Effects of mechanical vibration on designed steel-based plate geometries: behavioral estimation subjected to applied material classes using finite-element method

A research subject in structural engineering is the problem of vibration under a loading object. The twodimensional (2D) model of a structure under loading is an example. In general, this case uses an object that is given a random frequency, which then causes various changes in shape depending on the frequencymodel. To determine the difference in performance by looking at the different forms of each mode, modal analysis with ANSYS was used. The samples to be simulated were metal plates with three variations of the model, namely, a virgin metal plate without any holes or stiffness, plates with given holes, and metal plates with stiffness on one side. The model was simulated with modal analysis, so that 20 natural frequencies were recorded. The sample also used different materials: lowcarbon steel materials (AISI 304), marine materials (AISI 1090), and ice-class materials (AR 235). Several randomfrequency models proved the deformation of different objects. Variations of sheet-metal designs were applied, such as pure sheet metal, giving holes to the sides, and stiffening the simulated metal sheet.


Introduction
In recent years, structural engineering has become a popular topic of discussion. The problem of the vibration of a structure under loading or a moving force, which is included in the problem of moving loading, has become a research topic because vibration is a common case of movingload problems, such as in railroads, overhead cranes, ballistic systems, magnetic discs, cable lines, building plates, and bridges. To achieve certain objectives, simulated twodimensional modeling and efficient analytical methods are required for the reliable design and maintenance of a structure. In addition, this modeling and these methods can be used to accurately predict the vibration characteristics of structures. In general, structures under moving loads are modeled as one-(1D) or two-dimensional (2D) structures.
In this regard, several analytical methods were proposed for the vibration analysis of the problem of moving loads in one-and two-dimensional models. Some of them are the modal-analysis [1][2][3][4], integral-transformation (ITM) [5][6][7], Galerkin [8,9], differential-quadrature [10,11], finitedifference [12,13], finite-element method (FEM) [14][15][16][17][18], dynamic-stiffness [19,20], and frequency-domain spectralelement (SEM) [21][22][23][24][25] methods. The finite-element method (FEM) is a numerical procedure that can be applied to solve a wide variety of problems in engineering and science. In general, this method is used to solve steady, transient, and linear and nonlinear problems in electromagnetic fields, structural analysis, and fluid dynamics [1]. The finite-element method has the main advantage of being able to handle all kinds of geometries and nonhomogeneous materials without the need to change computer-code formulations. The idea of this method is to break the problem into a large number of planes, each with a simple geometry, to facilitate problem solving. As a result, the domain breaks down into a number of small elements, and the problem goes from small but difficult to solve to large and relatively easy to solve. Through the process of discretization, linear algebra problems are formed with many unknowns. In an electromagnetic case, a discretization scheme, as implied by FEM, which implicitly combines most of the theoretical features of the analyzed problem, is the best solution for obtaining accurate results in problems with complex, nonlinear geometries [2,3]. This method can also be used for complex differential equations that are very difficult to solve.
To investigate the vibrational characteristics of complex structures subjected to moving loads, FEM is extensively used. The shape function used in FEM is independent of the vibration frequency of a structure. FEM generally requires the very fine discretization of structures to obtain reliable solutions, especially in the high-frequency range [26][27][28]. This paper analyzes a series of designed plate geometries that are set as representative of structural components. Several steel classes are applied to the plates in a finite-element simulation as consideration of various structural types, e.g., marine and offshore structures, and inland infrastructure. Vibrational analysis is the focus of the current work, which is directed to quantify the natural frequency characteristics of the designed plates.

