Showing papers in "Archive of Applied Mechanics in 2008"
TL;DR: In this article, the governing equation of motion of gradient elastic flexural Kirchhoff plates, including the effect of in-plane constant forces on bending, is explicitly derived, and the resulting partial differential equation in terms of the lateral deflection of the plate is of the sixth order instead of the fourth, which is the case for the classical elastic case.
Abstract: The governing equation of motion of gradient elastic flexural Kirchhoff plates, including the effect of in-plane constant forces on bending, is explicitly derived. This is accomplished by appropriately combining the equations of flexural motion in terms of moments, shear and in-plane forces, the moment–stress relations and the stress–strain equations of a simple strain gradient elastic theory with just one constant (the internal length squared), in addition to the two classical elastic moduli. The resulting partial differential equation in terms of the lateral deflection of the plate is of the sixth order instead of the fourth, which is the case for the classical elastic case. Three boundary value problems dealing with static, stability and dynamic analysis of a rectangular simply supported all-around gradient elastic flexural plate are solved analytically. Non-classical boundary conditions, in additional to the classical ones, have to be utilized. An assessment of the effect of the gradient coefficient on the static or dynamic response of the plate, its buckling load and natural frequencies is also made by comparing the gradient type of solutions against the classical ones.
152 citations
TL;DR: In this article, the authors investigated the problem of the unsteady mixed convection peristaltic mechanism with temperature-dependent viscosity with thermal diffusion and diffusion-thermo effects.
Abstract: We investigate the problem of the unsteady mixed convection peristaltic mechanism. The flow includes a temperature-dependent viscosity with thermal diffusion and diffusion-thermo effects. The peristaltic flow is between two vertical walls, one of which is deformed in the shape of traveling transversal waves exactly like peristaltic pumping and the other of which is a parallel flat plate wall. The equations of momentum, energy, and concentration are subject to a set of appropriate boundary conditions by assuming that the solution consists of two parts: a mean part and a perturbed part. The solution of the perturbed part has been obtained by using the long-wave approximation. The mean part has been solved and coincides with the approximation of Ostrach. The mean part (zeroth order), the first order, and the total solution of the problem have been evaluated numerically for several sets of values of the parameters entering the problem. The skin friction, and the rate of heat and mass transfer at the walls are obtained and illustrated graphically.
111 citations
TL;DR: In this paper, an extension of Zhilin's direct approach to plates made of functionally graded materials is presented, where a conceptional different direction is connected with the direct approach in the plate theory.
Abstract: The classical plate theory can be applied to thin plates made of classical materials like steel. The first theory allowing the analysis of such plates was elaborated by Kirchhoff. But this approach was connected with various limitations (e.g., constant material properties in the thickness direction). In addition, some mathematical inconsistencies like the order of the governing equation and the number of boundary conditions exist. During the last century many suggestions for improvements of the classical plate theory were made. The engineering direction of improvements was ruled by applications (e.g., the use of laminates or sandwiches as the plate material), and so new hypotheses for the derivation of the governing equations were introduced. In addition, some mathematical approaches like power series expansions or asymptotic integration techniques were applied. A conceptional different direction is connected with the direct approach in the plate theory. This paper presents the extension of Zhilin’s direct approach to plates made of functionally graded materials.
96 citations
TL;DR: In this paper, the axisymmetric frictionless contact problem of a functionally graded coated half-space is investigated by using the transfer matrix method and Hankel integral transform technique, the problem is reduced to a Cauchy singular integral equation.
Abstract: The main interest of this study is a new method to solve the axisymmetric frictionless contact problem of functionally graded materials (FGMs). Based on the fact that an arbitrary curve can be approached by a series of continuous but piecewise linear curves, the FGM is divided into a series of sub-layers with shear modulus varying linearly in each sub-layer and continuous at the sub-interfaces. With this model, the axisymmetric frictionless contact problem of a functionally graded coated half-space is investigated. By using the transfer matrix method and Hankel integral transform technique, the problem is reduced to a Cauchy singular integral equation. The contact pressure, contact region and indentation are calculated for various indenters by solving the equations numerically.
83 citations
TL;DR: In this article, free vibration analysis of a rotating, double-tapered Timoshenko beam that undergoes flap-wise bending vibration is performed, where the parameters for the hub radius, rotational speed, shear deformation, slenderness ratio, and taper ratios are incorporated into the equations of motion.
