Order Reduction of Parametrically Excited Linear and Nonlinear Structural Systems

Transcription

1 Venkatesh Deshmukh 1 Eric A. Butcher ffeab@uaf.ediu Department of Mechanical Engineering, University of Alaska Fairbanks, Fairbanks, AK S. C. Sinha Nonlinear Systems Research Laboratory, Department of Mechanical Engineering, Auburn University, Auburn, AL Order Reduction of Parametrically Excited Linear and Nonlinear Structural Systems Order reduction of parametrically excited linear and nonlinear structural systems represented by a set of second order equations is considered. First, the system is converted into a second order system with time invariant linear system matrices and (for nonlinear systems) periodically modulated nonlinearities via the Lyapunov-Floquet transformation. Then a master-slave separation of degrees of freedom is used and a relation between the slave coordinates and the master coordinates is constructed. Two possible order reduction techniques are suggested. In the first approach a constant Guyan-like linear kernel which accounts for both stiffness and inertia is employed with a possible periodically modulated nonlinear part for nonlinear systems. The second method for nonlinear systems reduces to finding a time-periodic nonlinear invariant manifold relation in the modal coordinates. In the process, closed form expressions for true internal and true combination resonances are obtained for various nonlinearities which are generalizations of those previously reported for time-invariant systems. No limits are placed on the size of the time-periodic terms thus making this method extremely general even for strongly excited systems. A four degree-of-freedom mass- spring-damper system with periodic stiffness and damping as well as two and five degree-of-freedom inverted pendula with periodic follower forces are used as illustrative examples. The nonlinear-based reduced models are compared with linear-based reduced models in the presence and absence of nonlinear resonances. DOI: / Introduction Many structural systems with rotating components, periodic inplane loads, or systems described by nonlinear differential equations expanded about a periodic solution admit a set of second order differential equations with time-periodic coefficients as the governing equations of motion. The problem of characterizing the motion of a structural system with time-periodic mass, damping, and stiffness matrices by a model which has lower dimension than the original model is considered. For such systems, it is often possible to compute the response of the system in terms of a subset of the original coordinates. For time-invariant systems, the task of order reduction of linear and nonlinear structural systems has received a considerable amount of attention 1 6. In order to obtain accurate reduced order nonlinear models for nonlinear time-invariant systems, a technique known as nonlinear normal modes NNMs is used to describe the motion as nonlinear functions of a subset of all the natural coordinates or degrees of freedom of the system. The method is well known from its reformulation by Shaw and Pierre in the earlier part of the past decade 7. The NNMs are defined by them as motions occurring on invariant manifolds in the phase space, which are tangent to the corresponding eigenvectors of the linearized system at the equilibrium position. Based on this concept, NNM-based reduced models may be obtained both in state space and structural second order forms by approximating the invariant manifolds by polynomials as in center manifold theory. Unfortunately these techniques cannot be applied directly to time periodic systems. However, using a bounded invertible transformation called the Lyapunov-Floquet L-F transformation 8, 1 Current address: Department of Mechanical Engineering, Villanova University, Villanova, PA Corresponding author. Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received January 6, 004; final manuscript received December 1, 005. Assoc. Editor: D. Dane Quinn. the original time periodic system can be converted to a similar one with a completely time invariant linear part, after which various order reduction techniques can be applied. Using this methodology, reduced order models for linear 9 and nonlinear 10,11 time periodic systems in the state space form have been obtained by employing the L-F transformation. The problems become tractable due to the fact that such a transformation makes the linear part of the system equation time invariant. In the case of nonlinear time-periodic systems, this enables a time-periodic invariant manifold on which the motion in a particular mode occurs to be approximated similar to time-periodic center manifold theory 1 or nonlinear order reduction of systems with periodic external forcing via periodically varying invariant manifolds 13. The objective of the present manuscript is to propose methodologies to obtain reduced order models of periodic parametrically excited linear and nonlinear systems in structural form without placing any restrictions on the intensity of the parametric excitation terms. It is in the same vein as