Search results for: Finite element method

Authors: Djarot B. Darmadi

Abstract:

The wider growing Finite Element Method (FEM) application is caused by its benefits of cost saving and environment friendly. Also, by using FEM a deep understanding of certain phenomenon can be achieved. This paper observed the role of material properties and volumetric change when Solid State Phase Transformation (SSPT) takes place in residual stress formation due to a welding process of ferritic steels through coupled Thermo- Metallurgy-Mechanical (TMM) analysis. The correctness of FEM residual stress prediction was validated by experiment. From parametric study of the FEM model, it can be concluded that the material properties change tend to over-predicts residual stress in the weld center whilst volumetric change tend to underestimates it. The best final result is the compromise of both by incorporates them in the model which has a better result compared to a model without SSPT.

Keywords: Residual stress, ferritic steels, SSPT, coupled-TMM.

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Authors: M. Brandt, A. Peniak, J. Makarovič, P. Rafajdus

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This paper deals with a novel approach of power transformers diagnostics. This approach identifies the exact location and the range of a fault in the transformer and helps to reduce operation costs related to handling of the faulty transformer, its disassembly and repair. The advantage of the approach is a possibility to simulate healthy transformer and also all faults, which can occur in transformer during its operation without its disassembling, which is very expensive in practice. The approach is based on creating frequency dependent impedance of the transformer by sweep frequency response analysis measurements and by 3D FE parametrical modeling of the fault in the transformer. The parameters of the 3D FE model are the position and the range of the axial short circuit. Then, by comparing the frequency dependent impedances of the parametrical models with the measured ones, the location and the range of the fault is identified. The approach was tested on a real transformer and showed high coincidence between the real fault and the simulated one.

Keywords: Fault, finite element method, parametrical model of transformer, sweep frequency response analysis, transformer.

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Authors: Aki Happonen, Adrian Burian, Erwin Hemming

Abstract:

Fixed-point simulation results are used for the performance measure of inverting matrices using a reconfigurable processing element. Matrices are inverted using the Cholesky decomposition algorithm. The reconfigurable processing element is capable of all required mathematical operations. The fixed-point word length analysis is based on simulations of different condition numbers and different matrix sizes.

Keywords: Cholesky Decomposition, Fixed-point, Matrixinversion, Reconfigurable processing.

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Authors: Kuan-Liang Liu, Eric Lee, Jung-Jyh Lee, Jyh-Ping Hsu

Abstract:

The electrical interaction between two axisymmetric spheroidal particles in an electrolyte solution is examined numerically. A Galerkin finite element method combined with a Newton-Raphson iteration scheme is proposed to evaluate the spatial variation in the electrical potential, and the result obtained used to estimate the interaction energy between two particles. We show that if the surface charge density is fixed, the potential gradient is larger at a point, which has a larger curvature, and if surface potential is fixed, surface charge density is proportional to the curvature. Also, if the total interaction energy against closest surface-to-surface curve exhibits a primary maximum, the maximum follows the order (oblate-oblate) > (sphere-sphere)>(oblate-prolate)>(prolate-prolate), and if the curve has a secondary minimum, the absolute value of the minimum follows the same order.

Keywords: interaction energy, interaction force, Poisson-Boltzmann equation, spheroid.

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Authors: Sumachaya Harnsukworapanich, Tetsuo Ichimori

Abstract:

The apportionment method is used by many countries, to calculate the distribution of seats in political bodies. For example, this method is used in the United States (U.S.) to distribute house seats proportionally based on the population of the electoral district. Famous apportionment methods include the divisor methods called the Adams Method, Dean Method, Hill Method, Jefferson Method and Webster Method. Sometimes the results from the implementation of these divisor methods are unfair and include errors. Therefore, it is important to examine the optimization of this method by using a bias measurement to figure out precise and fair results. In this research we investigate the bias of divisor methods in the U.S. Houses of Representatives toward large and small states by applying the Stolarsky Mean Method. We compare the bias of the apportionment method by using two famous bias measurements: the Balinski and Young measurement and the Ernst measurement. Both measurements have a formula for large and small states. The Third measurement however, which was created by the researchers, did not factor in the element of large and small states into the formula. All three measurements are compared and the results show that our measurement produces similar results to the other two famous measurements.

Keywords: Apportionment, Bias, Divisor, Fair, Simulation

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Authors: S. Barabas, A.Fota.

