Differential Equations Of Motion Research Articles

Axial force contribution to the out-of-plane response of horizontally curved beams under a moving mass excitation

Horizontally curved beams subjected to a moving mass do experience a time-varying axial force due to the in-plane centrifugal action of the inertial accelerations. The current study aims at investigating the contribution of the induced axial force to the out-of-plane dynamics of horizontally curved beams under the excitation of a moving mass. The governing differential equations of motion are established to account for the action of the axial force, and also the effect of mass inertia, Coriolis and centrifugal forces introduced by a moving object. In the course of dealing with the semi-analytical approach, a new system of ordinary differential equations is distilled utilizing the transition matrix method in order to be amenable for numerical computations. The veracity of the solution has been corroborated through comparison with the results attained by the previous studies addressed in the literature. Moreover, the efficiency of the proposed solution is demonstrated through conducting an extensive parametric study. In that regard, the dynamic impact factor spectra have been obtained for various design parameters including the geometrical and dynamic properties of the system. The findings reveal that, in general, the dynamic impact factor is highly affected by the mass and velocity of the moving object. Furthermore, the results indicate that the inclusion of the axial force can result in a reduction in the dynamic impact factor spectra, which is shown to be more substantial for the response evaluation of short-span horizontally curved beams excited by a high-velocity heavy mass. In this particular situation, the mitigating effect of the axial force becomes comparable to that of the foundation damping. This paper concludes with an investigation on the influence of the axial force on the time-history patterns, which is demonstrated to attenuate the response amplitudes at both the forced and free vibration phases.

Flexural-torsional-coupled vibration analysis of Euler-Bernoulli beam by using the differential transform method

This study introduces an analytical solution method which is called the differential transform method (DTM) to analyze the free vibration response of a beam which features material coupling between flapwise bending and torsional vibrations. The governing differential equations of motion are solved by applying DTM. The mode shapes and the natural frequencies are obtained and plotted for bending and torsional behavior, separately. Application of the model solved by DTM is investigated for a typical cantilever aircraft wing when flexural and torsional motions are coupled and the bending-torsion coupling effect is ignored, separately, using a numeric computing environment, MATLAB.

Effects of receiving position and shell material on in-band full-duplex underwater communications' self-interference signal.

Despite the recent intensive research on adaptive algorithms for self-interference (SI) cancellation (SIC) in in-band full-duplex (IBFD) underwater acoustic communication (UWAC), there has been relatively little exploration of how the IBFD-UWAC modem shell affects the SI signal. This paper analyzes the effects of the shell material and the near-end receiver position on the SI signal. The analysis is done with a two-dimensional finite-element model in a free-field simulation environment, which combines the differential equation of motion and the time-dependent solver. The SI signal strength around the modem shell in the far-field conditions is obtained. The simulation and pool experiment results both show that (i) the strength of the received SI signal is lowest when the near-end receiver is on a line extending from the shell's geometric center perpendicularly to its central axis and (ii) a shell material with a high elastic coefficient is more conducive to suppressing the SI signal. A pool experiment showed that changing the spatial position of the near-end receiver and the shell material from aluminum to stainless steel enhanced the SIC performance of the IBFD-UWAC system by at least 12 and 4 dB, respectively.

Nonlinear dynamics of traveling beam with longitudinally varying axial tension and variable velocity under parametric and internal resonances

In this investigation, the nonlinear dynamics of an axially accelerating viscoelastic beam on a pully mounting system have been analyzed. The axial tension of the beam is modeled as a function of the traveling velocity, support stiffness parameter as well as spatial coordinate. Geometric cubic nonlinearity in the equation of motion is due to elongation in the neutral axis of the beam. The integro-partial differential equation of motion of the axially accelerating beam associated with the simply supported end conditions is solved analytically by adopting the direct perturbation method of multiple time scales. As a result, a set of complex variable modulation equations is generated, which governs the modulation of amplitude and phase. This set of modulated equations is numerically solved to explore the influence of the support stiffness parameter upon the stability and bifurcation of the beam which has not been addressed in the existing literature. Apart from this, the impact of fluctuating velocity component, viscoelastic coefficient, longitudinal stiffness parameter, internal and parametric frequency detuning parameters on the stability and bifurcation analysis is studied, revealing significant dynamic characteristics of the traveling system. The fourth-order Runge–Kutta method is applied is to find the dynamic solution of the system. The system displays stable periodic, quasi-periodic, and mixed-mode dynamic responses along with the unstable chaotic behavior for a specific set of system parameters. The results obtained through an analytical–numerical approach may help the design and operation of an axially moving beam.