Abstract: In this study, free vibration analysis of a rotating, double-tapered Timoshenko beam that undergoes flapwise bending vibration is performed. At the beginning of the study, the kinetic- and potential energy expressions of this beam model are derived using several explanatory tables and figures. In the following section, Hamilton’s principle is applied to the derived energy expressions to obtain the governing differential equations of motion and the boundary conditions. The parameters for the hub radius, rotational speed, shear deformation, slenderness ratio, and taper ratios are incorporated into the equations of motion. In the solution, an efficient mathematical technique, called the differential transform method (DTM), is used to solve the governing differential equations of motion. Using the computer package Mathematica the effects of the incorporated parameters on the natural frequencies are investigated and the results are tabulated in several tables and graphics.
62 citations
TL;DR: In this paper, a cylindrical cavity is modeled as a ramp-type heating of its internal boundary, which is assumed to be traction free, and the results are presented graphically for different values of the thermal relaxation times using the three different theories of generalized thermoelasticity.
Abstract: Thermoelastic interactions in an infinite orthotropic elastic medium with a cylindrical cavity are studied. The cavity surface is subjected to ramp-type heating of its internal boundary, which is assumed to be traction free. Lord–Shulman and Green–Lindsay models for the generalized thermoelasticity theories are selected since they allow for second-sound effects and reduce to the classical model for an appropriate choice of the parameters. The temperature, radial displacement, radial stress, and hoop stress distributions are computed numerically using the finite-element method (FEM). The results are presented graphically for different values of the thermal relaxation times using the three different theories of generalized thermoelasticity. Excellent agreement is found between the finite-element analysis and analytical and classical solutions.
51 citations
TL;DR: In this paper, a general treatment of the transient thermoelastic stresses in a rotating nonhomogeneous anisotropic solid under compressive initial stress is presented, and the system of fundamental equations is solved by means of a boundary element method (BEM).
Abstract: The present paper presents a general treatment of the transient thermoelastic stresses in a rotating nonhomogeneous anisotropic solid under compressive initial stress. The system of fundamental equations is solved by means of a boundary element method (BEM) and the numerical calculations are carried out for the temperature, displacement components and stress components. The results indicate that the effects of inhomogeneity and initial stress are very pronounced.
50 citations
TL;DR: In this paper, the effects of surrounding elastic medium, which is considered as a spring, defined by the Winkler model, and van der Waals forces from adjacent nanotubes are taken into account.
Abstract: In this study, free vibration of simply supported multi-walled carbon nanotubes (CNTs) embedded in an elastic medium was investigated by using the generalized shear deformation-beam theory (GSDBT). The effects of surrounding elastic medium, which is considered as a spring, defined by the Winkler model, and van der Waals forces from adjacent nanotubes are taken into account. Third-order shear deformation (TOSD) theory is used to study free vibration of a multi-walled carbon nanotube embedded in an elastic medium. Unlike Timoshenko beam theory, TOSD theory satisfies zero traction boundary conditions on the upper and lower surface of the structures, so there is no need to use a shear correction factor. Free vibration frequencies and amplitude ratios were obtained for various sides to thickness ratios and elastic medium effects and results are compared with previous studies. The results showed that significant difference exist between TOSD and Euler beam theory. It is also interesting to note that, although frequency parameter is increasing by increasing stiffness of embedded medium, amplitude ratios are insensitive to stiffness of embedded elastic medium.
47 citations
TL;DR: In this paper, an analytical solution of the plane constrained shear problem for single crystals exhibiting similar features is obtained and the comparison with the discrete dislocation simulation is provided, and the analytical solution is shown to have similar energy and dissipative thresholds for dislocation nucleation, Bauschinger translational work hardening, and size effect.
Abstract: Berdichevsky and Le have recently found the analytical solution of the anti-plane constrained shear problem within the continuum dislocation theory (CMT, Contin. Mech. Thermodyn. 18:455-467, 2007). Interesting features of this solution are the energetic and dissipative thresholds for dislocation nucleation, the Bauschinger translational work hardening, and the size effect. In this paper an analytical solution of the plane constrained shear problem for single crystals exhibiting similar features is obtained and the comparison with the discrete dislocation simulation is provided.
47 citations
TL;DR: In this article, the authors investigate the generation of thermal stresses in a nonhomogeneous anisotropic solid cylinder rotating about the z-axis at a constant angular velocity in the presence of a magnetic field.
