Vibration Equation Research Articles

Symplectic superposition method for the free-vibrating problem of sigmoid functionally graded material rectangular thin plates clamped at four edges

The symplectic superposition method (SSM) is implemented to investigate the free vibration problem of sigmoid functionally graded material (FGM) rectangular thin plates clamped at four edges. Firstly, the vibration equation of the sigmoid FGM rectangular thin plate is transformed into Hamiltonian canonical equations. Then based on the analysis of boundary conditions of the plate, the vibration problem of the sigmoid FGM rectangular thin plate clamped at four edges is decomposed as two sub-problems simply supported on two opposite edges. And then based on the symplectic elasticity method (SEM), the series expansion solutions for the two sub-vibrating problems are obtained. Afterward, by superposing the series expansion solutions for the two sub-vibrating problems, a symplectic superposition solution of the freely vibrating sigmoid FGM rectangular thin plates clamped at four edges is obtained. Finally, we apply the obtained symplectic superposition solution to research the frequencies and vibration mode functions of some specific sigmoid FGM rectangular thin plates, and the impact of the aspect ratio a/b and material gradient index k on the vibration frequencies of the sigmoid FGM rectangular thin plates is studied.

Non-linear vibration and bifurcation analysis of Euler-Bernoulli beam under parametric excitation

This paper presents an analysis of the non-linear vibrations of beams, which play a crucial role in various industrial and construction structures. Understanding the transverse vibrations of beams and accurately determining their frequency response is essential for achieving optimal design and structural performance. The novelty of this study lies in conducting a transverse non-linear vibration analysis of a three-dimensional beam while considering the effect of mid-plane elongation. By incorporating this aspect into the analysis, the study aims to provide deeper insights into the dynamic behavior of beams subjected to non-linear effects. A multiple-time scale approach has been adopted to conduct this research. To verify the accuracy of the method as well as the accuracy of the outcomes gained from this method, a contrast has been made with the 4th-order Runge-Kutta technique, which indicates that the results obtained are acceptable. The frequency response of the beam indicates the presence of a phenomenon of splitting into two non-linear branches during the three-dimensional vibrations of the beam, as well as a hardening state in the frequency response as a result of stretching the middle plane of the beam. Furthermore, a parametric study was conducted in which different parameters were examined to determine the starting point of non-linear bifurcation. As a result, the damping coefficient and resonance deviation parameter are two factors that affect the preference for critical bifurcation over safe bifurcation. Furthermore, the stretching of the middle plane results in a higher non-linear term coefficient in the vibration equations of the beam, which increases the oscillation frequency of the beam.

Inverse problem of determining the time- dependent beam stiffness coefficient in the beam vibration equation using the finite difference method

In this study, we consider the inverse problem of determining the time-dependent beam stiffness coefficient q(t) and transverse bending vibrations u(x,t) of a homogeneous beam using the finite difference method. The problem is governed by a fourth-order partial differential equation, which is discretized in space and time using numerical techniques to obtain an accurate and stable solution. An additional condition in the form of an integral is also included to determine q(t). The results obtained for different discretization grid points are compared, and a simple test example is presented to demonstrate the accuracy and agreement of the numerical solutions with analytical solutions. The paper provides a comprehensive overview of the methodology used to solve the inverse problem, and presents a detailed analysis of the results obtained.

INVERSE BOUNDARY PROBLEM FOR THE EQUATION OF FORCED VIBRATIONS OF A CANTILEVER BEAM WITH INTEGRAL CONDITIONS

The paper studies the solvability of an inverse boundary value problem with an unknown time-dependent coefficient for the equation of forced vibrations of a cantilever beam with the integral. Bending transverse vibrations of a homogeneous beam under the action of an external force in the absence of rotational motion during bending are described by a fourth-order differential equation. The purpose of the work is to determine the unknown coefficient and solve the problem under consideration. THIS problem under consideration is reduced to an auxiliary equivalent problem. Next, the existence and uniqueness of a solution to the equivalent problem is proved using the contraction mapping principle. As a result, using equivalence, the uniqueness of the existence of the classical solution is proved.

