Fundamental Natural Frequency Research Articles

Mechanical and vibrational behaviors of bilayer hexagonal boron nitride in different stacking modes

Hexagonal boron nitride (h-BN) is a semiconductor material with a wide band gap, which has great potential to serve as a nanoresonators in microelectronics and mass and force sensing fields. This paper investigates the mechanical properties and natural frequencies of bilayer h-BN nanosheets under five different stacking modes, which have been rarely studied, using molecular dynamics simulations. The mechanical properties, including Young’s modulus, the ultimate stress, ultimate strain, Poisson’s ratio and shear modulus, are studied for all five stacking modes. And the effects of strain rate, crystal orientation and temperature to bilayer h-BN nanosheets’ tensile properties have also been studied. Our findings suggest that bilayer h-BN nanosheets are basically an anisotropic material whose tensile properties vary substantially with stacking modes and temperature. Moreover, the natural frequencies are proposed in an explicit form based on the nonlocal theory. The differences of the fundamental natural frequencies among different stacking modes are affected by the constraint condition of bilayer h-BN sheet. The theory results match well with the simulation results. These findings establish elementary understandings of the mechanical behavior and vibration character of bilayer h-BN nanosheets under five different stacking modes, which could benefit its application in advanced nanodevices.

Experimental and Numerical Studies on the Vibration of Hybrid Composite with an Edge Crack

The hybrid composite plate is increasingly used in marine, robotic arm, and other applications. Edge crack is the greatest common fault in the structural systems during its working time. In this research, the successful fabrications of a hybrid composite of epoxy-based composites reinforced with bamboo and glass fibers have been done to examine the vibration characteristics under free vibration. First-order shear deformation theory is used to study the fundamental natural frequencies of asymmetrically hybrid composite beam with edge cracks for cantilever beam boundary conditions. Both numerical (FEM) using ABAQUS software and experimental investigations are done for the vibration analysis of unidirectional bamboo/glass fiber epoxy hybrid composite (BGHC) beam with edge cracks using fracture mechanics theory. It is observed that implications edge crack with different crack depth and various fiber weight ratio parameters have a major effect on vibration analysis. Also, it is clear that the natural frequencies of the hybrid composite are significantly affected due to the fiber weight ratio and crack depth of the hybrid composite materials. So, crack lengths are playing a vital role in the dynamic or vibration characteristics of the hybrid composite materials. The results of natural frequencies have been shown that they are decreased with an increase in crack depth and an increase in the bamboo fiber weight. Thus, BGHC-2 made from 30% bamboo and 10% glass fiber can be a viable candidate for applications that will produce good vibration properties. Generally, the first-third natural frequency of the hybrid composite beam decreases within the increase of bamboo fiber and decreases in glass fiber. The composite materials having 60% of epoxy with 20% bamboo and 20% glass fiber, and 60% of epoxy with 10% glass and 30% bamboo fibers, are observed with a gradual decrease in the values of natural frequencies with respect to increasing in crack depth. A fine agreement was accomplished between the simulation and experimental results. Therefore, the present approach can be used to identify cracks for mechanical health monitoring by linking the variation in natural frequencies of the hybrid composite beams.

LIQUEFACTION-INDUCED DAMAGE IN THE CITIES OF ISKENDERUN AND GOLBASI AFTER THE 2023 TURKEY EARTHQUAKE

<p>On February 6th 2023 two major earthquakes struck southeast Turkey, M7.7 Kahramanmaras and M7.6 Elbistan, respectively. Unfortunately, due to the impact of these catastrophic events more than 50 000 casualties and 35 000 collapsed buildings have been reported since then. The aim of the study is to demonstrate preliminary site response analysis and assessment of re-liquefaction potential of sites which have been affected by the earthquakes – especially the cities of Iskenderun and Golbasi. Both site-specific areas have clear evidences of liquefaction and lateral spreading events which imply the focus of the presented paper. A series of geophysical MASW and microtremor tests have been performed in order to determine shear wave velocities up to depth of 30 m as well as the fundamental natural frequency of the soil deposits. </p> <p>Moreover, samples have been collected from sand and silt ejecta in order to evaluate some basic physical properties – grain-size curves, specific gravity and plasticity parameters. On the basis of the obtained data seismic classification of the investigated sites according to current design codes has been made and in-depth distance to relatively stiff layer has been assumed. For the sake of evaluating risk of re-liquefaction the widely-used simplified stress-based approach to triggering assessment has been adopted considering some rules of the thumb (e.g., sieve analysis and plasticity properties evaluation). Lastly, post-liquefaction reconsolidation settlement and lateral displacement have been determined in terms of future earthquakes</p>