Abstract: In the present paper, we investigate the generation of thermal stresses in a nonhomogeneous anisotropic solid cylinder rotating about the z-axis at a constant angular velocity in the presence of a magnetic field. The governing equations are solved numerically using the boundary-element method (BEM) and numerical results are obtained for the variation of the temperature, displacements, and stresses along x-axis. The effect of nonhomogeneity is investigated.
43 citations
TL;DR: In this article, the problem of radiation (both heave and sway) of water waves by a submerged sphere in deep as well as in uniform finite depth water with an ice cover was formulated using the multipoles method, with the ice cover being modeled as an elastic plate of very small thickness.
Abstract: Using the multipoles method, we formulate the problems of radiation (both heave and sway) of water waves by a submerged sphere in deep as well as in uniform finite depth water with an ice-cover, with the ice-cover being modelled as an elastic plate of very small thickness. In each case this leads to an infinite system of linear equations which are solved numerically by standard techniques. The added-mass and damping coefficients for a heaving and swaying sphere are obtained and depicted graphically against the wave number for various values of the radius of the submerged sphere and flexural rigidity of the ice-cover to show the effect of the presence of ice-cover on these quantities. When the flexural rigidity is taken to be zero, the numerical results for the added-mass and damping coefficient for water with a free surface are recovered.
TL;DR: In this paper, the boundary-layer flow of a viscous fluid over a nonlinear axisymmetric stretching sheet is analyzed and the partial differential equations governing the flow are reduced to an ordinary differential equation.
Abstract: An analysis is performed for the boundary-layer flow of a viscous fluid over a nonlinear axisymmetric stretching sheet. By introducing new nonlinear similarity transformations, the partial differential equations governing the flow are reduced to an ordinary differential equation. The resulting ordinary differential equation is solved using the homotopy analysis method (HAM). Analytic solution is given in the form of an infinite series. Convergence of the obtained series solution is explicitly established. The solution for an axisymmetric linear stretching sheet is obtained as a special case.
TL;DR: In this article, the analytical solution for a radially piezoelectric functionally graded rotating hollow shaft is presented, where the variation of material properties is assumed to follow a power law along the radial direction of the shaft.
Abstract: In the present paper, the analytical solution for a radially piezoelectric functionally graded rotating hollow shaft is presented. The variation of material properties is assumed to follow a power law along the radial direction of the shaft. Two resulting fully coupled differential equations in terms of the displacement and electric potential are solved directly. Numerical results for different shaft geometries with different profiles of inhomogeneity are also graphically displayed.
TL;DR: In this article, the stiffness of a bolt is calculated as a function of the elastic energy in the structure, whereby the definition of the displacements related to the stiffness is circumvented.
Abstract: Bolt connections are among the most important connections used in structures. The stiffnesses of the bolt and of the connected members are the primary qualities that control the lifetime of the connection. The stiffness of the bolt can be estimated rather easily, in contrast to the member stiffness, but with finite element (FE) and contact analysis, it is possible to find the stiffness of the member. In the case of many connections and for practical applications, it is not suitable to make a full FE analysis. The purpose of the present paper is to find simplified expressions for the stiffness of the member, including the case when the width of the member is limited. The calculation of the stiffness is based on the FE, including the solution to the contact problem, and we express the stiffness as a function of the elastic energy in the structure, whereby the definition of the displacements related to the stiffness is circumvented. The contact analysis is performed using a method where iterations are not necessary, and the results are compared to alternative available results. New practical formulas for the stiffnesses are suggested.
TL;DR: In this article, objectivity and material frame-indifference are systematically discussed, because changing the observer and changing the motion of a material with respect to an observer independent standard frame of reference have to be distinguished carefully.
Abstract: In this paper objectivity and material frame-indifference are systematically discussed, because changing the observer and changing the motion of a material with respect to an observer independent standard frame of reference have to be distinguished carefully. Objectivity and observer invariance of the physical laws and of the constitutive mappings are introduced. Semi-objectivity and objectivity of different time derivative operators are investigated. As examples, changing the observer in liquid crystal theory and changing the motion in linear heat conducting materials is considered.
TL;DR: In this article, a submerged cavitating jet has been used for the study of influences of material, exposure time and working fluid temperature on the erosion process, and the results of erosion are analyzed in detail.
