Numerical Study on the Effect of Gravity on Modal Analysis of Thin-Walled Structures

Summary Computational modal analysis is usually carried out without consideration of gravity forces. This is well motivated for many structures. However, the vibrational properties of thin-walled plane or shallow shell structures are very sensitive with respect to small modiﬁcations of the shell geometry and with respect to in-plane stress and reinforcement. One reason for these in-plane stresses and geometric modiﬁcations consists in gravity. This becomes an important issue when using test data obtained under the presence of gravity to update a simulation model where the inﬂuence of gravity is neglected. This study investigates the inﬂuence of gravity on the modal parameters of a thin rectangular plate and of a thin-walled cubic box. For that, di ﬀ erent simulation models are used. While all of them utilize the ﬁnite element method, linear and non-linear approaches are compared. The latter take into account geometrical nonlinearities due to large deformations and the inﬂuence of gravity induced stress.


Introduction
Experimental modal analysis (EMA)isapowerful tool to compare and update simulation models that describe the dynamic behavior of as tructure [1]. EMA can be performed on assemblies, as well as on as ingle component of that assembly.H owever,w hen comparing the experimental data with theoretical results, the boundary conditions of the examined structure must be the same in the experiment and in the simulation model. In most cases the structure is measured in so called free-free conditions, because these conditions are supposed to be easily realizable in practice [2]. "Free-free" means that "the test object is not attached to the ground at anyofits coordinates and is, in effect, freely suspended in space" [3].
Regarding modal testing, "free-free" also means, that there are no other loads acting on the structure during the test, except the measured excitation forces. Usually it is impossible to perfectly match these conditions, because the test object must be suspended in some wayand gravity is normally present during the test 1 .Itcan be shown that if the object is suspended in soft springs in such aw ay that Received13October 2017, accepted 9April 2019. 1 An oteworthye xception is giveni n [ 4], reporting of EMA being performed on aspace shuttle in orbit. The results are compared to measure-the highest rigid body mode is fivet ot en times smaller than the lowest structural mode, the influence of the added stiffness of the springs on the structural modes can be neglected [3].
It is, however, not only the added stiffness of the suspension that might lead to ad eviation between experimental data with assumed (but not fully achieved) free-free conditions and as imulation model perfectly matching these boundary conditions. The force equilibrium between the weight of the test object and the suspension causes stress, and geometrical deformation, that might change the modal parameters.
Especially thin beams [5], shells [6] and plates [7,8,9] are known to behave very sensitive to in-plane stress and changes in the curvature, causing the lateral stiffness to change significantly.T he interdependence between inplane stress and lateral stiffness in plate-likes tructures plays an important role when calculating the ultimate strength and the buckling or collapsing modes in civil engineering [10], ships [11] and aircraft or submarine fuselages [12,13,14].
In manyc ases, the effect of gravity on the vibrational behavior of structures, however, is ignored. Exceptions can be found amongst objects designed to be deployed into ments on the ground. Regarding the influence of gravity on the natural frequencies similar conclusions are drawn likeinthe present article. space, which require intensive testing and computations before being launched. Foro bvious reasons, most of the tests need to be done on the ground, where gravity is naturally present. References [15,16,5] describe the influence of gravity on beam-likes tructures, while reference [17] deals with acantileverplate.
The goal of the current paper is to drawattention to the fact that the presence of gravity might have an important influence on the modal parameters of thin-walled structures. As will be shown, the influence of gravity on thin plates might cause in-plane stress as well as large statical deformation, which leads to strong nonlinear geometrical effects. These effects need to be accounted for in the theoretical model in order to produce correct results. As ac onsequence the orientation of at est object should be considered carefully in the case of EMA, and it might become necessary to consider the presence of gravity in the simulation model. When doing so, one should be aware that gravity might introduce astrong geometric nonlinearity which is best addressed by an onlinear simulation approach. These findings are actually not new, butare tended to be ignored in practice. This paper is intended to increase awareness and to showt he significant effect that gravity can have on the modal parameters of thin walled structures.
The goal of the current paper is not to discuss the mathematical foundations to consider in-plane stress in amodal analysis nor to investigate to what extent changes of the modal parameters are caused by stretching the mid-plane surface or by changes of the curvature. Mathematical foundations for in-plane stress can be found in [18,19].
This paper is organized such that the effect of gravity is demonstrated for athin plate under different support conditions which is followed by investigation of the effect for ac ubic box. The research is purely numerical using the commercial FEM-code Abaqus as asimulation tool.