Supplementation of deep neural networks with simplified physics-based features to increase accuracy of plate fundamental frequency predictions

To improve predictive machine learning-based models limited by sparse data, supplemental physics-related features are introduced into a deep neural network (DNN). While some approaches inject physics through differential equations or numerical simulation, improvements are possible using simplified relationships from engineering references. To evaluate this hypothesis, thin rectangular plates were simulated to generate training datasets. With plate dimensions and material properties as input features and fundamental natural frequency as the output, predictive performance of a data-driven DNN-based model is compared with models using supplemental inputs, such as modulus of rigidity. To evaluate model accuracy improvements, these additional features are injected into various DNN layers, and the network is trained with four different dataset sizes. When evaluated against independent data of similar features to the training sets, supplementation provides no statistically-significant prediction error reduction. However, notable accuracy gains occur when independent test data is of material and dimensions different from the original training set. Furthermore, when physics-enhanced data is injected into multiple DNN layers, reductions in mean error from 33.2% to 19.6%, 34.9% to 19.9%, 35.8% to 22.4%, and 43.0% to 28.4% are achieved for dataset sizes of 261, 117, 60, and 30, respectively, demonstrating potential for generalizability using a data supplementation approach. Additionally, when compared with other methods—such as linear regression and support vector machine (SVM) approaches—the physics-enhanced DNN demonstrates an order of magnitude reduction in percentage error for dataset sizes of 261, 117, and 60 and a 30% reduction for a size of 30 when compared with a cubic SVM model independently tested with data divergent from the training and validation set.

Multi-scale design of composite material structures for maximizing fundamental natural frequency

This study deals with the topology optimization (TO) of linear elastic composite structures considering fundamental natural frequency maximization. The concept of functionally graded structures (FGS) is used for anisotropic composite material structure design to improve the structural dynamic performance. The rational approximation of material properties (RAMP) method was employed as an alternative to the solid isotropic material with penalization (SIMP) method to improve the convergence stability for eigen-frequency problem. In the SIMP method, the occurrence of local eigen-modes at low frequencies in the void region causes ill-convergence, whereas the RAMP method prevents local eigen-modes from occurring. The method of moving asymptotes is used as an optimizer. Numerical verification was performed for three structural types: cantilever beam, double-clamped beam, and four-pinned beam structure. In addition, for cantilever beam design, bi-objective optimization for maximizing fundamental natural frequency as well as minimizing structural compliance was performed using the adaptive weighting method that quantitatively reflects the designer's preference by adjusting the weight for each objective function at each iteration.

Numerical study on train-induced vibrations: A comparison of timber and concrete buildings

This paper deals with the prediction of train-induced building vibrations by using a numerical framework. The framework is based on a sub-structure approach, where a sequence of different models are used. The free-field ground vibrations and the track receptance are calculated using 2.5D technique where the railway track is represented by finite elements that couple to a dynamic stiffness of the underlying soil, which in turn is obtained from the Green’s function of a horizontally layered half-space using a layer transfer matrix approach. A planar multi-body model of the train, coupled to the track receptance, is used for calculating the train–track interaction forces as the train runs over an uneven rail. Finally, the building response to the incident wavefield is calculated using a 3D finite element model, accounting for the soil dynamic stiffness.The framework is used to evaluate the vibrations in two buildings with identical layout, one lightweight wooden building and one heavyweight concrete building, due to a passenger train passing by at two different speeds. It was found that the difference in response between the two buildings were small. Compared to the incident wavefield, an amplification of the response inside the building was found in frequency bands around the fundamental natural frequencies of the slabs; however for higher frequencies and in terms of the 1 s running RMS velocity the building response was reduced. Further, it was found that accounting for soil-structure-interaction, as opposed to simply enforcing the free-field displacements at the building foundations, significantly reduced the building response in terms of 1 s RMS velocity.

