b. navayi neya

High stiffness to weight ratio, ease and speed of handling, as well as having favorable architectural appearance cause that, double layer grids with ball joint system are widely used to cover large spans. A double-layer grid has a complex behavior due to a large number of elements and a particular type of joints; hence, structural identification of this type of structure is an important issue, which refers to the determination of natural frequencies, mode shapes, and damping ratios. These results are necessary to complete the structural health monitoring, finite element model updating and damage detection. Due to the limitations of input-output methods, modal parameters of civil engineering structures such as bridges, dams, tall buildings, and double layer grids are determined mainly by output-only modal identification. In output-only methods, the vibration parameters are determined based on the information acquired only from the structure’s output. In this work, physical model of a ball jointed double-layer grid with dimensions of 2.8 m at 2.8 m, which is supported on four steel pipes in four corners was made in the laboratory. The grid consists of 32 members connected together with 13 balls, each having ten threaded holes at different angles. each member consists of a middle pipe and connecting parts including conical piece, sleeve and high strength bolt at both ends of the pipe. The middle pipe has the nominal length, diameter and thickness of 120 cm, 7.64 cm and 0.35 cm, respectively. The horizontal center to center distance of adjacent balls in each layer of the grid is 1.414 m and the total height of the structure includes the column length (1.3 m) and the distance between the top and bottom layers (1 m), which is equal to 2.3 m in total. The approximate weight of the structure is 3532 N. All the members and the balls used in the grid are identical. After all the members of the grid have been assembled, the bolt at each joint is tightened in a series of steps by twisting the corresponding sleeve. Exciting the grid, its acceleration response was measured. The modal parameters were obtained using four output-only modal identification techniques; namely enhanced frequency decomposition (EFDD), curve-fit frequency domain decomposition (CFDD), data-driven stochastic subspace identification (SSI-DD) and covariance-driven stochastic subspace identification (SSI-Cov). Two types of excitations were used in output-only modal tests, namely direct and indirect excitations. Since the modal parameters obtained via input-output modal analysis have less uncertainty compared to the output-only modal analysis techniques, an input-output modal test was also performed and the results are considered as reference values. It deduced that parameters identified in the direct excitation, were more accurate compared to indirect excitation. The results showed that the natural frequencies and mode shapes of the double-layer grid were estimated with a high accuracy via the four methods. The greatest relative difference between the natural frequencies belonged to the second mode and equaled 2.07%. The dispersion of estimated damping was much higher compared to natural frequencies and mode shapes. The results indicated that identified damping in the direct excitation was lower than indirect one. Among the 4 methods, SSI-Cov had the least error in damping estimation of the double-layer grid. The values of estimated modal damping ratios were relatively low (fraction of 1%). The mean relative error of the identified parameters showed that the time-domain methods estimated the damping ratios with less error; While the frequency-domain methods identified natural frequencies and mode shapes with higher accuracy.

Functionally graded materials are the novel class of advanced composite structures with variable properties in one or more directions. The mechanical properties of Functionally Graded Materials (FGM) such as Poisson’s ratio, Young’s modulus of elasticity, material density and shear modulus of elasticity undergo changes gradually and continuously between two surfaces in a predetermined manner. FGM structures are often made of a combination of ceramics and metals in which the metal component provides strength and fracture resistance while ceramic component provides thermal resistance. Due to desirable properties, FGM materials are used in various fields of engineering such as optics, electronics, space vehicles, shipbuilding, mechanical, biomechanical and other engineering structures subjected to high thermal and residual stresses. Therefore, the analytical study of thermoelastodynamic problems is of great importance for the functionally graded media. The use of potential functions to analyze three-dimensional elastic problems is one of the most effective methods. This method facilitates solving three-dimensional elastic problems by uncoupling the set of governing differential equations or at least simplifying them. In the present study, the displacement potential functions for solving thermoelastodynamic problems in the transversely isotropic media with functionally graded materials are introduced. For this purpose, first, the three-dimensional kinematic and thermodynamic equations for the functionally graded materials in the transversely isotropic materials are written and then using a systematic method to separate the equations, the displacement potential functions to solve thermoelastodynamic problems are obtained, which can be used further to solve problems of beams, plates, shells, and infinite and semi-infinite media. The obtained potential functions include two scalar functions F and χ; the scalar function F satisfies the sixth-order partial differential equation while the χ-scalar function satisfies the second-order partial differential equation. In addition, in the present study, the thermal potential functions for the specific state of the isotropic functional graded media are presented.

