Abstract
The stiffness of boundary conditions in mechanical structures is difficult to represent. A method is proposed to model the mechanical performance of the clamped boundary condition based on thinlayer element. Firstly, the contact surface of clamped boundary condition is parameterized by using isotropic thinlayer element to model the normal and tangential contact stiffness. Secondly, material parameter of thinlayer element is identified by using experimental modal data, parameter identification is transformed as an optimization problem. Experimental investigation is undertaken to verify the proposed method by employing an aluminum honeycomb panel, the numerical model of which is constructed by using the equivalent theory. Thinlayer elements with different properties are used to simulate the mechanical properties in different area of the boundary conditions, and the experimental modal data is adopted to identify the material parameters. Results show that the width to thickness ratio of the thin layer element has a great influence on the identification results.
Highlights
 Stiffness of the boundary condition is characterized by using thinlayer element.
 Material parameter of thinlayer element is identified by using experimental modal data.
 The width to thickness ratio of the thin layer element has great influences.
1. Introduction
Investigations on modeling of boundary conditions have attracted interests of researchers [1, 2]. To determine the stiffness of clamped boundary conditions, contact models [3, 4] can reflect the contact mechanism through analysis the asperities behavior. In order to obtain the quantitative nonlinear stiffness, identification of nonlinearities is always adopted [5]. However, the contact analysis for boundary conditions will be time consuming when apply to modeling of complex structures.
Identification of the boundary conditions from experimental data is an effective solution to improve the modeling accuracy. Pabst and Hagedorn [6] proposed an inverse method to identify the boundary stiffness and damping parameters of a viscoelastically clamped beam by using modal data. Yang et al. [7] constructed a numerical model to solve multivariables inverse problems for a beam with elastic boundary support. Wang [8] extended the crossmodel crossmode method for model updating by incorporating the connection flexibility and boundary conditions. Yang et al. [9] proposed an approach using frequency response functions to identify the joint parameters including the rotational stiffness. Baruh and Boka [10] modeled boundary conditions using axial and torsional springs. Boundary stiffness are simplified as spring models in the above researches, modeling difficulties will arise when apply spring models to complex structures, because spring element should be allocated in each boundary constraints, and the deficiency is obvious for modeling large contact surface.
Parameterization plays an important role in the stiffness identification of boundary conditions [11, 12]. Mottershead et al. [13] indicated that parameterization is critical in updating of boundary conditions, the boundary stiffness in a cantilever was modeled as the effective length of elements closest to the joint. The above approximate stiffness characterization methods are the alternative parametrized models of boundary stiffness, and modeling complexities still exist. In order to solve this problem, thinlayer element has been introduced to model the dynamics of mechanical joints [1416], The concept of thinlayer element was first proposed by Desai [17] who applied to the contact analysis in geotechnical engineering, and has been widely applied in equivalent modeling of joints, The normal and tangential stiffness of the contact surface can be accurately described by the thinlayer elements [18].
Based on the basic theory of thinlayer element, this paper parameterized the contact surface of the clamped boundary. The elastic parameters of the thinlayer element are identified by using the experimental modal data. Furthermore, the influence of the weighted matrix and the ratio coefficient of the thinlayer element are also investigated.
2. Basic theory
2.1. Thinlayer element
The size of thinlayer element is ${l}_{1}$×${l}_{2}$×$d$, according to the principle of virtual displacement derived the following equation:
${l}_{1}$ and ${l}_{2}$ are the lengths of the thinlayer elements in the $x$ and $y$ directions in the local coordinate system, and $d$ is the $z$direction size. Stiffness matrix [$K$] of thinlayer element is calculated by isoparametric transformation:
Fig. 1 shows the isoparametric transformation of thinlayer element. Where $\xi $, $\eta $, $\zeta $ are natural coordinate symbols, [$J$] is the Jacobi ratio matrix, and when the coordinates of the natural and local coordinates are consistent, which has the following form:
The following numerical expression for stiffness matrix [$K$] of thinlayer element can be obtained by the calculation of twonode Gaussian integration method:
where ${w}_{\xi}$, ${w}_{\eta}$, and ${w}_{\zeta}$ are the Gaussian integral weight function.
It is assumed that the thickness $d$ of the thinlayer element is much smaller than the feature sizes ${l}_{1}$ and ${l}_{2}$. In this case, the inplane strain components (${\epsilon}_{x}$, ${\epsilon}_{y}$, ${\gamma}_{xy}$) and the stress components (${\sigma}_{x}$, ${\sigma}_{y}$, ${\tau}_{xy}$) of the element are ignored. Which leads to the strain component ${\epsilon}_{x}={\epsilon}_{y}={\gamma}_{xy}\approx $ 0. Therefore, only three strain components at the Gaussian point of the thinlayer element are not zero. The strain component can be simplified to $\epsilon ={\left[{\epsilon}_{z}{\gamma}_{yz}{\gamma}_{zx}\right]}^{T}$. According to the previous analysis, assuming that the normal and tangential contact properties of the interface are independent and the two tangential contact properties are consistent.