Fluid-solid interaction simulations of an aeroelastic square prism in sinusoidal oscillatory flows

This study numerically investigates the aerodynamic and aeroelastic characteristics of a square prism (aeroelastic model) and wind field around it in sinusoidal oscillatory flows (SOFs). The reliability of the fluid-solid interaction (FSI) simulation is validated by a free vibration test and wind tunnel tests in smooth flow and SOF. The effects of the amplitude and frequency of SOFs are studied at the mean wind speed of vortex-induced resonance. The results show that increasing the amplitude and frequency of SOFs will amplify the root mean square (RMS) along-wind and across-wind base shear forces of the aeroelastic model but decrease the RMS across-wind displacement at the top of the aeroelastic model. The spectral analysis of the base shear forces indicates that the influence of vortex shedding on the across-wind base shear force is reduced by either increasing the amplitude or increasing the frequency of SOFs. The mean and instantaneous wind fields around the aeroelastic model in SOFs and smooth flow are compared, and the wake characteristics of the aeroelastic model in SOFs are analysed by dynamic mode decomposition. It is observed that when the frequency of SOFs is 1.5 times as large as the fundamental natural frequency of the aeroelastic model, the regular vortex shedding process is substantially affected.

Vibration Performance Evaluation of Cold-Formed Steel and Plywood Composite Floors

In this study, experimental natural frequencies and mid-span deflections were used to assess the tested composite floors against vibration serviceability limit states adopted by different standards such as AS3623. It was found that existing design criteria have predicted inconsistent results. Further, the finite element (FE) model was created, updated, and validated against the experimental modal parameters. The validated numerical model examined how various design parameters, such as material properties, cross-section geometry, and achieved a degree of composite action, have influenced the vibration properties such as natural frequencies and mode shapes. It was shown that as the added mass became more dominant than the stiffness enhancement, the fundamental natural frequency of the floor system dropped. This was evident from the results of the plywood slab thickness and the added permanent load parameters. On the other hand, the fundamental natural frequency increased with thicker and deeper joist sections since they made the floor system stiffer.

The effect of the viscoelastic support and GRPL-reinforced foam material on the thermomechanical vibration response of piezomagnetic sandwich nanosensor plates

This paper investigates the vibration characteristics of a sandwich nanosensor plate composed of piezoelectric materials, specifically barium and cobalt, in the upper and lower layers, and a core material consisting of either ceramic (silicon nitride) or metal (stainless steel) foams reinforced with graphene (GPRL). The study utilized the novel sinosoidal higher-order deformation theory and nonlocal strain gradient elasticity theory. The equations of motion for nanosensor sandwich graphene were derived using Hamilton's principle, considering the thermal, electroelastic, and magnetostrictive characteristics of the piezomagnetic surface plates. These equations were then solved using the Navier method. The core element of the sandwich nanosensor plate can be represented using three distinct foam variants: a uniform foam model, as well as two symmetric foam models. The investigation focused on analyzing the dimensionless fundamental natural frequencies of the sandwich nanosensor plate. This analysis considered the influence of three distinct foam types, the volumetric graphene ratio, temperature variation, nonlocal parameters, porosity ratio, electric and magnetic potential, as well as spring and shear viscoelastic support. Furthermore, an analysis was conducted on the impact of the metal and ceramic composition of the central section of the sandwich nanosensor plate on its dimensionless fundamental natural frequencies. In this context, the use of ceramic as the central material results in a mean enhancement of 33% in the fundamental natural frequencies. In contrast, the incorporation of graphene into the core material results in an average enhancement of 27%. The thermomechanical vibration behavior of the nanosensor plate reveals that the presence of graphene-supported foam and a viscoelastic support structure in the core layer leads to an increase in thermal resistance. This increase is dependent on factors such as the ratio of graphene, porosity ratio of the foam, and parameters of the viscoelastic support. Metal foam or ceramic foam has been found to enhance thermal resistance when compared to solid metal or ceramic core materials. The analysis results showed that it is important to take into account the temperature-dependent thermal properties of barium and cobalt, which are piezo-electromagnetic materials, and the core layer materials ceramics and metal, as well as the graphene used to strengthen the core. The research is anticipated to generate valuable findings regarding the advancement and utilization of nanosensors, transducers, and nano-electromechanical systems engineered for operation in high-temperature environments.