Parameter influence analysis of stochastic resonance and stochastic P-bifurcation for the shape-memory alloy laminate

This paper investigates the transverse vibration of an axially moving shape-memory alloy (SMA) fiber hybrid laminations under the combined action of transverse harmonic excitation and stochastic disturbance. Considering the shape memory alloy (SMA) fiber volume fraction random field, the kinetic energy and strain potential energy of SMA laminates are solved, and the axial motion equation of SMA laminates is derived according to Hamilton principle, a non-dimensional differential equation of the transverse vibration of the axially moving SMA laminated plates is obtained by adopting the Galerkin integral method. Considering the volume-fraction stochastic field, the stochastic natural vibration of a SMA laminate is analyzed. Considering the stochastic resonance of the SMA laminate under the action of Gaussian white noise and harmonic excitation, the effects of geometric parameters, physical parameters of materials, and temperature on the stochastic resonance behavior of the SMA laminate is analyzed. Finally, the effect of the stochastic P-bifurcation phenomenon of the parameters on the steady-state response of the SMA laminate is analyzed.

Frequency Design of a Vertical Cantilever Beam Carrying Tip Mass

Previous research has revealed that vertical cantilever beam structures, after considering the effects of gravity, can exhibit a 1:3 internal resonance between two transverse modes. However, whether a similar phenomenon would still occur in the system when a proof body is attached to the free end of cantilever beams, and how to perform parameter design to deliberately induce internal resonance are still open issues. The present study conducts research on the natural frequencies of a vertical cantilever beam with a tip mass. Within the framework of the Euler–Bernoulli beam theory, Hamilton’s principle is utilized to derive the integral-partial differential equation of free vibration of the system. The corresponding frequency equation is then solved using the Chebyshev pseudospectral method, with an algebraic equation of the natural frequency obtained. Moreover, an explicit analytical expression for the fundamental frequency is obtained using parameter fitting, which is then used to reveal the relationship among parameters under the critical state of buckling and evaluate the extent of gravity’s impact on the fundamental frequency. The study finally investigates the ratio of the first two natural frequencies. A parameter design method is proposed, which derives a parametric curve in terms of the fundamental frequency, ensuring a 1:3 ratio between the first two natural frequencies. The free vibration time history at the midpoint of the beam is acquired by directly solving the system’s partial differential equation through the finite element method. Fourier transformation of this time history verifies that the fundamental frequency aligns with a predetermined value, and the ratio between the first two natural frequencies is 1:3, thus confirming the efficacy of the proposed method. It is found that the addition of a tip mass provides more flexibility in adjusting parameters to achieve internal resonance, which facilitates the miniaturization of actual structures.

Free vibration of cracked FGM Mindlin plate in fluid

The free vibration of cracked functionally graded material (FGM) plate in fluid is investigated in this paper. The material components are supposed following the power law and formulated by the Voigt model. The Mindlin plate theory (MPT) including the shear deformation effect is established and the physical neutral plane is adopted to simplify the derivation due to the in-plane displacements of this plane assumed as zero. A massless linear rotational spring model (LRSM) is established to simulate the through-width surface crack. The hydrodynamic pressure at the fluid-plate interface is derived by the potential flow theory, and is regarded as the added mass of cracked FGM plate during the analyses of fluid-plate coupled vibration. The derivation and discretization of vibrational equations for cracked FGM plates in fluid are respectively achieved though the Hamilton’s principle and differential quadrature (DQ) method. The frequencies and mode shapes are iteratively calculated with these values of the vacuum case. The influences of immersed depth, fluid density, gradient index, crack depth, crack location, length-to-width ratio, length-to-thickness ratio, and boundaries on vibration characteristics are conducted in the section of numerical results. The present vibration analyses of cracked structures can be applicable to avoid structural resonance and failure.

