Vibration Equation Research Articles

Dynamic characteristics of novel damping sandwich composite open cylindrical shell: From theory to simulation

Based on the first-order shear deformation shell theory, the free vibration model of Multilayer Damping Sandwich Composite Open Cylindrical Shell (MDSCOCS) under different lay-up angles is established. The Navier double series and Rayleigh-Ritz method are used to solve the free vibration equation under four sided simply supported boundary conditions. The theoretical solutions are compared with the existing literature and finite element simulation results respectively to verify the validity of theoretical calculation model. The influences of the thickness, position, shell opening angle, composite lay-up angle and other factors on the natural frequency and loss factor of the structure under different number of core layers are investigated in detail. The analysis shows that the damping loss factor of the composite structure is greatly improved by the addition of multilayer damping sandwich, and the dynamic performance of the composite structure is optimized, compared with that of single-layer damping sandwich composite structure. It provides a theoretical basis for the design of multilayer damping sandwich composite open cylindrical shell.

Analytical solution for the free transverse vibration of an elastically connected annular plate system with discontinuities

Due to the performance requirements, structural discontinuities are inevitable in engineering structures. Free vibration of an elastically connected annular plate system with discontinuities has not yet been reported in the literature. In this study, an analytical method is developed for the free vibration of an annular plate system with discontinuities along its radius. The annular plates are connected continuously using an elastic layer. The coupled vibration equations of the annular plate system were decoupled into a series of uncoupled general “vibration” equations which are then transferred into a symplectic dual system. The general “vibration” state can then be analytically described in terms of waves utilizing the elastic wave theory. Using the analytical wave modes and satisfying the compatibility and boundary conditions at the discontinuities, a frequency equation can be obtained analytically. In numerical examples, the free vibrations of systems consist of two and three annular plates were investigated. The accuracy of the proposed method is verified by comparison with literature and finite element method (FEM).

Voltage-controlled non-axisymmetric vibrations of soft electro-active tubes with strain-stiffening effect

Material properties of soft electro-active (SEA) structures are significantly sensitive to external electro-mechanical biasing fields (such as pre-stretch and electric stimuli), which generate remarkable knock-on effects on their dynamic characteristics. In this work, we analyze the electrostatically tunable non-axisymmetric vibrations of an incompressible SEA cylindrical tube under the combination of a radially applied electric voltage and an axial pre-stretch. Following the theory of nonlinear electro-elasticity and the associated linearized theory for superimposed perturbations, we derive the nonlinear static response of the SEA tube to the inhomogeneous biasing fields for the Gent ideal dielectric model. Using the State Space Method, we efficiently obtain the frequency equations for voltage-controlled small-amplitude three-dimensional non-axisymmetric vibrations, covering a wide range of behaviors, from the purely radial breathing mode to torsional modes, axisymmetric longitudinal modes, and prismatic diffuse modes. We also perform an exhaustive numerical analysis to validate the proposed approach compared with the conventional displacement method, as well as to elucidate the influences of the applied voltage, axial pre-stretch, and strain-stiffening effect on the nonlinear static response and vibration behaviors of the SEA tube. The present study clearly indicates that manipulating electro-mechanical biasing fields is a feasible way to tune the small-amplitude vibration characteristics of an SEA tube. The results should benefit experimental work on, and design of, voltage-controlled resonant devices made of SEA tubes.

A novel 3D train–bridge interaction model for monorail system considering nonlinear wheel-track slipping behavior