Benchmark solutions for stochastic dynamic responses of rectangular Mindlin plates

There exist some researches on the benchmark responses of moderately thick plates based on Mindlin theory under static loads. In practice, rectangular Mindlin plates are often subjected to various random excitations. However, the benchmark solutions of analytically stochastic dynamic responses for rectangular Mindlin plates under random excitations were not investigated. In this paper, analytical power spectral density functions and root mean squares for stationary and nonstationary stochastic responses of rectangular Mindlin plates are derived, which provide benchmark solutions for other numerical methods and experimental design. Firstly, based on the closed-form free vibration analyses of rectangular Mindlin plates with two opposite simply supported edges, exact natural frequencies and modal shapes are obtained. Then, the pseudo excitation method is introduced into the decoupled differential equations of motion. By combining the exact solutions of free vibration, the analytical method for random vibration analysis of elastic Mindlin plate is proposed, and the analytical stochastic responses are derived. Due to the low efficiency of the AM based on symbolic operation, the efficient discrete analytical method, which can ensure that the results are exact in the spatial domain, is further developed. Finally, the performance of the proposed approaches is verified by comparing with the commercial software, Monte Carlo simulation, and literature. The remarkable effects of spatial distributions of excitations on the stochastic responses of Mindlin plates are illustrated. The important influences of parameters including relative thickness, elastic modulus, and boundary constraint degree on responses are revealed.

A tristable nonlinear energy sink

The tristable characteristics are mostly used in the field of energy harvesting and vibration isolator, but are rarely devoted to in the field of vibration absorption. In this paper, a tristable nonlinear energy sink (TNES) model is proposed for vibration absorption. The TNES is constructed by fixing a magnet in the middle of the clamping pre-pressure beam, and setting a magnet on both sides of the fixed magnet. By adjusting the magnet position, the proposed TNES can be transformed into bistable NES (BNES) and monostable NES (MNES). The TNES is coupled with the linear oscillator (LO), and the differential equation of motion of the coupled system is obtained with the Hamiltonian principle. The approximate analytical method and the numerical method are mutually verified. The vibration reduction efficiency of the TNES under harmonic excitation is presented. The influence of the parameters of the TNES on the dynamics and vibration reduction efficiency is studied. The optimal parameters of the TNES are obtained by the particle swarm optimization (PSO) algorithm. Compared with the corresponding bistable NES and monostable NES, it shows that the tristable NES has obvious advantages in vibration suppression efficiency. The effects of the TNES barrier depths and the excitation amplitude on the vibration reduction efficiency are revealed. The TNES can dissipate energy through chaotic inter-well oscillation between three stable positions, and achieve the purpose of effectively suppressing vibration.

Dynamic Response and Riding Comfort Analysis of a Passenger-Train-Bridge System in Urban Rail Transit

To exactly study the riding comfort of passengers in urban rail transit trains, a coupling vibration model, including passengers, train, and bridge is proposed in this paper. According to the Lagrange Equation, the differential equations of motion of each part are derived, which are coupled with each other through displacement and dynamic interaction forces. A corresponding calculation program is compiled by the Fortran language. A representative steel-concrete composite girder bridge on Beijing Metro Line 5 is selected as the research background, the dynamic responses of passengers and train are simulated in detail, and the dynamic response differences between the passenger and carriage are compared. The passengers’ riding comfort is evaluated according to ISO-2631 Standard based on the simulated data. The result shows that passengers in the middle carriages of the train feel better than those in the head and tail ones. While in the same carriage, passengers on the middle seats feel better than those on both ends of it. If only vibration is considered, passengers will feel better in a fully loaded carriage than those with few passengers. The riding comfort of passengers will be gradually reduced with the increase of the train speed.

Vibration analysis of elastically supported graphene strengthened beams

Vibrations of a functionally graded graphene strengthened Euler-Bernoulli beam with a vertical spring support is investigated in this study. The main objective of this work is to examine how the vibrational behaviour of a graphene strengthened beam of different graphene distributions and boundary conditions is affected by the addition of an intermediate spring attached to it vertically. The distribution functions for the continuous variations of graphene nanoparticles' weight fraction in the thickness direction are defined such that the overall graphene amount in each beam of different distribution patterns is fixed. The reinforcement effects of graphene on the material properties of the beam structure are considered using a modified twodimensional Halpin-Tsai micromechanics model and a rule of mixtures. The deformation of the beam is modelled utilising the Euler-Bernoulli beam theory. The spring is introduced to the system by employing a Dirac delta function. The differential equation of motion is obtained utilising Hamilton's principle, which is then solved for natural frequencies using an assumed eigen-function method for different boundary conditions. The influences of the location and stiffness of the spring on the natural frequencies of beams reinforced with various graphene distributions are explored and discussed.

