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 fre- quencies 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 vibra- tion textbooks when not totally ignored. In this contribution, it is intended to investigate the effect of pre- stress 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. Dif- ferent 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
Transcript
Prestressed Modal Analysis Using FiniteElement Package ANSYS
R. Bedri and M.O. Al-Nais
College of Technology at Hail,P.O. Box. 1690 Hail, Saudi Arabia
Abstract. It is customary to perform modal analysis on mechanicalsystems without due regards to their stress state. This approach is ofcourse well accepted in general but can prove inadequate when dealingwith cases like spinning blade turbines or stretched strings, to name butthese two examples.
It is believed that the stress stiffening can change the response fre-quencies of a system which impacts both modal and transient dynamicresponses of the system. This is explained by the fact that the stressstate 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 playinginstrument is intimately related to the amount of tension put on its cords.It is also expected that the same bridge would have different dynamicresponses at night and day in places where daily temperature fluctuationsare severe.
These issues are unfortunately no sufficiently well addressed in vibra-tion textbooks when not totally ignored.
In this contribution, it is intended to investigate the effect of pre-stress on the vibration behavior of simple structures using finite elementpackage ANSYS. This is achieved by first performing a structural analysison a loaded structure then make us of the resulting stress field to proceedon a modal analysis.
In this investigation, we are concerned by the effect of pressure loads on thedynamic response of shell structures.
A modal analysis is first undertaken to ascertain for the eigen-solutions for anunloaded annulus shell using a commercial finite element package ANSYS ([1]).
In the second phase, a structural analysis is performed on the shell. Dif-ferent pressure loads are applied and the resulting stress and strain fields aredetermined.
In the third phase, these fields are then used as pre-stress and new modalanalyses are performed on the pre-loaded shell.
The details on geometry, boundary conditions, and loading conditions aredepicted 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 thedensity 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 relatesthe nine components of stress to the nine components of strain by the linearrelation:
σij = cijklekl (3)
where ekl are the infinitesimal strain components, σij are the Cauchy stresscomponents and cijkl are the material parameters.
Furthermore, for an isotropic material ([3]) (3) simplifies to
σij = 2µeij + λδijekk (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 elementtreatment.
(λ + µ)grad(divu) + µ∇2u + f = ρu (5)
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 Galerkinform ([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 linearoperator and (,) stands for the dot product.
The finite element solution: the differential equation is discretized into a seriesof finite element equations that form a system of algebraic equations to be solved:
[K]{u} = {F}
Prestressed Modal Analysis Using Finite Element Package ANSYS 173
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 bythe method of Frontal solver or by the method of Conjugate gradient solver.
Modal analysis consists in solving an associated eigenvalue problem in theform [
[k] − ω2[M ]] {u} = {0} (7)
where [K] is the stiffness matrix and [M ] is the consistent mass matrix that isobtained by
[M ] =∫
v
ρ[N ]T [N ]dv (8)
[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 prob-lem (modal analysis) and then of our boundary value problem (static analysis)is being defined: an annulus with internal radius r1 = 0.5m and external radiusr2 = 0.8m. A corresponding finite element model is obtained by meshing thegeometric model using 60 elements.
Element type chosen: ANSYS shell 63 see fig.(3) in appendix for ample descrip-tion.
Thickness:0.003mElastic properties:Youngs modulus of elasticity: 193 GPaPoissons ratio:0.29Material density: 8030 kg/m3Constraints: 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 withLanczos ([9]) extraction and expansion method. The eigen solutions obtainedare analyzed and presented in the general postprocessor. The first five modes ofvibration are tabulated in tables 1 and 2. The mode shapes are also included inthe appendix.
3.2 Static Analysis
Static type analysis is now selected. Ten different sets of pressure loads are beingapplied to the annulus on its outer boundary i.e., r = r2. Details can be seen on
174 R. Bedri and M.O. Al-Nais
tables 1 and 2. For each loading case, prestress effect is being activated in theanalysis option of the program. The resulting stress field is then applied whenit 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 isapplied as prestress to the shell structure through the activation of this optionin the subsequent modal analysis. This procedure is reproduced for the twentydifferent preloading cases.
4 Results
The results of the different analyses i.e., modal analysis of the stress free annulusthe static analysis and then the modal analysis of the preloaded structure, areall 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, somefurther calculations are done and presented in tables 3 and 4 in the appendix.
Plots of prestress level versus percent increase or decrease in frequencies areplotted respectively in figures 1 and 2.
5 Comments on Results
5.1. Prestress produces no effect on the mode shapes of vibration of the shellstructure.
5.2. By examining the results presented in tables 1 and 2, it is evident that thefrequencies are impacted by preloading. The effect of such preloading seemsto be more apparent on the first modes than on the higher ones. The plottedcurves 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 linearcorrelation between the prestress level and the percent frequency increase ordecrease for each mode of vibration.
5.4. Tensile preloading produces an increase in frequency whereas compressivepreloading 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 topreloading.
6.2. Prestressing seems to impact the dynamic behavior of the structure.6.3. Tensile prestress acts as a stiffener and enhances the dynamic character-
istics of the structure resulting in frequency increase. Whereas compressiveprestress has a converse effect on the structure by reducing its frequencies.
Prestressed Modal Analysis Using Finite Element Package ANSYS 175
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176 R. Bedri and M.O. Al-Nais
Appendix
Table 1. The first five modes against the tensile prestress levels