Abstract: Experimental setup with a submerged cavitating jet has been used for the study of influences of material, exposure time and working fluid temperature on the erosion process. Each of the parameters has been varied separately, and the results of erosion are analyzed in detail. Additionally, comparison of experiments with nitrated and non-nitrated material has been made in order to study the enhancement (mostly reflected as the prolonged incubation time) of erosion resistance achieved by nitrating the specimen surface.
TL;DR: In this paper, the dispersion equation for SH waves in a monoclinic layer over a semi-infinite elastic medium with an irregularity was discussed and compared for the half-space without any irregularity.
Abstract: The present paper discusses the dispersion equation for SH waves in a monoclinic layer over a semi-infinite elastic medium with an irregularity. In the absence of the irregularity, the dispersion equation reduces to standard dispersion equation for SH waves in a monoclinic layer over an isotropic semi-infinite medium. The dispersion curves for different size of the irregularity are computed and compared for the half-space without any irregularity. It can be seen that the phase velocity is strongly influenced by the wave number and the depth of the irregularity.
TL;DR: In this article, a new state space formulation for the free vibration of circular, annular and sectorial plates is established by introducing two displacement functions and two stress functions, which can be separated into two independent catalogues and two kinds of vibrations can be readily found.
Abstract: New state space formulations for the free vibration of circular, annular and sectorial plates are established by introducing two displacement functions and two stress functions. The state variables can be separated into two independent catalogues and two kinds of vibrations can be readily found. Expanding the displacements and stresses in terms of Bessel functions in the radial direction and trigonometric functions in the circumferential direction, we obtained the exact frequency equation for the free vibration for some uncommon boundary conditions. Numerical results are presented and compared with those of FEM to demonstrate the reliability of the proposed method. A parametric investigation is also performed.
TL;DR: In this paper, a uniform cantilever beam under the effect of a time-periodic axial force is investigated and the stability of the system is investigated by a numerical method based on Floquet's theorem and an analytical approach resulting from a first-order perturbation.
Abstract: A uniform cantilever beam under the effect of a time-periodic axial force is investigated. The beam structure is discretized by a finite-element approach. The linearised equations of motion describing the planar bending vibrations of the beam structure lead to a system with time-periodic stiffness coefficients. The stability of the system is investigated by a numerical method based on Floquet’s theorem and an analytical approach resulting from a first-order perturbation. It is demonstrated that the parametrically excited beam structure exhibits enhanced damping properties, when excited near a specific parametric combination resonance frequency. A certain level of the forcing amplitude has to be exceeded to achieve the damping effect. Upon exceeding this value, the additional artificial damping provided to the beam is significant and works best for suppression of vibrations of the first vibrational mode of the cantilever beam.
TL;DR: In this article, a non-reversible friction model is proposed based on experimental results, which has the capability to include both hysteresis in the presliding regime and velocity hystereis (frictional memory effect or frictional lag) in the sliding regime.
Abstract: In this paper, based on experimental results, a new non-reversible friction model is proposed. The model has the capability to include both hysteresis in the presliding regime and velocity hysteresis (frictional memory effect or frictional lag) in the sliding regime. In this model, a differential function, which varies between −1 and +1 and has the characters of the Bouc–Wen model, is used to take the place of a sign function. This solution smoothes the transition from presliding to sliding and the hysteresis in the presliding regime can be described. In the sliding regime, this model has the form of a combination of a general velocity-dependent model with another acceleration-dependent part. This form allows a good description of velocity hysteresis in the sliding regime. The parameter identification procedure for this new model is also presented in this paper. The simulation results show the practicability and accuracy of the model in describing friction force both in the presliding and the sliding regimes.
TL;DR: In this article, a non-zero traction condition is introduced in piezoelectric crack problems with the unknown Coulombic traction acting on the crack surfaces, and an analytical solution under this condition is obtained by means of the generalized Stroh formalism and by accounting for the permittivity of medium inside the crack gap.
Abstract: The non-zero traction condition is introduced in piezoelectric crack problems with the unknown Coulombic traction acting on the crack surfaces. An analytical solution under this condition is obtained by means of the generalized Stroh formalism and by accounting for the permittivity of medium inside the crack gap. As the crack in such materials can be thought of as a low-capacitance medium carrying a potential drop, the Coulombic traction always pulls the two opposite surfaces of the crack together. It is proved that under relatively larger mechanical loading and relatively smaller electrical field, the Coulombic traction may be negligible and the previous investigations under the traction-free crack condition may be accepted in a tolerant way, otherwise the Coulombic traction may lead to some erroneous results with over 10% relative errors. It is also shown that, unlike the traction-free crack condition, the applied electric field does change the Mode I stress intensity factor (SIF) for a central crack in an infinite plane piezoelectric material, and in this way may significantly influence piezoelectric fracture. It is also concluded that the variable tendencies of the normalized SIF and the ERR against the applied electric field depend on the mechanical loading levels. This load-dependence feature may lead to a transformation of the normalized SIF and the ERR from an even functional dependence to an odd functional dependence on the applied electric field.
