Nonlinear Modeling and Identification of an Aluminum Honeycomb Panel with Multiple Bolts

This paper focuses on the nonlinear dynamics modeling and parameter identification of an Aluminum Honeycomb Panel (AHP) with multiple bolted joints. Finite element method using eight-node solid elements is exploited to model the panel and the bolted connection interface as a homogeneous, isotropic plate and as a thin layer of nonlinear elastic-plastic material, respectively. The material properties of a thin layer are defined by a bilinear elastic plastic model, which can describe the energy dissipation and softening phenomena in the bolted joints under nonlinear states. Experimental tests at low and high excitation levels are performed to reveal the dynamic characteristics of the bolted structure. In particular, the linear material parameters of the panel are identified via experimental tests at low excitation levels, whereas the nonlinear material parameters of the thin layer are updated by using the genetic algorithm to minimize the residual error between the measured and the simulation data at a high excitation level. It is demonstrated by comparing the frequency responses of the updated FEM and the experimental system that the thin layer of bilinear elastic-plastic material is very effective for modeling the nonlinear joint interface of the assembled structure with multiple bolts.


Introduction
In the aeronautical and aerospace fields, a much more attention has been attracted by the assembled mechanical structures which consist of substructures or parts connected to each other through different types of connections [1][2][3][4][5].If the effects of the mechanical joints on the dynamics are neglected in the model of the mechanical structures, there may exist a significant error between the theoretical and experimental results, especially in some cases having large excitation or motion levels [6].The main reason is that most of the structure connections may cause the change of stiffness and damping.However, due to the complex dynamics characteristics of the assembled structures, it is difficult and challenging to model these structures accurately.An accurate model for joints is the key to the success of the dynamic analysis of assembled structures.Many researchers have focused on how to resolve such a problem using the theoretical or experimental methods [7][8][9][10].For instance, Kuether and Allen presented two model substructuring techniques for describing the nonlinear dynamic behavior of an assembled structure [11].Hammami et al. [12] presented a detailed review on the damping design of a jointed structure.Segalman [13] investigated a four-parameter model for laptype joints to capture the dissipation behavior of harmonically loaded experiments.Bograd et al. [14] gave an overview of various joint types, their characteristics, and the models used for their simulation as well as an extensive information of up-to-date literature associated with this topic.Jalali et al. [15] studied the weakly nonlinear systems of multiple degrees of freedom by describing functions inversion.Morelli [16] developed a new method for the structures to identify transfer function models from the noisy data.Marinone et al. [17] proposed a model modification response technique for computing the time response of a nonlinear system.Peter et al. [18] presented an integrated method for estimating linear as well as nonlinear system parameters of a structure based on its nonlinear normal modes (NNMs) and performed the Harmonic Balance Method (HBM) to numerically calculate the NNMs.Among the mechanical connections, bolted joints are applied in assembled mechanical structures very frequently.A number of researchers have paid attention in developing predictive models for bolted joints in different structures.For example, Song et al. [19] developed an adjusted Iwan model approach to simulate the nonlinear dynamic behavior of bolted joints in beam structures.Jalali et al. [20] presented a nonlinear model possessing a single degree of freedom in order to describe the governing equation of bolted joints interface between two beams.Ahmadian and Jalali [4] used a nonlinear spring to represent the softening phenomenon of the joint interface in the modeling of the joint which was in the mid span of an Euler-Bernoulli beam.Li et al. [21] proposed an identification method for the bolted joint in a cantilever beam of steel.Yang et al. [22] derived the identification equations and identified the joint parameters in the assemble beam by using the substructural synthesis method.Luan et al. [23] proposed a simplified nonlinear dynamic model, in which the mechanical properties of the bolted joint in a flange assembled structure were modeled by bilinear springs.Mayer and Gaul [24] studied the modeling of the nonlinear stiffness and damping mechanisms between jointed members in the built-up structures and showed a comparison between two different segment-to-segment contact elements, that is, the so-called TLEs and zero thickness elements.
It is noted that the previous researches mainly focused on the assembled structures consisting of several beams, pipes, or other simple parts connected through a small amount of bolts.Finite Element Models (FEMs) have been developed to describe the dynamic phenomena of assembled structures by treating each connected interface as a thin layer of a nonlinear elastic-plastic material [25][26][27].It has been demonstrated that a thin layer of a nonlinear material is effective to describe the nonlinearity due to bolted connections.Alamdari et al. [26] utilized a thin layer of solid elements to model the joint interface between two pipes.Iranzad and Ahmadian [27] adopted a thin layer interface model of virtual elastic plastic material to simulate nonlinear effect of bolted joints in the beams.
It is to be noted here that less attention has been given to model the panel structure with multiple bolted connections although much of the literature focused on modeling of the bolted beams and pipes.The aim of this work is to study the nonlinear dynamics modeling and parameter identification of an AHP with multiple bolted joints.For this purpose, a thin layer of nonlinear elastic-plastic material is used to characterize the nonlinear characteristics of the bolted connection interface.The linear material parameters of the panel are necessarily identified via experimental tests at low excitation levels, whereas the nonlinear material parameters of the thin layer are updated by using the genetic algorithm to minimize the residual error between the measured and the simulation data at a high excitation level.Finally, the efficacy of the proposed scheme is demonstrated via comparisons of the FEM and experimentally obtained results under another exciting level.