Literature review 2.1 Introduction to finite-element method
The finite-element method (FEM) is process analysis carried out by dividing a certain quantity into smaller pieces to obtain a large-class engineering analysis solution. Mathematically, FEM is an approximate approach that serves to solve field problems. The finite-element method is also commonly referred to as finite-element analysis (FEA) [16,23]. One of the reasons for using FEA is because it is a powerful computational technique used to solve various engineering problems, especially those with complex geometries that are subject to general boundary conditions. Analysis was carried out with field variables that varied from point to point. Thus, this analysis has an infinite number of solutions in the domain [23]. This makes the problem need fairly complex analysis. For that reason, using FEA is necessary. In reality, the system comprises a number of parts -expression elements -that are not known in terms of the assumed function estimates in each element. For each element, a variational or weighted residual method is applied to create a systematic approximate solution [26]. These functions are usually called interpolation functions, and are then entered in terms of field variables at certain points, called nodes.
The nodes that are formed are connected to the connected elements and are usually located along the boundary of the element. This method is used as an analytical tool to solve engineering problems because the degree of flexibility in distinguishing irregular domains from finite elements. In general, the use of FEA is carried out to create new product designs and improve the development of existing products. This makes FEA popular for researchers to verify proposed designs prior to a prototyping or construction process. The goal is to find out the feasibility of a design produced for an application. FEA is very widely used in the structural field [15]. Structural analysis consists of linear and nonlinear models. The linear model considers simple parameters and assumes that the material does not undergo plastic deformation, while the nonlinear model assumes that the structure undergoes plastic deformation as a result of the initial load. In general, FEA uses three types of analysis, namely, 1D modeling to solve beam, rod, and frame elements; 2D modeling, which is useful for solving field-stress and plane-strain problems; and 3D modeling, which is useful for solving complex solid structures.

Implementation of FE calculation
Several years earlier, solutions for plate deflection under different boundary conditions were proposed by several researchers. Bhattacharya investigated plate deflection under static and dynamic loading using new finite-difference analysis. The actual mode shape of the plates of each node was found in this study, and this approach provides a fourth-order biharmonic equation that varies from node to node [29]. A mathematical approach to the large deflection of rectangular plates was also presented by Defu and Sheikh [30], who analyzed on the basis of second-degree and fourth-order Von Karman partial differential equations. In their research, lateral deflection with respect to the applied load was found, which made this solution usable in the direct practical analysis of plates under different boundary conditions [30]. In the same field, Bakker et al. studied a forecast analysis method for large deflections of rectangular thin plates with simple supported boundary conditions under the action of transverse loads. This approach was taken to find initial deflection shape and the total plate. From this analysis, the deflection behavior of the plate size under transverse load can be expressed as a function of the plane stiffness of the plate before buckling [31]. Furthermore, research was carried out by Liew et al., who developed the differential quadrature method and the harmonic differential quadrature method for the static analysis of three-dimensional rectangular plates. A study conducted by Liew et al. aimed to find the bending and bending of plates that were supported and clamped only at boundary conditions [32].
In the utilization of FEA, some researchers made use of it in solving the perforated-plate problem. Jain analyzed the effect of the ratio of a perforated plate D/A (D is hole diameter, and A is plate width) on the stress concentration and deflection factors in isotropic and orthotropic plates under transverse static loading [33]. The point put forward by Jain (2009) was related to and supported by research by Chaudhuri (1987), who presented the theory of stress analysis using the rhombic array of alternating method for several circular holes [34][35][36]. In that study, the effect of stress concentration on perforated laminated plates using the finite-element method was investigated. Then, Paul and Rao presented a theory for the evaluation of stress concentration factors for thick laminated plate and fiber-reinforced plastic (FRP) with the help of the Lo-Christensen-Wu highorder bending theory under transverse loading [37].
Furthermore, in the last few decades, authors developed examining plates with stiffeners. Although rigid rectangular plates are thoroughly studied, the application of stiffeners to round plates is not very popular. The deflection in isotropic coiled plates subjected to two stiffening rings under static pressure was studied by Steen and Byklum. The used methodology was with clamped boundary conditions during analysis. Previously, Troipsky worked on rigid plates under bending, stability, and vibration. Pape and Fox presented an infinite series approach to the correctness of the plate aspect ratio using a rigid elastic beam [36][37][38].
Further development was carried out by Xuan et al., who performed element refinement using the boundary integral method. Das et al. developed a fairly general classical method that could be applied to all boundary conditions. Meanwhile, Jain in his research presented analysis of stress concentrations and deflections on isotropic and orthotropic rectangular plates with circular holes under transverse statistical loading. His research assessed three types of elements to solve the problem of square triggers with various boundary and load conditions [33,39,40].
The plasticity characteristics and external lateral loading of an elastoplastic plate were developed by Moshaiov and Vorus using an integral equation formulation for bending an elastoplastic plate. Formulations for Kirchhoff plate analysis with subregions by varying thickness for the boundary element method were also developed by researchers such as Paiva and Aliabadi. The development can be used to analyze the floor of a building in such a way that the integral equation of the curvature boundary of the points located at the zone interface is inferred very easily, which allows for the bending moment to be ob-tained [41,42]. FEM is a numerical or computational technique that can be used to solve different field variables such as displacement, stress, strain, temperature, and electric charge that have certain boundary conditions [43,44]. The application of the finite-element method is exemplified simply in Figure 1, where the physical (real) system is assumed by a mathematical model, and the discretization of FEA was applied to it. After a good representation of the real physical system into the mathematical model, and increased discretization, the FEM solution approaches the exact solution to the mathematical model, called convergence in FEM analysis. Hence, when used effectively, FEM can enable innovations that are not possible. Different static and dynamic engineering problems can be solved using the finite-element method, ranging from simple beam structural-stress analysis (1D) or complex machines with very large capacities (3D), to dynamic responses under mechanical, electrical, magnetic, or thermal differences. The use of the finite-element method is very widespread in various industrial fields, such as the aerospace, aeronautical, defense, consumer-product, and industrial-equipment industries. In addition, with the rapid development of different computer-aided design (CAD) software and advanced computing systems, the application of FEM has penetrated into materials science, biomedical engineering, medicine, biology, and physics [22]. Table 1 summarizes some examples of applications using FEM. Table 2 provides different FEM application disciplines with degrees of freedom (DOF) and loading vectors. In this table, DOF is an unknown parameter that must be solved using Equation (1) which is the global displacement matrix [23].

