem is reformulated in the state vector form (-6) the analytical solution (-9) is obtained. This solution relies on the fundamental matrix in terms of the matrix exponential function for the corresponding homogeneous differential system (-8). The matrix exponential function may be calculated analytically as indicated by (-55), but the solution requires all eigen-solutions to the system matrix A. Again, the calculation of these becomes

29 .3 Conclusions 9 prohibited for large systems. Hence, both analytical or semi-analytical solution approaches are out of the question for large degree-of-freedom systems. The state vector formulation (-6) directly admits the application of vectorial generalizations of standard ordinary differential equation solvers such as the Euler method, the extended Euler method, the various Runge-Kutta algorithms or the Adams-Bashforth/Adams-Moulton algorithm. As is the case for all conditional stable algorithms the numerical stability of these schemes is determined by the length of the time step in proportion to the eigenperiod of highest mode of the system. Hence, in order to insure stability for large scale systems excessive small time steps becomes necessary, which means that the high accuracy of some of these algorithms cannot be utilized. Consequently, there is a need for numerical matrix differential solvers for which the length of the time step is determined from accuracy rather than stability. These algorithms predict stable although inaccurate responses for the highest modes. Instead, the time step is adjusted to predict accurate results for the lowest modes, which carry the global response of the structure. The devise of such algorithms will be the subject of Chapter. System reduction schemes such as (-0), (-03) and (-) require a limited number of low-frequency eigen-pairs to be know. Since, the high frequency components have been filtered out the numerical integration of the modal coordinate differential equations (-00) and (-0) may be performed by standard ordinary differential solvers or by modification of the methods devised in Chapter. Hence, the primary obstacle in using these methods is the determination of the low frequency eigen-pairs. This problem will be the subject of the Chapters 3-7 of the book. Moreover, only solutions to the GEVP (-9) will be considered, i.e. the involved system matrices are assumed to be symmetric and non-negative definite.

30 30 Chapter INTRODUCTION.4 Exercises. Given the following mass- and stiffness matrices M = 0 0, K = (a.) Calculate the eigenvalues and eigenmodes normalized to unit modal mass. (b.) Determine two vectors that are M-orthonormal without being eigenmodes.. The eigensolutions with eigenmodes normalized to unit modal mass of a -dimensional generalized eigenvalue probem are given as λ 0 0 Λ = =, Φ = Φ () Φ () = 0 λ 0 4 (a.) Calculate M and K..3 Write a MATLAB program, which solves the undamped generalized eigenvalue problem for the vibrating string problem considered in Example. for both the finite difference and the finite element discretized equations of motion. The circular eigenfrequencies should be presented in ascending order of magnitude, and the related eigenmodes should be normalized to unit modal mass. (a.) Use the program to evaluate the 4 lowest circular eigenfrequencies of the string as a function of the number of elements n for both discretization methods, and compare the numerical results with the analytical solution (-7). (b.) Based on the obtained results suggest a mass matrix, which will do better..4 Write a MATLAB program, which solves the undamped and damped generalized eigenvalue problems considered in Example.6.

31 CHAPTER NUMERICAL INTEGRATION OF EQUATIONS OF MOTION This chapter deals with the numerical time integration in the finite interval t 0,t 0 + T of the initial value problem (-). The solution is searched for. The idea of the numerical integration scheme is to determine the solution of (-) approximately at the discrete instants of time t j = t 0 + j t, j =,,...,n, where t = T/n. To facilitate notations the following symbols are introduced x j = x(t j ), ẋ j = ẋ(t j ), ẍ j = ẍ(t j ), f j = f(t j ), j =0,,...,n ( ) Single step algorithms in numerical time integration in structural dynamics determines the displacement vector x j+, the velocity vector ẋ j+ and the acceleration vector ẋ j+ at the new time t j+, on condition of knowledge of x j, ẋ j, ẋ j at the previous instant of time, as well as the load vectors f j and f j+ at the ends of the considered sub-interval t j,t j+. In multi step algorithms the solution at the time t j+ also depends on one or more solutions prior to the time t j. Additionally, distinction will be made between single value algorithms, which solves solely for the displacement vector x j, and multi value algorithms, where the solution is obtained for a state vector encompassing the displacement vector x j, the velocity vector ẋ j, and in some cases even the acceleration vector ẍ j. Classical algorithms in numerical analysis such as the vector generalization of the Runge-Kutta methods may be used for the solution of (-). However, given that large scale structural models contain very high frequency components, these schemes become numerical unstable unless extremely small time steps are used. For this reason the devise of useful algorithms in structural dynamics is governed by different objectives than in numerical analysis, as will be further explained below. Newmark algorithms treated in Section. are probably the most widely used algorithms in structural dynamics for solving (-). The derived single step multi value formulation of the methods serves as a generic example for specification of accuracy, stability, and numerical damping. D.G. Zill and M.R. Cullen: Differential Equations with Boundary-Value Problems, 5th Ed. Brooks/Cole, 00. N.M. Newmark: A Method of Computation for Structural Dynamics. J.Eng.Mech., ASCE, 85(EM3), 959,