the order reduction problem of the time-invariant structural systems except one has to account for the time-periodic system matrices and periodically modulated nonlinearities. The two methodologies proposed in the present paper are as follows: First, the local description of the time-periodic system is transformed into a form in which the linear system matrices are completely time invariant. This is accomplished by using the L-F transformation. Then, the resulting system in second order form has constant linear matrices and for nonlinear systems periodically modulated nonlinear coefficients. The position vector of the transformed system is assumed to be a direct sum of master and slave coordinates and a transformation describing the slave states in terms of master states is constructed. In the first method, focus is on a Guyan-type reduction method that accounts for both stiffness and inertia in the linear kernal. Such a technique has been proposed for undamped time-invariant second order systems in Ref. 5. For nonlinear systems, the nonlinear transformation has an unknown time invariant linear part and periodically modulated nonlinearities of master coordinates 458 / Vol. 18, AUGUST 006 Copyright 006 by ASME Transactions of the ASME

2 with unknown Fourier coefficients. The unknowns are solved in a two-stage procedure. In the first stage, the linear time invariant transformation matrix is found. In the second stage, the unknown Fourier coefficients of the periodic nonlinear terms are found by solving harmonic balance equations. The slave coordinates in the transformed domain are eliminated and a reduced order model is obtained in terms of the master states. This reduced order model along with the order reduction transformation, is sufficient to reconstruct the evolution of all the natural coordinates in the original variables. The second method for nonlinear systems proceeds instead by decoupling the time-invariant linear system matrices after application of the L-F transformation and finding a time-periodic invariant manifold relation in a particular mode. This formulation also enables the derivation of nonlinear true internal and true combination resonance conditions, which are generalizations of those previously discussed for time-invariant structural systems 4 6. No specific norm bounds are placed on the sizes of the time periodic terms and this fact makes the scheme extremely useful for order reduction of nonlinear parametrically excited systems. Because the motion on the invariant manifolds in the transformed coordinates is not periodic in the original coordinates because of the L-F transformation, we classify the procedure as invariant manifold or nonlinear based and not NNM based. It is shown using numerical examples that the nonlinear-based reduced order description for nonlinear time periodic systems is more accurate that the linear-based order reduction for parameter sets that avoid the nonlinear resonances. The linear-based order reduction used here does not ignore the time periodicity in the linearized dynamics, includes the nonlinearity in the reduced model, and therefore forms a very fair means of comparison with the nonlinear-based order reduction. Three relevant numerical examples are worked to show the effectiveness of the proposed technique: a four degree-of-freedom mass-spring-damper system with periodic stiffness and damping, and two and five degree-offreedom inverted pendula with periodic follower forces. Problem Statement Consider a parametrically excited structural system given by M t ẍ + C t ẋ + K t x + f x,ẋ,t =0 1 where x t is an n-dimensional displacement vector and M t,c t, and K t are n n T-periodic mass, damping and stiffness matrices, respectively. The term f x,ẋ,t = f x,ẋ,t+t if it exists contains polynomial nonlinearities and satisfies f 0,0,t =0. The strength of the parametric excitation in M t,c t,k t, and f x,ẋ,t is not assumed to be small and therefore asymptotic methods like perturbation and averaging that assume a small periodic parameter cannot be applied. It is desired to use the L-F transformation and other techniques to obtain a reduced order model of the form Mˆ ÿ m + Ĉẏ m + Kˆ y m + fˆ y m,ẏ m,t =0 where y m is an m dimensional subset m n of the coordinates in a transformed domain and Mˆ, Ĉ, and Kˆ are constant matrices, which, through the L-F transformation, preserves the dominant dynamics of Eq. 1. In addition, for linear systems it is also possible to find a reduced order model in a subset x m t m n of the original coordinates x t as M t ẍ m + C t ẋ m + K t x m =0 which retains the dominant dynamics of 1. 