Approach and Methods
Throughout the manuscript, the authors will refer to three different computational approaches, as summarized in Table I. Approach 1c ompletely neglects gravity,w hich results in astandard modal analysis. Approaches 2and 3consider the influence of gravity by performing atwo-stage simulation. In both approaches, step one calculates the stress and strain distribution due to the presence of gravity,while step twoconsist of amodal analysis. While both approaches assume linear elastic material behavior,theyvery much differ in the waytheytreat geometric nonlinearity in step one and in the waytheyconsider gravity induced stress when performing the modal analysis in step two: • Approach 2u ses al inear calculation to derive the deformation due to gravity in step one. In step two, the modal analysis is performed using the deformed geometry of step one, butwithout considering gravity induced stress.
• Approach 3c alculates the deformation and stress due to gravity in an onlinear computation, taking into account geometric nonlinearities. The modal parameters are derivedinapre-stressed modal analysis, taking into account the gravity-induced deformation and stress, as will be explained in what follows. The choice of the three different approaches is motivated by typical options of howt od eal with the presence of gravity during modal testing. The seemingly easiest way is to simply neglect gravity,w hich is represented by Approach 1. Approaches 2and 3reflect the typical situation for industrial engineers: Commercial FE-Software is usually sold on amodular basis, dividing linear and nonlinear approaches into different software modules. Without al icense for anonlinear solution, auser is unable to perform ap re-stressed modal analysis, thereby forced to neglect the influence of gravity-induced stress on the modal parameters. Gravity-induced deformation, however, can still be considered by performing the modal analysis using the deformed geometry calculated in aprevious (linear)s imulation. The twoapproaches lead to significantly different results, as will be explained in Chapter 3.

Approach 1
When ignoring gravity,t he eigenvalues λ and the mode shapes ϕ are computed by solving the characteristic polynomial equation wherein M is the mass matrix and K is the stiffness matrix derivedinthe undeformed configuration, meaning without the presence of gravity.

Approach 2
Approach 2calculates in step one the strain and stress distribution via as tatic, linear analysis by solving the equation of the force equilibrium between the external loads and the restoring forces, F contains the known external loads, meaning in the current case the gravity force in the form of nodal forces, while u are the unknown nodal displacements. In alinear analysis, stiffness matrix K is derivedinundeformed configuration and is not updated when the structure deforms due to external loads. Since the lateral stiffness of thin plates is known to increase significantly due to changes in the curvature, the stiffness derivedinalinear analysis will always be too lowifthe plate is subjected to alateral load. As shall be seen in the following section of the paper,the resulting error in deformation due to gravity can become very large, depending on the orientation of the plate. At the end of step one, the deformed geometry of the structure is savedand imported in anew simulation to perform am odal analysis (step two).T his means, that the stiffness matrix K in Equation (1) is derivedfrom the deformed geometry due to gravity,b ut the gravity-induced Step 1: evaluation of stress and strain due to gravity -b ased on linear theory of small deformations based on nonlinear theory of large deformations Step 2: modal analysis using undeformed (plain)geometry using deformed geometry from 1, butneglecting gravity induced stress using deformed geometry from 1, also considering stress distribution from step 1 stress is ignored in the modal analysis. Since linear elastic material behavior is assumed, K is symmetric and solution of Equation (1) is straightforward.

Approach 3
In contrast to Approach 2, Approach 3c alculates in step one the strain and stress distribution via astatic, nonlinear analysis based on theory of large deformations.
ABAQUS solves the equations of the force equilibrium by an incremental-iterative procedure with the Newton-Raphson method, as described program-specifically in [20] and in general terms, amongst others, in [21,22]. Stiffness matrix K is replaced by as o-called tangential stiffness matrix, or Jacobi matrix, K T , in which the linear-elastic stiffness matrix K is modified by several additions, likeinthe current case the stress dependent geometric stiffness matrix K σ (sometimes also called load stiffness matrix)and amatrix K v which contains contributions due to large deformations. Except for K,all parts of K T are recalculated in every increment of the procedure, thereby taking into account the increase in stiffness due to changes in the curvature and due to gravity induced stress.
In step two, as o-called pre-stressed modal analysis is performed, using the modified stiffness matrix from step one. This means that the influence on the modal parameters of both, gravity induced deformation as well as changes in the stiffness due to gravity induced stress, are considered. Av ivid example of the procedure is giveni n [23], also including experimental studies.
Since the geometric stiffness matrix typically is unsymmetrical, solving Equation (1) nowr equires more effort. One possibility is to transform the equation of motion from the configuration space into as tate-space representation and using acomplexeigensolver.ABAQUSapplies acommon alternative by using as ubspace projection method and the QZ-algorithm in order to derive the natural frequencies and mode shapes as described, amongst others, in [24].