Quasi-canonization of motion equations of a two wheeled cart

The modeling of a two wheeled cart models based on the quasi-canonization method is studied in this paper . For a two wheeled cart model, a set of suitable quasi-coordinates and quasi-momentums are defined according to the geometric characteristics of its configuration space, so that the motion differential equations of the two wheeled cart model is written in the form of Hamilton.. s canonical equation. In this way, the nonholonomic constraining force disappears from the motion differential equations of the wheeled cart model. And the number of the motion differential equations is also reduced due to the nonholonomic constraints on the cart. Moreover, the quasi-canonization means that the wheeled cart model has a special symplectic structure described by quasi-coordinates and quasi-momenta. This lays a foundation for applying numerical calculation methods and control methods based on the symplectic geometry to the study of the wheeled cart model.

Nonlinear vibration analysis of functionally graded porous circular plates under hygro-thermal loading

In this work, the nonlinear vibrations of functionally graded (FG) porous circular plates under hygro-thermal loading is studied utilizing a numerical approach. Hygroscopic stresses generated due to the nonlinear rise in moisture concentration, even and uneven porosity distributions, and temperature dependency of material properties are all taken into account. Modified Voigt’s rule of mixture is applied to obtain the hygro-thermo-mechanical properties of the FG circular plate. All material properties are assumed to be temperature-dependent using the Touloukian formula. In order to obtain the equations of motion, the first-order shear deformation theory, von-Kármán geometrical non-linearity assumption, and hygro-thermal strains are considered concurrently. After deriving the equations of motion using Hamilton’s principle, the differential quadrature method and the Newmark-beta time integration scheme are employed in conjunction with an iterative approach to solve the set of nonlinear governing differential equations of motion. The effects of various parameters including temperature distribution, plate’s thickness, porosity volume fraction, moisture concentration, and FG index are studied on plate’s maximum non-dimensional lateral deflection.

Experimental investigation and model prediction of sliding bubble dynamics in vertical subcooled boiling flow

In the subcooled flow boiling, the behavior of sliding bubbles significantly affects the partition of wall heat flux, especially the transient heat flux in the sliding area. Therefore, an experimental investigation was conducted to examine sliding bubble characteristics in subcooled flow in a vertical rectangular channel in this paper. The characteristics of sliding bubbles have been recorded and analyzed from size and velocity distribution, growth rate, sliding velocity, velocity-diameter dependence, and coalescence critical velocity in detail. By comparing experimental data and growth models, we found that all the bubble growth model at the thermal control stage overestimates the growth rate of sliding bubble. During the sliding phase, the effect of condensation gradually increases and the contribution of the microlayer evaporation to the bubble growth is negligible. In addition, based on the force balance model, the differential equation of motion describing the sliding bubble is developed. The model considers the effects of buoyancy, flow drag, bubble volume expansion, and additional mass effects brought about by pressure gradients. With near-wall drag coefficient and local liquid velocity correction, the model agrees well with the experimental data and predicts diameter-velocity correlations well.

Finite strain-based theory for the superharmonic and subharmonic resonance of beams resting on a nonlinear viscoelastic foundation in thermal conditions, and subjected to a moving mass loading

We address the nonlinear free vibration, superharmonic and subharmonic resonance response of homogeneous Euler–Bernoulli beams resting on nonlinear viscoelastic foundations, under a moving mass and an abrupt uniform temperature rise. The nonlinear differential equation of motion stemming from the Hamiltonian principle and Finite Strain Theory is discretized according to a Galerkin decomposition method, and is solved by means of a multiple time scale method. A comparison between the Finite Strain theory and the Von-Karman approach is discussed, accounting for the effect of temperature rise, linear and nonlinear coefficients of the elastic foundation on the nonlinear vibration history and phase trajectory. At the same time, we check for the sensitivity of the frequency response of the system in superharmonic and subharmonic resonance for different input parameters, namely, location, velocity, and magnitude of the moving load, temperature rise and elastic foundation.