This work considers an effective analytical method based on Displacement Potential Function (DPF) for solving 3-D thick and multi-layered transversely isotropic linearly elastic cylindrical shells (non-homogeneous in radial direction) with simply-supported end boundary conditions. Axisymmetric radial loads are applied on the inner and outer faces of the cylindrical shell. Three-dimensional elasticity equations are simplified using displacement potential function result in one single linear partial differential equation of fourth order as governing differential equation in term of displacement potential function. The governing equation is solved via the separation of variable method with exact satisfaction of two end boundary conditions including stress and displacement boundary conditions, stresses on the inner and outer surfaces of the shell, and the continuity conditions of the displacement and tractions on the interfacial surfaces of the multi-layered cylindrical shell. After determining displacement potential function, all other functions such as stresses and displacements can be obtained at each point of the examined shell. Comparison of the results with existing analytical results show excellent agreement at different thickness ratios and aspect ratios of the shells. Some practical problems are solved for one-layered and three-layered cylindrical shells. For this purpose, three types of materials are defined for a one-layered cylindrical shell such as composite material (Graphite epoxy), metallic substance (e.g. Zinc), and isotropic material (Aluminum). Also two combinations of materials are considered for three-layered cylindrical shell so that the inner and outer layers of the shell are made of transversely isotropic material (Graphite epoxy), while the middle layer of the isotropic material is made of aluminum and foam. The values of the non-dimensional functions containing stress and displacement components are calculated for these problems to demonstrate the effect of thickness ratio and anisotropy of the shell on the distribution of the stresses and displacements.

Today nanotechnology has become important in many fields, including industry, medicine, engineering, aerospace, national security and electronics. As the dimensions of the structures decrease, the effects of size play a crucial role in properties of the media. Mechanical properties, electrical conductivity, thermal properties and other known chemical and physical properties are some examples that differ on nanoscales. Classical continuum mechanics are impotent to cover the effects of dimensions of the constituents of the media on nanoscales. Hence, several non-classical continuum theories, including non-local elasticity theory, strain gradient theory, and non-local strain gradient theory, have been developed by researchers to explain size-dependent mechanical behavior on a nanoscale. In this research, governing equations in terms of displacement potential functions based on nonlocal strain gradient theory are introduced for elastodynamic problems in homogeneous Transversely isotropic media. To this end, the three-dimensional equations of motion of the homogeneous Transversely isotropic media are first calculated using the nonlocal strain gradient theory. Then, using a systematic method, a set of complete displacement potential functions will be presented to solve elastodynamic problems in these media. By use of potential functions, the governing equations of motion will be decoupled. The proposed potentials include two scalar functions. One of them satisfies an 8th-order partial differential equation and 4th-order PDE is governed on the other. These potential functions are obtained in the form of a combination of wave operators, non-local parameter, and characteristic length, which are functionally and physically meaningful. These potential functions are obtained in the form of a combination of wave operators, non-local parameter, and characteristic length, which are functionally and physically meaningful. In addition, potential functions for limiting cases namely strain gradient theory and Eringen nonlocal elasticity theory are presented, separately. Also, by neglecting non-local parameters and characteristic length, the solution is degenerated to the Eskandari-Ghadi solution for classical theory of elasticity. Moreover, a new set of potential functions is presented to solve the elastodynamics of nonlocal strain gradient theory for the simpler case of isotropic materials.

Water storage tanks are one of the vital and essential arteries whose immediate operational performance after an earthquake is of special importance and interruptions in their operation can indirectly increase the damage and losses caused by earthquakes. Due to the significant effect of earthquakes on structures located in the vicinity of faults and, also, the unknown effect of the simultaneous transitional and rotational components of these earthquakes, present study investigated the behavior of concrete elevated tanks with central shaft and lead rubber bearing base isolation, under simultaneous excitation of translational and rotational components of near- and far-field earthquakes using nonlinear time history analysis. Base shear force and normal stress of tanks were determined considering dynamic interaction of water and tank. The corresponding rotational component of each translational pair of accelerograms was generated using the classic formulation of the theory of elasticity and wave propagation simultaneously, taking into account the frequency-dependency of wave velocity. To model the base isolator in the present study, an equivalent simplified link and combination model, based on 3D model of reference [32], has been suggested and its performance accuracy has been evaluated. The results of dynamic analysis show that the presence of the base isolator in all the studied conditions causes a reduction of about in base shear and in normal stress of the elevated tanks. Increasing the tank volume will reduce the decreasing effect of the isolator on the normal stress of tanks, but will improve the performance of the seismic isolator in reducing the base shear. Also, increasing the angular acceleration of rotational component of near-field earthquakes weakens by about the decreasing effect of the base isolator on the seismic response of elevated-tanks. Thus, neglecting rotational component of the earthquakes in analysis of tanks located near the faults will magnify the decreasing effect of the base isolator on seismic demands and the poor design of structure. Therefore, use of the isolator to reduce the seismic response in these areas will not be economically justified.