Field
Application examples Solid/structural mechanics Wind-turbine blade design optimization, structure-failure analysis, crash simulation, nuclear-reactor component integrity analysis, beam and truss design and optimization, limit-load analysis Heat conduction Combustion engine, cooling and casting modelling, electronic cooling modelling Acoustic flow Seepage analysis, aerodynamic analysis of cars and airplanes, air-conditioning modelling of buildings Electronics/ electromagnetics Electromagnetic-interference suppression analysis, sensor and actuator field calculations, antenna-design performance predictions In general, the value of stiffness and mass on a structural plate is known in the form of a matrix. Stiffness and mass matrices are denoted by [K] and [M], respectively; then, the equation of motion of the system can be expressed as in Equation (1).
where {ü} and {u} are the acceleration and displacement vectors, respectively, so that in free vibration, the values of the acceleration and displacement vectors can be formulated according to Equations (2) and (3) as follows [24].
where {u} is the amplitude of the vector {u}; thus, Equation (1) can be substituted into Equation (4) as follows [24].
To obtain the characteristic equation of an attached structural plate, Equation (5) is used [24].
FEM is most widely used in linear and nonlinear static analyses. The various types of static problems that the FEM solves in this area include the analysis of elastic, elastoplastic, and viscoplastic beams, frames, plates, shells, and solid structures. Typically, static analysis includes the analysis of stress, strain, and displacement under static loading for one-, two-, or three-dimensional problems. In this chapter, the general theory of elasticity is discussed. In addition, the general formulation of the stiffness of 1D, 2D, and 3D elements is discussed with the detailed formulation of the stiffness matrix of the beam elements. After that, general solution methods of static analysis are shown for linear and nonlinear problems. The global equation for static analysis is the same as that in Equation (1), and solving for the static problem is exactly the same as the one mentioned above in the general FEM procedure.
The main objective of any stress analysis is to find the distribution of displacement and stress under static loading and limit conditions. The following equations are fulfilled if there is an analytical solution to a particular problem based on elasticity theory. Table 3 shows the types of equations in 1D, 2D, and 3D problems.
Problems that exist in real life are often solved by taking a mathematical approach. In addition to a mathematical approach, either experimenting by producing a prototype model with a certain scale or the original is also a solution to finding the solution to a problem. Mathematical modeling is a process of interpreting existing problems with the aim of simplifying and accelerating problem solving. This makes mathematical modeling both challenging and demanding, as the use of mathematics and computers to solve real-world problems is extensive and affects all branches of learning. Problem solving with a mathematical modeling approach is possible with software. A powerful finite-element piece of software for analyzing different problems is ANSYS. Various problems can be overcome with ANSYS such as elasticity, fluid flow, heat transfer, and electromagnetism. Nonlinear and transient analyzes can also be completed with the software. In general, ANSYS analysis has the following steps for problem solving: First, modeling, a process that includes the definition of system geometry and the selection of material properties. In this step, the user can draw a 2D or 3D representation of the problem. Second, meshing. This step involves discretizing the model according to predefined geometric elements [22][23][24]. This is followed by the retrieval of the result and solution, and this step involves applying boundary and load conditions to the system, and resolving problems. Lastly, post-processing, which involves plotting the nodal solution (unknown parameter), which may be displacement, stress, or reactive force.

