Prestressed Modal Analysis Using Finite Element Package ANSYS R. Bedri and M.O. Al-Nais College of Technology at Hail, P.O. Box. 1690 Hail, Saudi Arabia Tel: +966 531 7705 ext 240 Fax: +966 531 7704 r
[email protected] Abstract. It is customary to perform modal analysis on mechanical systems without due regards to their stress state. This approach is of course well accepted in general but can prove inadequate when dealing with cases like spinning blade turbines or stretched strings, to name but these two examples. It is believed that the stress stiffening can change the response frequencies of a system which impacts both modal and transient dynamic responses of the system. This is explained by the fact that the stress state would influence the values of the stiffness matrix. Some other examples can be inspired directly from our daily life, i.e., nay guitar player or pianist would explain that tuning of his playing instrument is intimately related to the amount of tension put on its cords. It is also expected that the same bridge would have different dynamic responses at night and day in places where daily temperature fluctuations are severe. These issues are unfortunately no sufficiently well addressed in vibration textbooks when not totally ignored. In this contribution, it is intended to investigate the effect of prestress on the vibration behavior of simple structures using finite element package ANSYS. This is achieved by first performing a structural analysis on a loaded structure then make us of the resulting stress field to proceed on a modal analysis. Keywords: Pre-stress, Modal analysis, Vibrations, Finite elements, ANSYS.
1
Scope
In this investigation, we are concerned by the effect of pressure loads on the dynamic response of shell structures. A modal analysis is first undertaken to ascertain for the eigen-solutions for an unloaded annulus shell using a commercial finite element package ANSYS ([1]). In the second phase, a structural analysis is performed on the shell. Different pressure loads are applied and the resulting stress and strain fields are determined. Z. Li et al. (Eds.): NAA 2004, LNCS 3401, pp. 171–178, 2005. c Springer-Verlag Berlin Heidelberg 2005
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In the third phase, these fields are then used as pre-stress and new modal analyses are performed on the pre-loaded shell. The details on geometry, boundary conditions, and loading conditions are depicted in the procedure section.
2
Theory
The equation of motion ([3]) for a body is given in tensorial notation by ∇.σ + f = ρ
∂2u ∂t2
(1)
where σ represents the second order stress tensor, f the body force vector, ρ the density and u the displacement field. Expressed in indicial notation (1) can be recast as σji,j + fi = ρi,tt
(2)
From the theory of elasticity, we know that the generalized Hookes law relates the nine components of stress to the nine components of strain by the linear relation: σij = cijkl ekl (3) where ekl are the infinitesimal strain components, σij are the Cauchy stress components and cijkl are the material parameters. Furthermore, for an isotropic material ([3]) (3) simplifies to σij = 2µeij + λδij ekk
(4)
where µ and λ are the so called Lame constants. For boundary-value problems of the first kind, it is convenient and customary to recast (1) in terms of the displacement field u ¯, amenable to finite element treatment. u (5) (λ + µ)grad(divu) + µ∇2 u + f = ρ¨ These equations are the so called Navier equations of motion ([3]). For the 3-D elasticity problem, this equation becomes an elliptic boundary problem. We can recall that the it is possible to find a weak form or a Galerkin form ([4]) i.e., L(u, v) = (f, v) instead of Lu = f (6) where Lu = f is the generalization of the differential equation and L is a linear operator and (,) stands for the dot product. The finite element solution: the differential equation is discretized into a series of finite element equations that form a system of algebraic equations to be solved: [K]{u} = {F }
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where [K] is the stiffness matrix, {u} is the nodal displacement vector and {F } is the applied load vector. These equations are solved in ANSYS ([1]) either by the method of Frontal solver or by the method of Conjugate gradient solver. Modal analysis consists in solving an associated eigenvalue problem in the form [k] − ω ¯ 2 [M ] {u} = {0} (7) where [K] is the stiffness matrix and [M ] is the consistent mass matrix that is obtained by [M ] = ρ[N ]T [N ]dv (8) v
[N ] being the shape functions matrix. For the prestressed modal analysis, the stiffness matrix [K] is being corrected to take into account the stress field.