1/2 and 1/3 Subharmonic Resonance Behaviors of a Rotating FG-CNTRC Beam with Geometric Imperfections

This paper aims at investigating 1/2 and 1/3 subharmonic resonances of a rotating functionally graded carbon nanotube-reinforced composite (FG-CNTRC) beam in the presence of geometric imperfections. A general function is introduced to denote three types of imperfections containing sine, global, and local modes. At first, based on the von-Kármán geometric nonlinearity assumption, the forced vibration equation is set up. Then, the Galerkin method and multiple scale method are utilized to study subharmonic resonance behaviors. Finally, coupled effects of FG-CNTRCs, rotating motion, and geometric imperfections on subharmonic resonance responses are investigated. It is found that a specific range of the excitation amplitude and frequency motivates the subharmonic resonance behavior. The coupling of geometric imperfections and geometric nonlinearities generates the 1/2 resonance phenomena, and geometric nonlinearities result in the 1/3 resonance behavior. Both the imperfection mode and amplitude can affect the resonance response and resonance regions. Moreover, in contrast with the 1/3 subharmonic resonance, geometric imperfections produce greater influence on the 1/2 subharmonic resonance.

An analytical method for free vibrations of the fuel rod with non-uniform mass of small modular reactor

The intricate internal structure of fuel rods results in a non-uniform mass distribution, making it imperative to employ analytical methods for accurate assessment. The study utilizes Euler beam theory to derive the transverse vibration equation for beams with varying mass distribution. The approach involves transforming the non-uniform mass beam into a multi-segment beam with concentrated mass points. Modal function relationships between adjacent uniform segments are established based on continuous conditions at connection points. This transformation leads to the conversion of the variable coefficient differential equation into a nonlinear matrix equation. The Newton-Raphson method is then applied to calculate the characteristic equation and mode shapes, essential for determining natural frequencies. To validate precision, the results obtained are compared with those derived from the finite element method. Furthermore, the developed method is employed to assess the impact of gas plenum location and length on the natural frequency of fuel rods. The proposed methodology serves as a rapid design tool, particularly beneficial during the design phase of fuel rods with non-uniform mass distribution, aiding in configuring structural aspects effectively.

Coupled Vibration Analysis of Multi-Span Continuous Cable Structure Considering Frictional Slip

As important load-bearing structures, suspension cables have been widely used in suspension bridges, engineering ropeways, cable suspension systems and other special equipment. Their dynamic problems have always been a research hotspot. Especially for complex cable systems such as engineering ropeways and cable lifting equipment, there will be moving loads acting on multi-span continuous friction-slip cable structures, resulting in nonlinear coupled vibration. Therefore, few scholars have studied how to calculate the nonlinear coupling vibration effect between such moving loads and multi-span continuous cables considering friction slip. Therefore, this paper proposes the use of the combination of the direct stiffness method and the Newmark-β integration method to solve the nonlinear system of equations of motion, which can be derived from the coupled vibration response between the moving load and the main cable. The corresponding calculation program is prepared. Combined with the dynamic load test and simulation results of engineering cases, the correctness and reasonableness of the coupled vibration equations and the program can be verified through comparative analysis. The results show that the calculation results of the self-programmed program are in good agreement with the dynamic load test results, in which the maximum error of the vertical displacement in the span is −4.40% and 0.86%, and the error of the static calculation reaches −13.90%. The impact effect is more obvious when hoisting the weight out of the pulling cable, in which the impact coefficient of the main cable can be up to 2.0. The impact coefficient of the deviation of the cable tower is 4.0. During the traveling process of the moving load, the vertical downward deflection of the main cable at the action point is the largest, and the upward deflection is in the region of 0.2~0.8L from the action point.

The Approximate Analysis of Higher-Order Frequencies of Nonlinear Vibrations of a Cantilever Beam With the Extended Galerkin Method

Abstract The nonlinear vibrations of elastic beams with large amplitudes are frequently treated as a typical problem of an elastica. As the continuation of the analysis of the deformation of an elastica, the nonlinear vibration equation of the elastic beam in the rotation angle of the cross section has been established. Using the deformation function, the nonlinear equation with the inertia effect has been solved by the newly proposed extended Galerkin method (EGM). The solution to the vibration problem of the elastica is compared with earlier approximate solutions including the frequencies and mode shapes obtained by other methods, and the rotation angle and energy of each mode at the high-order frequency are also calculated. This solution procedure provides an alternative technique to the elastica problem by the EGM with possible applications to other nonlinear problems in many fields of science and technology.

