Transverse Vibration Analysis Research Articles

A semi-analytical method using auxiliary sine series for vibration and sound radiation of a rectangular plate with elastic edges

This paper proposes an efficient semi-analytical method using auxiliary sine series for transverse vibration and sound radiation of a thin rectangular plate with edges elastically restrained against translation and rotation. The formulation, constructed by two-dimensional sine and/or cosine series, can approximately express the bending displacement, and calculate vibration and sound radiation under excitation of point force, arbitrary-angle plane wave, or diffuse acoustic field with acceptable accuracy. It is also applied for baffled or unbaffled conditions. A post-process program is developed to predict vibrating frequencies and modes, mean square velocity spectrum, and sound transmission loss via reduced-order integrals of radiation impedances. The method is validated by experiment and simulation results, demonstrating accurate and efficient computation using a single program for transverse vibration and sound radiation of a plate under different elastic boundary conditions and different excitations. Formulas given in this paper provide a basis for the code development on transverse vibration and sound radiation analysis of thin plates.

Effect of axial rotation on free transverse vibration analysis of two-phase nanobeams

Effect of axial rotation on free transverse vibration analysis of two-phase nanobeams

Transverse vibration analysis and active disturbance rejection decoupling control of rain erosion whirling arm

The transverse vibration caused by the mass eccentricity of the whirling arm and the unbalanced magnetic pull of the motor rotor is a critical factor that affects the stability and safety of the rain erosion whirling arm. To investigate the transverse vibration characteristics of the rain erosion whirling arm during operation, a four-degree-of-freedom transverse vibration model of the rotor-rain erosion whirling arm is established using the lumped mass method. The differential equation of motion of the rotor-rain erosion whirling arm, considering the influence of mass eccentricity and unbalanced magnetic pull, is then established based on the Lagrange equation. This equation is numerically solved using the Runge-Kutta algorithm to investigate the axis trajectory and amplitude distribution of the rotor-rain erosion whirling arm. To reduce the fatigue damage caused by vibration, the active disturbance rejection decoupling control method is employed. Furthermore, the parameters of the extended state observer are adjusted using pole assignment and bandwidth. The simulation results reveal that the vibration amplitude of the rotating blade can be reduced to within 0.2 mm, referring to the BSS7393 test standard. Additionally, experimental research is conducted to compare the vibration of the rain erosion whirling arm before and after implementing the active disturbance rejection decoupling control. The results demonstrate that the active disturbance rejection decoupling control method effectively suppresses the vibration of the rain erosion whirling arm, illustrating its feasibility and effectiveness for system vibration control.

Theoretical and experimental transverse vibration analysis of a non-uniform composite helical tidal turbine foil

Recently, tidal energy has gained attention as an attractive renewable source of power generation. In this context, tidal turbine foils must operate in harsh conditions and extremely variable dynamic loads. Considering the expensive dynamic tests required for these large structures, developing validated dynamic models could be valuable for failure prediction and design purposes. In this paper, the free and forced vibration behaviour of the cantilevered part of a non-uniform helical tidal turbine foil made from the composite material is investigated theoretically and experimentally. In this regard, governing vibration equations of the foil with related boundary conditions are presented. The free vibration analysis of the foil is performed by solving a boundary-value problem, where a closed-form solution, based on the mode summation method, for the forced vibration of the foil is employed. Finally, the obtained numerical results (natural frequency and dynamic displacements) are compared with the corresponding results from experiments to validate the vibration equations and the employed solution. Acceptable agreement between numerical and experimental data confirms that the vibration model presented in this study can be employed to predict the dynamic response, parametric study, optimization, and design of the tidal foil without performing costly experiments.

Numerical Analysis of Flow-Induced Transverse Vibration of a Cylinder with Cubic Non-Linear Stiffness at High Reynolds Numbers

Numerical calculations were performed to study the vortex-induced vibration (VIV) of a circular cylinder, which was elastically supported by springs of linear and cubic terms. These simulations were conducted at high Reynolds numbers ranging from 4200 to 42,000. To simulate the cylinder’s motion and the associated aerodynamic forces, Computational Fluid Dynamics were employed in conjunction with dynamic mesh capabilities. The numerical method was initially verified by testing it with various grid resolutions and time steps, and subsequently, it was validated using experimental data. The response of cubic nonlinearities was investigated using insights gained from a conventional linear vortex-induced vibration (VIV) system. This 2D study revealed that both the amplitude and frequency of vibrations are contingent on the flow velocity. The highest output was achieved within the frequency lock-in region, where internal resonance occurs. In the case of a hardening spring, the beating response was observed from the lower end of the initial branch to the upper end of the initial branch. The response displacement amplitude obtained for the linear spring case was 27 mm, whereas in the cubic nonlinear case, the value was 31.8 mm. More importantly, the results indicate that the inclusion of nonlinear springs can substantially extend the range of wind velocities in which significant energy extraction through vortex-induced vibration (VIV) is achievable.