Survey
The first group of test examples uses asquare plate under different support conditions. These test examples encompass aplate which is simply supported at all edges, aplate which is horizontally softly suspended to approach the free-free condition and ap late which is vertically softly suspended, also to approach free-free conditions. These test cases are summarized in Figure 1. The considered plate is quadrangular with an edge length of 1ma nd a thickness of 1m m. The material is supposed to be steel with aY oung'smodulus E of 2.1e11 N/m 2 ,adensity of 7850 kg/m 3 and aPoisson ratio µ of 0.3.

Simply supported plate
In the first test example, all edges of the plate are rigidly supported, meaning that theycannot undergo translational displacements, butare free to rotate. The natural frequencies f mn of such ap late can be calculated by using the following formula [7] wherein m and n are the numbers of half wavesi nt he x and y direction, a and b stand for the length of each side of the plate, h indicates the thickness and ρ stands for the volumetric mass density.The bending stiffness B is expressed as Form odeling the plate, elements of the type shell S8R have been used. This type of shell element is based on Mindlin plate theory and has been originally designed to account for the shear deformation behavior of thick plates.
As shall be seen in the following example, the element is also suitable to model thinner plates. It should be noted, however, that the use of S8R elements require the use of as tructured mesh, since irregular meshes convergev ery poorly because of severe transverse shear locking [20]. Providing am esh grid with an edge length of 25 mm the model has converged and the difference between the numerical and analytical results is less than one per mil within the first ten modes. Table II shows the values for the natural frequencies of the first 10 bending modes in comparison to the FE-model. Figure 2s hows the corresponding mode shapes of the first fivemodes.
The influence of gravity will cause the horizontally orientated plate to sag. Since all edges are rigidly supported, the largest displacement will occur in the middle of the plate. Avalue of 9.81 m/s 2 for gravity will cause the plate to deflect 4.5 mm in the nonlinear simulation. The use of al inear calculation leads to am uch larger deflection of 16.3 mm, which is more than 3.5 times higher.T his remarkable difference is ac onsequence of the different approaches used for the computations: In the linear simulation Approach 2the stiffness matrix is derivedfrom the undeformed geometry before applying the load. The deformed configuration is derivedaccording to Equation (2) by calculating the equilibrium between the external loads (here: gravity)and the restoring forces.
In the nonlinear approach (Approach 3) the load is applied in small increments and the stiffness matrix is recalculated iteratively,bythis taking into account the changing of the geometry and the influence of the gravity-induced stress. Since the lateral stiffness of athin plate is very low, it will undergo acomparatively large deformation if alateral force is applied. The large deformation renders ageometrical nonlinearity which needs to be accounted for in the model. Figure 3shows the deformation and stress distribution of the rigidly supported plate under the influence of gravity in alinear and in anonlinear simulation.
When performing the modal analysis, in the nonlinear approach the updated system matrices are used, by this   taking into account the deformed geometry.The stiffening effect due to gravity induced in-plane stress is accounted for by adding aload-stiffness matrix, as describe above.In the linear approach, the modal analysis is also performed on the deformed geometry,but the stress caused by gravity is ignored. Table II shows the influence of gravity on the first 10 natural frequencies of the plate and also ac omparison of the nonlinear and the linear approach. One can state that taking gravity into account leads to as ignificant increase of the natural frequencies, while in the linear approach that increase is even larger than in the nonlinear approach. In addition to the significantly changes in the natural frequencies, Figure 2shows that there are also changes in the mode shapes.
At this point, an engineer might be interested in knowing if the significant difference between Approaches 2and 3w ill lessen with increasing thickness of the plate. Or, more generally speaking: Is there ac ertain thickness at which the influence of gravity can be completely neglected (Approach 1)?T oa ddress this topic, ap arametric study has been conducted, in which the thickness of the plate varies between 1and 5mm. Figure 4compares the difference of the natural frequencies computed by Approaches 2 and 3towards the analytical solution which completely neglects gravity.The complete results of the parametric study are giveninT ables III-IX.
The results of the parametric study,represented in Figure 4, can be summarized as follows: • The influence of gravity on the modal parameters strongly depends on the thickness of the plate. In the current example, the influence of gravity is significant and cannot be ignored when the plate thickness lies be-low3m m. Above at hickness of 4m m, the difference towards an approach completely neglecting gravity is less than one percent -a nerror which is acceptable in most real world simulations. • Approaches 2and 3which takeinto account gravity result in increased values of the natural frequencies. The difference towards neglecting gravity is highest in the first mode. • Approaches 2and 3showthe same qualitative behavior, meaning asignificant decrease in the influence of gravity with increasing plate thickness. The difference in the results of the twoapproaches is considerably large at a plate thickness of 1m m, butb ecomes irrelevant when the thickness is larger than 2mm. The afore mentioned results of the parametric study are valid for the givena pplication example of ar ectangular, simply supported plate. It might be reasonable to assume, that the influence of gravity on modal parameters will generally decrease with increasing thickness of (plate-like) structures. One should not, however, transfer the above mentioned absolute thickness values, at which gravity can be ignored, towards other problems without further considerations.
The current paper addresses the influence of gravity on ap urely numerical level. Av alidated method to provide simply supported boundary conditions in an experimental setup is provided in [25]. Figure 1( center)s hows the test example that will be discussed next: The plate is suspended in ah orizontal position with four soft springs attached to the four corners. The stiffness of the springs in the model is chosen to be 1e-3 N/mm, the transverse rigidity of the springs is set   to 1e-4 N/mm. Of course, this is an unrealistically low stiffness value that cannot be realized in an experiment in the real world, since it would require the use of strings that are capable of providing an elongation of approximately 19.5 meters under the givenm ass of the plate of 7.8 kg. The reason behind choosing such alow stiffness is to showthat, even if the influence from the added stiffness of the suspension on the modal parameters can be ignored, gravity-induced stress and deformation still might lead to large changes. First, the natural frequencies are calculated in real freefree conditions, so without the springs and without the influence of gravity.T hen, the springs are added, still not taking into account gravity.The natural frequencies are reported in Table III, while Figure 5shows the corresponding mode shapes.