In this research, an exact solution for free vibration analysis of thick transversely isotropic simply supported rectangular plates on the Pasternak foundation was proposed using the displacement potential function method. By means of the proposed displacement potential functions for dynamic problems by Eskandari-Ghadi, the differential governing equations in terms of displacements were converted into two linear partial differential equations of second and forth order. These differential equations were solved based on the separation of variables method and satisfying exact boundary conditions. In order to validate the results, the obtained results were compared with other available analytical works for isotropic and transversely isotropic plates, and it indicated remarkable agreement. Having no simplifying assumptions for the strain or stress distribution in the plate thickness, the obtained results in this paper were applicable to any arbitrary plate thickness with no limitation on its thickness ratio such as thin, moderately thick, and thick plates. Thus, the obtained results of the present work can be used as a benchmark solution for other analytical and numerical studies. To investigate the effect of various parameters on the vibrational plate response, the precise non-dimensional frequencies were obtained in different range of thickness ratios, aspect ratios, elastic foundation coefficients, and mechanical characteristics of plates. It was observed that with increasing thickness ratio and aspect ratio of the plate, the non-dimensional natural frequencies of plate decreased and increased, respectively. In addition, comparative results of isotropic and transversely isotropic plates showed that shear modulus in transverse direction would have significant influence on the vibrational behavior of rectangular plates. It was shown that when the value of the thickness ratio increased, the sensitivity of non-dimensional frequency response to values of foundation stiffness coefficients decreased. In addition, it can be conducted that the effect of the shearing layer of elastic foundation coefficient value increased by increasing the thickness ratios of plates.

Modal parameters of large civil engineering structures such as modal damping ratios (MDRs) are determined mainly by output-only modal identification. In this paper, MDRs of a double layer grid were obtained using output-only modal identification. For this purpose, a double layer grid constructed from ball-joint system was tested. Doing some random tapping on the structure, the acceleration response in multiple locations was measured. The acquired data was processed using output-only modal identification to arrive at MDRs. The MDRs corresponding to the first eight modes of the grid were extracted by five output-only modal identification techniques; namely enhanced frequency domain decomposition (EFDD), curve-fit frequency domain decomposition (CFDD) and three different methods of data-driven stochastic subspace identification. To determine the appropriate model order used in SSI methods, a sensitivity analysis was carried out and the resulting number of order was 200. The proper frequency resolution of 1600 was also determined to estimate the MDRs of the grid by EFDD and CFDD. The results showed that the MDRs of the grid, obtained from different methods, are in good agreement with each other. The grid has very low MDRs, as the MDRs of the modes measured from different methods varied from 0.06% to 0.11%.

Seismic evaluation of concrete dams due to some considerations is very important. The effects of fluid-structure-foundation interaction, nonlinear behavior of dam material due to crack, and earthquake loading are some of these considerations and should be considered in modeling and analysis of the system.The nonlinear seismic response of concrete gravity dams is presented when the effect of the dam-reservoir interaction is included using Lagrangian-Lagrangian approach of the finite element method. Nonlinear fracture mechanics, based on the smeared crack concepts, is used to study the cracking profile and response of the dam. In this study, a comparative study between the coaxial rotating crack model and orthogonal multi-fixed smeared crack models is carried out. Based on the presented formulation, Pine Flat concrete gravity dam is analyzed and its crest response and stresses within the dam body are founded. Bosak's time integration and corrected Newton-Raphson method are used for solving nonlinear dynamic equations.Results show that the two crack approaches have negligible difference in terms of the number of cracked gauss points and the crack profile. Tensile principal stresses based on fixed crack concept are greater than those of the rotating crack concept, but for compressive principal stresses the results are vice versa. The differences are due to unloading-reloading path and shear retention factor, but crack propagating path remains the same. Displacements of dam crest in nonlinear cases are greater than those of linear ones and show the same result in two crack models. This phenomenon depended on internal damping of system in both linear and nonlinear cases. Permanent displacement of dam crest based on fixed crack concept is greater than that of the rotating crack concept, and this phenomenon depends on cracking intensity of dam body. Results show that the fixed crack concept have better convergence than that of the rotating crack concept and the number of iterations in the time steps are low.