Variable speed operation of the train cause easily the wheel-track slipping phenomenon, inducing strong nonlinear dynamic behavior of the suspended monorail train and bridge system (SMTBS), especially under an insufficient wheel-track friction coefficient. To investigate the coupled vibration features of the SMTBS under variable speed conditions, a novel 3D train–bridge interaction model for the monorail system considering nonlinear wheel-track slipping behavior is developed. Firstly, based on the D’Alembert principle, the vibration equations of the vehicle subsystem are derived by adequately considering the nonlinear interactive behavior among the vehicle components. Then, a high-efficiency modeling method for the large-scale bridge subsystem is proposed based on the component mode synthesis (CMS) method. The vehicle and bridge subsystems are coupled with a spatial wheel-track interaction model considering the nonlinear wheel-track sliding behavior. Furtherly, by a comprehensive comparison with the field test data, the effectiveness of the proposed method is verified, as well as the reasonable modal truncation frequencies of the bridge subsystem are determined. On this basis, the dynamics performances of the SMTBS are evaluated under different initial braking speeds and wheel-track interfacial adhesion conditions; besides, the nonlinear wheel-track slipping characteristics and their influences on the vehicle–bridge interaction are also revealed. The analysis results indicate that the proposed model is reliable for investigating the time-varying dynamic features of SMTBS under variable train speeds. Both the axle load transfer phenomenon and longitudinal slip of the driving tire would be easy to appear under the braking condition, which would significantly increase the longitudinal vehicle–bridge dynamic responses. To ensure a good vehicle–bridge dynamics performance, it is suggested that the wheel-track interfacial friction coefficient is larger than 0.35.

Exact closed-form solution for vibration characteristics of multi-span beams on an elastic foundation subjected to axial force

This paper investigates the vibration characteristics of multi-span beams resting on an elastic foundation and subjected to axial forces. A comprehensive analytical expression of the dynamic response of multi-span beams on an elastic foundation that is developed to address various boundary conditions. The vibration equation is derived by employing Newton's second law. By Laplace transformations and the Green's function method, the solution of this governing equation can be obtained. Subsequently, a unified description is implemented for distinct types of boundary conditions using matrix representations. The correctness is verified through reference results and finite element methods (FEM). The effects of different parameters such as support stiffness, foundation elastic and shear layer stiffness, and axial force on the vibration characteristics are analyzed. This study demonstrates two findings: First, there are two thresholds for support stiffness, and the stiffness value is divided into three intervals. In the same interval, multi-span beams show the same properties. Second, for a rigidly supported multi-span beam, the critical axial force with a natural frequency of zero is just the corresponding Euler's buckling load; for elastically supported multi-span beams, the critical axial force falls between the Euler’s buckling load corresponding to single-span and multi-span beams.

Investigation on the free vibration characteristics of vertical quasi-zero stiffness isolation system considering the disc spring's loading position

This study investigates a quasi-zero stiffness (QZS) isolation system designed for vertical vibration control in isolation structures. The QZS characteristics are achieved by connecting a negative stiffness disc spring in parallel with the linear positive stiffness spiral spring. Initially, the force-displacement relationship of the disc spring is established, taking into account the loading position, and validated through static experiments. Subsequently, the free vibration equations of the QZS system are derived. The Jacobi elliptic functions and the Newton-harmonic balance (NHB) method are then employed to study its free vibration characteristics without considering damping. The results demonstrate that the natural vibration period of the QZS system is related to the nondimensional amplitude and the height of the loaded inner cone. Furthermore, the second-order analytical approximate solution obtained via the NHB method exhibits remarkable accuracy. Finally, the critical damping ratio of the damped QZS isolation system is investigated using a numerical approach. The findings indicate that the critical damping ratio is only correlated with the nondimensional amplitude. This study addresses the gaps in existing research and provide recommendations for the parametric design of vertical QZS isolation system.

Theoretical and CFD investigation on nonlinear dynamics of a cable under vortex-induced vibration with different aerodynamic shapes

This paper investigates the effect of the aerodynamic shape on nonlinear dynamics of a cable induced by vortex-induced vibration (VIV) with theoretical and computational fluid dynamics (CFD) method. Firstly, a two-dimensional (2-D) CFD model is established and calculated by using the shear stress transport (SST) k-ω model to supply the necessary drag and lift coefficients for perturbation analysis. Secondly, the planar transverse vibration equation of the cable is established and the wind load is described by the van der Pol wake oscillator. Then, the above two equations are discretized by the Galerkin method, and the modulation equations are obtained by the multiscale method. The results are verified by Runge-Kutta method. Thereafter, three different aerodynamic shapes of the cable are considered and five wind attack angles are defined to extensively investigate the influence on nonlinear dynamic behaviors. The results show that the variation of the cable cross-section significantly changes its aerodynamic performance, and the aerodynamic coefficient of the asymmetric structure is very sensitive to the wind attack angle. Arranging aerodynamic measures (such as soft tail) in the downwind of the cable can effectively reduce the aerodynamic coefficient and dynamic response of the structure.