3 Lyapunov-Floquet Transformation A very important tool for obtaining reduced order models in case of time periodic systems is the L-F transformation. A brief 3 description of the L-F transformation is given in the following. Consider a linear time periodic system in the state space with period T given by u = A t u + f u,t 4 If t is the state transition matrix STM for the linear part of 4, then it satisfies t = A t t 5 According to the Floquet theory 14, the STM can be factored as t = L t e Ct = Q t e Rt 6 where the T-periodic matrix L t and constant matrix C are generally complex while the T-periodic matrix Q t and constant matrix R are real. Eigenvalues of C or R determine the local stability in all hyperbolic noncritical cases. Specifically, the eigenvalues i = i ± i i /T of C are related to the Floquet multipliers i as i =ln i, i =Arg i where i 0,,,...,n for asymptotic stability. The eigenvalues of R are similarly found except that i =arctan Im i /Re i so that they are equivalent to the characteristic exponents only when Re i 0. Now, a state transformation called the L-F transformation u = L t 7 converts system 4 into a similar one with a completely time invariant generally complex linear part given by v = Cv + ğ,t 8 Moreover, the T-periodic transformation matrix Q t produces a real system matrix R. Note that the invariant form in 8, along with the transformation 7, is exact both in terms of response and stability regardless of the size of the periodic part of A t. A technique for computing the L-F transformation for a given parameter set is found in Ref Guyan-Type Order Reduction Before a reduced order system can be obtained, Eq. 1 must be transformed into a form with time-invariant linear matrices by employing the L-F transformation. For this purpose, the system described by Eq. 1 is temporarily transformed to state space form so that the L-F transformation can be computed, and then another canonical transformation back into second order form shown in 1 is employed. Assuming the T-periodic L-F transformation matrix and real constant matrix are partitioned as Q t = Q 11 t Q 1 Q 1 t Q t R = R 11 R 1 R 1 R 9 this results in the second order system ÿ + C ẏ + K y + f y,ẏ,t =0 where assuming R 11,R 1 are invertible C = R 1 1 R 11 R 1 I + R R 1 1 R 1 11 R 1 K = R 1 1 R 11 R 1 R 1 R 1 11 R 1 R f y,ẏ,t = f y,ẏ,t C f 1 y,ẏ,t f 1 y,ẏ,t f y,ẏ,t = f 1 y,ẏ,t R 11 1 R 1 I 0 R 1 1 R 1 11 R 1 1 Q 11 t Q 1 Q 1 t Q t 0 M t f x y,ẏ,t,ẋ y,ẏ,t,t Journal of Vibration and Acoustics AUGUST 006, Vol. 18 / 459

3 ẋ y,ẏ,t = x y,ẏ,t Q 11 t Q 1 Q 1 t Q t R 11 1 R 1 I 0 R 1 y ẏ f 1 y,ẏ f 1,t 1 R 1 11 R 1 while retaining only those orders of nonlinear terms which were originally present in 1. Equation 10 should be used to eliminate ÿ when computing f 1 y,ẏ,t. Note that the periodic terms now appear only as coefficients to the nonlinear polynomial terms while the linear matrices are time invariant. The order reduction transformation is based on the undamped linear kernel, however, so that any linear damping if it exists will be neglected temporarily. Now, the n dimensional displacement vector y of the transformed system 10 is assumed to have m n dimensional master degrees of freedom y m, which will be retained in the reduced model and s=n m dimensional slave degrees of freedom y s will be eliminated. Equation 10, neglecting linear damping, is thus written as ÿ m ÿ s K 11 K 1 K 1 K y m y s f 1 y m,y s,ẏ m,ẏ s,t f y m,y s,ẏ m,ẏ s,t =0 1 It is suggested that the slave degrees of freedom are to be eliminated by the master-slave relationship y s = T y m + g y m,ẏ m,t 13 such that g has same or higher degree of nonlinearity as f and g 0,0,t =0. Both of these functions are periodic in time. The nonlinear term in transformation 13 has the form g y m,ẏ m,t = i,ñ g i,ñ t y m1 n1 y mm n m ẏ m1 n m+1 ẏ mm n m ē i ñ = n 1,...,n m,n m+1,...,n m ; n k = r 14 k The greatest value of r is the highest order of the nonlinear terms retained in f and each g i,ñ t is a Fourier series with unknown coefficients. The operator T and the unknown Fourier coefficients of g are computed by deriving a condition for 1 to admit a transformation of the form 13. These conditions are given in Ref. 5 as T = K 11 T K 1 1 K 1 T K g + K 11 T K 1 g = T f f 1 15a 15b Equation 15a is first solved via iteration for T and then 15b is solved for the Fourier coefficients of g by substituting 14 into 15b and employing harmonic balance. As discussed in Ref. 5, the matrix T is an extension of the traditional Guyan reduction matrix 1, which accounts for both stiffness and inertia effects and thus allows the reduced model to exactly preserve the linear undamped eigenstructure projected onto the y m coordinates. Alternatively, if m linear mode shapes 1,,..., m associated with the retained modes are arranged as = 1... m = s m, where submatrices s and m are s m and m m, respectively, then the transformation matrix is given by T = s m 1. The reduced order model is then obtained by applying a transformation y = I T y m to Eq. 10 to yield Eq. where 0 g y m,ẏ m,t y m + G y m,ẏ m,t 16 Mˆ = T, Kˆ = T K, Ĉ = T C, fˆ = T G + C Ġ + K G + f y m + G, ẏ m + Ġ,t, 17 This reduced model preserves the dominant dynamics of 1 by retaining the eigenvalues with the greatest negative real part or, for undamped systems, the lowest imaginary part. As discussed in Sec. 3, however, the imaginary parts and hence the form of the reduced model depends upon whether a T- ort-periodic L-F transformation was utilized. The slave states are, of course, given by 13. The state vector for the original system 1 is reconstructed from the reduced model using x ẋ = Q 11 t 1 R 1 Q 1 Q 1 t Q t R 11 I 0 R 