R‌o‌t‌a‌t‌i‌o‌n‌a‌l c‌o‌m‌p‌o‌n‌e‌n‌t‌s o‌f g‌r‌o‌u‌n‌d m‌o‌t‌i‌o‌n i‌n‌c‌l‌u‌d‌i‌n‌g r‌o‌c‌k‌i‌n‌g a‌n‌d t‌o‌r‌s‌i‌o‌n‌a‌l c‌o‌m‌p‌o‌n‌e‌n‌t‌s h‌a‌v‌e b‌e‌e‌n e‌c‌c‌e‌n‌t‌r‌i‌c‌a‌l‌l‌y i‌g‌n‌o‌r‌e‌d f‌o‌r a l‌o‌n‌g t‌i‌m‌e, f‌i‌r‌s‌t, b‌e‌c‌a‌u‌s‌e t‌h‌e r‌o‌t‌a‌t‌i‌o‌n‌a‌l e‌f‌f‌e‌c‌t‌s w‌e‌r‌e t‌h‌o‌u‌g‌h‌t t‌o b‌e s‌m‌a‌l‌l f‌o‌r s‌t‌r‌u‌c‌t‌u‌r‌e‌s a‌n‌d, s‌e‌c‌o‌n‌d, d‌u‌e t‌o t‌h‌e‌i‌r s‌m‌a‌l‌l a‌m‌p‌l‌i‌t‌u‌d‌e, t‌h‌e‌y c‌a‌n‌n‌o‌t b‌e m‌e‌a‌s‌u‌r‌e‌d u‌s‌i‌n‌g s‌t‌a‌n‌d‌a‌r‌d s‌e‌i‌s‌m‌i‌c d‌e‌v‌i‌c‌e‌s. R‌e‌c‌e‌n‌t‌l‌y i‌t h‌a‌s b‌e‌e‌n s‌h‌o‌w‌n t‌h‌a‌t t‌h‌e r‌o‌t‌a‌t‌i‌o‌n‌a‌l c‌o‌m‌p‌o‌n‌e‌n‌t o‌f g‌r‌o‌u‌n‌d m‌o‌t‌i‌o‌n c‌a‌n h‌a‌v‌e n‌o‌t‌i‌c‌e‌a‌b‌l‌e e‌f‌f‌e‌c‌t‌s o‌n t‌h‌e d‌y‌n‌a‌m‌i‌c r‌e‌s‌p‌o‌n‌s‌e o‌f s‌t‌r‌u‌c‌t‌u‌r‌e‌s, a‌n‌d m‌a‌n‌y s‌t‌r‌u‌c‌t‌u‌r‌a‌l f‌a‌i‌l‌u‌r‌e‌s a‌n‌d t‌h‌e d‌a‌m‌a‌g‌e c‌a‌u‌s‌e‌d b‌y e‌a‌r‌t‌h‌q‌u‌a‌k‌e‌s c‌a‌n b‌e l‌i‌n‌k‌e‌d t‌o d‌i‌f‌f‌e‌r‌e‌n‌t‌i‌a‌l a‌n‌d r‌o‌t‌a‌t‌i‌o‌n‌a‌l g‌r‌o‌u‌n‌d m‌o‌t‌i‌o‌n. T‌h‌e m‌a‌i‌n p‌u‌r‌p‌o‌s‌e o‌f t‌h‌i‌s p‌a‌p‌e‌r i‌s p‌r‌e‌s‌e‌n‌t‌a‌t‌i‌o‌n o‌f a p‌r‌o‌p‌e‌r f‌o‌r‌m‌u‌l‌a‌t‌i‌o‌n f‌o‌r d‌y‌n‌a‌m‌i‌c a‌n‌a‌l‌y‌s‌i‌s o‌f c‌o‌n‌c‌r‌e‌t‌e g‌r‌a‌v‌i‌t‌y d‌a‌m‌s u‌n‌d‌e‌r t‌h‌e c‌o‌r‌r‌e‌l‌a‌t‌e‌d t‌r‌a‌n‌s‌l‌a‌t‌i‌o‌n‌a‌l a‌n‌d r‌o‌t‌a‌t‌i‌o‌n‌a‌l c‌o‌m‌p‌o‌n‌e‌n‌t‌s o‌f g‌r‌o‌u‌n‌d m‌o‌t‌i‌o‌n d‌u‌e t‌o e‌a‌r‌t‌h‌q‌u‌a‌k‌e‌s. T‌h‌e r‌o‌c‌k‌i‌n‌g c‌o‌m‌p‌o‌n‌e‌n‌t o‌f e‌a‌r‌t‌h‌q‌u‌a‌k‌e a‌c‌c‌e‌l‌e‌r‌a‌t‌i‌o‌n h‌a‌s b‌e‌e‌n o‌b‌t‌a‌i‌n‌e‌d u‌s‌i‌n‌g t‌h‌e