Analyzing the Finite Element Dynamics of Nonlinear In-Plane Rods by the Method of Proper Orthogonal Decomposition

Transcription

1 COMPUTATIONAL MECHANICS New Trends and Applications S. Idelsohn, E. Oñate and E. Dvorkin (Eds.) c CIMNE, Barcelona, Spain 1998 Analyzing the Finite Element Dynamics of Nonlinear In-Plane Rods by the Method of Proper Orthogonal Decomposition Ioannis T. Georgiou 1,, and Jamal Sansour 2 1 Special Project for Nonlinear Science, Code Naval Research Laboratory, Washington DC Darmstadt University of Technology, Fachbereich Mechanik, Hochschulstr. 1, Darmstadt, Germany Key words: Nonlinear rods, finite element dynamics, chaos, proper orthogonal decomposition, active degrees-of-freedom, invariant manifolds. Abstract: This study concerns the analysis of spatio/temporal solutions of the finite element projection of infinite-dimensional nonlinear dynamical systems, Cosserat continua, modeling in-plane rods. The finite element solution is analyzed by applying POD (proper orthogonal decomposition) techniques. We find that the forced regular and chaotic response of straight in-plane rods and shallow in-plane arches is dominated by a few active degrees-of-freedom. 1 1 Research Scientist, SAIC-Science Applications International Corporation, McLean, VA

2 1 Introduction Nonlinear partial differential equations in structural dynamics can be reduced to a finite set of coupled oscillators by successive projections onto the basis of functions composed of the spatial shapes of the normal modes of the l inearized system. The resulting coupled set of oscillators can be analyzed by applying the theory of normal modes of oscillation 1,2,3 and the theory of geometric singular perturbations and invariant manifolds 4,5 to determine the active degrees-of-freedom. Given the fact that normal modes and their realization as two-dimensional invariant manifolds in phase space play a fundamental role in the analysis of coupled oscillators, we would like to explore the possibility to introduce these concepts in the analysis of the finite element dynamics of highly nonlinear coupled partial differential equations describing the motions of continua in solid mechanics. Finite element methodologies can take into account almost any type of nonlinearity. This is in contrast with the modal decomposition approach where one takes into account simple nonlinearities, for instance, quadratic an cubic. The basic problem now is to extract from the finite element solution of a dynamical system, whose nature of exact nonlinearities has not been sacrificed, the essential characteristics of the dynamics such as active degreesof-freedom. And somehow bring use the notion of invariant manifolds of motion to give definite meaning to the geometric structure of active degrees-of-freedom in phase space. This work attempts to address the issue of active degrees-of-freedom of the dynamics of Cosserat continua. These continua model rods and shells by taking into account the exact nature of geometric nonlinearities. Recently finite element schemes have been developed to solve this interesting class of infinite dimensional dynamical systems 6,7,8. We develop a method based on proper orthogonal decompositions 9,10,11 to analyze the finite element dynamics of in-plane rods and arches modeled as Cosserat continua. The method identifies the active degrees-of-freedom. 2 Nonlinear Rods We are interested in analyzing the dynamics of elastic rods. A rod can be modeled as onedimensional Cosserat continuum. When restricted to move in a plane, such a continuum is characterized by the axial displacement field u 1, the transverse displacement field u 2,and the rotation field ω. These fields are measured with respect to a reference configuration B, parametrized by arch length s, with boundary B. Letα denote the angle formed by the tangent vector of the configuration B and an horizontal axis. It has been shown that the following kinematic relations 7 : U 1 = cos(ω)+cos(α + ω) u 1 s +sin(α + ω) u 2 s, 2