Abstract:

In this paper is study the possibility of successfully implementing of hollow roller concept in order to minimize inertial mass of the large bearings, with major results in diminution of the material consumption, increasing of power efficiency (in wind power station area), increasing of the durability and life duration of the large bearings systems, noise reduction in working, resistance to vibrations, an important diminution of losses by abrasion and reduction of the working temperature. In this purpose was developed an original solution through which are reduced mass, inertial forces and moments of large bearings by using of hollow rollers. The research was made by using the method of finite element analysis applied on software type Solidworks - Nastran. Also, is study the possibility of rapidly changing the manufacturing system of solid and hollow cylindrical rollers.

Keywords: Large bearings, Von Mises stress, hollow rollers, flexible manufacturing system

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Authors: Maryam Khazaei Pool, Lori Lewis

Abstract:

This is a summary of articles based on higher order B-splines methods and the variation of B-spline methods such as Quadratic B-spline Finite Elements Method, Exponential Cubic B-Spline Method Septic B-spline Technique, Quintic B-spline Galerkin Method, and B-spline Galerkin Method based on the Quadratic B-spline Galerkin method (QBGM) and Cubic B-spline Galerkin method (CBGM). In this paper we study the B-spline methods and variations of B-spline techniques to find a numerical solution to the Burgers’ equation. A set of fundamental definitions including Burgers equation, spline functions, and B-spline functions are provided. For each method, the main technique is discussed as well as the discretization and stability analysis. A summary of the numerical results is provided and the efficiency of each method presented is discussed. A general conclusion is provided where we look at a comparison between the computational results of all the presented schemes. We describe the effectiveness and advantages of these methods.

Keywords: Burgers’ Equation, Septic B-spline, Modified Cubic B-Spline Differential Quadrature Method, Exponential Cubic B-Spline Technique, B-Spline Galerkin Method, and Quintic B-Spline Galerkin Method.

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Authors: Tomoaki Hashimoto

Abstract:

Recent technological advance has prompted significant interest in developing the control theory of quantum systems. Following the increasing interest in the control of quantum dynamics, this paper examines the control problem of Schrödinger equation because quantum dynamics is basically governed by Schrödinger equation. From the practical point of view, stochastic disturbances cannot be avoided in the implementation of control method for quantum systems. Thus, we consider here the robust stabilization problem of Schrödinger equation against stochastic disturbances. In this paper, we adopt model predictive control method in which control performance over a finite future is optimized with a performance index that has a moving initial and terminal time. The objective of this study is to derive the stability criterion for model predictive control of Schrödinger equation under stochastic disturbances.

Keywords: Optimal control, stochastic systems, quantum systems, stabilization.

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Authors: I. Made Gatot Karohika, Shigeyuki Haruyama, Ken Kaminishi, Oke Oktavianty, Didik Nurhadiyanto

Abstract:

Contact width and contact stress are important parameters related to the leakage behavior of corrugated metal gasket. In this study, contact width and contact stress of three-layer corrugated metal gasket are investigated due to the modulus of elasticity and thickness of surface layer for 2 type gasket (0-MPa and 400-MPa mode). A finite element method was employed to develop simulation solution to analysis the effect of each parameter. The result indicated that lowering the modulus of elasticity ratio of surface layer will result in better contact width but the average contact stresses are smaller. When the modulus of elasticity ratio is held constant with thickness ratio increase, its contact width has an increscent trend otherwise the average contact stress has decreased trend.

Keywords: Contact width, contact stress, layer, metal gasket, corrugated, simulation.

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Authors: M. Ntšolo, D. Kalumba, N. Lefu, G. Letlatsa

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Rock mass damage due to shear tectonic activity has been investigated largely in geoscience where fluid transport is of major interest. However, little has been studied on the effect of shear zones on rock mass behavior and its impact on stability of rock slopes. At Letšeng Diamonds open pit mine in Lesotho, the shear zone composed of sheared kimberlite material, calcite and altered basalt is forming part of the haul ramp into the main pit cut 3. The alarming rate at which the shear zone is deteriorating has triggered concerns about both local and global stability of pit the walls. This study presents the numerical modelling of the open pit slope affected by shear zone at Letšeng Diamond Mine (LDM). Analysis of the slope involved development of the slope model by using a two-dimensional finite element code RS2. Interfaces between shear zone and host rock were represented by special joint elements incorporated in the finite element code. The analysis of structural geological mapping data provided a good platform to understand the joint network. Major joints including shear zone were incorporated into the model for simulation. This approach proved successful by demonstrating that continuum modelling can be used to evaluate evolution of stresses, strain, plastic yielding and failure mechanisms that are consistent with field observations. Structural control due to geological shear zone structure proved to be important in its location, size and orientation. Furthermore, the model analyzed slope deformation and sliding possibility along shear zone interfaces. This type of approach can predict shear zone deformation and failure mechanism, hence mitigation strategies can be deployed for safety of human lives and property within mine pits.