32 3 Chapter NUMERICAL INTEGRATION OF EQUATIONS OF MOTION High frequencies and mode shapes of the spatially discretized equations (-) does not represent the behavior of the underlying physical problem very well. The corresponding modal components merely behave as numerical noise at the top of the response carried by the lower frequency modes. For this reason it is desirable to filter these components out of the response. In numerical time integrators in structural dynamics this is achieved by the introduction of numerical (artificial) damping, which are affecting merely the high frequency modes. However, it turns out that numerical damping cannot be introduced in the Newmark algorithms without compromising the accuracy of the response of the lower modes. Several suggestions to remedy this problem have been suggested. Here, we shall consider the so-called generalized alpha algorithm suggested by Chung and Hulbert, 3 which seems to be the most favorable single step single valued algorithm for this purpose. The outline of the text relies primarily on the monographs of Hughes 4, 5 and Krenk. 6. Newmark Algorithm The Newmark family consists of the following equations Mẍ j+ + Cẋ j+ + Kx j+ = f j+, j =,...,n ( ) ( ( ) x j+ = x j + ẋ j t + β ẍ j + β ẍ j+ ) t ( 3) ( ( ) ẋ j+ = ẋ j + γ )ẍj + γ ẍ j+ t ( 4) (-) indicates the differential equation at the time t j+, which is required to be fulfilled for the new solution for ẍ j+, ẋ j+, x j+. (-3) and (-4) are approximate Taylor expansions, which have been derived in Box.. The parameters β and γ determines the numerical stability and accuracy of the algorithms. The Newmark family contains several wellknown numerical algorithms as special cases. Examples are the central difference algorithm treated in Example., which corresponds to (β,γ) =(0, ), and the Crank-Nicolson algorithm treated in Example.3, which corresponds to (β,γ)=(, ). 4 There are several implementations of the methods. The most useful is the following single step single value implementation. At first, define the following predictors 3 J. Chung and G.M. Hulbert: A time Integration Algorithm for Structural Dynamics with Improved Numerical Dissipation: The Generalized α Method. Journal of Applied Mechanics, 60, 993, T.J.R. Hughes: The Finite Element Method. Linear Static and Dynamic Finite Element Analysis. Printice-Hall, Inc., T.J.R. Hughes: Analysis of Transient Algorithms with Particular Reference to Stability Behavior. Chapter incomputational Methods for Transient Analysis. Vol. in Computational Methods in Mechanics, Eds. T. Belytschko and T.J.R. Hughes, North-Holland, S. Krenk: Dynamic Analysis of Structures. Numerical Time Integration. Lecture Notes, Department of Mechanical Engineering, Technical University of Denmark, 005.

33 . Newmark Algorithm 33 x j+ = x j + ẋ j t + ( β ) t ẍ j ( 5) x j+ = ẋ j + ( γ ) t ẍ j ( 6) (-5) and (-6) specify predictions (preliminary solutions) for x j+ and ẋ j+ based on the information available at the time t j. The idea of the algorithm is to insert (-3) and (-4) into (-). Given that the solution is required to fulfill the equations of motion at the time t j+, and using (-5) and (-6), the following equations are obtained for the new acceleration vector in terms of known solution quantities from the previous time and the load vector f j+ ( ) M + γ tc + β t K ẍ j+ = f j+ C x j+ K x j+ ( 7) Next, based on the solution for ẍ j+ obtained from ( 7) corrected (new) solutions for ẋ j+ and x j+ may be obtained from (-3) and (-4). These may be written as x j+ = x j+ + β t ẍ j+ ( 8) ẋ j+ = x j+ + γ t ẍ j+ ( 9) To start the algorithm the acceleration ẍ 0 at the time t 0 is needed. This is obtained from the equation of motion Mẍ 0 = f 0 Cẋ 0 Kx 0 ( 0) The algorithm has been summarized in Box.. In stability and accuracy analysis a single step multi value formulation for the state vector made up of the displacement and velocity vectors is preferred. In order to derived this, eqs. (-3) and (-4) are multiplied with M. Next, the accelerations are eliminated by means of the differential equations at the times t j and t j+, leading to Mx j+ = Mx j + M tẋ j + ( ( ) ) β (fj ) ( ) Cẋ j Kx j + β fj+ Cẋ j+ Kx j+ t Mẋ j+ = Mẋ j + ( ( )( ) ( ) ) γ fj Cẋ j Kx j + γ fj+ Cẋ j+ Kx j+ t M + β t K β t C x j+ = γ t K M+ γ t C ẋ j+ M ( β) t K tm ( β) t C ( γ) t K M ( γ) t C x j ẋ j + ( β) t β t ( γ) t γ t f j f j+