3 U 2 = sin(ω) sin(α + ω) u 1 s +cos(α + ω) u 2 s, K = ω (1) s provide a natural measure of strain in the continuum deformed by the distributed axial force P 1, transverse force P 2 and in-plane moment M. Letn 1, n 2,andmbethe forces and moment conjugate to the strain measures (1) in the sense that δψ int = n 1 δu 1 + n 2 δu 2 + mδk where Ψ int is the strain energy. Let T and Ψ ext denote respectively the the kinetic energy and the work of the external forces. The motion of the continuum is governed by the principle of virtual work: J δ (T Ψ int +Ψ ext ) = + ρ (Aü 1 δu 1 + Aü 2 δu 2 + I ωδω) ds B (n 1 δu 1 + n 2 δu 2 + mδk) ds B (P 1 δu 1 + P 2 δu 2 + Mδω) ds B + (P 1 δu 1 + P 2 δu 2 + Mδω) B =0, (2) where A, I denote respectively the cross-section and its second moment. Note that the external forces may depend on the velocity field to account for dissipation. We focus our attention on the class of continua with linear elastic material behavior, that is, n 1 Ψ = EA(U 1 1), U 1 n 2 Ψ = GAU 2, U 2 m Ψ = EIK (3) K where E, andg denote respectively the modulus of elasticity, and the shearing modulus of elasticity. The above linear constitutive relations introduce through the kinematic strain relations (1) the exact nature of geometric nonlinearity in the functional J. The functional J provides the most primitive description of the dynamics of the motion of the continuum. First, it can be used to derive a set of coupled partial differential 3

4 equations to be integrated to obtain the spatio-temporal dynamics of the fields u 1, u 2 and ω. These equations are nonlinear and can be tackled analytically to some extend by perturbation methods provided that the nonlinearities are weak and are approximated by the leading terms of a polynomial expansion. The classical approach is to turn the simplified nonlinear equations into a set of coupled oscillators by a modal decomposition (Galerkin projection). This approach leads to a phase space setting and one can used ideas from singular perturbations and global invariant manifolds (normal modes) to explore the dynamics of the coupled oscillators 4,13. Second, the functional J can be used to derive a finite element representation of the equations of motion. This avenue of numerical analysis avoids almost all simplifications regarding the nonlinearities in a modal decomposition. Recently, efficient finite element discretizations have been developed where the exact nature of geometric nonlinearity is kept intact 7. The trade off is now that the dynamics are available in numerical form. The advantage, however, is that no restrictions have been applied by assuming a priory knowledge of normal modes. The purpose of this work is to report some preliminary results in an ongoing research effort to develop systematic methods to analyze the finite element dynamics of nonlinear continua. In particular, we present a method to determine the active degrees-of-freedom that govern the dynamics of a continuum given a spatio/temporal solution of the motion obtained by a finite element solution scheme. The method combines tools from finite element analysis and nonlinear dynamical systems. 3 The Finite Element Dynamics Let N denote the number of modes in a finite element discretization of the continuum. A typical finite element discretization accounts to a projection of the infinite system onto a 6N-dimensional phase space. A finite element projection has the following general format: MÜ(t)+C U(t)+K(U(t)) = F(t), (4) where U denotes the spatial discretized of the vector field {u 1 (s, t),u 2 (s, t),ω(s, t)}, that is, U(t) (u 1 (t), u 2 (t),ω(t)), u 1 (t) (u 11 (t),u 12 (t),u 13 (t),...,u 1N (t)) T, u 2 (t) (u 21 (t),u 22 (t),u 23 (t),...,u 2N (t)) T, ω(t) (ω 1 (t),ω 2 (t),ω 3 (t),...,ω N (t)) T (5) 4

5 The triple {u 1k (t),u 2k (t),ω k (t)} represents the dynamics of the k-th node. Furthermore, M is the mass matrix, C is the dissipation matrix, K is a matrix-valued nonlinear function of U, and F denotes the modal forces and moments. To integrate the above equation in time, we apply an implicit mid-point integration scheme 7. Such an integration scheme casts the above equation into the following general compact format: [ 2 M ( t) U 2 ] 2 t U m + C 1 t U + K(U m U) =F m. (6) where U m+1 = U m + U, (7) U m (u 1 (t m ), u 2 (t m ),ω(t m )), t m = m t, m =1, 2,...,M. (8) The initial conditions are U 0 (u 1 (0), u 2 (0),ω(0)), U 0 ( u 1 (0), u 2 (0), ω(0)). (9) The last relation constitutes a nonlinear equation for the increment U where t is the time step. It is solved by Newtons method. From the standpoint of a nonlinear dynamical system, the above equation is a map representation of the dynamics of a discrete set of material points of the continuum. The issue here is to extract basic characteristics of the dynamics of the original infinite system (2) from its finite element dynamics generated by equation (6). 4 Proper Orthogonal Decomposition The finite element solution is not based on any priory knowledge of normal modes as is usually the case with a typical modal decomposition. Just as experimental data, the finite element dynamics contains valuable information on the temporal and spatial characteristics of the degrees-of-freedom that govern the dynamics. There is thus a natural need to analyze the finite element dynamics by applying methods that resemble spectral methods such as modal analysis of linear problems. Methods that resemble to a modal analysis are those based on proper orthogonal decompositions of data. Just as in the modal analysis of a linear partial differential equation, 5