Modal analysis and random vibration
The natural or resonant frequency is a fundamental resonance mode of a vibrating system. The natural frequency can also be defined as the frequency that vibrates when under a small impact load for a certain period of time. Important parameters that can represent the dynamic response of a test object in a vibration test are the natural frequency and mode shape of the object. To identify a dynamic characteristic such as natural frequency, the shape of the mode can be established using modal analysis [45,46]. Modal analysis is based on the fact that the vibrational response of a linear time variant dynamic system can be expressed as a linear combination of a series of simple harmonic motions called natural vibration mode. The natural mode of vibration is generally determined entirely by its physical properties such as mass, strength, damping, and spatial distribution; this is also because the natural mode of vibration is inherent in dynamic systems [47][48][49][50]. Modal analysis is often used to abstract the modal parameters of a system, including natural frequency, mode shape, and modal damping ratio. This parameter depends only on the system itself, but dominates the system response to excitation. Modal analysis is the fundamental response of analysis and is of increasing concern. Experimental modal analysis (EMA) can estimate the natural frequency, mode shape, and damping value of a measured structure [51]. The finite-element (FE) model structure can be validated by comparing the experimental natural frequency and mode shape with the results of FE model analysis. Modal analysis is a technology of linear analysis that is used to determine the natural-frequency structure and vibration modes. In modal analysis, the only effective load is the zero-displacement limit. If a certain degree of freedom (DOF) determines a nonzero displacement limit, the program replaces the degrees of freedom with a zero displacement limit [52,53].
Modal analysis is based on several basic equations with an overview of geometric problems, as shown in Figure 2 below. where E is Young's modulus, ρ is the density of the material, and v is Poisson's ratio. The plate geometry image has infinite degrees of freedom due to the determination of the location of each particle on the plate body, which has unlimited coordinates. In this case, deflection and linearity are assumed to be small values, so the equation for the bending motion of the plates can be found through Equation (6) below [54].
where w is the transverse deflection, ∇ 2 is Laplacian, and Equation (8) is the plate flexural rigidity. Deflection is a function of x and y spatial coordinates, and from time function t. If the equation of motion is filled with form deflection functions, then the equation of motion can be written as follows.
where ∅ (x, y) is a function satisfying the plate boundary conditions, and q (t) is the corresponding general coordinate. By adding the equations of motion and separating the variables, Equations (10) and (11) can be obtained [54].
where ω is the natural frequency. The solutions of Equations (10) and (11) are solutions to infinite sets of eigenvalues and functions (or eigenvectors). In this case, the boundary conditions determine the natural frequency, and the initial conditions determine the contribution of each mode according to the vibration of the plate. To determine the natural frequency, several methods can generally be used, such as the Rayleigh, Rayleigh-Ritz, and FEM methods, shown in Equations (12), (14), and (15), respectively [54].
where Vmax is the maximal velocity that can be formulated according to Equation (13), a is an unknown constant, and λn is the frequency parameter. In Equation (14), Rayleigh-Ritz, the natural frequency is written in the form of ωn [54].
where ω is the natural circular frequency and f is the natural frequency. In Equation (15) of the FEM method, the equation of motion for the structure is formulated according to Equations (1-5).
Random-vibration analysis is used to determine the response of the structure under random loading. ANSYS uses the power spectral density (PSD) spectrum as analysis of random vibrations from the load input. Power spectral density is a probability statistical method and the rootmean-square value of a random variable, including the measurement of random-vibration energy and frequency information. The power spectrum can be in the form of displacement, velocity, acceleration or power spectral density, and in other forms [55][56][57][58][59].