Use of complex notation for solving vibrations equations

Complex numbers and complex functions are frequently utilized in solving equations of motion in free and harmonic responses of vibration systems. However, certain fundamental details are sometimes overlooked and many books and articles may present differential equations with inconsistencies, which can confuse meticulous readers. This confusion often arises because one side of the equations is in real form, while the other side is expressed in complex form. To address this issue, the present article offers examples from literature to highlight the perplexing situation and provides a didactic methodology, aimed at enhancing clarity and ensuring that readers are left with a thorough understanding of the subject matter.

STODOLA-VIANELLO ITERATION METHOD FOR FREE TORSIONAL VIBRATION ANALYSIS OF MONOSYMMETRIC BOX-BEAM BRIDGES

Box-beam bridges are used in large spans and wider decks due to their high strength and greater torsional and flexural stiffness. They are usually prone to vibration due to moving vehicular traffic. Their eigenfrequency analysis is a crucial aspect of their design in order to ensure that the natural frequency is not close to the excitation frequency to avert resonance failures. The free torsional vibration equation is a fourth order partial differential equation (PDE) with variable parameters. For prismatic cross-sections, homogeneous and isotropic materials the governing PDE have constant parameters. This paper explores the Stodola-Vianello iteration method (SVIM) for solving the PDE for isotropic, homogeneous prismatic box-beams. Harmonic response is assumed, decoupling the PDE to two equations, one in terms of time and the second an ordinary differential equation (ODE) in terms of space coordinates. The Stodola-Vianello method is used by the method of four successive integrations to express the ODE as an algebraic iteration problem with four constants of integration. The four boundary conditions are used to solve for the four constants of integration, thus making the problem determinate. Application of the boundary conditions results in the full determination of the iteration equations. For simply supported boundaries studied in the work, a trigonometric buckling shape function that satisfies all the boundary conditions is employed in the SVIM formula to obtain the next bucking modal shape function. The requirement for convergence is then used to establish the characteristic buckling equation from which the eigenvalues are obtained. The solution to the characteristic buckling equation gave the exact mathematical expression for the natural torsional frequency at the nth vibration mode. The frequencies are determined for the first eight vibration modes and compared with previously obtained values. It was found that the natural frequencies are identical with previously found values of the exact natural frequencies. The natural frequency obtained is exact because it satisfies the PDE and the boundary conditions at all points in the domain.

On the issues of criticism and further development of the national regulatory framework for the calculation of structures for seismic impacts

A scientific analysis of the provisions of the studies [1–4] devoted to the critical analysis of the state of regulatory documentation on the calculation of structures for earthquakes has been carried out. In these works, it is argued that the developers of domestic standards made a serious methodological mistake when dynamic coefficients, rather than reaction spectra, were used when setting the initial seismic information. It is believed that in foreign norms, as well as in the monographs of famous foreign scientists, the ground parts of structures during earthquakes are not affected by any external forces. This incompleteness of representations is present in the equation of horizontal vibrations in translated and domestic works and monographs, but in them, unlike the analyzed works, it is not stated that during earthquakes the ground parts of structures are not affected by any external forces. The reasons for the inadequacy of mathematical models of the interaction of foundations and structures are revealed and the equations of translational and rocking plane-parallel vibrations of a rigid structure on a pliable base are given, clearly taking into account the effect of gravity.

Simulation and experimental study on transverse nonlinear vibration of axially moving yarn

This article focuses on the nonlinear dynamics of transverse vibration of the axially moving yarn system. In this article, the motion of yarn in actual production is analyzed, and the transverse nonlinear vibration equation of the viscoelastic axially moving yarn system with Kelvin model is established by using Newton’s second law. In the modeling process, the viscoelasticity of yarn material and the periodic fluctuation of tension in the process of movement are considered. The partial differential control equations of the system are transformed into ordinary differential equations by the Galerkin method. In the first three modes, when the speed is low, the related trial functions occupy the main part, and with the increase of speed, the trial functions of other orders occupy more proportion in the modal functions. Through numerical analysis, it is found that when the moving yarn is subjected to external excitation, the lateral vibration displacement has an increasing trend, and it is found that the nonlinear dynamic phenomenon alternates with the change of system parameters, such as single periodic motion, double periodic motion, and multiple periodic motion. Finally, an experimental platform for lateral vibration of yarn transmission was built and its working principle introduced. The lateral vibration of axially moving yarn was studied from the experimental point of view. It was proved that there are abundant phenomena of periodic motion, period doubling motion, and chaotic motion in the axially moving yarn system, which provides an experimental reference for theoretical research of lateral nonlinear vibration of axially moving yarn.