6 we expand the semi-discretized vector filed U as follows: U(t) = k=1 λ k A k (t)φ k, Φ k (Φ 1k, Φ 2k, Φ 3k.) T (10) The difference now is that we have a set of highly nonlinear coupled partial differential equation and its finite element solutions. In contrast to the modal analysis of linear problems, we do not substitute the above expansion in the equations of motion, but we require that the modal vectors Φ k and and their companion amplitudes A k satisfy the following orthonormality relations: 1 N Φ 1 m Φ n = δ mn, A m (t)a n (t)dt = δ mn (11) T 2 T 1 T 1 where denotes the inner vector product. The above orthonormality conditions give rise to the following eigenproblem: T2 1 C(t, τ)a k (τ)dτ = λ m A k (t) (12) T 2 T 1 T 1 where the kernel C(t, τ) is identified with the temporal autocorrelation of the vector field, that is, T2 C(t, τ) 1 U(t) U(τ). (13) N Since C(t, τ) =C(τ,t), we have 0 λ k. The scalar quantity defined by 1 E T 2 T 1 is the energy of the motion; it satisfies T2 T 1 C(t, t)dt (14) K E = λ k =1. (15) k=1 Since the energy is normalized at 1, we have 0 λ k 1. Once the eigenproblem (12) has been solved, the PO modes are computed by the projection: λ k Φ k λ k (Φ 1k, Φ 2k, Φ 3k ) T = 1 T 2 T 1 T2 T 1 A k (t)u(t)dt (16) We say that mode Φ m is more energetic than mode Φ n if their corresponding energy fractions (eigenvalues) satisfy λ m >λ n. The expansion is optimal in the sense the energy fraction contained in a fixed number of modes is equal or larger than the energy contained in the same number of modes of any other basis expansion, for instance, the 6

7 Fourier modes of the linearized system. Because of this optimality, the expansion is called proper orthogonal decomposition. Historically, this expansion was introduced in statistical mechanics 10,11 and fluid mechanics 9, and also is known as the K-L expansion. It has been applied to vibro-impact problems 12. Recently, it has been combined with ideas from geometric singular perturbations, invariant manifolds, and inertial manifolds to develop a systematic study of large scale nonlinear dynamical problems in mechanics 13,14. 5 POD Analysis of In-Plane Rods A POD analysis of a typical finite element solution, that is, a discretized spatio/temporal record of data, consists of computing the discrete analogue of the autocorrelation operator C, then computing its eigenvalues λ k and its eigenvectors A k, and the PO modes using the discrete analogue of projection (16). First we perform a POD analysis of the finite element dynamics of an in-plane straight rod, that is, an in-plane elastica shown in Fig. 1. The boundary conditions are as follows: u 1 (s =0,t) = u 10, u 2 (s =0,t)=0, M(s =0,t)=0, u 1 (s = L, t) = 0, u 2 (s = L, t) =0, M(s = L, t) =0. (17) The boundary axial displacement u 10 can be controlled to cause buckling. The finite element dynamics of the rod have been explored in reference 7. The rod responds to a spatially uniform time-harmonic load with regular and chaotic motions. Figure 2 shows the Poincare section of a typical chaotic motion. The POD analysis of several periodic, quasi-periodic, and chaotic motions with both local and global attractors reveals that the dynamics are governed by a single PO mode or active degree-of-freedom. Figure 3 depicts the axial, transverse, and rotational components of this single PO mode. Note that all symmetries of the problem and boundary conditions are reflected in the symmetries of the mode. We find that the transverse component of the single PO mode coincides with the fundamental transverse spatial mode of the linearized motion. It is well known that the dynamics of a buckled in-plane elastica are governed by a single oscillator, the Duffing oscillator. This conclusion is based on the modal analysis of an integro-partial differential equation with cubic nonlinearities. Here we obtain the same result by including the whole geometric nonlinearity. However, we obtain additional information on the dynamics of the axial and rotational fields. Since the fundamental frequencies of the axial and rotational motions are higher than the fundamental frequency of the transverse motion, we have a case of soft (low frequency)- stiff (high frequency) dynamical system 4. Ideas from geometric singular perturbation theory, and invariant manifolds of morion 4 reveal that the temporal dynamics of the active degree-of-freedom take place on 7