Fig. 1Isoparametric change of thinlayer element
In the finite element analysis, according to the basic theory of thinlayer element, the constitutive equation of the material will be transformed to the following form:
In this equation, the normal elastic constant and the tangent elastic constant are correlated. The selection of different thickness of thin layer element have a great influence on the calculation results. Desai et al. [17] recommended the width to thickness ratio for obtaining the exact result, the ratio is expressed as:
Calculating the contact stiffness is one of the key problems on thinlayer element for simulating the bolt connected structures. When the contact surface maintains a linear bonding state. The constitutive relation of the material can be integrated into the thinlayer element and the finite element dynamic analysis of the connection structure can be carried out. However, there are some shortcomings that need to be tested separately for the contact stiffness in different directions, and the requirements for the test device and the test method.
2.2. Parameter identification
In this paper, material parameter identification of thinlayer elements is transformed as an optimization problem. The parameters to be identified are the elastic modulus $E$ and the shear modulus $G$ of the thinlayer element. And the minimum value of the objective function is expressed:
where ${\epsilon}_{f}={z}^{m}{z}^{a}\left(p\right)$ is the discrepancy between the experimental data and the numerical results. $W$ is the weighted matrix, which reflects the relative weight of modal parameters. In general, the expression of $W$ is the identity matrix $\mathbf{I}$ or ${\left[\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\left({z}^{m}\right)\right]}^{2}$. Until the numerical modal data satisfy the requirement of precision, the accurate contact surface material parameters are obtained. The procedure of the above optimization problem is shown in Fig. 2.
Fig. 2The flow chat of elastic parameters identification
3. Case studies: a clamped honeycomb panel
An aluminum honeycomb sandwich panel is clamped on the fixed stage is shown in Fig. 3. The contact surface is modeled by thinlayer elements of three different properties. The dimension of the honeycomb sandwich panel is 400 mm×400 mm×16.3 mm. Panel and core materials are aluminum, the elastic Modulus ${E}_{s}$ is 68 GPa and density is 2700 kg/m^{3}.^{}Adhesive material is epoxy resin which elastic Modulus ${E}_{s}$ is 5 GPa and density is 1500 kg/m^{3}.
Fig. 3The clamped honeycomb panel
a)
b)
3.1. Equivalent modeling
Neglecting lateral shearing stress, the honeycomb core is equivalent to an orthotropic material of uniform thickness. As a result, the face sheets are modeled as the upper and the lower panels, the core is equivalent to a solid structure. The interlaminar shear effect caused by the weak layer has an important influence on the macroscopic mechanical properties of the sandwich material. Therefore, the influence of the adhesive layer on the dynamic characteristics of the honeycomb sandwich composite can’t be neglected. Fig. 4 shows the finite element model of honeycomb plate. Clamped boundary condition is modeled by isotropic thinlayer elements.
3.2. Parameter identification
The influence of different weighting matrices on the results of parameter identification is studied by establishing a thinlayer element with $R=$ 100. After the identification, the accuracy of the inherent frequency is greatly improved. The error of the first three order natural frequency is less than 2.63 %, the convergence error can be effectively reduced by ${W}_{\epsilon}={\left[\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\left({z}^{m}\right)\right]}^{2}$.
Based on the above results, parameter identification are carried out when the $R$ of thin layer elements is 10, 50, 100, 500, respectively. The convergence error is reduced with the increase of the width and thickness ratio. However, the more iterations are needed for convergence. Table 2 is the comparison of calculation natural frequency and parameter identification results of thinlayer element under different $R$.
In this case, the stiffness properties of the contact surface are determined by the elastic parameters and the thickness of the thin layer element.
Fig. 4Numerical model of the clamped honeycomb plate
Fig. 5Frequency convergence under different R: a) R= 10, b) R= 50, c) R= 100, d) R= 500
a)
b)
c)
d)
Table 1Comparison of natural frequency
Mode  ${f}^{m}$ / Hz  Initial  ${W}_{\epsilon}={\left[\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\left({z}^{m}\right)\right]}^{2}$  
${f}^{a}$ / Hz  Error (%)  ${f}^{a}$ / Hz  Error (%)  
1  49.59  71.94  45.07  49.27  –0.64 
2  199.8  218.09  9.15  205.05  2.63 
3  411.93  473.83  15.3  416.16  1.03 
Table 2Comparisons of calculated frequency errors under different R
Mode  ${f}^{m}$ / Hz  $R=$ 10  $R=$ 50  $R=$ 100  $R=$ 500  
${f}^{a}$ / Hz  Error (%)  ${f}^{a}$ / Hz  Error (%)  ${f}^{a}$ / Hz  Error (%)  ${f}^{a}$ / Hz  Error (%)  
1  49.59  42.53  –14.23  49.30  –0.56  49.27  –0.64  49.32  –0.54 
2  199.8  200.53  0.36  205.10  2.65  205.05  2.63  205.14  2.67 
3  411.93  414.37  0.59  414.70  0.68  416.16  1.03  414.47  0.62 
4. Conclusions
An approach on identifying the stiffness of the clamped boundary condition is proposed in this paper. The contact surface is modeled based on the theory of thinlayer element, and modal parameters are used to identify the material parameters of thinlayer element. Three kinds of thin layer elements with different properties are used to simulate the mechanical change of the boundary caused by the bolt. The effect of the weighted matrix and the proportion of $R$ on the convergence results are discussed.
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About this article
The work described in this paper was supported by the National Natural Science Foundation of China (No. 11602112).