Keywords: Numerical modeling, open pit mine, shear zone, slope stability.

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Authors: G. F. Guimarães, A. R. de Faria, R. R. Rego, A. L. R. D’Oliveira

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One of the most relevant phenomena in manufacturing is the residual stress state development through the manufacturing chain. In most cases, the residual stresses have their origin in the heterogenous plastification produced by the processes. Although a few manufacturing processes have been successfully approached by numerical modeling, there is still lack of understanding on how these processes' interactions will affect the final stress state. The objective of this work is to analyze the effect of the grinding procedure on the residual stress state generated by a quasi-static indentation. The model consists in a simplified approach of shot peening, modeling four cases with variations in indenter size and force. This model was validated through topography, measured by optical 3D focus-variation. The indentation model configured with two loads was then exposed to two grinding procedures and the result was analyzed. It was observed that the grinding procedure will have a significant effect on the stress state.

Keywords: plasticity, residual stress, finite element method, manufacturing

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Authors: M. H. Kazemi, D. Salac

Abstract:

A semi-implicit phase field method for droplet evolution is proposed. Using the phase field Cahn-Hilliard equation, we are able to track the interface in multiphase flow. The idea of a semi-implicit finite difference scheme is reviewed and employed to solve two nonlinear equations, including the Navier-Stokes and the Cahn-Hilliard equations. The use of a semi-implicit method allows us to have larger time steps compared to explicit schemes. The governing equations are coupled and then solved by a GMRES solver (generalized minimal residual method) using modified Gram-Schmidt orthogonalization. To show the validity of the method, we apply the method to the simulation of a rising droplet, a leaky dielectric drop and the coalescence of drops. The numerical solutions to the phase field model match well with existing solutions over a defined range of variables.

Keywords: Coalescence, leaky dielectric, numerical method, phase field, rising droplet, semi-implicit method.

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Authors: Guang Zeng, Jin Huang, Zicai Li

Abstract:

In this paper, we research the standard 13-point difference schemes for solving the biharmonic equation. Heuristic method is applied to judging the stability of multi-level difference schemes of the biharmonic equation. It is showed that the standard 13-point difference schemes are stable.

Keywords: Finite-difference equation, computational stability, hirt method.

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Authors: Khaled Moaddy

Abstract:

In this paper, we present a fast and accurate numerical scheme for the solution of a Laplace equation with Dirichlet boundary conditions. The non-standard finite difference scheme (NSFD) is applied to construct the numerical solutions of a Laplace equation with two different Dirichlet boundary conditions. The solutions obtained using NSFD are compared with the solutions obtained using the standard finite difference scheme (SFD). The NSFD scheme is demonstrated to be reliable and efficient.

Keywords: Standard finite difference schemes, non–standard schemes, Laplace equation, Dirichlet boundary conditions.

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Authors: Hitham M. Tlilan

Abstract:

The effect of notch depth on the elastic new strainconcentration factor (SNCF) of rectangular bars with single edge Unotch under combined loading is studied here. The finite element method (FEM) and super position technique are used in the current study. This new SNCF under combined loading of static tension and pure bending has been defined under triaxial stress state. The employed specimens have constant gross thickness of 16.7 mm and net section thickness varied to give net-to-gross thickness ratio ho/Ho from 0.2 to 0.95. The results indicated that the elastic SNCF for combined loading increases with increasing notch depth up to ho/Ho = 0.7 and sharply decreases with increasing notch depth. It is also indicated that the elastic SNCF of combined loading is greater than that of pure bending and less than that of the static tension for 0.2 ≤ ho/Ho ≤ 0.7. However, the elastic SNCF of combined loading is the elastic SNCF for static tension and less than that of pure bending for shallow notches (i.e. 0.8 ≤ ho/Ho ≤ 0.95).