Theoretical and Simulation Finite Element Modal Analysis of Rotating Cantilever Beam

The finite element method is used to carry out modal analysis of rotating cantilever beam. The virtual work method is then used to derive the stiffness matrix of a rotating beam element. The stiffness matrix of a rotating beam element is simply seen to be sum of the stiffness matrix of the non-rotating beam and an incremental stiffness matrix induced by rotation. This work presents a novel generalized incremental stiffness matrix that takes care of any element at any distance from the rotation axis. The established rotational stiffness matrix is then used together with consistent mass matrix (not affected by rotation) to carry out modal analysis of rotating cantilever beam. Theoretical computations were validated by simulations from ANSYS. It is seen that the contribution to modal frequencies and shapes of the blade of rotation via incremental stiffness matrix is very marginal at low rotational speeds. For exampleit is seen for a numerical case study that five-element model computes slight increase in fundamental natural frequency due to rotation relative to that of the non-rotating blade as 0.000195% for rotational speed of 300rpm. It is also seen that ten times increase in speed leads to about hundred times increase in rotational contribution meaning that it becomes more imperative to model rotation as speed of the blade increases. Points are also made regarding application to avoiding resonance of rotating cantilever beam.

Dynamic Stability of a Three-Layer Beam – Generalisation of the Sandwich Structure Theory

Abstract The work focuses on the dynamic stability problem of a simply supported three-layer beam subjected to a pulsating axial force. Two analytical models of this beam are developed: one model takes into account the non-linear hypothesis of cross-section deformation, and the other takes into account the standard “broken line” hypothesis. Displacements, strains and stresses for each model are formulated in detail. Based on the Hamilton principle, equations of motion are determined for each of these models. These systems of two differential equations for each model are approximately solved with the consideration of the axial pulsating force, and the fundamental natural frequencies, critical forces and the Mathieu equation are determined. Detailed studies are performed for an exemplary family of beams. The stable and unstable regions are calculated for the three pulsating load cases. The values of fundamental natural frequencies and critical forces of exemplary beams calculated from two models are compared.

Quantum computing for solid mechanics and structural engineering – A demonstration with Variational Quantum Eigensolver

Variational quantum algorithms exploit the features of superposition and entanglement to optimize a cost function efficiently by manipulating the quantum states. They are suitable for noisy intermediate-scale quantum (NISQ) computers that recently became accessible to the worldwide research community. Here, we implement and demonstrate the numerical processes on the 5-qubit and 7-qubit quantum processors on the IBM Qiskit Runtime platform. We combine the commercial finite-element-method (FEM) software ABAQUS with the implementation of Variational Quantum Eigensolver (VQE) to establish an integrated pipeline. Three examples are used to investigate the performance: a hexagonal truss, a Timoshenko beam, and a plane-strain continuum. We conduct parametric studies on the convergence of fundamental natural frequency estimation using this hybrid quantum-classical approach. Our findings can be extended to problems with many more degrees of freedom when quantum computers with hundreds of qubits become available in the near future.

Advanced Frequency Study of Thick FGM Cylindrical Shells by Using TSDT and Nonlinear Shear

For the advanced frequency study of thick functionally graded material (FGM) circular cylindrical shells, it is interesting to consider the extra effects of nonlinear coefficient term in third-order shear deformation theory (TSDT) of displacements on the calculation of varied shear correction coefficient. The formulation for the advanced nonlinear shear correction coefficient are based on the energy equivalence principle. The values of nonlinear shear correction coefficient are usually functions of nonlinear coefficient term of TSDT, power-law exponent parameter and environment temperature. The free vibration frequencies of thick FGM circular cylindrical shells are investigated with the simply homogeneous equation by considering that simultaneous effects of the TSDT, the nonlinear shear correction coefficient of transverse shear force and the two direction of mode shapes. The novelty is more important and reasonable for the thick FGM circular cylindrical shells, especially for the ratio of length to thickness is five by considering the effects of advanced nonlinear shear correction coefficient and nonlinear coefficient term of TSDT on the advanced calculation of fundamental first natural frequencies.