Nonlinear Dynamics Modeling
As shown in Figure 1, this study is concerned with the assembled structure consisting of an AHP and a mounting base connected to each other via multiple bolts.The AHP has a length of 1000 mm, a width of 1000 mm, and a thickness of 30 mm.The volume and weight of the AHP are 0.03 m 3 and 5.80 kg, respectively.The AHP is fixed at the bottom by using 26 bolts (13 bolts for each side).The pretightening torque of the bolts is set to be about 2 Nm.
The nonlinear characteristics of the bolted joints, such as micro/macroslip, may have a significant effect on the dynamical behavior of assembled structures.Therefore, it is of practical importance to accurately model the nonlinear characteristics of the bolted joints interface.The equation of motion for a nonlinear system with multiple degrees of freedom under harmonic excitation can be expressed as follows [28]: where x and ẋ are the vectors of displacements and velocities, F NL (x, ẋ ) represents the nonlinear internal force which is a function of the displacements and velocities, f is the amplitude vector of the excitation forces,  is the excitation frequency, and M, C, and K denote the mass, damping, and stiffness matrix, respectively.Quite different from the cases of linear systems, it is well known that the frequency response of such a nonlinear system depends on the levels of the excitation forces due to the presence of the nonlinear internal force.In this study, the finite element method is adopted to model the assembled structure where the connection interface with multiple bolts is treated as a thin layer of bilinear elastic-plastic material so as to characterize the nonlinear behavior of the bolted joints.Figure 2 shows a sketch of the thin layer which connects Structure A (the AHP) to Structure B (the mounting base).
The thickness of the thin layer is much less than the geometric sizes in the other two directions.Following [24], the following assumptions are considered to model the constitutive relation of the thin layer.The thickness of the thin layer element is approximately equal to zero; consequently, the normal contact behavior of thin layer element is decoupled from the tangential behavior.Take the case of linear material, for example.The constitutive relation of the thin layer element can be written as [24] where , , and  represent the contact normal and the local and -tangential directions, respectively;   and   represent the normal stress and strain component of the interface;   and   represent the tangential stress; and   and   represent the tangential strain.In the case of linear material,  11 denotes Young's modulus, and  55 =  66 is the shear modulus, which can describe the normal and tangential stiffness of the thin layer element.
In the case of the nonlinear material, the constitutive relation has a more complex form.In this study, a bilinear elasticplastic model under the assumption of isotropic hardening and Von Mises yield criterion is used to describe the stressstrain relationship of a thin layer element [29].Elastic-plastic behavior of an isotropic hardening is described by using the three parameters, that is, Young's modulus, yielding point, and hardening modulus.Figure 3 shows the bilinear stressstrain behavior for one-dimensional uniaxial case [30], where the material initially deforms with Young's modulus   as its slope, until the stress level reaches the uniaxial yield stress   , and then the behavior of material can be described by using hardening modulus   .
As shown in Figure 3,   ,   , and   represent Young's modulus, yield stress, and hardening modulus, respectively.In the case of one-dimensional uniaxial case, the constitutive equation can be written as where  is the stress and  is the strain.The FEM of the AHP with multiple bolts is created in MSC/NASTRAN [29].Figure 4 shows the FEM of the structure.Eight-node solid elements are adopted to create the AHP.The material of the model is assumed to be an isotropic and described using three material parameters, that is, Young's modulus, Poisson's ratio, and shear modulus.The elastic-plastic material properties of the thin layer can be defined by using MAT1 and MAST1 cards of NASTRAN.The thickness of the thin layer is selected by considering the following dimensionless parameter: where  1 and  2 are length and width of the thin layer element and  is the thickness of the thin layer element.To facilitate numerical computation, it is suggested by [31] to select  ∈ [10,100].In this work, the thickness of the thin layer is chosen such that  = 100.