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.

Transverse free vibration analysis of thin sectorial plates by the weak form quadrature element method

The weak form quadrature element method is applied to free vibration analysis of thin sectorial plates with arbitrary vertex angles and boundary conditions. To tackle the strong stress singularity around the vertex, analytical displacement descriptions are introduced into the inner sectorial subdomain, while the outer annular subdomain is modeled by a single weak form quadrature thin plate element. The continuity on the interface between the two subdomains is enforced afterwards. Eventually, a generalized eigenvalue formulation is established after introducing Hamilton’s principle. The first six non-dimensional frequency parameters for various vertex angles and boundary conditions are obtained and compared with available results. Several typical free vibration modes are plotted. The accuracy, convergence rate, and computational cost of the present formulation are discussed at length.

Analysis of transverse free vibrations of rotating beams with variable cross-section using the point collocation method

In this article, a study of transverse free vibrations of rotating cantilever beams with variable cross-sectional area is carried out using the point collocation method. The study considers four thickness variation functions, including linear, parabolic, sinusoidal, and exponential types, for the beam’s cross-sectional area changes. The point collocation method uses Gauss-Legendre points to solve the equation of motion and determine the natural frequencies. To enhance the convergence rate, the initial guess function is based on the precise mode shapes of the stationary uniform beam. The effect of section functions and various rotational speeds on the structure’s natural frequencies is investigated. The outcomes of the investigation are compared with 3D finite element simulations and related published results. Through the analysis, the results show that the point collocation method is highly accurate, with a maximum error of 0.0025% compared to 3D finite element simulations. Moreover, the natural frequencies of the rotating cantilever beams decrease as the section functions become steeper. Finally, the increase in rotational speed causes the natural frequencies to increase in a non-linear manner, with the maximum increase rate occurring at a specific rotational speed.

Mass lumping schemes fitted to MLS-based numerical manifold method in vibration of plates with cutouts using CPT and FSDT

It is critical to accurately extract the natural frequencies in the transverse vibration of plates using the classical plate theory (CPT) and the first-order shear deformation plate theory (FSDT). The moving least squares (MLS) based numerical manifold method (MLS-based NMM) in conjunction with a suitable diagonally lumped mass matrix is a reasonable choice because it can naturally treat plates with cutouts and easily formulate an H2 regular approximation based on MLS interpolation, as well as mitigate shear locking issues for vibration analysis of thin plates based on FSDT. Moreover, the mass lumping techniques involving the row-sum method, the diagonal scaling method, and the mathematically rigorous manifold-based method are extended and derived in the unified framework of MLS-based NMM for transverse vibration analysis of plates. Furthermore, the positive-definiteness of the lumped mass matrix (LMM) generated by the manifold-based mass method is explained in MLS settings, and the row-sum method is proven to be a special case of the manifold-based method. A comprehensive comparative study of different LMMs is performed in accordance with the consistent mass matrix (CMM) on extensive numerical benchmarks. Numerical results demonstrate the convergence properties of LMMs compared with CMM in MLS-based NMM for plate vibration based on CPT and FSDT. It can be observed that LMMs yield comparable performance compared with CMM. The row-sum method obtains accurate results in most cases but can not guarantee the positivity. In contrast, the manifold method can guarantee the positive-definiteness of LMM and is more accurate than the diagonal scaling method in most cases.

Dynamic analysis of a plate system coupled through several nonlinear spring-mass couplers

Nowadays, plates have been widely used in various engineering fields, where nonlinear additional equipment is designed to control the vibration of plates. However, the current studies mainly concentrate on the single-plate system with nonlinear additional equipment. Seldom studies use nonlinear couplers to connect the plate system, making the working mechanism of nonlinear couplers on the vibration responses of the plate system unclear, limiting the application of nonlinear couplers on vibration control of the plate system. To explore the potential application of nonlinear couplers, this work introduces nonlinear spring-mass couplers to connect the plate system, where the transverse vibration analysis model of the plate system coupled through nonlinear couplers is established. The Galerkin truncation method (GTM) is used to predict vibration responses of the plate system coupled through nonlinear couplers. After ensuring the validation of numerical results, forced vibration responses of the plate system coupled by nonlinear couplers are investigated. According to numerical results, responses in the vibration of the plate system are influenced by nonlinear couplers greatly. Choosing suitable parameters for nonlinear couplers plays an important role in the vibration control of the plate system. Under a certain parameter range of nonlinear couplers, responses in vibration present complex characters. Under complex responses, the target energy transfer is present in the vibration responses of the plate system.