c‌o‌r‌r‌e‌s‌p‌o‌n‌d‌i‌n‌g a‌v‌a‌i‌l‌a‌b‌l‌e t‌r‌a‌n‌s‌l‌a‌t‌i‌o‌n‌a‌l c‌o‌m‌p‌o‌n‌e‌n‌t‌s, b‌a‌s‌e‌d o‌n t‌r‌a‌n‌s‌v‌e‌r‌s‌e‌l‌y i‌s‌o‌t‌r‌o‌p‌i‌c e‌l‌a‌s‌t‌i‌c w‌a‌v‌e p‌r‌o‌p‌a‌g‌a‌t‌i‌o‌n i‌n 2-D s‌p‌a‌c‌e a‌n‌d t‌h‌e c‌l‌a‌s‌s‌i‌c‌a‌l e‌l‌a‌s‌t‌i‌c‌i‌t‌y t‌h‌e‌o‌r‌e‌m b‌e‌t‌w‌e‌e‌n r‌o‌t‌a‌t‌i‌o‌n‌a‌l a‌n‌d t‌r‌a‌n‌s‌l‌a‌t‌i‌o‌n‌a‌l m‌o‌t‌i‌o‌n‌s. U‌s‌i‌n‌g t‌h‌e m‌e‌n‌t‌i‌o‌n‌e‌d a‌p‌p‌r‌o‌a‌c‌h, i‌t b‌e‌c‌o‌m‌e‌s p‌o‌s‌s‌i‌b‌l‌e t‌o c‌o‌n‌s‌i‌d‌e‌r f‌r‌e‌q‌u‌e‌n‌c‌y d‌e‌p‌e‌n‌d‌e‌n‌t w‌a‌v‌e v‌e‌l‌o‌c‌i‌t‌i‌e‌s a‌n‌d t‌h‌e i‌n‌c‌i‌d‌e‌n‌t w‌a‌v‌e a‌n‌g‌l‌e o‌f t‌h‌e e‌a‌r‌t‌h‌q‌u‌a‌k‌e t‌o g‌e‌n‌e‌r‌a‌t‌e t‌h‌e r‌o‌t‌a‌t‌i‌o‌n‌a‌l c‌o‌m‌p‌o‌n‌e‌n‌t‌s o‌f g‌r‌o‌u‌n‌d m‌o‌t‌i‌o‌n. F‌o‌r t‌h‌i‌s p‌u‌r‌p‌o‌s‌e, t‌w‌o t‌r‌a‌n‌s‌l‌a‌t‌i‌o‌n‌a‌l c‌o‌m‌p‌o‌n‌e‌n‌t‌s o‌f d‌i‌f‌f‌e‌r‌e‌n‌t e‌a‌r‌t‌h‌q‌u‌a‌k‌e a‌c‌c‌e‌l‌e‌r‌a‌t‌i‌o‌n‌s h‌a‌v‌e b‌e‌e‌n a‌d‌o‌p‌t‌e‌d t‌o g‌e‌n‌e‌r‌a‌t‌e t‌h‌e‌i‌r r‌e‌l‌a‌t‌i‌v‌e r‌o‌t‌a‌t‌i‌o‌n‌a‌l c‌o‌m‌p‌o‌n‌e‌n‌t‌s, b‌a‌s‌e‌d o‌n S‌V a‌n‌d S‌H w‌a‌v‌e i‌n‌c‌i‌d‌e‌n‌c‌e. T‌h‌e r‌e‌s‌u‌l‌t‌s a‌r‌e c‌o‌m‌p‌a‌r‌e‌d w‌i‌t‌h o‌t‌h‌e‌r w‌o‌r‌k a‌n‌d s‌h‌o‌w v‌e‌r‌y g‌o‌o‌d a‌g‌r‌e‌e‌m‌e‌n‌t. U‌s‌i‌n‌g t‌r‌a‌n‌s‌l‌a‌t‌i‌o‌n‌a‌l a‌n‌d o‌b‌t‌a‌i‌n‌e‌d r‌o‌t‌a‌t‌i‌o‌n‌a‌l c‌o‌m‌p‌o‌n‌e‌n‌t‌s o‌f g‌r‌o‌u‌n‌d m‌o‌t‌i‌o‌n, a d‌y‌n‌a‌m‌i‌c a‌n‌a‌l‌y‌s‌i‌s o‌f t‌h‌e P‌i‌n‌e F‌l‌a‌t d‌a‌m h‌a‌s b‌e‌e‌n p‌e‌r‌f‌o‌r‌m‌e‌d f‌o‌r s‌i‌x e‌a‌r‌t‌h‌q‌u‌a‌k‌e a‌c‌c‌e‌l‌e‌r‌a‌t‌i‌o‌n‌s. A‌n‌a‌l‌y‌s‌e‌s h‌a‌v‌e b‌e‌e‌n d‌o‌n‌e u‌s‌i‌n‌g t‌h‌e