34 34 Chapter NUMERICAL INTEGRATION OF EQUATIONS OF MOTION z j+ = Dz j + E j ( ) where x j z j = ẋ j M + β t D = K γ t K M + β t E j = K γ t K β t C M+ γ t C β t C M+ γ t C β) t C M ( t K tm ( ( γ) t K M ( γ) t C ( t β t ( γ) t γ t f j f j+ ( ) D denotes the so-called amplification matrix. The bar indicates that this is an approximation to the exact amplification matrix, which has has been derived in Example.. Box.: Newmark algorithm Given the initial displacement vector x 0 and the initial velocity vector ẋ 0 at the time t 0. Calculate the initial acceleration vector ẍ 0 from Mẍ 0 = f 0 Cẋ 0 Kx 0 Repeat the following items for j =0,,...,n. Calculate predictors for the new displacement and velocity vectors ( ) x j+ = x j + ẋ j t + β t ẍ j x j+ = ẋ j + ( γ ) t ẍ j. Calculate new acceleration vector from ( ) M + γ tc + β t K ẍ j+ = f j+ C x j+ K x j+ 3. Calculate new displacement and velocity vectors x j+ = x j+ + β t ẍ j+ ẋ j+ = x j+ + γ t ẍ j+

35 . Newmark Algorithm 35 Box.: Derivation of (-3) and (-4) Based on conventional integration theory the following identities may be formulated x(t j+ )=x(t j )+ ẋ(t j+ )=ẋ(t j )+ tj+ t j tj+ t j ẋ(τ)dτ ẍ(τ)dτ Integration by parts of the first relation provides ( 3) (tj+ x(t j+ )=x(t j ) τ ) tj+ ẋ(τ) + tj+ ( tj+ τ ) ẍ(τ)dτ t j t j x j+ = x j + t ẋ tj + tj+ t j ( tj+ τ ) ẍ(τ)dτ ( 4) The indicated derivation is due to Krenk. 6 (-4) may interpreted as a truncated Taylor expansion, where the integrals represent the remainder. Correspondingly, the nd equation in (-3) is written as ẋ j+ = ẋ j + tj+ t j ẍ(τ)dτ ( 5) Next, the integrals in (-4) and (-5) are represented by the following linear combinations of the value of the acceleration vector at the end of the integration interval tj+ t j tj+ ( tj+ τ ) ẍ(τ)dτ t j ẍ(τ)dτ ( γ ) t ẍ j + γ t ẍ j+ ( ) β t ẍ j + β t ẍ j+ ( 6) It is seen that the result in (-6) becomes correct in case of constant acceleration, where ẍ(τ) ẍ j = ẍ j+. In any case the values of β and γ reflect the actual variation of the acceleration during the interval. If ẍ(τ) is assumed to be constant and equal to the mean of the end-point values, one obtains (β,γ) =(, ), whereas a linear variation between 4 the end-point values provides (β,γ)=(, ). 6 The modal expansion (-97) defines a one-to-one transformation from the physical to the modal coordinates. Hence, the time integration may equally well be performed on the differential equations for the modal coordinate equations. It follows that the synthesized motion (-97) becomes

36 36 Chapter NUMERICAL INTEGRATION OF EQUATIONS OF MOTION numerical unstable, if the integration of just one of the modal coordinates render into instability. Similarly, the accuracy of the synthesized motion is determined by the accuracy of those modal coordinates, which are retained in the truncated modal expansion (-0). On condition of the modal decoupling condition (-94) the time integration of the modal coordinates is reduced to the integration of n decoupled single-degrees-of-freedom systems. Since the stability and accuracy analysis of a SDOF system can be performed analytically, the important role of the modal decomposition assumption in the stability and accuracy analysis of numerical time integrators becomes clear. In this respect let q(t) denote an arbitrary of the n modal coordinates, and ζ, ω 0 and F (t) the corresponding modal damping ratio, undamped circular eigenfrequency and modal load. On condition that the eigenmodes have been normalized to unit modal mass, the differential equation of the said modal coordinate reads q(t)+ζω 0 q(t)+ω0 q(t) =F (t) ( 7) The corresponding Newmark integration of (-7) is given by (-5), using M =, C = ζω 0, K =ω0 and f(t) =F (t) in (-6), resulting in the system matrices z j = q j q j +β t D = ω0 ζβω 0 t γω0 t +ζγω 0 t = D D D D where E j = = ( β) ω 0 t t ( β) ζω 0 t ( γ)ω0 t ( γ)ζω 0 t ( +β t ω0 ζβω 0 t β) t β t γω0 t +ζγω 0 t ( γ) t γ t E E E E F j F j+ F j F j+ ( 8) D = +γζκ +(β )κ +(β γ)ζκ 3 +γζκ + βκ D = +(γ )ζκ +(β γ)ζ κ t +γζκ + βκ D = + (β γ)κ κ +γζκ + βκ t D = +(γ )ζκ +(β γ)κ (β γ)ζκ 3 +γζκ + βκ ( 9)