8 a three-dimensional inertial manifold, that is, a time dependent two-dimensional invariant manifold. Physically, this means that the axial and rotational fields are slaved to the transverse field. Next we perform a POD analysis of the dynamics of an in-plane curved rod, more specifically, the shallow arch shown in Fig. 1 with boundary conditions, u 1 (s =0,t)=0, u 2 (s =0,t)=0, M(s =0,t)=0, u 1 (s = L, t) =0, u 2 (s = L, t) =0, M(s = L, t) =0. (18) The finite element dynamics of this continuum have been explored in reference 7. When a time harmonic point load is applied at its middle point, the arch responds with periodic motions and above some critical forcing amplitude and appropriate forcing frequency it experiences a period doubling cascade giving rise to a local and global chaotic attractors. Figure 4 shows a Poincare section of a typical chaotic motion. We find that regular and chaotic motions are characterized by identical PO modes. Most the energy is contained is one mode. This is true for a symmetric load and in the range of parameters we have examined. Figure 5 shows the axial, transverse, and rotational components of the dominant PO mode. Clearly the symmetries of the mode reflect the symmetry of the load as well as the symmetry of the boundary conditions. In contrast to the fact that the single PO mode of the rod coincides with the fundamental spatial mode of the linearized problem, both PO modes of the shallow arch are composed of several symmetric linear modes. This simply reflects the fact that the POD expansion is optimal. For nonsymmetric loads, The dynamics range from regular to chaotic, A POD analysis reveals that the dynamics are governed by three PO modes. Figure 6 shows the components of the dominant PO mode for a chaotic motion due to a harmonic load applied at the 7-th node. One needs several linearized modes (Fourier modes) to compose the PO modes. This again is a sign of the optimal character of the expansion. Now returning to the continuum, we conjecture that a typical motion of it is uniquely characterized by the sequence {λ k,a k (t), Φ k (s)} K k=1.givenk, the scalar λ k represents the fraction of energy contained in the k-th PO mode whose temporal dynamics are given by A k (t) and its spatial distribution is given by Φ k (s). 6 Summary One-dimensional and two-dimensional continua such as rods and shells can be modeled as Cosserat continua. This method of modeling takes into account the exact nature of geometric nonlinearities. In this work we apply the method of proper orthogonal decomposition to analyze the finite element solution of one-dimensional Cosserat continua. 8