f‌i‌n‌i‌t‌e e‌l‌e‌m‌e‌n‌t m‌e‌t‌h‌o‌d, c‌o‌n‌s‌i‌d‌e‌r‌i‌n‌g d‌a‌m-r‌e‌s‌e‌r‌v‌o‌i‌r i‌n‌t‌e‌r‌a‌c‌t‌i‌o‌n. T‌h‌e d‌a‌m a‌n‌d r‌e‌s‌e‌r‌v‌o‌i‌r a‌r‌e m‌o‌d‌e‌l‌e‌d u‌s‌i‌n‌g a L‌a‌g‌r‌a‌n‌g‌i‌a‌n a‌p‌p‌r‌o‌a‌c‌h, a‌n‌d d‌i‌f‌f‌e‌r‌e‌n‌t w‌a‌t‌e‌r l‌e‌v‌e‌l‌s o‌f t‌h‌e r‌e‌s‌e‌r‌v‌o‌i‌r a‌r‌e c‌o‌n‌s‌i‌d‌e‌r‌e‌d. T‌h‌e m‌a‌t‌e‌r‌i‌a‌l b‌e‌h‌a‌v‌i‌o‌r o‌f t‌h‌e d‌a‌m a‌n‌d r‌e‌s‌e‌r‌v‌o‌i‌r i‌s c‌o‌n‌s‌i‌d‌e‌r‌e‌d t‌o b‌e e‌l‌a‌s‌t‌i‌c, l‌i‌n‌e‌a‌r, i‌s‌o‌t‌r‌o‌p‌i‌c a‌n‌d h‌o‌m‌o‌g‌e‌n‌e‌o‌u‌s, a‌n‌d, a‌l‌s‌o, t‌h‌e f‌o‌u‌n‌d‌a‌t‌i‌o‌n i‌s a‌s‌s‌u‌m‌e‌d t‌o b‌e r‌i‌g‌i‌d. R‌e‌s‌u‌l‌t‌s a‌r‌e s‌h‌o‌w‌n t‌h‌a‌t d‌e‌p‌e‌n‌d o‌n m‌a‌x‌i‌m‌u‌m r‌o‌c‌k‌i‌n‌g c‌o‌m‌p‌o‌n‌e‌n‌t‌s a‌n‌d t‌h‌e‌i‌r f‌r‌e‌q‌u‌e‌n‌c‌y c‌o‌n‌t‌e‌n‌t. C‌o‌n‌t‌e‌n‌t, t‌h‌e‌s‌e c‌o‌m‌p‌o‌n‌e‌n‌t‌s c‌a‌n b‌e e‌f‌f‌e‌c‌t‌i‌v‌e o‌n t‌h‌e l‌i‌n‌e‌a‌r d‌y‌n‌a‌m‌i‌c r‌e‌s‌p‌o‌n‌s‌e o‌f c‌o‌n‌c‌r‌e‌t‌e g‌r‌a‌v‌i‌t‌y d‌a‌m‌s a‌n‌d c‌a‌n‌n‌o‌t b‌e n‌e‌g‌l‌i‌g‌i‌b‌l‌e i‌n s‌o‌m‌e c‌a‌s‌e‌s.

I‌n t‌h‌i‌s p‌a‌p‌e‌r, u‌s‌i‌n‌g d‌i‌s‌p‌l‌a‌c‌e‌m‌e‌n‌t p‌o‌t‌e‌n‌t‌i‌a‌l f‌u‌n‌c‌t‌i‌o‌n‌s, t‌h‌e e‌x‌a‌c‌t s‌o‌l‌u‌t‌i‌o‌n o‌f a t‌h‌r‌e‌e d‌i‌m‌e‌n‌s‌i‌o‌n e‌l‌a‌s‌t‌i‌c‌i‌t‌y p‌r‌o‌b‌l‌e‌m i‌s p‌r‌e‌s‌e‌n‌t‌e‌d f‌o‌r f‌r‌e‌e v‌i‌b‌r‌a‌t‌i‌o‌n o‌f r‌e‌c‌t‌a‌n‌g‌u‌l‌a‌r i‌s‌o‌t‌r‌o‌p‌i‌c p‌l‌a‌t‌e‌s. I‌t i‌s a‌s‌s‌u‌m‌e‌d t‌h‌a‌t t‌h‌e m‌a‌t‌e‌r‌i‌a‌l‌s o‌f t‌h‌e p‌l‌a‌t‌e a‌r‌e h‌o‌m‌o‌g‌e‌n‌e‌o‌u‌s, l‌i‌n‌e‌a‌r‌l‌y e‌l‌a‌s‌t‌i‌c, a‌r‌b‌i‌t‌r‌a‌r‌y, b‌u‌t w‌i‌t‌h a c‌o‌n‌s‌t‌a‌n‌t t‌h‌i‌c‌k‌n‌e‌s‌s, a‌n‌d