3
Procedure
In the preprocessor of ANSYS ([1]) geometric modelling of our eigenvalue problem (modal analysis) and then of our boundary value problem (static analysis) is being defined: an annulus with internal radius r1 = 0.5m and external radius r2 = 0.8m. A corresponding finite element model is obtained by meshing the geometric model using 60 elements. Element type chosen: ANSYS shell 63 see fig.(3) in appendix for ample description. Thickness:0.003m Elastic properties: Youngs modulus of elasticity: 193 GPa Poissons ratio:0.29 Material density: 8030 kg/m3 Constraints: mixed type boundary conditions For r = r1 , the six translations and rotations are being set to zero, i.e. u ¯ = {0, 0, 0, 0, 0, 0}. 3.1
Modal Analysis: Stress Free Modal Analysis
In the solution processor of ANSYS, modal analysis type is first chosen with Lanczos ([9]) extraction and expansion method. The eigen solutions obtained are analyzed and presented in the general postprocessor. The first five modes of vibration are tabulated in tables 1 and 2. The mode shapes are also included in the appendix. 3.2
Static Analysis
Static type analysis is now selected. Ten different sets of pressure loads are being applied to the annulus on its outer boundary i.e., r = r2 . Details can be seen on
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tables 1 and 2. For each loading case, prestress effect is being activated in the analysis option of the program. The resulting stress field is then applied when it comes to performing subsequently modal analysis on the annulus. 3.3
Modal Analysis with Prestress Effect
Once the stress field is being established from the above static analysis, it is applied as prestress to the shell structure through the activation of this option in the subsequent modal analysis. This procedure is reproduced for the twenty different preloading cases.
4
Results
The results of the different analyses i.e., modal analysis of the stress free annulus the static analysis and then the modal analysis of the preloaded structure, are all summarized and displayed in tabular form see tables 1 and 2 in the appendix. To ascertain the effect of the prestress level on the modes of vibration, some further calculations are done and presented in tables 3 and 4 in the appendix. Plots of prestress level versus percent increase or decrease in frequencies are plotted respectively in figures 1 and 2.
5
Comments on Results
5.1. Prestress produces no effect on the mode shapes of vibration of the shell structure. 5.2. By examining the results presented in tables 1 and 2, it is evident that the frequencies are impacted by preloading. The effect of such preloading seems to be more apparent on the first modes than on the higher ones. The plotted curves of figures 1 and 2 are here to corroborate these conclusions. 5.3. A closer look at these curves discloses that there seems to be a linear correlation between the prestress level and the percent frequency increase or decrease for each mode of vibration. 5.4. Tensile preloading produces an increase in frequency whereas compressive preloading results in a decrease in frequency.
6
Conclusions
Three pieces of conclusions can be inferred from this study: 6.1. The mode shapes of vibration of the structure are not sensitive to preloading. 6.2. Prestressing seems to impact the dynamic behavior of the structure. 6.3. Tensile prestress acts as a stiffener and enhances the dynamic characteristics of the structure resulting in frequency increase. Whereas compressive prestress has a converse effect on the structure by reducing its frequencies.
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References 1. ANSYS, Users manual, revision 5.6, Swanson Analysis Inc., Nov. 1999. 2. Reddy, J.N.: An introduction to the finite element method , McGraw-Hill, New York 1984. 3. Reddy, J.N., energy and variational methods in applied mechanics, John Wiley, New York 1984. 4. Zienkiewicz, O.C., and Taylor, R.L., The finite element method , 4th ed., McGrawHill, New York, 1989. 5. Bathe, K.J.. Finite element procedures in engineering analysis, Prentice-Hall, Englewood Cliffs, N.J., 1982. 6. Cheung, Y.K. and Yeo, M. F., a practical introduction to finite element analysis, Pitman, London, 1979. 7. Rao, S.S., The finite element method in engineering, Pergamon Press, Oxford, 1982. 8. Meirovitch, L., Elements of vibration analysis, McGraw-Hill, New York, 1975. 9. Lanczos, C, the variational principles of mechanics, the university of Toronto press, Toronto, 1964. 10. Hutchinson, J.R., Axisymmetric Vibrations of a Solid Elastic Cylinder Encased in a Rigid Container, J. Acoust. Soc. Am., vol. 42, pp. 398-402, 1967. 11. Reddy, J.N., Applied Functional Analysis and Variational Methods in Engineering, McGraw-Hill, New York, 1985. 12. Sandbur, R.B., and Pister, K.S., Variational Principles for Boundary Value and Initial-Boundary Value Problems in Continuum Mechanics, Int. J. Solids Struct., vol. 7, pp. 639-654, 1971. 13. Oden, J.T., and Reddy, J.N., Variational Methods in Theoretical Mechanics, Springer-Verlag, Berlin, 1976. 14. http://caswww.colorado.edu/courses.d/AFEM.d/ 15. http://caswww.colorado.edu/courses.d/IFEM.d/ 16. Liepins, Atis A., Free vibrations of prestressed toroidal membrane, AIAA journal, vol. 3, No. 10, Oct. 1965, pp. 1924-1933. 17. M. Attaba, M.M. Abdel Wahab, Finite element stress and vibration analyses for a space telescope, ANSYS 2002 Conference, Pittsburg, Pennsylvania, USA, April 22-24, 2002. 18. Alexey I Borovkov, Alexander A Michailov, Finite element 3D structural and modal analysis of a three layered finned conical shell, ANSYS 2002 Conference, Pittsburg, Pennsylvania, USA, April 22-24, 2002. 19. Sun Libin, static and modal analysis of a telescope frame in satellite, ANSYS 2002 Conference, Pittsburg, Pennsylvania, USA, April 22-24, 2002. 20. Erke Wang, Thomas Nelson, structural dynamic capabilities of ANSYS, ANSYS 2002 Conference, Pittsburg, Pennsylvania, USA, April 22-24, 2002.