Keywords: Bar, notch, strain, tension, bending

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Authors: Swarniv Chandra, Basudev Ghosh

Abstract:

Using the quantum hydrodynamic (QHD) model for quantum plasma at finite temperature the modulational instability of electron plasma waves is investigated by deriving a nonlinear Schrodinger equation. It was found that the electron degeneracy parameter significantly affects the linear and nonlinear properties of electron plasma waves in quantum plasma.

Keywords: Amplitude Modulation, Electron Plasma Waves, Finite Temperature Model, Modulational Instability, Quantum Plasma.

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Authors: Marcelo Hayashi, João Lima, Bruno Chieregatti, Ernani Volpe

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The adjoint method has been used as a successful tool to obtain sensitivity gradients in aerodynamic design and optimisation for many years. This work presents an alternative approach to the continuous adjoint formulation that enables one to compute gradients of a given measure of merit with respect to control parameters other than those pertaining to geometry. The procedure is then applied to the steady 2–D compressible Euler and incompressible Navier–Stokes flow equations. Finally, the results are compared with sensitivities obtained by finite differences and theoretical values for validation.

Keywords: Adjoint method, optimisation, non–geometric sensitivities, boundary conditions.

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Authors: S. M. Abdur Razzak, M. A. Rashid, Y. Namihira, A. Sayeem

Abstract:

A simple microstructure optical fiber design based on an octagonal cladding structure is presented for simultaneously controlling dispersion and leakage properties. The finite difference method with anisotropic perfectly matched boundary layer is used to investigate the guiding properties. It is demonstrated that octagonal photonic crystal fibers with four rings can assume negative ultra-flattened dispersion of -19 + 0.23 ps/nm/km in the wavelength range of 1.275 μm to 1.68 μm, nearly zero ultra-flattened dispersion of 0 ± 0.40 ps/nm/km in a 1.38 to 1.64 μm, and low confinement losses less than 10-3 dB/km in the entire band of interest.

Keywords: Finite difference modeling, group velocity dispersion, optical fiber design, photonic crystal fiber.

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Authors: Indu Saini, Phool Singh, Vikas Malik

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A novel method based on Genetic Algorithm to solve the boundary value problems (BVPs) of the Falkner–Skan equation over a semi-infinite interval has been presented. In our approach, we use the free boundary formulation to truncate the semi-infinite interval into a finite one. Then we use the shooting method based on Genetic Algorithm to transform the BVP into initial value problems (IVPs). Genetic Algorithm is used to calculate shooting angle. The initial value problems arisen during shooting are computed by Runge-Kutta Fehlberg method. The numerical solutions obtained by the present method are in agreement with those obtained by previous authors.

Keywords: Boundary Layer Flow, Falkner–Skan equation, Genetic Algorithm, Shooting method.

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Authors: Brajesh Kumar Jha, Neeru Adlakha, M. N. Mehta

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Calcium [Ca2+] is an important second messenger which plays an important role in signal transduction. There are several parameters that affect its concentration profile like buffer source etc. The effect of stationary immobile buffer on Ca2+ concentration has been incorporated which is a very important parameter needed to be taken into account in order to make the model more realistic. Interdependence of all the important parameters like diffusion coefficient and influx over [Ca2+] profile has been studied. Model is developed in the form of advection diffusion equation together with buffer concentration. A program has been developed using finite volume method for the entire problem and simulated on an AMD-Turion 32-bit machine to compute the numerical results.

Keywords: Ca2+ profile, buffer, Astrocytes, Advection diffusion, FVM

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Authors: Naveed Ahmed, Gunar Matthies

Abstract:

We deal with the numerical solution of time-dependent convection-diffusion-reaction equations. We combine the local projection stabilization method for the space discretization with two different time discretization schemes: the continuous Galerkin-Petrov (cGP) method and the discontinuous Galerkin (dG) method of polynomial of degree k. We establish the optimal error estimates and present numerical results which shows that the cGP(k) and dG(k)- methods are accurate of order k +1, respectively, in the whole time interval. Moreover, the cGP(k)-method is superconvergent of order 2k and dG(k)-method is of order 2k +1 at the discrete time points. Furthermore, the dependence of the results on the choice of the stabilization parameter are discussed and compared.

