Effect of Mass Matrix Formulation Schemes on Dynamics of Structures

Transcription

1 Effect of Mass Matrix Formulation Schemes on Dynamics of Structures Swapan Kumar Nandi Tata Consultancy Services GEDC, 185 LR, Chennai , India Sudeep Bosu Tata Consultancy Services GEDC, 185 LR, Chennai , India Abstract Finite element modeling for dynamic analysis of large and complex structures such as ship hulls, offshore structures, aerospace structures etc, introduces a very large computational overhead due to presence of large number of elements and nodes. Proper representation of mass and inertia properties is critical for accuracy and reliability of results. Different mass representation schemes have influences on assembling time, storage space of matrices, and solution time. Errors in natural frequencies and mode shapes precipitate to significant errors in dynamic response analysis. To arrive at efficient error analysis procedures for vibration analysis, detailed studies on performance of the various mass lumping schemes are necessary. Investigations conducted in the present study are aimed at understanding the influence of the mass lumping and its distribution based on accuracy of the results of the dynamic analysis. Detailed studies have been conducted to investigate the performance of the existing methods of representation of mass that are generally employed for free vibration and dynamic response analysis of structures using finite element method (FEM). A number of configuration for free vibration analysis of beams, plates, and plates under plane stress state have been solved using different finite elements, mesh configurations schemes in ANSYS. Results of some configuration are validated with analytical solution wherever possible, results for other configuration are compared with results obtained from the computer program given by K. J. Bathe - modified to accept banded matrices apart from skyline form. The investigation also includes shape distortion and its effect on vibration characteristics. The studies show that the mass lumping scheme used has significant influence over the accuracy of the results of dynamic analysis. Introduction The main stumbling block in dynamic analysis of large structure is the computational effort that is required for their solution. The formulation and subsequently the solution of the dynamic equation greatly burden the software. Proper representation of mass and inertia properties in finite element analysis is essential for accuracy and reliability of results. Consistent mass matrix formulation is a well-established way of representing element mass properties. Such matrices have non-zero off diagonal terms and are sparsely populated. Dynamic analysis involving large number of dynamic degrees-of-freedom (dof) using consistent mass formulation will require higher computational effort. Alternately, using diagonal lumped mass formulations not only reduces the computational effort significantly but also simplifies the program coding. In this context, there is considerable scope and need to study the influence of different mass schemes/formulations on the quality of solution in vibration/dynamic analysis. Free vibration or dynamic response analysis is an order of magnitude more complex than the static analysis as it involves considerable computational effort. As, previously done with static analysis [1], attempts have been made to identify the most potential sources of error that influence the quality of solution in vibration analysis. Errors in natural frequencies and mode shapes will contribute significantly to errors in dynamic response analysis. It may be recognized that errors in the representation of mass as well as geometry and refinement of finite element mesh of the structure will significantly influence the quality or reliability of the solution. These aspects call for utmost care in modeling the structure and its associated attributes. This can be achieved to a great degree of satisfaction through reliable error analysis procedures and adaptive refinements. To arrive at efficient error analysis procedures for vibration analysis, it is necessary to study