Topology optimization of extruded beams modeled with the XFEM for maximizing their natural frequencies

In this paper, an efficient topology optimization approach is developed for maximizing the fundamental natural frequency of extruded beams. Mass fraction and static compliance bounds are defined using inequality-type constraints in the optimization problem. An XFEM approach, previously proposed by the authors for analyzing beam elements, is extended herein to compute the natural frequencies of the beam. The method allows for 3D modeling of beams with a significant reduction in the number of degrees of freedom and therefore also yields efficient optimization procedure. This reduction is made possible by incorporating global enrichment functions in the longitudinal direction, which enables a significant reduction in the number of elements in that direction without loss of accuracy. A nonlinear optimization problem is formulated using continuous density-based design variables that represent the material distribution in the beam’s cross-section. The optimization problem is then solved using a gradient-based approach with analytical sensitivities. The well-known Solid Isotropic Material with Penalization (SIMP) method is used to acquire discrete solutions. We study the optimal design of short and long beams. It is shown that for short beams, localized vibration modes appear within the cross-section, leading to a significant distortion deformation mode of the cross-section. The optimized design of the long beam shows global deformation modes with an increase of 15% in the fundamental frequency compared with a non-optimized design consisting of a hollow rectangular cross-section with the same mass.

Quantifying the effect of end support restraints on vibration serviceability of mass timber floor systems: Testing

Design of mass timber floor systems is commonly governed by vibration serviceability due to high stiffness-to-weight ratio and low inherent damping of timber. Research and design practice have shown that static deflection under a concentrated load and fundamental natural frequency can be effective and robust indicators for vibration performance of mass timber floors. These design parameters are normally calculated by assuming simply supported conditions in existing design methods. However, such an assumption deviates from actual floor supports, especially in platform-framed buildings, and in-situ end support restraints have been widely recognized as a significant factor affecting the vibration performance. The purpose of the study is to quantify the influence of end support restraints on vibration serviceability of mass timber floors in platform construction through a comprehensive experimental program and analytical treatment. This paper is the first part and focus specifically on the experimental work on cross-laminated timber (CLT) panels. In particular, extensive laboratory tests have been conducted on different CLT floor panels with various end support restraints induced by top loads, self-tapping screws and steel angle brackets. The fundamental natural frequency and mid-span deflection under a concentrated load were measured for each end support configuration. The rotational restraint stiffness was determined by comparing results of restrained supports to those of simple supports and represented as end fixity factors. The analysis of test results shows that the CLT floor-to-wall connection exhibited inherent non-linear behaviour and such characteristic was more significant for higher top loads. Compared with screws and brackets, the top loads dominated the partially restrained effect but such dominance gradually diminished for lower-level top loads. In addition, support wall thickness notably impacts the support restraint. It was then suggested that the clear span could be used to determine deflection and frequency in the design, but further investigation is needed.

An exact method for free vibration of beams and frameworks using frequency-dependent mass, elastic and geometric stiffness matrices

The frequency-dependent mass, elastic and geometric stiffness matrices of an axially loaded Bernoulli-Euler beam are developed through rigorous application of symbolic computation. These three matrices are related to the corresponding dynamic stiffness matrix so that free vibration analysis of axially loaded beams and frameworks can be carried out in an exact manner by applying the Wittrick-Williams algorithm as solution technique. Representative results from the proposed theory are presented for different boundary conditions of beams and frameworks, carrying tensile and compressive loads. Comparative results from finite element method are also presented. The duality between the free vibration and buckling problems is captured in that when the compressive load in a beam or frame approaches its critical buckling load, the fundamental natural frequency tends to zero and thus, buckling can be thoughtfully interpreted as free vibration at zero frequency. The investigation has opened the possibility of including damping in free vibration analysis of beams and frameworks when applying the dynamic stiffness method.

Robustness evaluation of the damped outrigger structure

Robustness is defined as the insensitivity of the structure to uncertainties like earthquakes, fire, cyclones, explosions, tsunamis, etc. Robustness analysis of Kalman observer with robust controller for damped outrigger structure is studied to analyze its design performance in the presence of various effects like an earthquake. This is a novel innovative study undertaken in evaluating the robustness of the proposed controller to signify its performance in the field of structural control. The damped outrigger structure is modeled using the finite element approach by finding mode shapes, fundamental natural frequency, and period. The Kalman observer is modeled according to the requirement of the structure with the Riccati equation. The robust proportional–integral–derivative controller is designed according to the input disturbance with Ziegler–Nichols ultimate gain approach. The issue of deterioration in the system performance due to saturation was observed during the analytical investigation of the robust proportional–integral–derivative controller with Kalman observer, which has been addressed by the anti-windup approach. The robustness index of the structure is calculated using sensitivity and complementary sensitivity. The maximum amplitude ratio of the sensitivity for viscous damper-controlled structure is 1.4723 and the value decreases for the other controllers, with a minimum value of 1.0 for the proposed anti-windup robust proportional–integral–derivative controller with Kalman observer. Respectively, the percentage overshoot for the uncontrolled case is 23.4% that values decreasing for other controlled cases, with a minimum of 7.8% for proposed anti-windup robust proportional–integral–derivative controller with Kalman observer. This robustness index and performance indices discriminate the significant robust performance of the proposed robust proportional–integral–derivative controller with Kalman observer-based damped outrigger structure in comparison with other controlled and uncontrolled cases.