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Appendix
Table 1. The first five modes against the tensile prestress levels Mode 0 1 2 3 4 5
2.602 2.634 2.656 2.790 2.839
Tensile Prestress in N/m 103 2.103 4.103 8.103 16.103 32.103 64.103 Frequency in kHz 2.603 2.604 2.606 2.610 2.618 2.633 2.664 2.635 2.636 2.638 2.642 2.649 2.665 2.695 2.657 2.658 2.660 2.664 2.672 2.688 2.718 2.791 2.792 2.794 2.798 2.805 2.821 2.852 2.840 2.841 2.843 2.847 2.855 2.870 2.901
105 2.105 3.105 2.698 2.729 2.752 2.885 2.935
2.789 2.821 2.844 2.977 3.027
2.876 2.904 2.932 3.065 3.115
Table 2. The first five modes against the compressive prestress levels Mode 0 1 2 3 4 5
2.602 2.634 2.656 2.790 2.839
Pressure Prestress in N/m 103 2.103 4.103 8.103 16.103 32.103 64.103 Frequency in kHz 2.601 2.600 2.598 2.594 2.587 2.571 2.539 2.633 2.632 2.630 2.626 2.618 2.602 2.570 2.655 2.654 2.652 2.648 2.641 2.625 2.593 2.789 2.788 2.786 2.782 2.774 2.758 2.726 2.838 2.837 2.835 2.831 2.823 2.807 2.775
105 2.105 3.105 2.502 2.533 2.556 2.690 2.739
Fig. 1. % Frequency increase versus Prestress
2.396 2.428 2.450 2.585 2.634
2.285 2.316 2.338 2.474 2.523
Prestressed Modal Analysis Using Finite Element Package ANSYS Table 3. The percent frequency increase against prestress levels Mode
1 2 3 4 5
Tensile Prestress in N/m 103 2.103 4.103 8.103 16.103 32.103 64.103 % increase in frequency 0.038 0.077 0.154 0.307 0.615 1.191 2.383 0.038 0.076 0.152 0.304 0.569 1.177 2.316 0.038 0.075 0.151 0.301 0.602 1.205 2.184 0.036 0.072 0.143 0.287 0.538 1.111 2.222 0.035 0.070 0.141 0.282 0.563 1.092 2.184
105 2.105 3.105 3.689 3.607 3.614 3.405 3.381
7.187 7.099 7.078 6.702 6.622
10.530 10.250 10.391 9.857 9.722
Table 4. The percent frequency decrease against prestress levels Mode
1 2 3 4 5
Pressure Prestress in N/m 103 2.103 4.103 8.103 16.103 32.103 64.103 % decrease in frequency 0.038 0.077 0.154 0.307 0.576 1.191 2.421 0.038 0.076 0.152 0.304 0.607 1.215 2.430 0.038 0.075 0.151 0.301 0.565 1.167 2.372 0.036 0.072 0.143 0.287 0.573 1.147 2.294 0.035 0.070 0.141 0.282 0.563 1.127 2.254
105 2.105 3.105 3.843 3.834 3.765 3.584 3.522
7.917 7.821 7.756 7.348 7.221
Fig. 2. % Frequency Decrease versus Pressure Level
12.183 12.073 11.973 11.326 11.131
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Fig. 3
Fig. 4