2 the performance of mass lumping schemes in detail. Errors that significantly influence the quality of solution in vibration/dynamic analysis are identified and discussed in the paper. Different schemes that are employed to represent mass in free vibration and dynamic analysis of structures by using FEM are described. Numerical studies have been conducted on beams, plates and structural members in plane stress state to investigate the performance of different mass representation schemes. The investigations conducted in the present study are aimed at understanding the influence of the mass lumping and its distribution on the accuracy of the results of the vibration/dynamic analysis Literature Review It is well known that the accuracy of the eigenvalues and mode shapes obtained by using FEM is dependent on the quality of both the mass matrix and stiffness matrix. In other words, the finite element discretization of the problem significantly influences the quality of solution. Static analysis errors can be attributed to errors in stiffness matrix by non-satisfaction of overall equilibrium of stress in the domain. Several persons worldwide have studied methodologies of predicting errors and improving the performance of the solution in linear static analysis in detail. The other matrix that comes into play for dynamic analysis is the mass matrix. The foremost aim while assembling the mass matrix of a structure is to preserve its mass completely. It was observed by Iyer [1] that preservation of mass alone is not sufficient to ensure the quality of solution. This clearly indicates that appropriate distribution of mass has significant influence on the quality of the solution. The method employed for representing mass in finite element analysis has significant influence on the accuracy of solutions by free vibration and dynamic analysis. As mentioned earlier, use of diagonal lumped mass matrices simplifies the program coding, resulting in significant reduction in computational effort. In recent years there has been partial return to diagonal mass lumping as many investigators found that the use of consistent mass did not always lead to improved accuracy justifying the additional computational effort involved. This aspect was demonstrated by Clough [2] and Washizu [3] using simple elements for which lumping procedures are obvious. However, the extension of the lumping approximations to the higher order elements raised many questions. In this context, Key and Beisinger [4] presented a method for deriving a diagonal mass matrix from the standard consistent mass matrix for elements with linear or cubic displacement functions. Hinton, et al [5] obtained more accurate solution for plate vibration problems by adopting a mass lumping scheme based on the introduction of an area of influence for the shape functions. The effect of co-ordinate transformation on the lumped mass matrix has been demonstrated and some approximations to make the matrix diagonal without affecting the results considerably are also presented A comprehensive study has been made by N R Iyer [6] on different mass matrix representation schemes. Different Mass Formulation Schemes The governing equation of motion for a structure subjected to dynamic forces can be expressed in matrix form as. M x+ C x+ Kx = F( t) where M, is the mass matrix; C, damping matrix; and K, stiffness matrix; x, x and x are the generalized acceleration, velocity and displacement vectors, respectively; and F(t), is the applied time dependent force vector. In finite element analysis the overall matrices for the structure are formed by assembling the relevant element matrices. The stiffness matrix of an element depends on the type of element used to idealize the structure. The scope of the present investigation is limited to undamped vibration analysis and hence, details of formulation/representation of damping matrix are not reviewed here. It is well known from the literature1-6 that different methodologies exist to formulate element mass matrices. The familiar and preferred approach is to represent mass either in consistent form or in lumped form. The consistent mass matrix is constructed using the interpolation function used to describe the displacement field and is thus consistent with that of the stiffness matrix. The element mass matrix formulated in this manner generally. (1)

3 results in populated form. The lumped mass matrix is formed by lumping of mass at the nodes of the element. This produces a diagonal mass matrix. The advantage of having diagonal mass matrix in an eigenvalue analysis is the ease and reduction of computational effort as well as requirement of lesser storage space. However, it is a general belief that consistent mass matrix leads to more accurate solution. In spite of this, diagonal or lumped mass matrices are employed widely because of the lesser computational effort it requires and the use of efficient time integration schemes, such as the explicit method of analysis. Eigenvalue analysis is used to obtain the dynamic characteristics of a system in terms of its natural frequencies (eigen values) and associated free vibration modes (eigenvectors or eigenmodes). The eigenmodes may then be used to decouple the governing equation of motion for dynamic analysis. For free vibration analysis the equation reduced to, M x+ Kx = 0 (2) The equation can be simplified to the generalized eigen value equation as shown below, [ λm ] φ = 0 K (3) where λ=ω 2, the roots of the equation; vector ф, the eigenvector; and ω, a constant representing the frequency of vibration in radians/sec, also known as eigenvalue. There are n such roots, where n is the number of dof of the structure. The stiffness matrix is positive definite when the structure is constrained against all possible rigid body motions and is positive semidefinite when structure is unconstrained against some or all of the rigid body modes. If the mass matrix is not positive definite then the number of eigenvalues will be less than n. It is evident from the equation (3), that the eigens obtained will depend on the composition and distribution of the mass matrix M. This work is aimed at finding out the effect that the composition and distribution of M will have on the eigen values. Different mass matrix assembly schemes are, Consistent Mass Matrix (CMM) If the shape functions used to describe the variation of acceleration field over the element are consistent with those used to describe the displacement variation, then the corresponding mass matrix is known as the consistent mass matrix. Although, this is demonstrated for plates/shells, the procedure is general and equally applicable for other finite element types such as beams, plane stress/strain, solids, etc. By D Alembert s principle, the lateral and transverse accelerations u, v and w produce inertia forces- ρ u, ρ v and ρ w respectively, where ρ is the mass density per unit area of the plate and t b is the plate thickness. Using the principle of virtual work, the nodal forces due to inertia may be expressed as, A T Ff = N ρndaδ = MCMM δ (4) where N, is the element shape function matrix; and M CMM,the consistent mass matrix. The mass matrices can be evaluated with a suitable numerical integration rule.