9 The solution has been obtained by a finite element procedure and a symplectic implicit integration scheme. We demonstrated the method by analyzing the dynamics of in-plane straight rod, and an in-plane shallow arch. The dynamics of the in-plane rod, buckled or not, forced by a timeharmonic and spatially uniform transverse load, are governed by one-degree-of-freedom system. The dynamics of the shallow arch loaded at its middle point with a time-harmonic force are dominated by one-degree-of-freedom system. Whenever the load is not applied at the middle point, the dynamics are governed by three active-degrees-of-freedom. A thorough analysis of the dynamics of the in-plane Cosserat continua will be published in the near future. References [1] R. M. Rosenberg, On the Existence of Normal Mode Vibrations of Nonlinear Systems With two degrees of Freedom, Quarterly of Applied Mathematics, Vol. XXII, No. 3, , (1964). [2] S. W. Shaw and C. Pierre, Normal Modes for Nonlinear Vibratory Systems, J. Sound and Vibration, Vol. 164, pp , (1993). [3] A. F. Vakakis, L. I. Manevitch, Yu. V. Mikhlin, V. N. Pilipchuck and A. A. Zevin, Normal Modes and Localization in Nonlinear Systems, New York: Wiley Interscience, (1996). [4] I. T. Georgiou, A. K. Bajaj and M. Corless, Slow and Fast Invariant Manifolds, and Normal Modes in a Two-Degree-Of-Freedom Structural Dynamical System with Multiple Equilibrium States, Int. J. of Non-Linear Mechanics, Vol. 33, No. 2, pp , (1998) [5] I. T. Georgiou and I. B. Schwartz, I. B., 1996, The Slow Invariant Manifold of a Conservative Pendulum-Oscillator System, International Journal of Bifurcation and Chaos 6, pp , (1996). [6] C. Sansour and H. Bednarcyck, The Cosserat surface as shell model, theory and finite element formulation, Computer Methods in Applied Mechanics and Engineering 120, pp. 1-32, (1995). [7] C. Sansour, J. Sansour and Wriggers, A Finite Element Approach to the Chaotic Motion of Geometrically Exact Rods Undergoing In-Plane Deformations, Nonlinear Dynamics 11, pp , (1996). 9

10 [8] C. Sansour, P. Wriggers and J. Sansour, Nonlinear Dynamics of Shells: Theory, Finite Element Formulation, and Integration Schemes, Nonlinear Dynamics, to appear, (1997) [9] L. Sirovich, Turbulence and the Dynamics of Coherent Structures, Pt. I, Coherent Structures, Quart. Appl. Math., Vol. 45, , (1987). [10] Loève, M Probability theory. Princeton, New Jersey: D. Van Nostrand. [11] J. L. Lumley Stochastic Tools in Turbulence. Academic, [12] J. P. Cusumano, Sharkady and B. W. Kimble, Experimental Measurements of Dimensionality and Spatial Coherence in the Dynamics of a Flexible-Beam Impact Oscillator, Phil. Trans. R. Soc. Lond. A 347, , (1994). [13] I. T. Georgiou and I. B. Schwartz, Dynamics of Large Scale Coupled Structural/Mechanical Systems: A Singular Perturbation/Proper Orthogonal Decomposition Approach, submitted, (1996). [14] I. T. Georgiou and I. B. Schwartz, E. Emaci, A. Vakakis, 1997, Interaction Between Slow and Fast Oscillations in an Infinite Degree-of-Freedom Linear System Coupled to a Nonlinear Subsystem: Theory and Experiment, submitted, (1997). 10

11 Figure 1: Schematics of a straight rod and a shallow arch. The left end of the rod is kept under constant in time compression. The structural parameters for the rod are length L = 10, modulus of elasticity E = , Poison s ratio ν =0.2, density ρ = , damping d = , cross-section A =1.0. The structural parameters for the arch are L = 70, E = , ν =0.3, ρ = ,radiusR = 400, thickness t =1,and height h =

12 600 Chaotic attractor Transverse velocity Transverse displacement Figure 2: Structure of the Poincare section of the dynamics of the node at the middle point of the rod loaded as shown in Fig

13 0.50 PO mode axial transverse rotational Rod length (s) Figure 3: The three components of the single PO mode characterizing the chaotic motion whose Poincare section is shown in Fig

14 2000 Chaotic attractor Transverse velocity Transverse displacement Figure 4: Structure of the Poincare section of the dynamics of the node at the middle point of the arch loaded as shown in Fig

15 0.50 PO mode 0.25 axial transverse rotational Arch axial span (s) Figure 5: The three components of the dominant PO mode characterizing the chaotic motion whose Poincare section is shown in Fig. 4. The arch is loaded as shown in Fig

16 0.50 PO mode axial rotational transverse Arch axial span (s) Figure 6: The thee components of the most energetic PO mode characterizing the chaotic motion of the asymmetrically loaded arch. 16