Transverse Vibration Analysis of a Self-Excited Beam Subjected to Delayed Distributed and a Singular Load Using Differential Transformation Method

Transverse Vibration Analysis of a Self-Excited Beam Subjected to Delayed Distributed and a Singular Load Using Differential Transformation Method

Comment on “Transverse vibration analysis of single-layered graphene sheet under magneto-thermal environment based on non-local plate theory [J. Appl. Phys. 116(16), 164303 (2014)]” and “Nonlocal elasticity based magnetic field affected vibration response of double single-walled carbon nanotube systems [J. Appl. Phys. 111(11), 113511 (2012)

The impact of a time-varying external magnetic field on carbon nanotubes has been studied using Maxwell’s equations in the mentioned articles. However, the Maxwell’s equations are erroneously presented in the papers. The incorrect Maxwell’s equations, their derivatives, and the Lorentz force equations have all been examined in this work. In this article, we have endeavored to ensure that the equations are formatted correctly and have made an effort to do so. We have made an effort to intuitively correct the equations’ errors. We did not comment on the other portions of the mentioned articles; instead, we restricted ourselves to the examination of the impact of Maxwell’s equations and Lorentz force equations.

GMLS-based numerical manifold method in mechanical analysis of thin plates with complicated shape or cutouts

We proposed a generalized moving least squares (GMLS) based numerical manifold method (NMM) and applied to investigate mechanical behaviors of thin plates with various shape and cutouts. The influence domains of nodes in generalized moving least squares (GMLS) interpolation are employed to construct the mathematical cover, which is specifically fitted for the fourth-order problems with rotational degrees of freedom. Moreover, the GMLS-based NMM can naturally treat plates with arbitrary complex geometry and openings without resorting to those complicated criteria and tailored operations while simplifying the physical cover generation process. In numerical study, a suitable mass lumping scheme is formulated and advised for transverse vibration analysis. Bending, vibration and buckling behaviors of thin plates with different shapes and cutouts are studied. Numerical results manifest that GMLS-based NMM can accurately model transverse mechanical behaviors.

Analytical Wave Propagation Method for Free and Forced Transverse Vibration Analysis of a System of Multiple Elastically Connected Beams

An analytical wave propagation approach is developed in this paper for the free and forced vibration of a system of multiple elastically connected beams for the first time. The beams of the system are continuously joined by a massless, linear, elastic layer which can be regarded as continuous spring. The coupled partial differential equations governing the vibration of the multi-beam system are established and decoupled by using a technic developed based on matrix theory. For the decoupled equations, a general “vibration” state is introduced into the symplectic dual system. By solving the symplectic eigenproblem and utilizing the wave propagation theory, the general “vibration” state can be analytically described in symplectic space. By using these analytical expressions and satisfying the physical boundary conditions of the system, the natural frequencies, mode shapes and forced responses can be obtained analytically and explicitly. In the numerical examples, free and forced transverse vibration of the two- and three-beam system with various combinations of boundary conditions are considered. The effectiveness of the present method is validated by comparing the present results with the analytical results from the literature and the results calculated by the finite element method.

Transverse vibration analysis of double-walled carbon nanotubes in an elastic medium under temperature gradients and electrical fields based on nonlocal Reddy beam theory

This work studies the mechanical behavior of double-walled carbon nanotubes (DWCNTs) under temperature gradients and electrical fields based on the nonlocal reddy beam model, focusing on the free vibration of DWCNTs. The DWCNTs include the interaction of van der Waals (vdW) forces between the outer and inner tubes, and they are supposed to be embedded in a Winkler-Pasternak foundation. Eringen’s nonlocal elasticity theory is taken into account to consider the small-scale effects. The governing equations are achieved according to Hamilton's principle and then discretized and solved using the generalized differential quadrature method. The impact of various parameters, such as nonlocal parameters, length and ratio of thickness-to-radius, Winkler, Pasternak coefficients and piezoelectric effects and radius to length ratio, has been studied. The outcomes show that with increasing outer radius of the nanotube, the first dimensionless frequency decreases. In addition, the results illustrate that when the length of the nanotube and the Winkler spring constant increase, the initial frequencies are increased. Moreover, the desirable value for ratios of L/r2 of DWCNTs in order to obtain the minimum and maximum dimensionless natural frequencies for Winkler spring model Kw is approximately 0.7 and 6.3, respectively, whereas this value for the shear constant of Pasternak Gp is about 1.25 and 4.75.