Design sketch
The sheet-metal scheme used in this study is shown in Figure 3. Figure 3a shows a metal-sheet scheme without stiffener or holes, Figure 3b is a metal sheet with holes, and Figure 3c is a metal sheet with a given stiffener. This was performed to determine the effect of having a stiffener and holes in the metal sheet with the same material. In this case, piercing a hole into the metal sheet obviously makes it the lightest material compared to the others, while the metal sheet with two stiffeners in a cross-sectional manner is the heaviest material. However, as Figure 3 shows, stiffener application was carried out with very small dimensions, so that the weight difference between the three material variations was not too far apart.
The three variations of the design sketch were simulated to determine which metal sheet was more effective from various perspectives, such as material weight, performance, and level of difficulty in manufacturing. The performance of each metal sheet was examined using finite-element analysis, which was randomly assigned a frequency.

Materials
The material to be used was one type of low-carbon steel material (AISI 304), marine material (AISI 1090), and ice class material (AR 235). Each type of material was simulated in the same way to examine the difference in results against a given random frequency. Table 5 shows the properties of each material. The properties of each material were then set in the ANSYS software according to the material data sheet.

Low-carbon steel
Carbon steel is a type of alloy steel consisting of iron (Fe) and carbon (C) elements, where iron is the basic element and carbon is the main alloying element. In the steelmaking process, the addition of other chemical elements, such as sulfur (S), phosphorus (P), silicon (Si), manganese (Mn), and other chemical elements is in accordance with the desired properties of the steel. Low-carbon steel is steel with carbon-element content in a steel structure of less than 0.3% C. This low-carbon steel has high toughness and ductility, but low hardness and wear resistance. In general, this type of steel is used as raw material for the manufacture of building structural components, building pipes, bridges, car bodies, and others.

