Vibration Equation Research Articles

A theoretical calculation method for critical air velocity to prevent methane draft pressure-caused airflow reversion based on oscillation theory

Methane draft pressure is a secondary disaster for mine ventilation following coal and gas outbursts, which poses a long-term threat to coal mining and workers’ safety. To study the law of methane draft pressure-caused airflow variation, oscillation theory is introduced in this study. The vibration equation is used to deduct the basic equation of airflow oscillation in parallel upward ventilated airways, which regards the entire methane and air as one vibrating object in the loop of the parallel airways. The theoretical critical air velocity to prevent airflow reversion is calculated by the airflow oscillation equation using numerical methods. Furthermore, through dimensional analysis, a formula is given to calculate the critical velocity directly. The theoretical results were compared with experiments and good agreements were obtained, which verifies the rationality of the assumptions in the theoretical deduction process. The conclusion of this study indicates that, in a short time, oscillation theory can be used to predict the critical air velocity to prevent airflow reversion, and be feasible in revealing the essence of methane draft pressure-caused airflow reversion.

Nonlinear vibration of porous truncated conical shell under unified boundary condition and mechanical load

The aim of this article is to research nonlinear vibration responses of a metal porous truncated conical shell undergoing mechanical harmonic loads in thickness direction. The artificial spring technique is employed to unify boundary condition which can arrive at arbitrary classical boundary condition in real-word by changing the stiffness of springs. Utilizing Hamilton's principle, theoretical formulations of the conical shell are developed in view of FSDT and von-Kármán geometric relation. Afterwards, by employing generalized differential quadrature method (GDQM), the governing equation of nonlinear vibration in the form of partial differential equation is truncated and discretized into an ordinary differential equation which is a high-dimensional nonlinear dynamic system described in physical space rather than the mode space. On this basis, the analysis of effect of exciting frequency on the amplitude is carried out for three different distributing types of porous truncated conical shells by numerical integration. The frequency-amplitude responses of the system are depicted utilizing the sweeping frequency. The bifurcation diagrams, time series and phase diagrams are utilized to analyze the nonlinear dynamics of the porous truncated conical shell under unified boundary condition. It is found that the multiple solutions high-incidence area is near the 1st primary resonance, and vibration amplitude is dominated by the 1st order primary resonance and the 3rd one. The hardening-spring nature decreases for Type3, Type2 and Type1 successively.

A symplectic analytical wave propagation model for forced vibration of thin plate with acoustic black hole indentation

In recent years, acoustic black holes (ABHs) have been revealed as an effective bending wave control technology and widely used to regulate the vibration and noise reduction performance of plate structure. However, the isolation effect of most traditional ABH plate can only be aroused at frequencies above the cut-off frequency. In order to widen the frequency range of the ABH effect, this paper embedded single and multiple ABH indentations at the end of plate. For the forced vibration response of the ABH indentation plate, an analytical wave propagation model was proposed based on the symplectic method. The governing equations of bending vibration of thin plate was first introduced into the symplectic dual system, then the symplectic analytical solution of wave propagation parameters and waveforms were obtained. A Gaussian-discretization scheme was used to discretize the ABH indentation, whereas the proposed method was used to evaluate the wave amplitude. The vibration response was finally obtained using the superposition principle and wave propagation, reflection, and transmission relationships. The effectiveness of the proposed method was verified using the finite element results. The results showed that the ABH indentation plate, especially the plate with multiple ABH indentations, had significant vibration reduction performance. Considering that embedding ABH indentation can weaken the structural rigidity of the plate, the ABH indentation plate with reinforcing stiffeners was then proposed to alleviate this limitation, and the influence of the width and quantity of reinforcing stiffeners on the vibration response was studied. The results indicated that the increase of width will to some extent weaken the ABH effect, and the quantity of stiffeners had a significant impact on the ABH effect.

Loaded Euler-Bernoulli beam with the distributed hysteresis properties

In this article, we propose a new perspective mathematical model of the beam with the distributed hysteresis properties. Hysteresis properties are formalized within two approaches: phenomenological (Bouc-Wen model) and design (Prandtl-Ishlinskii model). The equations for the beam vibrations are obtained using the well-known Hamilton approach. The dynamical response of the beam with distributed hysteresis is considered under various types of external load, such as impulse, periodic, and a seismic load. Numerical simulations show that the hysteresis beam is more “resistant” to external loads than the classical Euler-Bernoulli beam. Particularly, with the same types of the external load, the amplitude of oscillations of the hysteresis beam as well as its energy characteristics are lower than those of the classical one. These results may find some applications in the field of the design of earthquake-resistant constructions and buildings.