Keywords: Convection-diffusion-reaction equations, stabilized finite elements, discontinuous Galerkin, continuous Galerkin-Petrov.

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Authors: Betül Gezer, Ahmet Tekcan, Osman Bizim

Abstract:

In elliptic curve theory, number of rational points on elliptic curves and determination of these points is a fairly important problem. Let p be a prime and Fp be a finite field and k ∈ Fp. It is well known that which points the curve y2 = x3 + kx has and the number of rational points of on Fp. Consider the circle family x2 + y2 = r2. It can be interesting to determine common points of these two curve families and to find the number of these common points. In this work we study this problem.

Keywords: Elliptic curves over finite fields, rational points on elliptic curves and circles.

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Authors: E. Abedi, A. Ahmadi Nadooshan, S. Salehi

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Modeling and vibration of a flexible link manipulator with tow flexible links and rigid joints are investigated which can include an arbitrary number of flexible links. Hamilton principle and finite element approach is proposed to model the dynamics of flexible manipulators. The links are assumed to be deflection due to bending. The association between elastic displacements of links is investigated, took into account the coupling effects of elastic motion and rigid motion. Flexible links are treated as Euler-Bernoulli beams and the shear deformation is thus abandoned. The dynamic behavior due to flexibility of links is well demonstrated through numerical simulation. The rigid-body motion and elastic deformations are separated by linearizing the equations of motion around the rigid body reference path. Simulation results are shown on for both position and force trajectory tracking tasks in the presence of varying parameters and unknown dynamics remarkably well. The proposed method can be used in both dynamic simulation and controller design.

Keywords: Flexible manipulator, flexible link, dynamicmodeling, end point.

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Authors: Z. Veselý, M. Honner, J. Mach

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The aim of the performed work is to establish the 2D and 3D model of direct unsteady task of sample heat treatment by moving source employing computer model on the basis of finite element method. Complex boundary condition on heat loaded sample surface is the essential feature of the task. Computer model describes heat treatment of the sample during heat source movement over the sample surface. It is started from 2D task of sample cross section as a basic model. Possibilities of extension from 2D to 3D task are discussed. The effect of the addition of third model dimension on temperature distribution in the sample is showed. Comparison of various model parameters on the sample temperatures is observed. Influence of heat source motion on the depth of material heat treatment is shown for several velocities of the movement. Presented computer model is prepared for the utilization in laser treatment of machine parts.

Keywords: Computer simulation, unsteady model, heat treatment, complex boundary condition, moving heat source.

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Authors: Daniel Tarrazó-Serrano, Sergio Pérez-López, Sergio Castiñeira-Ibáñez, Pilar Candelas, Constanza Rubio

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Zone Plates (ZPs) are used in many areas of physics where planar fabrication is advantageous in comparison with conventional curved lenses. There are several types of ZPs, such as the well-known Fresnel ZPs or the more recent Fractal ZPs and Fibonacci ZPs. The material selection of the lens plays a very important role in the beam modulation control. This work presents a comparison between two Fresnel ZP made from different materials in the ultrasound domain: Polylactic Acid (PLA) and brass. PLA is the most common material used in commercial 3D-printers due to its high design flexibility and low cost. Numerical simulations based on Finite Element Method (FEM) and experimental results are shown, and they prove that the focusing capabilities of brass ZPs and PLA ZPs are similar. For this reason, PLA is proposed as a Magnetic Resonance Imaging (MRI) compatible material with great potential for therapeutic ultrasound focusing applications.

Keywords: Fresnel zone plate, magnetic resonance imaging polylactic acid, ultrasound focusing.

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Authors: Jeries J. Abou-Hanna

Abstract:

Several techniques exist for determining stress-intensity factors in linear elastic fracture mechanics analysis. These techniques are based on analytical, numerical, and empirical approaches that have been well documented in literature and engineering handbooks. However, not all techniques share the same merit. In addition to overly-conservative results, the numerical methods that require extensive computational effort, and those requiring copious user parameters hinder practicing engineers from efficiently evaluating stress-intensity factors. This paper investigates the prospects of reducing the complexity and required variables to determine stress-intensity factors through the utilization of the stress gradient and a weighting function. The heart of this work resides in the understanding that fracture emanating from stress concentration locations cannot be explained by a single maximum stress value approach, but requires use of a critical volume in which the crack exists. In order to understand the effectiveness of this technique, this study investigated components of different notch geometry and varying levels of stress gradients. Two forms of weighting functions were employed to determine stress-intensity factors and results were compared to analytical exact methods. The results indicated that the “exponential” weighting function was superior to the “absolute” weighting function. An error band +/- 10% was met for cases ranging from a steep stress gradient in a sharp v-notch to the less severe stress transitions of a large circular notch. The incorporation of the proposed method has shown to be a worthwhile consideration.