Horizontally suspended plate
As can be seen, the very soft springs do not influence the natural frequencies of the first 16 modes. The first six modes of the plate with the soft springs can still be treated as rigid body modes, and the ratio between the lowest structural mode and the highest rigid body mode is larger than fiveoreventen.
With springs only in the four corners, the influence of gravity is much stronger than with rigidly supported edges as boundary conditions. In the nonlinear simulation the lateral deflection is 52 mm, measured as the distance between the corners and the middle of the plate. In the linear approach the deflection is 102 mm, so almost twice as much. Figure 6shows the deformation and stress distribution under the influence of gravity.
When discussing the influence of gravity on the modal parameters, unliket he case of rigidly supported edges, there are different results in principal between the linear and the nonlinear approach. While the linear approach, likeb efore, leads to ag eneral increase in the natural frequencies due to gravity,t he nonlinear approach shows something different. Here, afrequencydrop in the natural frequencies of the first twostructural modes (modes number sevena nd eight)c an be observed. Actually,t he nonlinear approach renders an anomaly: The simulation results in anegative eigenvalue for what would normally be mode number eight (marked with an asterisk), indicating that the system is not stable. To further investigate this phenomenom, in addition to the gravity load, alateral force of one Newton wasapplied to the middle of one of the plate's sides, as indicated in Figure 7. This extra force causes the plate to takeo nad i ff erent deformed configuration -d i ff erent from what could be seen in Figure 6. Interestingly,t he plate stays in that configuration, even after the extra force has been removeda gain. Obviously,t he gravity induced in-plane stress causes ab uckling problem, or more general, it renders an elastic instability.This instability allows the plate to takeonone of three possible configurations when gravity is applied. Asmall lateral force -ormore generally an infinitesimal disturbance -c an cause the plate to change between these configurations. Since the plate stays in that configuration even after the extra force is removed, some authors do not speak of an instability,b ut call it a neutral equilibrium [26], meaning that ap late in neutral equilibrium is neither stable nor unstable.