The interaction between cross-flow induced vibration and convection heat transfer in tube bundle at subcritical Reynolds number

This work numerically investigates the interaction between flow-induced vibration (FIV) and forced convection heat transfer in tube bundle with a pitch ratio of 1.5 at subcritical Reynolds number (Re = 5 × 103-3 × 104). The range of reduced velocity is 0.53 ≤ Vr ≤ 2.71. The mass-damping parameter of coupling system is 0.07. The three-dimensional refined FIV simulation considering fluid viscous damping has been validated by the literature experiment to capture three continuous FIV mechanisms, namely turbulence-induced vibration, vortex-induced resonance and fluidelastic instability. The heat transfer and pressure drop in tube bundle are validated by the experiment. The coupling of Navier-Stokes equations, energy equations and vibration equations is solved by under-relaxation algorithm. The results of vibration response, the change of damping for the coupling system, flow field and temperature field, heat transfer and pressure drop are presented and delineated in this paper. The results show that for the low velocity turbulence-induced vibration, the interaction between FIV and flow heat transfer is negligible due to the weak fluid–solid interaction. In vortex-induced resonance region, the wall heating makes each row tube be out of phase by affecting the vortex shedding and reduces the positive feedback of resonance, which reduces the amplitude. The wall heating reduces the damping of coupling system, which causes an earlier transition from vortex-induced resonance to fluidelastic instability and a faster increase of amplitude in fluidelastic instability. In both vortex-induced resonance and fluidelastic instability, the vibration of tube bundle dominates the vortex shedding by the natural frequency, and further dominates the change of heat transfer. The best performance of enhanced heat transfer induced by FIV is located in vortex-induced resonance region, in which the time–space averaged Nusselt number is increased by 8.83%. Due to the redistribution of flow field and the elimination of deflective flow pattern, compared with vortex-induced resonance, the increasing ratio of heat transfer induced by FIV is lower and the increasing ratio of pressure drop induced by FIV is higher in the fluidelastic instability.

Size-dependent coupled bending-torsional analysis of piezoelectric micro beams

Nowadays, piezoelectric micro-beam structures are used in various industries, and knowing and analysis of these smart microstructures is important. However, in the modeling of these electromechanical microbeams, the torsion during bending is generally omitted for simplicity. Considering that in some cases, twisting and bending cannot be omitted and must be modeled simultaneously. Therefore, this article presents a coupled bending-torsion model for the first time for static deflection and free vibration analysis of piezoelectric microbeams with nonclassical continuum theories. The governing equations of the motion and boundary conditions of the Timoshenko beam at the micro-scale have been obtained using size-dependent electromechanical theory and Hamilton’s principle. In addition, the Laplace transform is used to solve the static equations, while the generalized differential quadratic (GDQ) method is used to discretize and solve the free vibration equations. In the following, the influence of various parameters, such as the effect of size and geometric conditions, on the static displacements and the natural frequency of the microbeam is discussed.

Blow-Up of the Solution to the Equation for Nonlinear Beam Vibrations with Allowance for Transverse Deformation Effects

Blow-Up of the Solution to the Equation for Nonlinear Beam Vibrations with Allowance for Transverse Deformation Effects

Blow-Up of the Solution to the Equation for Nonlinear Beam Vibrations with Allowance for Transverse Deformation Effects

Beam vibrations with allowance for deformation effects in the transverse direction are modeled using a nonlinear Sobolev-type differential equation, for which the Cauchy problem is investigated in the space of continuous functions. Conditions for solution blow-up on a finite time interval are considered.

Vibration stability of a lifting pipe transporting solid–liquid two phase flow

A new mathematical model is developed to investigate the vibration stability of a lifting pipe induced by solid–liquid two phase flow. Three different slip models and the momentum conservation theorem are introduced to describe the slip characteristics of solid particles in two phase flow, which are coupled with the pipe vibration equations to establish a new governing equation and solved using the harmonic differential quadrature method. The mathematical model is validated by published data for the single-liquid flow case. Numerical results show that the slip model has noticeable effects on two phase flow-induced pipe vibration performance. The transport concentration of solid particles has a significant effect on the stability characteristic; however, the effect of particle size is very limited. The pipe flow mass ratio and the gravity coefficient are also key parameters affecting two phase flow-induced pipe vibration performance.