Layerwise formulation of poroelastic composite plate under pre-buckling and thermal shock loading

The present article scrutinizes the coupled time-variant poro-thermoelasticity behavior of functionally-graded graphene reinforced (FG-GPLR) composite plates embedded in an elastic foundation, intending to provide innovative solutions to preserve operating mechanisms toward harmful impacts of thermal shock load. The nanocomposite plate is initially stressed in three different orientations and subjected to thermal shock loading. The temperature distribution is acquired by employing the celebrated energy balance equation. The analytical solving approach is utilized for differential motion equations of the system based upon substitution of the equivalent Fourier series expansion for each state variable. Eventually, mapping the structure's response from Laplace to the time domain is carried out by a modified version of the celebrated Dubner and Abates' approach. Convergence of the outcomes regarding time is monitored and certified. The acquired results record the impact of varied parameters like length to thickness ratio of the structure, distribution pattern, and weight-content of GPL on the coupled time-variant poro-thermoelastic response of the FG-GPLR composite plate. One of the main conclusions of this survey is that increasing the weight-content of GPL in the way of locating more GPL near the upper and bottom surface of the plate applies enhancing impacts on the system's resistance against detrimental consequences of thermal shocks.

Sticky Brownian Rounding and its Applications to Constraint Satisfaction Problems

Semidefinite programming is a powerful tool in the design and analysis of approximation algorithms for combinatorial optimization problems. In particular, the random hyperplane rounding method of Goemans and Williamson [ 31 ] has been extensively studied for more than two decades, resulting in various extensions to the original technique and beautiful algorithms for a wide range of applications. Despite the fact that this approach yields tight approximation guarantees for some problems, e.g., Max-Cut , for many others, e.g., Max-SAT and Max-DiCut , the tight approximation ratio is still unknown. One of the main reasons for this is the fact that very few techniques for rounding semi-definite relaxations are known. In this work, we present a new general and simple method for rounding semi-definite programs, based on Brownian motion. Our approach is inspired by recent results in algorithmic discrepancy theory. We develop and present tools for analyzing our new rounding algorithms, utilizing mathematical machinery from the theory of Brownian motion, complex analysis, and partial differential equations. Focusing on constraint satisfaction problems, we apply our method to several classical problems, including Max-Cut , Max-2SAT , and Max-DiCut , and derive new algorithms that are competitive with the best known results. To illustrate the versatility and general applicability of our approach, we give new approximation algorithms for the Max-Cut problem with side constraints that crucially utilizes measure concentration results for the Sticky Brownian Motion, a feature missing from hyperplane rounding and its generalizations.

Three-to-One Internal Resonance of L-Shaped Multi-Beam Structure with Nonlinear Joints

In this paper, a reduced-order analytical model for an L-shaped multi-beam structure with nonlinear joints is presented to investigate the nonlinear responses of the system with three-to-one internal resonances conditions. Firstly, the global mode shapes are used to obtain an explicit set of nonlinear ordinary differential equations of motion for the system. Then, the first two natural frequencies of the system are calculated to determine the specific tip mass that results in three-to-one internal resonance. Subsequently, an approximation of the analytical solution of the dynamic model with two-degree-of-freedom is derived by using the multi-scale method. The accuracy of the approximation solution is verified by comparing it with the numerical solution obtained from the original motion equations. Based on the nonlinear dynamical model obtained by this paper, the frequency response curves are given to investigate the nonlinear dynamic characteristic of the L-shaped multi-beam structure with nonlinear joints. The results show that the nonlinear stiffness of the joints has a great influence on the nonlinear response of the system with three-to-one internal resonance conditions.

Tests of gravitational-wave birefringence with the open gravitational-wave catalog

We report the results of testing gravitational-wave birefringence using the largest population of gravitational-wave events currently available. Gravitational-wave birefringence, which can arise from the effective field theory extension of general relativity, occurs when the parity symmetry is broken, causing the left- and right-handed polarizations to propagate following different equations of motion. We perform Bayesian inference on the 94 events reported by the 4th-Open Gravitational-wave Catalog (4-OGC) using a parity-violating waveform. We find no evidence for a violation of general relativity in the vast majority of events. However, the most massive event, GW190521, and the second most massive event, GW191109, show intriguing non-zero results for gravitational-wave birefringence. We find that the probability of association between GW190521 and the possible electromagnetic (EM) counterpart reported by Zwicky Transient Facility (ZTF) is increased when assuming birefringence. Excluding GW190521 and GW191109, the parity-violating energy scale is constrained to $M_\mathrm{PV} > 0.05$ GeV at $90\%$ credible interval, which is an improvement over previous results from twelve events by a factor of five. We discuss the implications of our results on modified gravity and possible alternative explanations such as waveform systematics. More detections of massive binary black hole mergers from the upcoming LIGO/Virgo/KAGRA run will shed light on the true origin of the apparent birefringence.