Model Identification
In the FEM of the assembled structure, the AHP is treated as a linear plate of homogeneous, isotropic material.The mass density of the plate is determined using the volume and weight of the AHP.The equivalent Young's modulus and Poisson's ratio are set to be guessed values and then updated by using the SOL200 procedure in NASTRAN to minimize the error between the first three-order natural frequencies of FEM and real structure.Accordingly, the objective function can be expressed as follows: where  FEM  and  Test  are numerically and experimentally obtained th order natural frequencies and   is the weight coefficient.It is noteworthy that the location and magnitude of the applied exciting forces are identical with those in the experimental test.
Moreover, the nonlinear material parameters of the thin layer, that is, Young's modulus   , yielding point   , and hardening modulus   , are updated by minimizing the error between the responses of numerical simulation based on FEM and the experimental test.The dynamic responses of the FEM with harmonic excitation are obtained in the time domain by using nonlinear transient module of NAS-TRAN and then transferred into the frequency domain using MATLAB programming environment.The three material parameters of the thin layer are chosen as design variables and are then updated to minimize the following objective function: where  FEM (  ) and  Test (  ) are numerical and experimental responses and  is the number of the selected frequency points.In other words, the objective function of ( 6) is a measure of the difference between numerical and experimental responses in the frequency domain.The nonlinear optimization problem can be cast as finding X so as to minimize the objective function (X) subject to possible inequality and equality constraints: It is worthy to note here that the parameter updating of the thin layer material is nonlinear by nature and cannot be solved using the off-the-shell modules in NASTRAN.Thus, the genetic algorithm optimizer of MATLAB is used for the optimization task through the real-time data exchange with the NASTRAN.Figure 5 depicts the main procedure of the optimization task: an input file (panel.bdf)with model information is generated using a preprocessing modeling software, and then NASTRAN is called with the input file for a series of nonlinear transient analyses of the structure subject to exciting forces with different frequencies; the dynamic responses obtained using NASTRAN and experimental testing are converted into the frequency domain through the application of the fast Fourier transform method; the genetic algorithm optimizer of MATLAB takes the frequency response data for the instantaneous computation of the objective function and the constraints and updates the design variables, namely,   ,   , and   , so as to reduce the constraint violation or to improve the performance index until a prespecified optimization criterion is satisfied.