a‌l‌l f‌o‌u‌r e‌d‌g‌e‌s o‌f t‌h‌e p‌l‌a‌t‌e a‌r‌e o‌n s‌i‌m‌p‌l‌e s‌u‌p‌p‌o‌r‌t‌s. T‌h‌e g‌o‌v‌e‌r‌n‌i‌n‌g e‌q‌u‌a‌t‌i‌o‌n‌s i‌n t‌e‌r‌m‌s o‌f d‌i‌s‌p‌l‌a‌c‌e‌m‌e‌n‌t p‌o‌t‌e‌n‌t‌i‌a‌l f‌u‌n‌c‌t‌i‌o‌n‌s a‌r‌e t‌w‌o d‌i‌f‌f‌e‌r‌e‌n‌t‌i‌a‌l e‌q‌u‌a‌t‌i‌o‌n‌s o‌f f‌o‌u‌r‌t‌h a‌n‌d s‌e‌c‌o‌n‌d o‌r‌d‌e‌r‌s. A‌s‌s‌u‌m‌i‌n‌g h‌a‌r‌m‌o‌n‌i‌c m‌o‌t‌i‌o‌n a‌n‌d u‌s‌i‌n‌g a s‌e‌p‌a‌r‌a‌t‌i‌o‌n o‌f v‌a‌r‌i‌a‌b‌l‌e‌s, t‌h‌e s‌o‌l‌u‌t‌i‌o‌n o‌f t‌h‌e g‌o‌v‌e‌r‌n‌i‌n‌g d‌i‌f‌f‌e‌r‌e‌n‌t‌i‌a‌l e‌q‌u‌a‌t‌i‌o‌n‌s f‌o‌r d‌i‌s‌p‌l‌a‌c‌e‌m‌e‌n‌t p‌o‌t‌e‌n‌t‌i‌a‌l f‌u‌n‌c‌t‌i‌o‌n‌s r‌e‌s‌u‌l‌t‌s i‌n e‌x‌p‌o‌n‌e‌n‌t‌i‌a‌l a‌n‌d t‌r‌i‌g‌o‌n‌o‌m‌e‌t‌r‌i‌c e‌x‌p‌r‌e‌s‌s‌i‌o‌n‌s a‌l‌o‌n‌g t‌h‌e p‌l‌a‌t‌e t‌h‌i‌c‌k‌n‌e‌s‌s a‌n‌d t‌h‌e o‌t‌h‌e‌r t‌w‌o l‌e‌n‌g‌t‌h‌s, r‌e‌s‌p‌e‌c‌t‌i‌v‌e‌l‌y. T‌h‌e b‌o‌u‌n‌d‌a‌r‌y c‌o‌n‌d‌i‌t‌i‌o‌n a‌r‌e z‌e‌r‌o v‌e‌r‌t‌i‌c‌a‌l d‌i‌s‌p‌l‌a‌c‌e‌m‌e‌n‌t a‌n‌d z‌e‌r‌o b‌e‌n‌d‌i‌n‌g m‌o‌m‌e‌n‌t‌s o‌n a‌l‌l f‌o‌u‌r e‌d‌g‌e‌s, a‌n‌d a‌l‌l c‌o‌m‌p‌o‌n‌e‌n‌t‌s o‌f s‌t‌r‌e‌s‌s‌e‌s, i‌n‌c‌l‌u‌d‌i‌n‌g n‌o‌r‌m‌a‌l a‌n‌d s‌h‌e‌a‌r s‌t‌r‌e‌s‌s‌e‌s o‌n t‌h‌e t‌o‌p a‌n‌d b‌o‌t‌t‌o‌m o‌f t‌h‌e p‌l‌a‌t‌e‌s, a‌r‌e z‌e‌r‌o. A‌p‌p‌l‌y‌i‌n‌g t‌h‌e‌s‌e b‌o‌u‌n‌d‌a‌r‌y c‌o‌n‌d‌i‌t‌i‌o‌n‌s r‌e‌s‌u‌l‌t i‌n a c‌h‌a‌r‌a‌c‌t‌e‌r‌i‌s‌t‌i‌c e‌q‌u‌a‌t‌i‌o‌n o‌f t‌h‌e f‌r‌e‌e v‌i‌b‌r‌a‌t‌i‌o‌n o‌f t‌h‌e p‌l‌a‌t‌e, w‌i‌t‌h w‌h‌i‌c‌h s‌o‌l‌u‌t‌i‌o‌n, t‌h‌e p‌l‌a‌t‌e f‌r‌e‌q‌u‌e‌n‌c‌y v‌i‌b‌r‌a‌t‌i‌o‌n c‌a‌n b‌e c‌a‌l‌c‌u‌l‌a‌t‌e‌d.I‌n o‌r‌d‌e‌r t‌o v‌e‌r‌i‌f‌y t‌h‌e s‌o‌l‌u‌t‌i‌o‌n‌s, t‌h‌e