К ТОЧНОМУ РЕШЕНИЮ ЗАДАЧИ О РАСЧЕТЕ БАЛОЧНОЙ ПЛИТЫ И ПЛАСТИНКИ НА УПРУГОМ ОСНОВАНИИ ВИНКЛЕРА

The paper examines the problems of calculating a beam slab axisymmetrically loaded with round and rectangular plates on an elastic Winkler foundation in the traditional formulation without taking into account shear stresses and one-sided connections in the contact zone between the structure and the elastic foundation. The deflections of the structure are represented as a series of eigenfunctions of the differential equation of bending vibrations of a structure with free edges. The external load is also arranged in a series according to its own functions. Substituting these expansions into the differential equation for the deflection of a structure on the Winkler base and equating the coefficients for the same eigenfunctions, we find an expression for the deflections of the structure. Next, the forces in the structure are determined from the deflections. Solutions are given to problems of bending of a beam slab axisymmetrically loaded with round and square plates partially loaded with a uniformly distributed load.

Size-dependent nonlinear forced vibration of microbeam using two-phase local/nonlocal viscoelastic integral model with thermal effects

Combination strain-driven (εD) and stress-driven (σD) two-phase local/nonlocal integral models (TPNIM) with classic Kelvin-Voigt viscoelastic model, εD- and σD-two-phase local/nonlocal viscoelastic integral models (TPNVIM) are proposed to study the viscoelastic nonlinear forced vibration behaviors of microbeams in thermal environment. Based on von-Karman large deformation theory and the Hamilton’s principle, the differential governing equations of motion and boundary conditions are derived for nonlinear forced vibration of microbeam. Several non-dimensional variables are introduced to simplify the mathematical formulation. Nonlocal viscoelastic equations in integral form are transferred unitedly into equivalent differential form with constitutive constraints. The axial force of nonlocal microbeam with immovable ends is derived explicitly. Vibration frequency and corresponding linear vibration mode shape are derived through Laplace transformation for linear thermo-elastic vibration of microbeam using εD- and σD-TPNIMs. The Ritz-Galerkin approach is utilized to derive the reduced differential governing equation for nonlinear vibration of microbeam, which has exact same form as single-degree-of-freedom oscillator with cubic nonlinearity when material viscosity is neglected. The method of multiple scales (MMS) based on trigonometric functions is utilized to solve the reduced nonlinear differential equation approximately. The frequency-amplitude responses are addressed for primary and super-harmonic resonances of microbeam. The effects of viscous coefficient, amplitude of excitation force and environmental temperature variation as well as nonlocal length parameter on the nonlinear vibration behavior of microbeam are investigated numerically for different boundary conditions.

The harmonic resonance and singularity analysis of bifurcation for the magnetized elastic plate with action of time-varying magnetic potential

This paper deals with the superharmonic resonance of a ferromagnetic thin rectangular plate in the air-gap magnetic field excited by armature magnetic potential. Electromagnetic forces applied on the plate by the air-gap magnetic field cause the plate to transversal vibrate, which affects the air-gap magnetic field in turn, resulting the vibration and magnetic field coupling. According to the basic electromagnetic field theory and considering the magnetoelastic coupling effect, the air-gap magnetic field intensity is obtained by solving the Laplace's equation satisfied the air-gap magnetic boundary conditions. The electromagnetic force model of soft ferromagnetic plates is determined based on theories of electromagnetic and elasticity. According to the large deflection theory of plates, basic energy relationships and variational equations of the elastic plate are given. Eventually, the nonlinear magnetoelastic vibration equation of ferromagnetic thin plates is derived using Hamilton's principle and Galerkin method. The multi-scale method is used to solve the superharmonic resonance to obtain the amplitude-frequency response equation and the stability discriminant of solutions. The topological analysis of amplitude-frequency equation is carried out using singularity theory, and the bifurcation characteristics of systems on physical parameter planes in different regions are obtained according to the transition set. The correctness of analytical solutions is verified by comparison with numerical solutions. Through numerical calculations, curves of the static deflection and equivalent magnetic force of plate with parameters are given, and the amplitude curves, dynamic phase plane trajectories and time history diagrams of system response with changes in electromagnetic and structural parameters are plotted. Results show that both the decrease of armature magnetic potential amplitude and the increases of plate thickness and initial air-gap thickness reduce the static deflection. The increase of armature magnetic potential amplitude increases the equivalent magnetic force. As the decrease of initial air-gap thickness and the increases of armature magnetic potential amplitude and excitation force amplitude, the amplitudes of the upper branch and lower branch curves representing stable solutions decrease and increase, respectively, and the single-value solution region increases.