Marine materials
Marine materials generally use high-strength low-alloy steel materials. In this case, the material used is a type of HSLA steel material, namely, AR 235 Abrasion Resisting Steel. Abrasion Resisting Steel AR 235 is medium carbon and high manganese. This type of steel is tough and is wearresistance ductile enough to allow for specific machining. The chemical composition of this type of steel gives a Brinell hardness of about 235 under as-roll conditions and tensile strength of about 115,000 psi. Abrasion-resistant steel has 2 to 10 times the service life of mild steel when used in scraper blades, mixers, manure-moving equipment, loaders, conveyors, scoop gear, pull-conveyor bottoms, spouts, shovels, mine screens, troughs, hoppers, truck body dumps, concrete buckets, fan blades, ore jumps, tail sluices, bucket lips, rock screens, loading channels, agitator oars, dredge pump liners, grinding pans, liner plates, and sluice gates. This steel, if not too cold, takes a 90 ∘ bend to a reasonable radius with a thickness of up to about 3/8 inches without any fracture by being gently bent (in degrees) until forming is complete. For more difficult forming and all heavier gauge forming, the gauge must be heated and shaped while hot. Heat forming should take place at an ambient temperature of 1500 ∘ F. If the steel is allowed to cool slowly, its abrasive resistance qualities are not lost, nor are there any cracks or distortion. High-carbon steels such as this grade should not be worked by any method when it is very cold. This grade, due to its high hardness and toughness, is rather difficult for machines to work with. However, ordinary high-speed tools are more than capable of machining when needed.

Ice-class materials
Ice-class material is generally applied in the shipping sector, and is also known as polar-class material. This material is usually used for ship hulls with certain requirements. The layered structure is a rigid plate element that is in contact with the hull of the ship and is subject to ice loads. This requirement applies to an inboard level that is lower than the height of the adjacent truss or parallel supports, or 2.5 times the depth of the frame cutting the layered structure. The thickness of the lining and the slightest amount of installed stiffening is such that the level of attachment of the required ends for the shell frame must be ascertained. Class associations have various rules covering requirements and qualifications on ships intended to operate in icy waters, a set of rules made up of several ice notations. Meanwhile, notation generally depends on the geographic area of operation, and differs with respect to operational capability and structural strength. Depending on the ship's destination and area of operation, the owner is responsible for selecting the appropriate ice-class notation on the ship. Initially, the elastic approach and structural capacity definition put forward by the Finnish-Swedish Ice Class Rules (FSICR) regarding ships were used to ensure safe operations in the Baltic Sea during winter. In this case, prescribed notations are those used to determine the minimal requirements for engine power and ice reinforcement for the ship, assuming that icebreaker assistance is available when needed. Initially the regulations were aimed at operating merchant ships crossing ice routes. All polar-class vessels require effective protection against abrasion caused by ice and against corrosion. The most important part of this protection is applied to all surface coatings of polar-class ships. Polar-class ves-sels have a minimal corrosion/abrasion addition tc = 1.0 mm applied to all internal structures within the hull area reinforced by ice, including layered members adjacent to shells, and rigid nets and flanges.

Geometry and boundary condition
Three different sheet-metal designs with the same material were simulated against a given random vibration. Further- more, 20 modes of total deformation for each sheet were seen in value, but not all forms of the total deformation were shown. Two form modes of total deformation were shown. This mode was taken because it represents a change in the shape of each design. Each metal-sheet design had the same main dimensions of 180 × 180 mm. The first design was a metal sheet without any variation in either holes or stiffener, the second design was a perforated metal sheet with 16 holes and a diameter of 15 mm in each hole, and the third design was a sheet metal with two stiffeners as shown in Figure 3a-c with a thickness of 2 mm. The total deformation of each metal sheet was obtained through modal analysis simulation in ANSYS. In addition to using modal analysis, a harmonic-response simulation  model was also established in ANSYS to obtain the frequency value for the amplitude of the metal sheet. In performing the harmonic-response simulation process, one side of the metal sheet was set as the fixed mode, and the sheet metal was given a constant force of 300 N. Figure 4a shows the dimensions of the design sketch used in the simulation, while Figure 4b is a design application model represented by a design scheme.