Keywords: Fracture mechanics, finite element method, stress intensity factor, stress gradient.

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Authors: R. Haoui

Abstract:

The aim of this work is to determine the supersonic nozzle profiles used in propulsion, for the launchers or embarked with the satellites. This design has as a role firstly, to give a important propulsion, i.e. with uniform and parallel flow at exit, secondly to find a short length profiles without modification of the flow in the nozzle. The first elaborate program is used to determine the profile of divergent by using the characteristics method for an axisymmetric flow. The second program is conceived by using the finite volume method to determine and test the profile found connected to a convergent.

Keywords: Characteristic method, nozzles, supersonic flow, propellers.

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Authors: Shabbir Chowdhury, Japatosh Mondal

Abstract:

Photonic Crystal Fiber (PCF) uses are no longer limited to telecommunication only rather it is now used for many sensors-based fiber optics application, medical science, space application and so on. In this paper, the authors have proposed a microstructure PCF that is designed by using Finite Element Method (FEM) based software. Besides designing, authors have discussed the necessity of the characteristics that it poses for some specified applications because it is not possible to have all good characteristics from a single PCF. Proposed PCF shows the property of ultra-high birefringence (0.0262 at 1550 nm) which is more useful for sensor based on fiber optics. The non-linearity of this fiber is 50.86 w^-1km^-1 at 1550 nm wavelength which is very high to guide the light through the core tightly. For Perfectly Matched Boundary Layer (PML), 0.6 μm diameter is taken. This design will offer the characteristics of Nonzero-Dispersion-Shifted Fiber (NZ-DSF) for 450 nm waveband. Since it is a software-based design and no practical evaluation has made, 2% tolerance is checked and the authors have found very small variation of the characteristics.

Keywords: Chromatic dispersion, birefringence, NZ-DSF, FEM, non-linear coefficient, DCF, waveband.

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Authors: Petru Manescu, Joseph Azencot, Michael Beuve, Hamid Ladjal, Jacques Saade, Jean-Michel Morreau, Philippe Giraud, Behzad Shariat

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Organ motion, especially respiratory motion, is a technical challenge to radiation therapy planning and dosimetry. This motion induces displacements and deformation of the organ tissues within the irradiated region which need to be taken into account when simulating dose distribution during treatment. Finite element modeling (FEM) can provide a great insight into the mechanical behavior of the organs, since they are based on the biomechanical material properties, complex geometry of organs, and anatomical boundary conditions. In this paper we present an original approach that offers the possibility to combine image-based biomechanical models with particle transport simulations. We propose a new method to map material density information issued from CT images to deformable tetrahedral meshes. Based on the principle of mass conservation our method can correlate density variation of organ tissues with geometrical deformations during the different phases of the respiratory cycle. The first results are particularly encouraging, as local error quantification of density mapping on organ geometry and density variation with organ motion are performed to evaluate and validate our approach.

Keywords: Biomechanical simulation, dose distribution, image guided radiation therapy, organ motion, tetrahedral mesh, 4D-CT.

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Authors: Ali Raad Hassan

Abstract:

In this paper, an (irregular) case relating to base circle, root circle, and pressure angle has been discussed and a computer programme has been developed to simulate and plot spur gear tooth profile, including involute and trochoid curves based on the formulation of rack cutter using different values of pressure angle and profile shift factor and it gave the values of all important geometric parameters. The results showed the flexibility of this approach and versatility of the programme to draw many different cases of spur gear teeth of any module, pressure angle, profile shift factor, number of teeth and rack cutter tip radius. The procedure developed can be extended to produce finite element models of heretofore intractable geometrical forms, to exploring fabrication of nonstandard tooth forms also. Finite elements model of these irregular cases have been built using above programme, and modal analysis has been done using ANSYS software, and natural frequencies of these selected cases have been obtained and discussed.

Keywords: involute, trochoid, pressure angle, profile shift factor, natural frequency

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