Study on the magnetic levitation of a magnetic flea

We investigated the problem of ‘magnetic levitation’ originating from the International Young Physicists’ Tournament (IYPT). The problem was first investigated by a PRL paper (Baldwin et al.) in 2018, which states that the flea of a magnetic stirrer spinning fast enough in a liquid with a high viscosity coefficient can jump from the bottom and levitate stably. The magnetic force and gravity balance periodically. This phenomenon includes several concepts: magnetic dipoles, rigid-body rotation, fluid mechanics and magnetic levitation. They are more or less unfamiliar to undergraduate students. However, the movement of the flea could be described with a concise forced vibration equation, which is familiar in textbooks. The phenomenon could be divided into two stages: synchronous movement and levitation state. The transition is the jumping of the flea. We demonstrated this process and presented several equations to build this physical model. The progression of the phenomenon is due to the increase in the drive magnet angular velocity called the drive velocity. We verified our theory by simulation and experiments. Several parameters are experimentally verified to influence the phenomenon. We also discussed the origin of dynamic stabilization, which would be slightly complicated but worthy for students. In short, we introduce an interesting problem originating from the PRL paper that can be easily achieved under laboratory conditions. We extend some content in a pedagogical way that would be helpful for students to understand the related physical concepts, such as the influence of the viscosity coefficient of the liquid on the flea’s motion, which is not discussed in the PRL paper.

Magneto-thermo-elastic coupling vibration and bifurcation characteristics of functionally graded rotating cylindrical shell

The magneto-thermo-elastic coupling vibration of rotating ferromagnetic functionally graded cylindrical shell is investigated in this paper. Based on the physical neutral surface theory and Donnell's theory, the expressions of kinetic energy and strain energy are obtained by introducing geometric nonlinearity. The magneto-thermo-elastic vibration equations of the rotating ferromagnetic functionally graded cylindrical shell in multi-physical field are derived via the Hamilton's principle. Considering the magnetization effect of ferromagnetic shell in magnetic field, the expression of magnetization force is obtained according to the nonlinear magnetic constitutive relation. For simply supported boundary conditions, Galerkin's method is used to obtain the discretized ordinary differential equations, and the static deflection of the shell under static load is solved. The steady-state response of the system during primary resonance and the stability of solutions are given by the multi-scale method and the Lyapunov stability theory. Through numerical examples, the characteristic curves and surfaces of amplitude are plotted, and the parameter ranges corresponding to regions of multi-valued solutions and stable solutions are determined. The system dynamics laws describing the coupling effect between multiple parameters such as force, magnetism, heat, and rotational speed are evaluated. The Global bifurcation diagrams and the maximum Lyapunov exponent spectrums under different control parameters are plotted to analyze the effect of parameter changes on the dynamic response of system. The results show that the vibration characteristics of the system are significantly influenced by the external physical fields, in addition to the shell size and material volume fraction. Slight changes in parameters can lead to violent changes in the dynamic response of the system, exhibiting intermittent, jumping, and chaotic characteristics.

Theoretical and experimental harmonic analysis of cracked blade vibration

Blade cracks pose deadly threats to aviation safety and have caused serious aviation accidents in recent years. In order to diagnose blade cracks at the early stage, the harmonic vibration of cracked blade was analyzed theoretically and experimentally. Firstly, a recursive form solution was deduced for the complex the nonlinear dynamic equation of blade vibration, revealing the close relationship between harmonic component power and crack parameters. Furthermore, the upper bound of the adjacent harmonic component power ratio was obtained by theoretical derivations. The results show that the harmonic power decreases as the harmonic component order increases, and the degree of attenuation is decided by the crack parameters. Therefore, a crack detection approach was proposed according to the power ratio of harmonic components. The advantage of this method is that the blades do not need to be in a resonant state and can process vibration data at all rotational frequencies. This improves data utilization and diagnostic robustness. The recommended method was validated by the simulation analysis of stainless-steel and titanium blades. Finally, a test bench for blade vibration was set up whose highlight was the use of optical sensors for non-contact measurements of blade vibration. The results of both simulation and testbed experiment were much consistent with the theoretical inference.

Free vibration of functionally graded graphene platelets reinforced composite porous L-shaped folded plate

This paper investigates the free vibration of a functionally graded graphene platelets reinforced composite (FG-GPLRC) porous L-shaped folded plate. According to the folded position, the L-shaped folded plate is divided into three rectangular plates. Based on Mindlin-Reissner plate theory and Hamilton’s principle, the governing equations for free vibration of each subelement are deduced. Utilizing the generalized differential quadrature element (GDQE) method, the governing equations for each subelement are discretized and numerically solved to obtain the natural frequency. The study focuses on discussing the influence of several factors, including porosity distribution, porosity coefficient, plate size, crank angle, boundary conditions, GPL weight fraction, and GPL pattern on the natural frequency. The findings reveal that the incorporation of GPL enhances the mechanical performance of the porous folded plate compared to a pure porous plate. Dispersion of more GPL near the upper and lower layers enhances the stiffness of the FG-GPLRC porous folded plate, leading to higher natural frequency. Additionally, variations in the crank angle have a significant impact on the natural frequency.