4 Consistent Diagonal Lumping (CDL) A consistent diagonal mass matrix is more sophisticated than a lumped mass matrix [6]. It can be derived from consistent mass matrix. Rotary inertia is ignored. The following procedure may be applied to compute the elements of the mass matrix: (i) Compute only the diagonal coefficient associated with translational dof of the consistent mass matrix; (ii) Compute S, the total mass of the element; (iii) Compute the number D by adding the diagonal coefficients of the consistent mass matrix associated with translations; and (iv) Scale the diagonal coefficients obtained in (i) by multiplying them by the ratio S/D. This preserves the translational mass of the element. Numerical Studies A number of example problems are selected to conduct free vibration analysis by employing different mass representation schemes. The objective of this investigation is to study the influence of different mass representation schemes on the accuracy of solution. This is achieved by studying free vibration problems of beams, plates in plane stress state and flexural state with different boundary conditions and geometry. Equation (3) is solved for eigenvalues (ω in rad/sec) and corresponding eigenvectors ф by using sub-space iteration technique. Sturm sequence check is performed to ensure that no eigenvalues are missed. The overall stiffness and mass matrices are assembled in banded form. The computer program given by Bathe [7], for matrices stored in skyline form has been suitably modified to take banded matrices as input. Different finite elements have also been used in some cases. During the analysis, preservation of total mass in each element is ensured. This study is intended to get a wider understanding of the influence of mass representation on the accuracy of solution. Keeping this in view, the following example problems are selected. Cantilever beam; Simply supported beam; Fixed-fixed beam; Inplane vibrations of cantilever square plate; Inplane vibrations of clamped square plate; Cantilever plate for shape distortion test; Vibration of Beams A two noded beam element (BEAM3) is employed to conduct free vibration analysis of slender beams. A straight beam with (i) simply supported ends; (ii) fixed-free ends; and (iii) fixed-fixed ends is modelled with ten elements. The attributes used in the analysis are: span, (L) = 10.0, cross sectional area, (A) = 1.0, mass density, ρ=1.0, modulus of elasticity, E = and moment of inertia, I= (all in consistent units). The mass representation schemes used are consistent mass matrix (CMM) and consistent diagonal lumping (CDL). The results for first four modes are compared with those given by Hurty and Rubinstein1 [8] in Figure 1. It may be noted that the frequencies obtained using CMM scheme and CDL scheme are in close agreement with the exact solution and computer program. However, the frequencies obtained from computer program by CMM scheme are generally higher than the exact solution, while the frequencies obtained from computer program using CDL are lower than the exact solution.

5 Figure 1. Flexural Vibration of Beams (natural frequencies in Hz) Inplane Vibration of Plates Cantilever square plate and clamped square plate examples are considered for inplane vibration analysis using eight-noded elements (PLANE82) with different finite element meshes and mass representations. There are two inplane translational dof per node. The geometry and other related information used in the analysis is given below. Length of square plate, L = 1.0; Poisson s ratio, ν= 0.30; Thickness of plate, t b = 0.10; Mass density, ρ=1.0; Modulus of elasticity, E = 1.0 (all in consistent units). Eigenvalues for the first six modes of inplane vibration of plates are extracted. The results of natural frequencies obtained by using different mass schemes are presented in Figure 2 and Figure 3 for cantilever square plate and clamped square plate, respectively. It may be noted from the results that, as expected, the natural frequency for mode pair (1, 2), that has symmetry, anti-symmetry, is the lowest for the clamped square plate problem. It can be seen from Figure 2 and Figure 3 that the frequencies obtained using CMM are higher, whereas frequencies obtained using CDL scheme are lower.

6 Figure 2. Inplane Vibration of Cantilever Square Plate (natural frequencies in Hz) Transverse Vibration of Plates Eight-noded plate elements (SHELL93) have been used to conduct free vibration analysis of plates of different shapes and boundary conditions. Shape distortion tests have also been carried out for different mass formulation schemes. For checking the performance of eight-noded elements with shape distortion, an example problem of cantilever square plate is chosen. Four types of mesh configurations with 2 2 idealization are used for the free vibration analysis. The finite element meshes are shown in Figure 1. The first four natural frequencies are computed. Comparison of the results obtained by employing different mass representation schemes in the present study are made by computing the percentage error with reference to the corresponding values obtained from Mesh I. These are presented in Figure 4.