TL;DR: In this paper, the electroelastic coupling interaction between multiple screw dislocations and a circular inclusion with an imperfect interface in a piezoelectric solid is investigated, and the explicit expressions of image forces exerted on the piezel dislocation are derived by means of the complex-variable method.
Abstract: The electroelastic coupling interaction between multiple screw dislocations and a circular inclusion with an imperfect interface in a piezoelectric solid is investigated. The appointed screw dislocation may be located either outside or inside the inclusion and is subjected to a line charge and a line force at the core. The analytic solutions of electroelastic fields are obtained by means of the complex-variable method. With the aid of the generalized Peach–Koehler formula, the explicit expressions of image forces exerted on the piezoelectric screw dislocations are derived. The motion and the equilibrium position of the appointed screw dislocation near the circular interface are discussed for variable parameters (interface imperfection, material electroelastic mismatch, and dislocation position), and the influence of the nearby parallel screw dislocations is also considered. It is found that the piezoelectric screw dislocation is always attracted by the electromechanical imperfect interface. When the interface imperfection is strong, the impact of material electroelastic mismatch on the image force and the equilibrium position of the dislocation becomes weak. Additionally, the effect of the nearby dislocations on the mobility of the appointed dislocation is very important.
TL;DR: In this paper, the flow of an incompressible couple stress fluid in an annulus with local constriction at the outer wall is considered, and the effects of couple stress fluids parameters α and σ, height of the constriction (e), and ratio of radii (k) on impedance and wall shear stresses are studied graphically.
Abstract: The flow of an incompressible couple stress fluid in an annulus with local constriction at the outer wall is considered. This configuration is intended as a simple model for studying blood flow in a stenosed artery when a catheter is inserted into it. The effects couple stress fluid parameters α and σ, height of the constriction (e), and ratio of radii (k) on the impedance and wall shear stresses are studied graphically. Graphical results show that the resistance to the flow as well as the wall shear stress increases as the ratio of the radii increases and decreases as the couple stress fluid parameters increases.
TL;DR: In this article, a general model of a Rub-impact rotor-bearing system with initial permanent bow is set up and the corresponding governing motion equation is given, and the nonlinear oil-film forces from the journal bearing are obtained under the short bearing theory.
Abstract: A general model of a rub-impact rotor-bearing system with initial permanent bow is set up and the corresponding governing motion equation is given. The nonlinear oil-film forces from the journal bearing are obtained under the short bearing theory. The rubbing model is assumed to consist of the radial elastic impact and the tangential Coulomb type of friction. Through numerical calculation, rotating speeds, initial permanent bow lengths and phase angles between the mass eccentricity direction and the rotor permanent bow direction are used as control parameters to investigate their effect on the rub-impact rotor-bearing system with the help of bifurcation diagrams, Lyapunov exponents, Poincare maps, frequency spectrums and orbit maps. Complicated motions, such as periodic, quasi-periodic even chaotic vibrations, are observed. Under the influence of the initial permanent bow, different routes to chaos are found and the speed when the rub happens is changed greatly. Corresponding results can be used to diagnose the rub-impact fault in this kind of rotor systems and this study may contribute to a further understanding of the nonlinear dynamics of such a rub-impact rotor-bearing system with initial permanent bow.
TL;DR: In this paper, the instability of multiwalled carbon nanotubes (MWCNTs) induced by the moving fluid inside is investigated based on an elastic beam model, and the effect of the van der Waals (vdW) interaction between tubes is investigated and it is found that the vdW interaction can enhance the stability of MWCNTs in general.
Abstract: Based on an elastic beam model, the instability of multiwalled carbon nanotubes (MWCNTs) induced by the moving fluid inside is investigated. At critical flow velocities, the MWCNTs become unstable and undergo pitchfork bifurcation and subsequently Hopf bifurcation. These critical velocities are found to increase very quickly with respect to decreasing inner radius and are inversely proportional to the length-to-outer-radius ratio. The effect of the van der Waals (vdW) interaction between tubes is investigated and it is found that the vdW interaction can enhance the stability of MWCNTs in general, but the vdW interaction reduces the stability capacity of MWCNTs with very small inner radius.