A two-degree of freedom mathematical modelling of a dental implant to estimate frequency and micro-displacement using electromagnetic RFA

Background/PurposeA dental implant is an orthodontic prosthesis inserted into the patient's jawbone by a surgical procedure to compensate for a lost tooth. Upon osseointegration (bone-implant fusion), the abutment and crown can be attached to the implant to complete the implantation process. To guarantee the successful implantation of dental prosthetics, it's vital to monitor the primary stability in micro-displacement. This research aims to create a two-degree-of-freedom (2 DOF) mathematical model of dental implant prosthetics, which will be used to evaluate the micro-displacement under electromagnetic excitation. This mathematical model development will verify the functionality and accuracy of the electromagnetic RFA technique by estimating the resonance frequency and associated micro frequency of dental implants. MethodsDental implant systems are evaluated according to the frequency of vibration corresponding to the maximum micro-displacement (micro-mobility). The stiffness and damping values of the mathematical model were determined using Finite Element Analysis (FEA). Using concepts and equations of vibration, the maximum displacement of the implant is determined. ResultsVariations in input frequency ranges have also been compared to observe the micro displacement and resonance frequencies. In MATLAB, an input frequency of 5 to 15 kHz was provided. The maximum micro displacements and the resonance frequencies were plotted in the time domain and frequency domain using Fast Fourier Transform (FFT) in MATLAB. ConclusionsThe findings of this research reveal that input frequency ranges of 5 to 15 kHz can be used to determine the resonance frequencies of a dental implant and the corresponding micro-displacements. there is a need in the medical field for a non-invasive device with a 2 DOF modeling concept that can detect even small changes in implant stability which ultimately determines both the short- and long-term clinical effectiveness of implant treatment.

Analytical solution for seismic response of vertical shaft with primary and secondary linings under SH wave excitation

This paper presents a novel mechanical model for seismic response analysis of vertical shafts. The model is based on elastic foundation beam theory and simplifies the shaft linings as parallel Euler Bernoulli beams. Their interaction is simulated using uniformly distributed springs. The governing differential equation for shaft vibration is derived for SH wave excitation. The seismic input motion is applied to the primary lining by converting the free field displacement into harmonic form. The closed-form solutions for seismic response of both primary and secondary linings are obtained via the transfer function method. The accuracy of analytical solution is validated by comparing with finite element analysis results. Additionally, the impact of foundation stiffness, secondary lining thickness, shaft outer diameter, and relative stiffness between linings on seismic response are investigated in the parametric analysis. The analytical solutions of the seismic response analysis of vertical shafts indicates that the relative stiffness between linings significantly affects the secondary lining's peak bending moment. Furthermore, higher stiffness of the elastic connection layer between the two linings can effectively reduce the peak displacement response of the secondary lining.

Fundamental solution in the coupled theory of thermoelastic nanoporous materials with triple porosity

AbstractIn this paper, the linear coupled theory of thermoelasticity for nanoporous materials with a triple porosity structure based on Darcy's law and the volume fraction concept is considered. The fundamental solution of the system of steady vibration equations is constructed by means of elementary functions. Finally, some basic properties of this solution are established.

Isogeometric vibration of the magneto-electro-elastic sandwich plate with functionally graded carbon nanotube reinforced composite core

This paper investigates the free vibration analysis of the magneto-electro-elastic (MEE) sandwich plates with functionally graded carbon nanotube reinforced composite (FG-CNTRC) core using isogeometric approach (IGA). The sandwich plate is composed of the homogeneous MEE face sheets and FG-CNTRC core with four types of carbon nanotubes (CNTs) distribution, including CNT-UD, CNT-O, CNT-V and CNT-X. The external electric voltage and magnetic potential are applied in the top and bottom layers of the MEE sandwich plate. Employing the refined plate theory (RPT) and Hamilton principle, the governing equation for free vibration of the MEE sandwich plate is derived. The IGA employs Non-Uniform Rational B-Splines (NURBS) basic functions to approximate the displacement fields and the magnetic and electric potentials in the RPT model. The study examines and discusses the impact of different factors on the frequency of the MEE sandwich plate, including parameters like CNTs distributions, CNTs volume fraction, external electric voltage and magnetic potential, and the geometrical parameter of the plate. This research has revealed several important discoveries regarding the fabrication of MEE sandwich structures.