1 1 R 11 y m + G y m,ẏ m,t 18 ẏ m + Ġ y m,ẏ m,t f 1 as well as 11 and 16 while retaining the same order of nonlinearity as in 1. If a linear-based order reduction transformation is applied such that g 0 in Eq. 13, then a nonlinear reduced order model of the form is obtained where fˆ is obtained by setting G =0 in 17 as 1 R 1 fˆ = T f y m, ẏ m,t 19 while the slave states are given by only the linear part of 13. This linear-based order reduction will be compared with the nonlinear-based one in Sec. 6. If Eq. 1 is linear, then we can obtain a reduced order model in a subset m of the original coordinates x t. For this purpose, Eq. 18 with G = f 1 =0 is written as xm x s ẋ m ẋ s = Q11 t Q1 t Q 1 t Q t R 11 1 R 1 0 R 1 1 t Qˆ t = Qˆ Qˆ 3 t Qˆ 4 t ym ẏ m I 1 R 1 y m ẏ m 1 R 11 0 where Qˆ i t are m m for,3 and s m for i=,4. The slave displacements and velocities in the original coordinates are found from 0 in terms of master displacements and velocities in the original coordinates as x s ẋ s = Qˆ t Qˆ 4 t Qˆ 1 t Qˆ 3 t 1 x m ẋ m = Q 11 t Q 1 t Q 1 t Q t x m ẋ m 1 where each block Q ij t is s m. The reduced model may be obtained by applying the transformations = I x Q 11 t x m 0 m = 1 t x m + t ẋ m Q 1 t ẋ ẋ = 0 Q 1 t x m to Eq. 1 to obtain Eq. 3 where I Q t ẋ m = 3 t x m + 4 t ẋ m 460 / Vol. 18, AUGUST 006 Transactions of the ASME

4 M t = 4 T t M t 4 t C t = T 4 t M t 3 t + 4 t + C t 4 t + K t t 3 K t = T 4 t M t 3 t + C t 3 t + K t 1 t For special linear periodic systems which can be decoupled via a time invariant transformation, as in the first example in Sec. 6, the order reduction transformation matrix in is exactly the same as the linear part of Eq. 13, i.e., x s =T x m. In that case, the reduced matrices in Eq. 3 are given by M t = T M t C t = T C t K t = T K t 4 where is defined the same as in Eq Invariant Manifolds and Derivation of Resonance Conditions Alternatively, in the absence of linear damping C =0, itmay be desirable to first apply a linear modal transformation to decouple K matrix in 10. Then the linear part of the master-slave relation in 13 vanishes T =0 and the order reduction transformation is purely nonlinear. This approach defines the time-varying invariant manifold in the linear modal coordinates and is used here to derive the resonance conditions. The occurrence of such a resonance depends upon the parameter values and the type of nonlinearities. There are three main types of resonances associated with parametrically excited nonlinear systems, e.g., a parametric resonance, b true internal resonance, and c true combination resonance. The following description in terms of Floquet exponents/multipliers assumes that the system does not have linear damping. a Parametric Resonance: In a single degree-of-freedom system, a parametric resonance occurs when the frequency of parametric excitation is / n times the natural frequency of the system, where n is an integer. The principal parametric resonance takes place when their ratio is n=1. In terms of the Floquet exponents or eigenvalues of the R matrix in Eq. 11, an even number of them are zero or real which corresponds to the same number of the Floquet multipliers being 1, 1 or opposite these values on the real axis. This is a linear case of instability and does not depend on the form of the nonlinearities. For a system with multiple degrees of freedom, sum and difference type of parametric resonances are also possible. b True internal resonance: A true internal resonance occurs when the eigenvalues of the R matrix in Eq. 11 in this case, purely imaginary Floquet exponents satisfy certain integral relationships for specific types of nonlinearities. This corresponds to the Floquet multipliers on the unit circle having rationally related subtended angles with ratios :1 quadratic case, 1:1, or 3:1 cubic case. If the state space matrix A t in Eq. 4 is written as A 0 A 1 t, then this resonance condition reduces to the familiar case of internal resonance defined in terms of the i in ẍ i i p i t x i + f i x,ẋ,t =0 as 0. For arbitrarily large, however, the definition of the true internal resonance above must be used. Although the system may be locally stable, this nonlinear resonance couples the associated modes such that it is impossible or at least inaccurate to isolate a single one in a reduced model. c True combination resonance: A true combination resonance occurs when one or more Floquet exponents which are purely imaginary in this case and the parametric forcing frequency satisfy special integral relationships for specific types of nonlinearities. As in case b, this reduces to the familiar case of combination resonance defined in terms of the parametric frequency and the i as above as 0, while for arbitrarily large the definition of the true combination resonance must be used. This is also a nonlinear resonance which prevents the accurate isolation of one of the modes associated with the resonance. We assume that the system is not in parametric resonance and is locally stable in the linear sense. If either true internal resonance or true combination resonance occurs in a parametrically excited system, then it