Sound absorption properties of the metamaterial curved microperforated panel

This paper proposes a metamaterial that utilizes resonators to enhance the sound absorption capacity of the curved microperforated panel (CMPP), and reliable theoretical analyses and experimental verification are comprehensively given. A metamaterial curved microperforated panel (MCMPP) absorber is formed by attaching local resonators on the CMPP. The displacements of the midplane of the curved panel are expanded into the improved Fourier series based on Spectral-Geometry Method (SGM), and the vibration equations of the MCMPP are derived by the Rayleigh-Ritz method. The sound absorption characteristics of the MCMPP are solved by the effective medium method and the acoustic-electric analogy method. Both a structure-acoustic coupling simulation model and an impedance tube test system are established to validate the accuracy of the present theoretical model. The essential parametric analyses are carried out in conjunction with mode splitting and acoustic-vibrational coupling, and the results show that the local resonance absorption peak within 1000 Hz increases from 1 to 5 with the absorption coefficients greater than 0.8 at these peaks under the installation options of the periodic gradient array and multiple resonances, realizing the effective absorption in multiple frequency bands. This paper provides theoretical support and experimental protocols for the design and application of the metamaterial curved microperforated panel.

Generalized frequency shift and attenuation in simply-supported micro/nano-beam resonators with thermoelastic dissipation

The low frequency shift, weak frequency attenuation, and large quality factor (Q) are strongly required for micro/nano-resonators with excellent operation performance. Therefore, the accurate predictions of frequency shift, frequency attenuation, and Q based on thermoelastic dissipation (QTED ) are significant when optimally designing micro/nano-resonators. In this work, employing the modified-couple-stress (MCS) and nonlocal-single-phase-lag (NSPL) models, the generalized analytical expressions of the frequency-shift ratio, frequency-attenuation ratio, and QTED are first proposed for the simply-supported micro/nano-beam resonator with two-dimensional (2D) heat conduction. In the modeling procedure, utilizing the Galerkin approach, the function of 2D fluctuation temperature is solved from the governing equation of NSPL thermoelasticity. For simplicity, the function of one-dimensional (1D) fluctuation temperature is presented directly. Various expressions of complex natural frequency are obtained from the governing equation of MCS vibration with thermoelastic coupling, and then the models of the frequency-shift ratio, frequency-attenuation ratio, and QTED are developed based on the relevant expressions of complex natural frequency. It is confirmed that the present developed models with generality can effectively degenerate into the classical models. In the simulation cases, silicon (Si) that is a representative semiconductor material is selected. Results reveal that the MCS size-dependence effect can affect the frequency shift more markedly than the NSPL and heat-conduction-dimension (HCD) effects. Moreover, for the beam with a small thickness and operated in a high-order mode, the NSPL and HCD effects will significantly determine behaviors of the frequency shift, frequency attenuation, and QTED .