Results and discussion
The natural frequency of each sheet-metal design was obtained through modal analysis in ANSYS. Natural frequencies are shown in Tables 6 and 7, which are graphs and values in Hz. There were differences in the natural-frequency values of each sheet-metal design. Some examples are natural frequencies in Modes 3 and 12 for sheet metals. In Mode 3, the sheet metal without holes and a stiffener had a natural-frequency value of 333.29 Hz, the sheet metal with holes had a natural-frequency value of 312.71 Hz, while the sheet metal with a stiffener had a natural-frequency value of 430.23 Hz. The sheet metal with stiffener application had the largest natural-frequency value in Mode 3. However, in Mode 12, the largest natural-frequency value was achieved by the sheet-metal design without holes and stiffener, and the natural-frequency value for the sheet metal with a stiffener was even smaller than the natural-frequency value of the sheet metal with holes. The natural-frequency value for the sheet metal without holes and stiffener reached 2201.5 Hz, the naturalfrequency value for the sheet metal with holes was 1751.4 Hz, and the sheet metal with a stiffener had the lowest value of 1732.7 Hz because Modes 3 and 12 had different deformations, which Table 8 shows. Table 8 also shows the contours of pressure distribution and shape that occurred in Modes 3 and 12 for each design. Figure 4 shows a graph of the frequency and amplitude that occurred in all the models with the same material: (a) virgin metal sheet, (b) sheet metal with holes, and (c) sheetmetal model with stiffener application. Figure 5 shows that the largest amplitude occurred in the metal-sheet model with a hole, while the sheet-metal model with a stiffener was the model with the lowest amplitude. The amplitudes of the models of the virgin metal sheet, metal sheet with holes, and metal sheet with stiffener were 188, 218, and 73.6 mm, respectively. Each model experienced maximal amplitude at different frequencies. For the virgin metal sheet, it occurred at a frequency of 52 Hz; for the metal sheet with holes, it occurred at a frequency of 50.5 Hz; and for the sheet metal with a stiffener, it occurred at a frequency of 82 Hz. Conversely, at the same frequency, amplitude was different. This shows that the three models had different natural frequencies due to design changes. Figure 6 is a picture of the total deformation shown on the basis of its contour with the same classification of material types, namely, marine material or AISI 1090. Figure 6a-c show that each model variation showed signifi-  stiffener occurred at the corners of both sides of the model; deformation in the middle did not look bad. In Figure 6b, the sheet-metal model with holes looks worse than all the other models, with the four worst deformation points in this model. Deformation concentrations occurred in almost every part of this model. This was due to the holes in this model, so that in the section of the path between the two holes, there was a fairly bad concentration of deformation. Figure 6c shows that the sheet-metal model with a stiffener was the best because the worst deformation only occurred on a small scale at the corners of both sides of the model. The center and whole of the model showed better deformation because there were two reinforcements along the sheet-metal model. As a result, the sheet metal became more rigid. Figure 7 shows the total deformation contours of the same model with different materials. The used material was ice-class material AR 235. Although the results were very However, when comparing all the materials that were simulated -low-carbon steel, marine material, and iceclass material in the same mode and variation -the iceclass material had the worst maximal deformation. The maximal deformation that occurred in the 12th mode of the virgin metal sheet model was 158.5, while the maximal deformation that occurred in the low-carbon steel and marine material with the same mode and model was 155.59 and 157.99, respectively. This was because of differences in material properties.

Conclusions
Modal-analysis simulation with the resulting random vibration showed that the different designs or models given to each metal sheet changed its characteristics and, especially, its natural frequency. As a result, the performance of the three models was different. In addition to design models that can change the characteristics of sheet metal, different types of materials also have different characteristics, which is indicated by the different natural frequencies for each material with the same simulation process and model. Good or bad performance depends on the application of the material and model. However, capital analysis using FEA or FEM proved that variation in the model was very significant. In this case, it was evidenced by the first natural mode frequency generated for each different model with the same material; the virgin metal sheet, hole, and stiffener models were 52.50, 50.57, and 82.02 Hz, respectively. Meanwhile, the frequency of the first natural mode in the same model with different materials, low-carbon steel, marine materials, and ice-class materials was 82.02, 84.94, and 85.21 Hz, respectively. Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.