may not be possible to use the nonlinear order reduction procedure to isolate one of the nonlinear modes associated with the resonance. This conclusion is not apparent in the procedure outlined in Sec. 4 due to the iterative procedure involved in the solution of Eq.. In order to determine the various resonance conditions and the time-varying invariant manifolds, we proceed in an alternate fashion as in the following. We start with a system of the form of Eq. 1 without linear damping and the L-F transformation is applied to obtain the form of Eq. 19 with constant stiffness matrix. The periodic terms that remain appear as coefficients to the nonlinear polynomials and have the parametric frequency. Applying a modal transformation to the system given by Eq. 1, the equivalent system in the modal space is given by ȳ m ȳ s m 0 0 s ȳ m ȳ s f m ȳ m,ȳ s,ȳ m,ȳ s,t f s ȳ m,ȳ s,ȳ m,ȳ s,t =0 5 where m, s are diagonal matrices with the entries given by the squares of the master m and slave s natural frequencies, respectively, which are also the magnitudes of the imaginary characteristic exponents recall Sec. 3. The resonance conditions for the original system 1 are derived from system 5 since the two systems are dynamically equivalent. It should be noted that the application of the nonlinear order reduction procedure to system 5 using Eqs. 13 and 14 results in a purely nonlinear masterslave relationship implying that the linear part of transformation 13 vanishes. Thus, using a purely nonlinear master-slave transformation given by ȳ s is computed as ȳ s = d dt g ȳ m ȳ m + g ȳ m ȳ s = g ȳ m,ȳ m,t ȳ m + t g = ȳ m g ȳ m ȳ m g ȳ m m ȳ m + g t ȳ m g ȳ m m ȳ m + g t m ȳ m + t g ȳ m ȳ m g ȳ m g ȳ ȳ m m ȳ m m ȳ m + g 6 t = s ȳ s f s ȳ m,g,ȳ m,ġ,t 7 where higher order terms have been neglected. This yields the required equation for g ȳ m,ȳ m,t. This equation must be solved in order to determine the Fourier coefficients of g i,ñ t which has complex and real Fourier expansions given by g i,ñ t = g i,ñ, e i t = i,ñ0 + i,ñ, s sin t + i,ñ, c cos t 8 where = /T is the parametric frequency. Substituting the complex real form of 8 into 14 and equating both sides of Eq. 7, one obtains a set of complex real linear algebraic equations for the unknown Fourier coefficients in Eq. 8 that are decoupled in the frequency. It is immediately observed that the solvability condition requires that the determinant of the coefficient matrix be nonzero. If the complex representation has coefficient matrix D, then the real one which is twice as large has coefficient matrix given by Journal of Vibration and Acoustics AUGUST 006, Vol. 18 / 461

5 Im D D = Re D 9 Im D Re D Therefore we require that det D 0. If det D =0 then various resonances between master and slave modes may occur and nonlinear order reduction is possible only by retaining all resonant modes in the multimode reduced model. It should be noted that for the time invariant case =0 and det D 0 =0 yields the resonance conditions for this case. At this point we consider various special forms of nonlinearities and determine the corresponding forms of the resonance conditions. First, consider reducing the system given by Eq. 5 to a single degree-of-freedom reduced model with quadratic nonlinearities of the form y m,ẏ m y m,ẏ m in g. The 3 3 complex coefficient matrix can be computed in terms of the master m and one of the slave natural frequencies s as = s m 4 i m m D 4i s 4 m 4i m i s m 30 while the real 6 6 coefficient matrix is given by 9. For =0, the :1 true internal resonance condition s = m is obtained by solving for s from the equation det D 0 =0. For 0, the true combination resonance conditions are obtained from the equation det D =0 as s = m ± and s =. For a single degree-of-freedom reduced model with cubic nonlinearities of the form y m 3,ẏ m y m,y m ẏ m,ẏ m 3 in g the complex 4 4 coefficient matrix is given as 3 m D = s 4 i m m 0 6i s 7 m 4 4i m m 6 4i s 7 m 6i m 0 i s 3 m 31 while the 8 8 real coefficient matrix is computed from 9. Substituting =0 in det D =0 gives the 3:1 and 1:1 true internal resonance conditions s =3 m, s = m. The combination resonance conditions for 0 are obtained as s = m ±, 3 m ±. It is necessary for the existence of a nonlinear-based reduced model that these resonance conditions are not satisfied. Even in the vicinity of the resonance relationships, the nonlinearbased reduced order model of the original systems performs poorly as compared to the linear based model. Consequently, an accurate reduced order model in the presence of these resonances requires that both the modes whose frequencies satisfy the above relationships to be retained in the multimode reduced model. For multimode reduced models, the resonance conditions can be computed by the same procedure. For a three degree-offreedom reduced order model with quadratic nonlinearities, the complex matrix for a two degree-of-freedom reduced order model is and is given as D = 1 m m1, m m where m1, m are the two master frequencies retained in the reduced model. The blocks 1 m1, 1 m are obtained by substituting m = m1 and m = m, respectively, in Eq. 30. The block matrix m1, m is