Effects of splitter plate and mass ratio on flow-induced vibration and heat transfer characteristics of a circular cylinder in turbulent flow: A numerical study

The two-degree-of-freedom (2DOF) flow-induced vibration (FIV) of a circular cylinder with heat transfer in turbulent flow is numerically investigated. The cylinder with an attached splitter plate is elastically supported and allowed to vibrate in both in-line and transverse directions. The unsteady RANS and energy equations along with the SST k-ω turbulent model are coupled with the equation of vibration to study the FIV of the cylinder. The effects of the attached plate length (L/D = 0.5, 1.0, and 1.5) and the mass ratio (mr=2 and 6.76) on the heat transfer rate, vortex shedding patterns, vibration trajectories, FFT, and cylinder displacements are studied in a wide range of Reynolds number (1156–9246) and reduced velocity (2–16). The results indicate that FIV can have either a positive or negative effect on heat transfer enhancement, depending on the splitter plate length and the type of cylinder vibration response. At high Reynolds numbers, the cylinder vibration response is observed in the galloping region, accompanied by a significant vibration amplitude. In this region, for L = 0.5D, vibration has a positive effect on heat transfer, while for other lengths, this effect is negative. For L = 1.5D and a reduced velocity of 5.5, contributions of vortex-induced vibration (VIV) and surface area in the heat transfer enhancement are 23.5 % and 44.3 %, respectively. By reducing the mass ratio from 6.76 to 2.6 for L = 0.5D, the heat transfer rate increases by approximately 40 % compared to a stationary cylinder.

Gyroscopically coupled in-plane and transverse vibrations of an annular free-clamped microplate

In this work, we construct and study a model of coupled plane-transverse oscillations of a circular thin plate with a concentric hole under the action of Coriolis and centrifugal inertia forces caused by the rotation of the system along an axis located in the plane of the plate. Equations of vibrations in partial derivatives are obtained using the variational principle of Hamilton - Ostrogradsky. Assuming the smallness of the angular velocity of rotation with respect to the frequency of the working skew-symmetric flexural form of the plate oscillations, an approximate analytical solution is found for both the radial and circumferential, and transverse components of the displacement field in the free oscillation mode. Using the Galerkin projection method, the problem was reduced to a system of two second-order linear differential equations for modal coordinates of mutually orthogonal basic skew-symmetric modes of the plate vibrations. It is found that the regime of initially excited harmonic oscillations in the presence of rotation is transformed into the regime of amplitude-modulated beats. Analytical expressions are found both for the frequency of the slow beat envelope and for the relative depth of their amplitude modulation. The fundamental possibility of determining the modulus of the projection of the angular velocity vector onto the plane of the plate from the measured value of the envelope frequency is shown. The problem of choosing the optimal geometric shape of the resonator from the point of view of maximizing the sensitivity of the system to changes in the value of the angular velocity of rotation is studied. The question of determining the direction of the projection of the angular velocity vector onto the plane of the plate from the measured depth of the amplitude modulation of the beat regime is considered.

Horizontal oscillations of the wood sawing support during the cutting process in a wood sawing line using a vertical bandsaw

The wooden trolley (in the sawing line using a vertical bandsaw) during work generally moves by iron wheels on two rails placed on the concrete floor. Due to the effect of noticeable cutting force applied by the saw blade and the vehicle's uneven weight on the wheels, it causes the vehicle to vibrate during the horizontal movement, affecting the stability of the wood sawing circuit. By modeling the problem of vibration of beams located on Winkler elastic foundations, subjected to mobile concentrated loads, the article has built a system of differential equations of vibration of two rails, giving an expression to calculate the vibration amplitude of the swan plane. The influence of the flexural rigidity EI of the rails and the elastic stiffness k of foundation on the vibration amplitude of the cutting plane is studied and analyzed. The use of the Dirac Delta function and the employed analytical solution allow the designers to evaluate the accuracy and reliability of the calculated results.