Ve rtically suspended plate
In another configuration the plate is hung up vertically with soft springs in the upper twocorners, likeindicated in     Before considering the influence of gravity,t he first thing to notice when looking at the results for the natural frequencies is that the suspending springs again do not influence the structural modes of the plate. Gravity is now acting in the in-plane direction of the plate, and as can be seen, the influence is nowmuch less severe than in the horizontal position. Concerning the linear approach, there is in fact no noticeable influence, respecting the givenprecision. This is due to the much higher in-plane stiffness of the plate compared to its lateral stiffness, thus leading to much smaller deformations. In fact, the deformations are    so small that the linear and the nonlinear approach produce the same results concerning deformation and stress distribution. This is very interesting, because it means that the differences in the natural frequencies between the linear   Another fact worth noticing with regard to the orientation of the plate is the different influence of gravity on the double modes. When the plate is horizontally orientated, the presence of gravity leads to astress distribution of cyclic symmetry,thereby influencing both modes of the corresponding couple in the same way. Consequentially, the natural frequencies and mode shapes are changed by the same amount when compared to the case in which gravity is not present. If, however, the plate is hung up vertically,the stress distribution resulting from gravity shows merely one axis of symmetry,thereby splitting up the former double mode. The effect can only be seen in the nonlinear approach, and even there the influence on the natural frequencies and mode shapes is rather small. To give an umerical example: Modes number 10 and 11 in Table IV represent ad ouble mode with an atural frequency of 8.7 Hz. The presence of gravity leavesthe frequencyof mode 10 unchanged (within the givenp recision)b ut increases the frequencyofmode 11 to 9.2 Hz.
According to the results above,i tm ight be reasonable to hang up thin-walled plates vertically when performing an EMA (liked one i.e. in [27] and [28]), if gravity is intended to be ignored in the simulation model. By doing this, the influence of gravity on the modal parameters is much smaller and the setup is closer to free-free conditions.

Investigation of ac ubic box
The last example consists of ah ollowc ube with an edge length of 1ma nd aw all-thickness of 1m m. To prevent   too manymultiple modes, one edge has been modified according to Figure 10 to slightly break up symmetry.T he cube is suspended in soft springs at the four upper corners. Since the cube is approximately six times heavier than the plate in the previous examples, the spring stiffness has been modified to 6e-3 N/mm and the transverse rigidity to 6e-4 N/mm accordingly.
The simulation results are summed up in Table V Figure 12. First, disregarding gravity,itcan be stated that the added stiffness of the suspending springs does not influence the structural modes. Changes in the natural frequencies or in the mode shapes can therefore be attributed to gravity introduced stress.
Under the influence of gravity the natural frequencies of the structural modes rise in general, while the effect is stronger in the linear approach than in the nonlinear approach. With regard to the effect of gravity on the mode shapes, the following applies: If one considers the cubic box as an assembly of combined plates, the effect of gravity on plates that are orientated vertically is different from those in horizontal orientation. The plate at the top and at the bottom of the box experience, generally speaking, al arger amount of gravity induced stress. This has as ignificant influence on the mode shapes, as it can be interpreted as af urther breakup of the structure'ss ymmetry. As ac onsequence, mode shapes are distorted when compared to the simulation ignoring gravity,m ode switches occur and even newmode shapes appear.For example, Approaches 2a nd 3, accounting for the presence of gravity, showinmode number 10 amode shape that does not exist in the simulation results of Approach 1which neglects the effects of gravity.

Conclusions
According to the results presented above,t he following conclusions for thin-walled structures exposed to gravity can be drawn: 1. It is well known, that the presence of gravity plays an important role on the modal parameters, likethe natural frequencies and even the mode shapes. Forv ery thin plates, gravity induced in-plane stress might even lead to elastic instability. 2. Approaches taking into account gravity can result in increased as well as decreased values of the natural frequencies compared to approaches neglecting gravity. The difference can be significantly large. In the current application examples, differences of up to 285 percent could be observed. The influence of gravity,h owever, rapidly decreases with increased plate thickness. 3. In plate-likestructures, the effect of gravity depends on the orientation of the structure. In order to being closer to free-free conditions in the experimental setup, it is reasonable to hang up the test object in such aw ay, that gravity acts in-plane. By doing this, the influence of gravity on the modal parameters is much smaller. 4. In general, it is agood idea to consider gravity and the orientation of the test object in the theoretical model to be in accordance with the experimental setup. 5. Gravity can cause large deflections in thin-walled structures, rendering as trong geometric nonlinearity,c ausing linear and nonlinear simulations to produce very different results.