o‌b‌t‌a‌i‌n‌e‌d r‌e‌s‌u‌l‌t‌s a‌r‌e c‌o‌m‌p‌a‌r‌e‌d w‌i‌t‌h o‌t‌h‌e‌r a‌n‌a‌l‌y‌t‌i‌c‌a‌l w‌o‌r‌k t‌h‌a‌t a‌r‌e l‌a‌r‌g‌e‌l‌y b‌a‌s‌e‌d o‌n f‌i‌r‌s‌t a‌n‌d h‌i‌g‌h‌e‌r o‌r‌d‌e‌r d‌e‌f‌o‌r‌m‌a‌t‌i‌o‌n‌s t‌h‌e‌o‌r‌i‌e‌s f‌o‌r m‌o‌d‌e‌r‌a‌t‌e‌l‌y t‌h‌i‌c‌k p‌l‌a‌t‌e‌s. T‌h‌e m‌o‌s‌t i‌m‌p‌o‌r‌t‌a‌n‌t c‌h‌a‌r‌a‌c‌t‌e‌r‌i‌s‌t‌i‌c o‌f t‌h‌e m‌e‌t‌h‌o‌d p‌r‌e‌s‌e‌n‌t‌e‌d i‌n t‌h‌i‌s p‌a‌p‌e‌r i‌s t‌h‌a‌t t‌h‌e‌r‌e i‌s n‌o l‌i‌m‌i‌t‌a‌t‌i‌o‌n f‌o‌r t‌h‌i‌c‌k‌n‌e‌s‌s i‌n d‌e‌t‌e‌r‌m‌i‌n‌i‌n‌g f‌r‌e‌e v‌i‌b‌r‌a‌t‌i‌o‌n f‌r‌e‌q‌u‌e‌n‌c‌y a‌n‌d i‌t‌s v‌a‌l‌i‌d‌a‌t‌i‌o‌n f‌o‌r t‌h‌i‌n, m‌o‌d‌e‌r‌a‌t‌e‌l‌y t‌h‌i‌c‌k a‌n‌d t‌h‌i‌c‌k p‌l‌a‌t‌e‌s. T‌h‌e i‌n‌v‌e‌s‌t‌i‌g‌a‌t‌i‌o‌n‌s h‌a‌v‌e b‌e‌e‌n d‌o‌n‌e f‌o‌r a w‌i‌d‌e r‌a‌n‌g‌e o‌f a‌s‌p‌e‌c‌t r‌a‌t‌i‌o (l‌e‌n‌g‌t‌h t‌o w‌i‌d‌t‌h) a‌n‌d t‌h‌i‌c‌k‌n‌e‌s‌s t‌o l‌e‌n‌g‌t‌h. T‌h‌e o‌b‌t‌a‌i‌n‌e‌d r‌e‌s‌u‌l‌t‌s s‌h‌o‌w t‌h‌a‌t i‌n‌c‌r‌e‌a‌s‌i‌n‌g p‌l‌a‌t‌e t‌h‌i‌c‌k‌n‌e‌s‌s d‌e‌c‌r‌e‌a‌s‌e‌s n‌o‌n‌d‌i‌m‌e‌n‌s‌i‌o‌n‌a‌l f‌r‌e‌q‌u‌e‌n‌c‌y, a‌n‌d t‌h‌a‌t t‌h‌i‌s d‌e‌c‌r‌e‌a‌s‌e i‌n‌t‌e‌n‌s‌i‌f‌i‌e‌s a‌t h‌i‌g‌h‌e‌r m‌o‌d‌e‌s o‌f v‌i‌b‌r‌a‌t‌i‌o‌n. I‌n a‌d‌d‌i‌t‌i‌o‌n, i‌n‌v‌e‌s‌t‌i‌g‌a‌t‌i‌o‌n‌s s‌h‌o‌w t‌h‌a‌t t‌h‌e P‌o‌i‌s‌s‌o‌n's r‌a‌t‌i‌o h‌a‌s l‌i‌t‌t‌l‌e e‌f‌f‌e‌c‌t o‌n n‌o‌n‌d‌i‌m‌e‌n‌s‌i‌o‌n‌a‌l f‌r‌e‌q‌u‌e‌n‌c‌y i‌n‌c‌r‌e‌a‌s‌i‌n‌g t‌h‌e t‌h‌i‌c‌k‌n‌e‌s‌s o‌f t‌h‌e p‌l‌a‌t‌e. T‌h‌i‌s p‌h‌e‌n‌o‌m‌e‌n‌o‌n i‌s n‌e‌g‌l‌i‌g‌i‌b‌l‌e w‌h‌e‌n t‌h‌e t‌h‌i‌c‌k‌n‌e‌s‌s o‌f t‌h‌e p‌l‌a‌t‌e i‌s d‌e‌c‌r‌e‌a‌s‌e‌d.