given as s m1 m s m1 i i m1 m m i m i m1 s m1 i i m m m1 s m1 i i m1 m m 33 Setting the determinant of the corresponding 0 0 real matrix to zero, the additional true internal and true combination resonance conditions are obtained as s = m1 ± m for =0 and s = m1 ± m ± for 0, respectively. Similarly, for a system with cubic nonlinearities, the resonances obtained for a two degree-of-freedom reduced order model include the additional true internal resonance conditions s = m1 ± m for =0 and the additional true combination resonance conditions s = m1 ± m ± for 0. For a three degree-of-freedom reduced order model with cubic nonlinearities, the additional true internal resonance conditions are obtained as s = m1 ± m ± m3 for =0 and the additional true combination resonance conditions as s = m1 ± m ± m3 ± for 0. As discussed in Ref. 4, these are all of the relevant cases for a system with quadratic and cubic nonlinearities. It should be noted that in all the above cases, det D 0 =0 yields the same true internal resonance conditions obtained in Refs. 4,6 for the case of timeinvariant nonlinear structural systems, if the Floquet exponents here are associated with the natural frequencies of the timeinvariant systems considered in those studies. As discussed previously, they are only equivalent as 0 for the state space matrix in the form A 0 A 1 t. Also, the resonance expressions derived here are special cases of the general case including linear damping derived in Ref. 11 by state space analysis. 46 / Vol. 18, AUGUST 006 Transactions of the ASME

6 Fig. 1 Reduced order trajectories for periodic mass-springdamper system with initial conditions on the first eigenmode Fig. Reduced order trajectories for periodic mass-springdamper system with initial conditions on the second eigenmode 6 Examples Example 1: Order Reduction of a Linear Four Degree-of- Freedom (4-DOF) System. Consider a four degree-of-freedom spring-mass-damper system with periodically varying spring and damping stiffnesses ẍ cos t sin t ẋ + cos t sin t x =0 34 The second order time-invariant system after the L-F transformation is given by ÿ ẏ y = A two degree-of-freedom reduced order model is obtained using the method in Sec. 4 as m ÿ ẏ m y m = where the matrix T = is employed in Eq. 13, while a single degree-of-freedom reduced order model is obtained as 7.361ÿ ẏ y 4 =0 37 by using the matrix T = T. Since both of the order reduction transformations are based on the undamped response, a comparison between the actual and the reconstructed trajectories of the reduced model in Eq. 36 shows more error in the second eigenmode than in the first as shown in Figs. 1 and. The higher error in the second mode can be explained by noting that the characteristic exponents of Eq. 34 eigenvalues of 35 are given by 0.038±0.600i, ±1.1370i, 0.138±1.1911i, 0.618±1.477i while the retained eigenvalues of the reduced model 36 are given by 0.038±0.600i, ±1.1398i. The first complex pair are also the eigenvalues of the reduced model 37. It can be seen that the reduced order models are less erroneous for lightly damped systems as the order reduction transformation is based on the undamped dynamics. For this special problem which can be decoupled in second order form via a time-invariant transformation, the matrix T for direct order reduction in the original coordinates is the same as above in the transformed coordinates. The reduced order mass, damping and stiffness matrices for a two degree-of-freedom reduced model Eq. 3 in the original coordinates are therefore found from Eq. 4 as Mˆ t = Ĉ t = cos t + 0. sin t Kˆ t = 1 + cos t + sin t Journal of Vibration and Acoustics AUGUST 006, Vol. 18 / 463

7 while a single degree-of-freedom reduced model in the original coordinates is given as 7.361ẍ cos t + 0. sin t ẋ cos t + sin t x 4 =0 39 Both of these reduced models exactly preserve the lowest Floquet exponents of the original system, which are the same as the eigenvalues in 36 and 37. Example : Linear Order Reduction of a Five-Mass Inverted Pendulum. Next consider the five mass inverted pendulum with periodic follower force in Fig. 3. The linearized system at x 1,...,x 5 = 1,..., 5 = 0,...,0 with m=1 is given as k 0 0 p k k p k 0 p k k p 0 k k p k p k = k k p k p ẍ k k 1 p 1 x For a particular normalized parameter set with values U i =0, i =1,...,5 no excitations, k =1.0, p = cos t, =, =1, the system has purely imaginary Floquet exponents given as ±0.3580i, ±0.916i, ±1.33i, ±1.3045i, ±1.3471i. After employing the L-F transformation, the time-invariant system in the second order form is given by ÿ y = = By computing T , a two degree-of-freedom reduced model in the transformed coordinates with y m = y 4 y 5 T is obtained as ÿ m y m = This model retains the lowest two and of the original five natural frequencies. The comparisons of the actual and reconstructed trajectories on the first and second eigenmodes in Figs. 4 and 5 show no error. This is expected since there is no damping. The trajectories of the reduced model in the original coordinates according to Eqs. 3 and 3 not shown explicitly Fig. 3 force Five mass inverted pendulum with periodic follower Fig. 4 Reduced order trajectories for five mass inverted pendulum with initial conditions on the first eigenmode 464 / Vol. 18, AUGUST 006 Transactions of the ASME