Innovation and Development of Pipa Art in Contemporary Plucked Instrument Performance

Abstract This study summarizes the direction of innovation in contemporary pipa performance techniques through theoretical analysis of pipa history, performance techniques, form, and production materials. The independent variable of string vibration during pipa performance is set, and the string vibration equation of pipa performance is deduced based on the actual vibration situation. Afterward, we collected and pre-processed the sound signals of the pipa performance and selected simulation experiments to analyze the string vibration and acoustic signals during the pipa performance. The results show that when the plucking force of 1 string is 0.8N, 1.6N, 2.4N, and 3.2N, the corresponding Y-direction displacements are 2.95mm, 6.37mm, 8.85mm, and 12.48mm, with the ratio of 1:2.16:3:4.23, and the maximum Y-direction displacements are monotonically and linearly positively correlated with the plucking force. In addition, the sound power of the two pipas gradually increased with the increase in frequency, which led to a dynamic range of more than 18.53 Hz when the frequency exceeded 8000 Hz. In this study, through the simulation of string vibration and acoustic signals of the pipa playing innovative techniques, the laws and characteristics of the pipa art can be recognized so as to strengthen the awareness of innovation and development of pipa playing and to make the art of pipa better developed and inherited.

The dropper stress characteristics under different moving load speeds for a high-speed railway

Dropper failure seriously threatens the operation safety of a high-speed railway. In this work, for a simple chain suspension catenary, one span with five droppers is performed to establish a model and thus the effects of the moving load speed on dropper stress are investigated. First, the partial differential vibration equation of dropper is obtained through the mechanical analysis and converted into the finite difference equation. Then, we consider contact line as a beam element to obtain its motion equation. Furthermore, the boundary and initial conditions of five droppers are determined. Finally, the stresses of five droppers are numerically calculated and the effects of the moving load speed on dropper stress are investigated by writing a MATLAB code. The results suggest that the dropper location significantly affects its stress. Compared with other droppers, droppers II and IV have much more severe vibration amplitudes. Different moving load speeds could cause different stress change of each dropper. With the increasing speed, dropper experiences longer bending compression stage and the bending amplitude increases. The impact of the moving load speed on dropper stress is significant.

Analyzing nonlinear oscillations with He's frequency-amplitude method and numerical comparison in jet engine vibration system

In this study, we describe and successfully solve a jet engine vibration equation using a straightforward tool known as the He's frequency-amplitude Method (HFAM). The jet engine vibration system demonstrates diverse applications across aerospace, power generation, industrial machinery, transportation, marine propulsion, energy optimization, defense, and aviation training. Utilizing HFAM, we derive periodic solutions in a general form for this system, considering various cases dependent on damping and driving forces. The obtained results highlight the effectiveness of HFAM as a distinct and straightforward technique for nonlinear equations. By comparing the solutions with numerical results obtained using the fourth-order Runge-Kutta method, we demonstrate the excellent accuracy of our solutions.

Виброактивность электромеханических устройств вследствие наличия технологических погрешностей

This article discusses a theoretical justification for the presence of vibration in operating modes of electromechanical devices due to technological errors occurring during the machining of parts and assembly of products. The reason for the presence of a dense spectrum of vibration caused by technological errors in the structural elements of ball bearings (outer and inner rings, balls, and cage) is also denoted. The analytical dependence of the vibration amplitude on the magnitude of technological errors is shown. It follows from the results of the analysis that for the most effective minimization of the vibration activity of an electromechanical device, it is necessary to use broadband damping, which is liquid–viscous in most cases. In addition to the above, it is advisable to apply damping in the cascade version. When solving the system of vibration equations for an electromechanical device, replacing ball bearings with sliding bearings eliminates the first four terms, which characterize a wide and dense vibration spectrum of the ball bearings in the operating mode of the electromechanical device.

The Logic and Practical Path of Higher Education Curriculum Reform Based on Cognitive Psychology

Abstract Based on cognitive psychology theory, this study explores the doctrinal logic and practical path of higher education curriculum reform to enhance the effectiveness of higher education programs, with particular attention to improving students’ psychological quality and learning effectiveness. The study uses the finite element method and digital waveguide technology to simulate and analyze the stringed musical instrument. Accurate simulation of the vibration characteristics of the musical instrument is realized through the vibration equations of an ideal string and the principles of the digital waveguide algorithm. The experimental results show that the applied simulation technique can effectively simulate the vibration characteristics of the resonance box of the musical instrument, such as the ideal string vibration and the cavity coupling effect. In addition, the study involves the mathematical expressions of forced vibration and resonance and the effects of various materials on the modal frequencies of the resonance box. The application of artificial intelligence technology in studying acoustic properties of musical instruments significantly improves the accuracy and efficiency of simulation. It provides essential theoretical support for the design and production of musical instruments.