TL;DR: In this article, the performance of a novel global collocation method for the eigenvalue analysis of freely vibrated elastic structures when either basis or shape functions are used to approximate the displacement field was investigated.
Abstract: This paper investigates the performance of a novel global collocation method for the eigenvalue analysis of freely vibrated elastic structures when either basis or shape functions are used to approximate the displacement field. Although the methodology is generally applicable, numerical results are presented only for rods in which one-dimensional basis functions in the form of a power series, as well as equivalent Lagrange, Bernstein or Chebyshev polynomials are used. The new feature of the proposed methodology is that it can deal with any type of boundary conditions; therefore, the cases of two Dirichlet as well as one Dirichlet and one Neumann condition were successfully treated. The basic finding of this work is that all these polynomials lead to results identical to those obtained by the power series expansion; therefore, the solution depends on the position of the collocation points only.
TL;DR: In this paper, the transversal forced vibrations of axially moving sandwich belts are described by coupled partial nonhomogeneous differential equations, and the partial differential equations are analytically solved.
Abstract: Based on the author’s previously published results for transversal free vibrations of axially moving sandwich belts described by coupled partial differential equations, which are derived and analytically solved, this paper contains new analytical results, for forced vibrations of the same system excited by transversal external excitation. The transversal forced vibrations of the axially moving sandwich belts are described by the coupled partial nonhomogeneous differential equations. The partial differential equations are analytically solved. Bernoulli’s method of particular integrals and Lagrange’s method of the variations of the constants are used.
TL;DR: In this paper, the existence and uniqueness theory for maximal monotone operators of the gephyroidal model is studied. But the convergence order of the scheme is not known.
Abstract: Persoz’s gephyroidal model, which consists of elementary rheological models (dry friction element and linear spring), can be covered by the existence and uniqueness theory for maximal monotone operators. Moreover, classical results of numerical analysis allow one to use a numerical implicit Euler scheme, with convergence order of the scheme equal to one. Some numerical simulations are presented.
TL;DR: In this article, a numerical approach is introduced to solve the viscoelastic flow problem of filling and post-filling in injection molding, where the governing equations are in terms of compressible, non-isothermal fluid, and the constitutive equation is based on the Phan-Thien-Tanner model.
Abstract: A numerical approach is introduced to solve the viscoelastic flow problem of filling and post-filling in injection molding. The governing equations are in terms of compressible, non-isothermal fluid, and the constitutive equation is based on the Phan–Thien–Tanner model. By introducing some hypotheses according to the characteristics of injection molding, a quasi-Poisson type equation about pressure is derived with part integration. Besides, an analytical form of flow-induced stress is also generalized by using the undermined coefficient method. The conventional Galerkin approach is employed to solve the derived pressure equation, and the ‘upwind’ difference scheme is used to discrete the energy equation. Coupling is achieved between velocity and stress by super relax iteration method. The flow in the test mold is investigated by comparing the numerical results and photoelastic photos for polystyrene, showing flow-induced stresses are closely related to melt temperatures. The filling of a two-cavity box is also studied to investigate the viscoelastic effects on real injection molding.
TL;DR: In this paper, numerical solutions of partial differential equations and an integro-partial differential equation were investigated to model transverse vibration of nonlinear strings via numerical solution of partial-differential equations.
Abstract: Modeling transverse vibration of nonlinear strings is investigated via numerical solutions of partial-differential equations and an integro-partial-differential equation. By averaging the tension along the deflected string, the classic nonlinear model of a transversely vibrating string, Kirchhoff’s equation, is derived from another nonlinear model, a partial-differential equation. The partial-differential equation is obtained via neglecting longitudinal terms in a governing equation for coupled planar vibration. The finite difference schemes are developed to solve numerically those equations. An index is proposed to compare the transverse responses calculated from the two models with the transverse component calculated from the coupled equation. A steel string and a rubber string are treated as examples to demonstrate the differences between the two models of transverse vibration and their deviation from the full model of coupled vibration. The numerical results indicate that the differences increase with the amplitude of vibration. Both models yield satisfactory results of almost the same precision for vibration of small amplitudes. For large amplitudes, the Kirchhoff equation gives better results.