8 Fig. 5 Reduced order trajectories for five mass inverted pendulum with initial conditions on the second eigenmode here since it was computed on a point by point basis are likewise identical to those of the full model. Fig. 6 Double inverted pendulum with periodic follower force Example 3: Nonlinear Order Reduction of a -DOF Inverted Pendulum. The final example is a double inverted pendulum with a periodic follower force as shown in Fig. 6. The local equations of motion expanded up to cubic terms about x 1,x = 1, = 0,0 with m=1 and b 1 =b =0 no damping are given by ẍ 1 0.5k p 0.5k p 5 k p 1.5 x 1 ẍ 0.5k 3 p x 0.5 ẋ1 + ẋ x 1 x p k x 1 x x 3 / x 1 x k p 4 x 1 + k 3+p x = ẋ 1 + ẋ x 1 x p k x 1 x x 3 /1 0.5 x 1 x k p 7 x 1 + k 5+p 3 x where the inverse mass matrix has already been premultiplied and p = p 1+ p cos t is the source of time periodic stiffness matrix and periodically modulated nonlinearities. For a particular normalized parameter set p 1=1, p =0.7, k =1., =1, =, it is found that the transformed nonlinear system is given by ÿ m ÿ s y m y s f 1 y m,y s,ẏ m,ẏ s,t f y m,y s,ẏ m,ẏ s,t =0 44 The master-slave frequencies of 44 are thus m = and s = with a ratio of , which is not in the vicinity of 1:1 or 3:1 true internal resonances. Because the process of computing the T-periodic L-F transformation requires normalization of the principal period to unity 8, f 1 and f contain periodically modulated cubic nonlinearities with frequency =. Thus it can be verified that the system is also not in true combination resonance, i.e., ±, ±. Application of the Guyan-based reduction method yields the master-slave relation which defines the time-periodic invariant manifold as y s = y m g g 30ci cos it y m g 1si sin it y m ẏ m g 10 + g 1ci cos it y m ẏ m g 03si sin it ẏ m where the Fourier coefficients g 30si, g 10, g 1ci, g 1si, g 030, g 03ci are zero. The nonlinear-based reduced order model is thus obtained as in Eqs. and 17 where fˆ y m,ẏ m,t contains periodically modulated cubic terms in y m and ẏ m. For comparison, a master-slave relation given by the linear part of 45 yields a linear-based reduced order model with fˆ y m,ẏ m,t given by Eq. 19 instead. Figure 7 a shows the comparison between the trajectories of the full model and from the nonlinear-based and linear-based reduced order models. It can be clearly seen that the nonlinear-based reduced order model produces a more accurate response than its linear counterpart and is also confirmed by Fig. 7 b, which compares the quantity Journal of Vibration and Acoustics AUGUST 006, Vol. 18 / 465

9 Fig. 8 The projection of the invariant manifold onto y m,y s space. The parametric forcing period of normalized transformed system is.0 Fig. 7 a Comparison of the actual, nonlinear reduced and linear reduced model trajectories for double inverted pendulum away from resonance. b Integral square error for gradually increasing initial conditions for double inverted pendulum away from resonance. N t f e 0 t dt = e t i t e t i 46 for the responses of linear-based and nonlinear-based order reduced models with e t being the error between the actual trajectory of the original system and the corresponding trajectories reconstructed from the two reduced models. It can be seen from Fig. 7 b that the nonlinear-based reduced order model is more accurate in terms of measure 46 than its linear counterpart as the size of the initial condition is increased. The initial conditions chosen for the simulations are on the true invariant manifold in 45 at time t=0. The projection of the invariant manifold on the y m,y s configuration space at different times during the normalized excitation period is shown in Fig. 8. For another parameter set p 1=1, p =0.7, k =1.0, =1, =, the transformed nonlinear system takes the form ÿ m ÿ s y m y s f 1 y m,y s,ẏ m,ẏ s =0 f y m,y s,ẏ m,ẏ s,t 47 with the master-slave frequencies as m =1.930 and s = N Their ratio of 1.06 is in the vicinity of 1:1 true internal resonance. The resonance effects can be seen even in the vicinity of the true internal resonance as verified from the trajectories of the actual system and those reconstructed using linear-based and nonlinearbased order reductions in Fig. 9 a. Figure 9 b shows the integral square error for the linear- and nonlinear-based reduced models and it can be seen that even in the vicinity of the resonance condition, the linear-based reduced model is marginally better than the nonlinear-based reduced model. For a third parameter set p 1=1,p =0.7,k =1.31, =, the transformed nonlinear system is given as ÿ m ÿ s y m y s f 1 y m,y s,ẏ m,ẏ s =0 f y m,y s,ẏ m,ẏ s,t 48 and the parametric frequency in the transformed domain is again normalized to. The eigenvalues of the time invariant part of system 48 are given as ±0.5381i and ±1.551i. Hence, the system is close to true combination resonance since = Figure 10 a shows the comparison between the actual trajectory of the full model and those reconstructed using linear-based and nonlinear-based reduced order models, respectively. It can be confirmed from Fig. 10 b, which plots measure 46 versus the size of the initial condition, that the linear-based reduced model does achieve more accurate results than the nonlinear-based reduced order model. 