Motion dynamics of a multicharging system in an electric field

In electrotechnical research there is a problem of analysis of the interaction of moving charged bodies on their trajectories. Its practical solution is possible only on the basis of an adequate mathematical model. To this end, we have adapted the law of force interaction of stationary charges by Charles Coulomb in the case of motion at all possible speeds. This takes into account the finite rate of propagation of the electrical interaction. Differential equations of motion of a closed system of charged moving bodies in their electric field are obtained. On this basis, the transients in a three-charge proton-electron system are simulated, such as the electromechanical equilibrium of an atom of a periodic table of elements. The simulation results are attached.

Energy harvesting using vibrating honeycomb sandwich panels with auxetic core and carbon nanotube-reinforced face sheets

The literature on energy harvesting using piezoelectric materials is firmly concentrated on structures like cantilever beams with piezoelectric layers. In the present research, a piezo harvester made of a honeycomb sandwich panel is studied. The modeled sandwich panel is made of an auxetic core and two carbon nanotube-reinforced face sheets. It will be shown that this harvester offers more flexibility in its stiffness, leading to better performance in harvested energy. The displacement field of the panel is formulated in a general form, which includes different types of shear deformable plate theories. Within these plate models, transverse shear stresses and rotary inertias are taken into account. A piezoelectric layer is integrated at the top of the honeycomb substructure, and vibrations are originated from an external harmonic source. The material properties of the carbon nanotubes vary lineally in the thickness direction. Hamilton’s principle is adopted to derive governing equations of motion, and then weighted residual method is applied to discretize the coupled differential electro-mechanical equations of motions. After validating the model, a detailed parametric study is carried out to investigate the impacts of different parameters such as auxetic cell characteristics, carbon nanotube volume fraction, carbon nanotube configuration, and geometrical dimensions on the electrical power and voltage of the harvester.

Exploration of radiative heat on the Jeffery fluid flow in a microchannel for the impact of suction/injection

AbstractThe purpose of the present paper is to investigate the flow and heat transfer of thermal radiation on the Jeffery fluid flow within a microchannel for the effects of the superhydrophobic surface (SHS) within suction/injection. The governing differential equations of motion and heat transfer are transformed into nonlinear coupled ordinary differential equations (ODEs) using appropriate similarity transformations. The ODEs are solved along with boundary conditions by adopting Runge–Kutta with shooting technique. Symbolic computational software such as MATLAB, the solver bvp4c syntax examines the behavior of the relevant physical parameters. However, some effective emerging parameters on the flow problem reveal that the microchannel walls within the suction/injection parameter increase the temperature, and the SHS is heated. In contrast, without slip, the opposite behavior is rendered. It is clearly shown that the velocity profile diminishes with increasing the Prandtl number. Furthermore, it is noticed that velocity decreases for increasing values of Hartmann number. Comparison with available results for particular cases is an excellent agreement.

Parametrically Excited Instability of Periodic Visco-Elastomer Sandwich Plate with Supported Masses Under Quadrilateral Longitudinal Harmonic Excitations

The parametrically excited instability of the spatially periodic visco-elastomer sandwich plate with supported masses under quadrilateral longitudinal temporal harmonic excitations is studied. The improvability of the parametrically excited instability by periodic distribution parameters is explored. The direct eigenvalue analysis approach for solving the parametrically excited stability problem of the periodic sandwich plate system under longitudinal harmonic excitations is proposed. The spatial periodic distribution of facial layer thickness and core layer modulus of the sandwich plate is considered. The non-linear partial differential equations of longitudinal and transverse coupling motions of the periodic visco-elastomer sandwich plate with supported masses under biaxial longitudinal boundary excitations are derived. The longitudinal displacements of the sandwich plate are separated into two parts and the longitudinal boundary excitations relevant to symmetric part are incorporated into the sandwich plate system. Then the partial differential equations with boundary excitations are converted into parametrically excited system equations and further converted into ordinary differential equations with time-varying parameters, which describe the parametrically excited vibration with multi-mode coupling of the periodic sandwich plate system. The fundamental perturbation solution to the equations is expressed as the product of periodic and exponential parts based on the Floquet theorem. The ordinary differential equations with harmonic parameters are converted into a set of algebraic equations using the harmonic balance method. Then the parametrically excited instability of the periodic sandwich plate system is determined directly by matrix eigenvalues. The overall instability characteristics of parametrically excited vibration with multi-mode coupling of the system under longitudinal harmonic excitations are illustrated by numerical results on unstable regions. The parametrically excited instability can be improved by the spatially periodic distribution of geometrical and physical parameters. The proposed approach is applicable to general sandwich structures with spatial distribution parameters in multi-mode-coupling parametrically excited vibrations for overall instability analysis on continuous frequency band.