Experimental Studies
Experimental test was conducted to investigate the dynamic characteristics of the AHP with multiple bolts.The testing system setup mainly consists of an electromagnetic vibration  exciter (S 51075-M from TIRA GmbH), a power amplifier (B&K2718 from Brüel & Kjaer), and a vibration test controller (Spider-81 from Crystal Instruments), as shown in Figure 6.
The input force and the dynamic responses of the system are measured using a force transducer (208C02 from PCB Piezotronics) and four acceleration sensors (333B52 from PCB Piezotronics), respectively, as shown in Figures 6 and 7.
Two different stages of experimental tests were carried out.In the first part, the AHP is excited by using random excitation (RMS 1 × 10 −4 ) and sinusoidal forces (0.5 N), respectively.The excitation time is set in the Spider-81 software system to be 10 minutes.Figure 8 describes the data of Frequency Response Function (FRF) acquired at sensor location A using the Engineering Data Management (EDM) software from Crystal Instruments.The experimental results obtained using the random and the sinusoidal excitations at  a small level are almost the same to each other since the assembled structure behaves linearly in the case of small excitation level.In addition, Table 1 gives the first three natural frequencies of the assembled structure obtained using the linear tests.The first group of the tests are performed for the linear identification of the AHP parameters, where the AHP is treated as a clamped plate of homogeneous, isotropic material.
The linear material parameters of AHP are updated by minimizing the objective function defined as (5) by using SOL 200 procedure of NASTRAN.The initial values of Young's modulus and Poisson's ratio are taken to be 4.50 Gpa and 0.33, respectively.Figures 9-11 show the variations of the objective function and the design parameters with the design cycles in the optimization process, respectively.Table 2 shows the updated results of material parameters and Table 3 shows the experimentally measured, initially estimated, and updated natural frequencies.
In the second part, the AHP was excited using sinusoidal forces at much larger levels so as to identify the nonlinear characteristics of the system.Two different levels of excitation forces (5 N and 8 N) were considered for the experimental tests with its testing frequency range set to be from 11 Hz to 15 Hz in the Spider-81 software system.Quite different from the case of a linear structure, Figure 12 shows the measured FRF and the natural frequency of the structure being decreased with the amplitude of the exciting force.
In this part, the parameters of Young's modulus   , yielding point   , and hardening modulus   are updated by minimizing the objective function defined as (6).The optimization process of the nonlinear material parameters is illustrated by Figure 5 and performed using the genetic algorithm optimizer of MATLAB through the real-time   data exchange with the NASTRAN solver.The results of optimization are shown in Table 4 and Figures 13 and 14.The responses of the updated FEM model well coincide with that of the experimentally measured responses, as shown in Figure 13.Finally, the proposed strategy is verified by comparing the frequency responses of the updated model  and the experimental test under another exciting level (8 N). Figure 14 demonstrates a remarkable agreement between the dynamic responses of the updated FEM model and the experimental test under 8 N excitation force.It confirms that the updated FEM model can well characterize the nonlinear behavior of the assembled structure with multiple bolts subject to higher levels of excitations.

Conclusions
The aim of this study is to investigate the nonlinear dynamics modeling and parameter identification of AHP with multiple bolts.It is shown by experimental tests that the dynamic responses of the AHP with multiple bolts under low and high excitation levels are linear and nonlinear, respectively.The equivalent linear material parameters of the panel are identified using the linear response at low excitation level.A thin layer of elastic-plastic material is used to characterize the bolted joints interface since the constitutive relation of such a material can describe the nonlinear behavior of the bolted interface.The parameters of nonlinear material of thin layer are updated by minimizing the error between FEM and experimentally measured results under a high excitation level.It is demonstrated that the frequency responses of the updated FEM model and experimental tests agree well with each other.The updated FEM model can be used to characterize the nonlinear behavior of the assembled structure with multiple bolts subject to higher levels of excitations.

Figure 1 :
Figure 1: (a) The AHP connected to a mounting base via multiple bolts.(b) A graphic illustration of the bolted connections.

Figure 2 :
Figure 2: The sketch of the thin layer of elastic-plastic material.

Figure 5 :
Figure 5: Flowchart of the optimization procedure.

Figure 6 :
Figure 6: A schematic view of the experimental setup.

Figure 7 :
Figure 7: Experimental system for vibration test of the AHP with multiple bolts.

Figure 8 :
Figure 8: Measured linear FRF at sensor location A.

9 Figure 9 :
Figure 9: Variation of Young's modulus with the design cycles.

Figure 10 :
Figure 10: Variation of Poisson's ratio with the design cycles.

Figure 11 :
Figure 11: Variation of the objective function with the design cycles.

Figure 12 :
Figure 12: Measured nonlinear FRF at sensor location A.

Table 1 :
Measured first three natural frequencies.

Table 2 :
Initial and updated material parameters.

Table 3 :
Experimentally measured, initial, and updated natural frequencies.

Table 4 :
Updated parameters of the thin layer.