Radial sandwich radial-bending composite transducer designed based on acoustic black hole theory

A novel radial sandwich radial bending composite transducer is proposed based on the increasing amplitude of bending waves in the structure of an acoustic black hole (ABH). The transducer consists of a radial sandwich circular ring transducer and an outer acoustic black hole structure. The existence of the ABH structure realizes the conversion between radial vibration and bending vibration of the transducer, and improves the acoustic radiation performance of the transducer. Based on the one-dimensional acoustic black hole bending vibration theory, the bending vibration equation of the circular ABH structure is given. An analytical model of the ABH structure bending vibration is established by using the method of geometrical-acoustics, and the eigen frequency of its bending vibration is given. The relationship between the electromechanical conversion performance and the size of the transducer is discussed by using the finite element method, and the results show that when the eigen frequency of the bending vibration of the ABH structure is close to the resonant frequency of the transducer, the electromechanical coupling coefficient of the transducer reaches a maximum value. The radiation sound pressure field, radiation sound intensity, and radiation directionality of the transducer in the air are also analyzed by the finite element method. The results indicate that the presence of the ABH structure can improve the electromechanical conversion performance of the bending vibration of the transducer, enhance the acoustic radiation performance of the transducer, make the transducer exhibit radiation directionality and the lower order of the bending vibration mode of the ABH structure, the stronger radiation directionality of the transducer towards the left side and the right side. Finally, in order to verify the accuracy of the simulation and theory, we made a prototype transducer and carried out the experimental test of transducer electrical impedance and vibration mode. The measured results are in good agreement with the simulation results. The displacement distribution of the radiation surface of the transducer at resonance frequency is measured, which verifies the bending vibration mode of the ABH structure. This type of transducer is expected to be used as acoustic projector in the air or water.

Virtual Simulation Research on Resonance Box Vibration Characteristics of College Symphony Orchestra Driven by Artificial Intelligence Technology

Abstract This study focuses on applying artificial intelligence techniques to virtually simulate the vibration characteristics of the resonance box of a college symphony orchestra to improve the understanding and optimization of the acoustic performance of the instrument. The study uses the finite element method and digital waveguide technology to simulate and analyze the vibration characteristics of the stringed resonating musical instrument, which is realized by the vibration equation of an ideal string and the principle of digital waveguide algorithm. The experimental results show that the applied simulation technique can effectively simulate the vibration characteristics of the resonance box of the musical instrument, such as the ideal string vibration and the cavity coupling effect. In addition, the study involves the mathematical expressions of forced vibration and resonance and the effects of various materials on the modal frequencies of the resonance box. The application of artificial intelligence technology in studying acoustic characteristics of musical instruments significantly improves the accuracy and efficiency of simulation. It provides essential theoretical support for the design and production of musical instruments.

Vibration-polynomial-based optimal trajectory planning for mobile concrete pumping equipment with state constraints

To improve the operating stabilities of the flexible hydraulic manipulator in concrete pumping equipment, a trajectory planning method based on optimal vibration energy is presented. First, predefined trajectories are generated by the quintic polynomial method, and kinematic inversions are established by unifying the velocity and angle constraints into the velocity level. Then, relationships between the boom vibration and the vibration energy are solved in modal space. Specifically, hydraulic actuators are regarded as spring-damping systems to calculate generalized forces into the vibration equation. The vibration equation can solve the modal coordinate and vibration model function, such that the elastic potential energy to cause vibration is completely described. Next, floating values are captured by a genetic algorithm to minimize vibration energy, and then introduced to the predefined trajectory to weaken the boom vibration. Last, the proposed trajectory planning method is verified by a concrete pumping spreader with a 13 m hydraulic manipulator.