Identifying the Influence of Stacking Sequence on Mechanical and Vibration Properties of Bamboo/Glass‐Epoxy Composites

AbstractThe consumption of renewable materials such as natural fiber reinforced composites is highlighted in many engineering applications because of their degradable and environmentally friendly properties. Composite with dedicated natural fibers is limited their use to semi‐structural applications because of their hydrophobic nature and instability under dynamic load. Present research work is attempted to develop woven bamboo (B)/Glass (G) hybrid epoxy composites. The dedicated (BBBB and GGGG) and hybridized composites with different stacking sequences fabricated through the compression molding process and their mechanical and dynamic mechanical properties are investigated. In mechanical properties, the flexural strength and flexural modulus for the GBBG layered composites are best at 210 and 6256 MPa respectively, close to the dedicated glass composite. Impact strength at around 342–368 J m−1 is observed for all hybrids which are extremely good when compared to pure bamboo composite. Experimental modal analysis is also executed to evaluate the dynamic properties as fundamental natural frequencies and the damping ratio. The GBGB composite possesses a fundamental frequency of 25.1 Hz and a damping ratio of 0.0301. The damping of hybrid composites is improved by 96% compared to GGGG stacked composites. The experimental results are well in agreement with the analytical results. These developed bamboo/glass hybrid composites can be a better alternative composite material in semi‐structural applications.

Improvement of the low-frequency sound insulation performance of hollow double-leaf panels by inserting a bending panel

Air-borne sound transmission loss of building materials generally follows the mass law. Transmission loss decreases at lower frequencies and is minimal at the fundamental natural frequency f0. On the other hand, transmission loss increases below f0, and is controlled by stiffness. Moreover, the stiffer the materials, the higher the f0. In the other words, even without increasing the mass, increasing the stiffness can improve low-frequency transmission loss. We focus on the fact that flexible lightweight panels are easily stiffened by bending deformation. We conduct a basic study on the effect of inserting a bending panel into hollow double-leaf panels. Experimental results on single panels showed a correspondence between the increase of f0 by bending deformation and the improvement of low-frequency sound insulation performance. In the case of the double-leaf panels with the lightweight bending panel inserted inside, the stronger the bending, the more low-frequency sound insulation performance tended to be improved.

A DFT-based finite element model to study the elastic, buckling and vibrational characteristics of monolayer bismuthene

PurposeIn this paper, based on the density functional theory (DFT) and finite element method (FEM), the elastic, buckling and vibrational behaviors of the monolayer bismuthene are studied.Design/methodology/approachThe computed elastic properties based on DFT are used to develop a finite element (FE) model for the monolayer bismuthene in which the Bi-Bi bonds are simulated by beam elements. Furthermore, mass elements are used to model the Bi atoms. The developed FE model is used to compute Young's modulus of monolayer bismuthene. The model is then used to evaluate the buckling force and fundamental natural frequency of the monolayer bismuthene with different geometrical parameters.FindingsComparing the results of the FEM and DFT, it is shown that the proposed model can predict Young's modulus of the monolayer bismuthene with an acceptable accuracy. It is also shown that the influence of the vertical side length on the fundamental natural frequency of the monolayer bismuthene is not significant. However, vibrational characteristics of the bismuthene are significantly affected by the horizontal side length.Originality/valueDFT and FEM are used to study the elastic, vibrational and buckling properties of the monolayer bismuthene. The developed model can be used to predict Young's modulus of the monolayer bismuthene accurately. Effect of the vertical side length on the fundamental natural frequency is negligible. However, vibrational characteristics are significantly affected by the horizontal side length.