7 Conclusions A technique for obtaining reduced order models of linear and nonlinear time-periodic parametrically excited structural systems in second order form is proposed. The L-F transformation converts the linear part of the system into a time-invariant form. The subsequent Guyan-type order reduction preserves the eigenstructure of the undamped system. The technique for nonlinear systems is an extension of the existing invariant manifold-based order reduction of nonlinear systems that has been applied in the literature 466 / Vol. 18, AUGUST 006 Transactions of the ASME

10 Fig. 9 a Comparison of the actual, nonlinear reduced and linear reduced model trajectories for double inverted pendulum in 1:1 true internal resonance. b Integral square error for gradually increasing initial conditions for double inverted pendulum in 1:1 true internal resonance. Fig. 10 a Comparison of the actual, nonlinear reduced, and linear reduced model trajectories for double inverted pendulum in true combination resonance. b Integral square error for gradually increasing initial conditions for double inverted pendulum in true combination resonance. for time-invariant nonlinear structural systems. The systems considered are not limited by the norm or the strength of the periodic terms. The known resonance conditions for the failure of the nonlinear reduced model are also derived by a novel approach. The true internal and true combination resonance conditions derived here are generalizations of those found previously for timeinvariant structural systems. It was shown that, in the vicinity of one of these resonances, the nonlinear coupling between these modes prevents the accurate isolation of one of them in the reduced model. The reduced models obtained via the proposed methodology were directly compared with linear-based reduced models and direct numerical integration of the full model in three examples. For linear systems it was shown that the reduced models in transformed or original coordinates are more accurate for light damping. In the case of nonlinear systems it was shown that the nonlinear-based reduced models are superior to the linearbased reduced models in the absence of resonance conditions, particularly as the size of the initial condition is increased. Thus, while the linear-based reduced model is immune to nonlinear resonance issues, its failure to incorporate the nonlinearities in the master-slave relation results in poor accuracy for higher amplitude oscillations, a situation for which the nonlinear-based methodology using the invariant manifolds is more suitable. Acknowledgment Financial support provided by the Air Force Office of Scientific Research under Grant No. F is gratefully acknowledged. References 1 Guyan, R. J., 1965, Reduction of Stiffness and Mass Matrices, AIAA J., 3, p Flax, A. H., 1975, Comment on Reduction of Structural Frequency Equations, AIAA J., 13 5, pp Downs, B., 1980, Accurate Reduction of Stiffness and Mass Matrices for Vibration Analysis and Rationale for Selecting Master Degrees of Freedom, ASME J. Mech. Des., 10, pp Shaw, S. W., Pierre, C., and Pesheck, E., 1999, Modal Analysis-Based Reduced-Order Models for Nonlinear Structures An Invariant Manifold Approach, Shock Vib., 31 1, pp Burton, T. D., and Rhee, W., 000, On the Reduction of Nonlinear Structural Dynamics Models, J. Vib. Control, 6, pp Peschek, E., Boivin, N., and Pierre, C., 001, Nonlinear Modal Analysis of Structural Systems Using Multi-Mode Invariant Manifolds, Nonlinear Dyn., 5, pp Shaw, S. W., and Pierre, C., 1993, Normal Modes for Nonlinear Vibratory Systems, J. Sound Vib., 164 1, pp Journal of Vibration and Acoustics AUGUST 006, Vol. 18 / 467

11 8 Sinha, S. C., Pandiyan, R., and Bibb, J. S., 1996, Liapunov-Floquet Transformation: Computation and Applications to Periodic Systems, J. Sound Vib., 118, pp Deshmukh, V., Sinha, S. C., and Joseph, P., 000, Order Reduction and Control of Parametrically Excited Dynamical Systems, J. Vib. Control, 6, pp Sinha, S. C., Redkar, S., Butcher, E., and Deshmukh, V., 003, Order Reduction of Nonlinear Time Periodic Systems Using Invariant Manifolds, Proceedings of DETC 03, DETC003/VIB-48445, CD-ROM. 11 Sinha, S. C., Redkar, S., and Butcher, E. A., 005, Order Reduction of Nonlinear Systems With Time Periodic Coefficients using Invariant Manifolds, J. Sound Vib., 84, pp Sinha, S. C., Butcher, E. A., and Dávid, A., 1998, Construction of Dynamically Equivalent Time Invariant Forms for Time Periodic Systems, Nonlinear Dyn., 16, pp Jiang, D., Pierre, C., and Shaw, S., 003, Nonlinear Normal Modes for Vibratory Systems Under Periodic Excitation, Proceedings of DETC 03, DETC003/VIB-48443, CD-ROM. 14 Yakubovich, V. A., and Starzhinskii, V. M., 1975, Linear Differential Equations With Periodic Coefficients, Wiley, New York, Parts I and II. 468 / Vol. 18, AUGUST 006 Transactions of the ASME