Solution of some half-strip problems in quadratures for the string vibration equation

Solution of some half-strip problems in quadratures for the string vibration equation

Simulation and modeling of grinding surface topography based on fractional derivatives

The grinding surface’s 3-D topography has self-affinity. To more accurately simulate this characteristic, this paper proposes for the first time to construct a grinding surface simulation model based on fractional derivatives, including: utilizing the Weierstrass-Mandelbrot function to simulate the point cloud of the grinding wheel surface; further deriving the trajectory equation based on the He’s fractal derivative; the semi-inverse method is employed to construct the fractional-order variational principle of the vibration equation. Experiments show that the simulation model designed in this paper can predict the surface height more accurately, and is better than the traditional model in the calculation of roughness parameters Sa and Sq. The model is helpful to thoroughly understand the formation mechanism of grinding surface topography, and provides simulation data guidance for machine vision inspection of grinding surface in micron scale.

Damage Detection of Gantry Crane with a Moving Mass Using Artificial Neural Network

Gantry cranes play a pivotal role in various industrial applications, and their reliable operation is paramount. While routine inspections are standard practice, certain defects, particularly in less accessible components, remain challenging to detect early. In this study, first a finite element model is presented, and the damage is introduced using random changes in the stiffness of different parts of the structure. Contrary to the assumption of inherent reliability, undetected defects in crucial structural elements can lead to catastrophic failures. Then, the vibration equations of healthy and damaged models are analyzed to find the displacement, velocity, and acceleration of the different crane parts. The learning vector quantization neural network is used to train and detect the defects. The output is the location of the damage and the damage severity. Noisy data are then used to evaluate the network performance robustness. This research also addresses the limitations of traditional inspection methods, providing early detection and classification of defects in gantry cranes. The study’s relevance lies in the need for a comprehensive and efficient damage detection method, especially for components not easily accessible during routine inspections.

A swarm-intelligence based formulation for solving nonlinear ODEs: γβII-(2+3)P method

A reverse engineering approach is taken to develop efficient numerical techniques for solving nonlinear ODEs through the numerical solution of a particular problem in engineering. The problem is the dynamic analysis of a mechanical system under a severely irregular seismic signal or earthquake load. It is a challenging problem which requires strong formulation to be tackled. Its solution is a rich source of information that could be exploited through artificial intelligence (AI) and statistical computation. This work devises a new strategy to conduct such a task and construct a series of 3∼5 points high-precision time integrators. The integrators belong to single-step two-derivative class of methods. The interconnectivity relationships between the weights of the integrators are formulated by linear regression and heuristics. These relationships make it possible to generate a wide spectrum of ODE-solving techniques in a continuous weighting space. In an innovative attempt, strong Hermite interpolators are constructed to couple with the integrators and improve the accuracy of the presented algorithms. They precisely evaluate the instep points of the integrator. All these issues are appropriately collected in the presented methodology. Four classes of weighting rules are presented for the integrators: 1) Symmetry weighting rule (SWR) 2) Consistency weighting rule (CWR), 3) Fundamental weighting rule (FWR), and 4) Auxiliary weighting rule (AWR). The first two rules are adopted from outward form of the well-configured quadratures available in literature. The third and the most important one, the so-called FWR, is disclosed here for the given integrator. It formulates the correlation between the weights of an optimal integrator. Finding the min-error solution of the vibration equation guides us to the FWR. Evolutionary grey wolf optimizer (GWO), accompanied with a statistical multilinear regression, extracts the FWR. This research pioneers the application of swarm-intelligence in formulation of the FWRs for the given two-derivative integrator. Finally, all the weighting rules and Hermite interpolators are combined into some unified algorithms so-called γβII−q+rP algorithms to efficiently cope with initial value ODEs. Great performance of the presented methods is demonstrated through the numerical examples.

Solution of Some Problems for the String Vibration Equation in a Half-Strip by Quadratures

Solution of Some Problems for the String Vibration Equation in a Half-Strip by Quadratures