Modeling, analysis, and control of shaft transverse vibration from rotating systems through active bearing concept

Rotating parts are widely applied to mechanical systems such as pumped-storage hydroelectricity, nuclear power plant, machining tools and so on. While operating, they can be easily damaged or destroyed by unbalanced mass, bending, torsion and misalignment. In order to solve this problem, rotor vibration control can be conducted through active bearing concept. In this work, active bearing system which consists of piezo actuators and rubber grommets is proposed and applied to a rotating system motivated from a pumped-storage hydroelectricity, for performing active vibration control. The main point of this paper is to prevent damage or failure caused by harsh transverse vibration through active bearings. First, the rotating system is modeled by transfer matrix method (TMM) based on Euler-Bernoulli beam theory and in order to check accuracy of this model, the responses of TMM are compared with the responses from the finite element method (FEM). For implementing active control in real time, normalized least mean square (NLMS) algorithm is utilized. The results show that the proposed active bearing concept shows great performance on the attenuation of shaft transverse vibration.

Gas Adsorption Response of Piezoelectrically Driven Microcantilever Beam Gas Sensors: Analytical, Numerical, and Experimental Characterizations.

This work presents an approach for the estimation of the adsorbed mass of 1,5-diaminopentane (cadaverine) on a functionalized piezoelectrically driven microcantilever (PD-MC) sensor, using a polynomial developed from the characterization of the resonance frequency response to the known added mass. This work supplements the previous studies we carried out on the development of an electronic nose for the measurement of cadaverine in meat and fish, as a determinant of its freshness. An analytical transverse vibration analysis of a chosen microcantilever beam with given dimensions and desired resonance frequency (>10 kHz) was conducted. Since the beam is considered stepped with both geometrical and material non-uniformity, a modal solution for stepped beams, extendable to clamped-free beams of any shape and structure, is derived and used for free and forced vibration analyses of the beam. The forced vibration analysis is then used for transformation to an equivalent electrical model, to address the fact that the microcantilever is both electronically actuated and read. An analytical resonance frequency response to the mass added is obtained by adding simulated masses to the free end of the beam. Experimental verification of the resonance frequency response is carried out, by applying known masses to the microcantilever while measuring the resonance frequency response using an impedance analyzer. The obtained response is then transformed into a resonance frequency to the added mass response polynomial using a polynomial fit. The resulting polynomial is then verified for performance using different masses of cantilever functionalization solution. The functionalized cantilever is then exposed to different concentrations of cadaverine while measuring the resonance frequency and mass of cadaverine adsorbed estimated using the previously obtained polynomial. The result is that there is the possibility of using this approach to estimate the mass of cadaverine gas adsorbed on a functionalized microcantilever, but the effectiveness of this approach is highly dependent on the known masses used for the development of the response polynomial model.

Transverse forced nonlinear vibration analysis of a double-beam system with a supporting nonlinearity

To exploit the potential application of supporting nonlinearity in marine engineering, an attempt is made to establish the transverse forced vibration analysis model of a double-beam system supported by a spring-mass system that is nonlinear. This kind of vibration system consists of two beam sections, boundary supports, a coupling component, and a nonlinear spring-mass arrangement. The variational approach and the generalized Hamiltonian concept are used to develop the governing equations of such a double-beam system. The Galerkin truncation method (GTM) is a technique for obtaining the governing equations’ residual equations. By solving the associated residual equations numerically, the nonlinear responses of the double-beam system can be figured out. The GTM has good solidity and correctness in the prediction of the vibration system’s forced transverse vibration. The dynamic responses of the double-beam structure supported by a spring-mass system that is nonlinear are subtle to their initial calculation values. Appropriate parameters of the nonlinear support will subdue the level of vibration at the boundary of the double-beam system. In contrast, unsuitable parameters of the nonlinear support motivate complex dynamic responses of the double-beam system and harmfully influence the vibration repression at the boundary of the vibration system.

Dynamic Instability Analysis of Transverse Vibrations of Functionally Graded Rectangular Plates under Moving Masses

Dynamic Instability Analysis of Transverse Vibrations of Functionally Graded Rectangular Plates under Moving Masses

Analytical and Numerical Analysis of Injection Pump (Stepped) Shaft Vibrations Using Timoshenko Theory

Abstract The free transverse vibrations of shafts with complex geometry are studied using analytical methods and numerical simulations. A methodology is proposed for evaluating the results of a natural transverse vibration analysis as generated by finite element (FE) models of a shaft with compound geometry. The effectiveness of the suggested approach is tested using an arbitrarily chosen model of the injection pump shaft. The required analytical models of the transverse vibrations of stepped shafts are derived based on the Timoshenko thick beam theory. The separation of variables method is used to find the needed solutions to the free vibrations. The eigenvalue problem is formulated and solved by using the FE representation for the shaft and for each shaft-simplified model. The results for these models are discussed and compared. Additionally, the usefulness of the Myklestad–Prohl (MP) method in the field of preliminary analysis of transverse vibration of complex shaft systems is indicated. It is important to note that the solutions proposed in this paper could be useful for engineers dealing with the dynamics of various types of machine shafts with low values of operating speeds.