7 Figure 3. Inplane Vibration of Clamped Square Plate (natural frequencies in Hz) Figure 4. Errors in the Natural Frequencies of Cantilever Plate

8 Figure 5. Distorted Mesh Configuration for Cantilever Square Plate Results From Figure 1, it can be observed that for flexural vibration of beams the error in the frequency obtained by using CMM and CDL schemes in ANSYS, with respect to the exact solution, is more when compared to results from computer program; whereas the results of CDL from ANSYS are more close to exact solution than CMM scheme from ANSYS. In case of inplane vibration of plates with different boundary conditions and with different meshes, it can be seen from Figures 2 and 3, that the frequencies obtained using CMM are higher as compared to mass lumping scheme. Similar trend is also found on results from computer program. Based on the studies carried out for transverse vibration with 2 2 distorted meshes for cantilever square plate, it is observed that the errors in the frequency are higher for solution obtained by using CMM scheme. A critical look at the performance of different mass representation schemes for different class of problems from Figure 2 to Figure 4 shows that the natural frequencies computed by using consistent mass without rotary inertia are higher than those obtained using lump mass representation scheme for both ANSYS and the computer program. However similar trend is not found in results from ANSYS for beams as shown in Figure 1, though results from computer program indicates the same trend as found in Figure 2 to Figure 4. Thus, for plate the consistent mass scheme and mass lumping scheme seem to form a band around the exact solution.

9 Conclusion Different mass representation schemes used in the finite element vibration analysis are described. These are, consistent mass formulation without rotary inertia and consistent diagonal lumping. It is observed that different mass representation schemes result in significant variations in the frequencies for a given stiffness formulation. A number of examples for free vibration analysis of beams, plates and plates under plane stress state have been solved using different finite elements, mesh configurations and boundary conditions employing different mass formulation schemes. The investigation also includes shape distortion tests. A useful and important observation is that although preservation of mass in any scheme is ensured in all the elements, the natural frequencies obtained are different. It may also be noted that there is a general consistency among the errors in the results obtained using each mass representation scheme. A careful study of the results indicates that the choice of mass representation scheme influences the quality of solution. The use of Consistent Diagonal Lumping (CDL) for plate elements results in the lowest eigenvalues as compared to Consistent Mass Matrix (CMM). On the other hand, among all other mass formulation schemes, the use of consistent mass formulation without rotary inertia yields highest eigenvalues for the same problem. Thus, the two mass formulation schemes mentioned above form a band around the exact solution. Prima facie, it would appear that consistent mass matrix scheme (CMM) is preferable over other scheme. This is because for plate frequencies from CMM scheme of ANSYS are very close to frequencies of CMM scheme of computer program; where as frequencies of CDL scheme of ANSYS are lower than frequencies of CDL scheme of computer program. References 1 R Iyer (1993) Error Estimation and Adaptive Refinements for Finite Element Analysis of Structures. Ph D Thesis, Indian Institute of Science, Bangalore. 2 R W Clough, R H Gallagher, et al (eds) (1971) Analysis of Structural Vibration and Response. Recent Advances in Matrix Methods of Structural Analysis and Design, Alabama Press, pp K Washizu (1971), R H Gallagher, et al (eds) (1971) Some Remarks on Basic Theory for Finite Element Method. Recent Advances in Matrix Methods of Structural Analysis and Design. Alabama Press, pp S W Key and Z E Beisinger (1971) Transient Dynamic Analysis of Thin Shells by Finite Element Methods. Proceedings of Third Conference Matrix Methods in Structural Analysis, Wright-Patterson Air Force Base, Ohio. 5 E Hinton, T A Rock and O C Zienkiewicz (1976) A Note on Mass Lumping and Related Process in FEM. International Journal of Earthquake Eng and Structural Dynamics, 4, pp N R Iyer, G S Pilani, and T V S R A Rao (2003) Influence of Mass Representation Schemes on Vibration Characteristics of Structures. The Institute of Engineers (India), Technical Journals: Aerospace Engineering, 84, pp K J Bathe (1990) Finite Element Procedures in Engineering Analysis. Prentice-Hall of India Pvt Ltd, New Delhi. 8 Hurty and Rubinstein (1967) Dynamics of Structures. Prentice-Hall of India Pvt Ltd, New Delhi.