Differential Equations Of Motion Research Articles

Influence of circuit on power generation performance of wave energy power generation device using oscillating water column technology

In order to study the influence of the load circuit on the system efficiency and power oscillation of the oscillating water column device, a unified modeling model of the oscillating water column device is built for simulation. According to the coupling relationship in various fields of the oscillating water column device, the differential equation of motion of the turbine generator set is established. The power plant components and systems are modeled by Modelica language. Based on this system, the system-level simulation is carried out and the load circuit is optimized. In addition, experimental tests of the traditional load circuit and the modified load circuit are carried out. Results indicate that the modified load circuit can increase the speed of the turbine to better follow the airflow and balance the system damping, thereby improving the system efficiency. Increasing resistance can reduce the fluctuation of output electric power. Different battery, resistance and generator parameters can cause differences in system efficiency. The research results provide a reference to improve the performance of oscillating water column device.

Study on vortex-induced multimodal coupled vibration of arch bridge suspenders

The objective of this paper was to investigate the 2-DOF suspender response under the condition of vortex-induced force and reveal the multimodal coupling mechanism. Firstly, an elastically mounted segmental model of suspender was established. Wind tunnel experiments were conducted to obtain the time history displacement signals in cross-flow and in-line directions, the empirical parameters were obtained by identify those signals. The nonlinear partial differential motion equation of suspender with two degrees of freedom was established based on Hamilton's principle, which considered the dynamic strain, bending stiffness and vortex-induced force of suspender. The Galerkin method was used to discretize the nonlinear partial differential equations into nonlinear motion governing equations, which considered the first three order modes. A numerical method was used to solve the motion governing equation, the time history displacement curve, phase diagram, amplitude spectrum and Poincare map of the multi-modal responses in the cross-flow and in-line directions were obtained. With the weak coupling effect of nonlinear stiffness and vortex-induced force, the cross-flow and in-line responses demonstrate period doubling bifurcation, quasi-periodic vibration, period doubling vibration, crises and transient chaotic phenomena. However, with the strong coupling effect, the crises and transient chaos in response amplitude disappear. More importantly, planar/non-planar multimodal internal resonance were found in this work. With the weak multimodal coupling effect in suspender response, the low-order modal response is dominant; as the multimodal coupling effect increases, the high-order modal response becomes predominant. In addition, the system energy not only transfers from high-order modal response to low-order modal response, but also the system energy transfer from low-order modal response to high-order modal response. These conclusions are useful for the design and practical engineering of suspenders, which have high theoretical research value in understanding the mechanism of multimodal coupling effects on suspender response.

RETRACTED: Analysis of governing equations of multibody dynamic systems through kinematic properties of joints and effects of kinematic chains on closed-form mechanism

In this investigation, the fundamental equations dictating the behavior of dynamic systems were ascertained by means of an innovative approach that amalgamated the kinematic properties of joints and the intricate kinematic chains of multibody systems into a series of governing equations. Subsequently, the equations that govern the behavior of multibody systems were converted into ordinary differential equations (ODEs) utilizing the calculus of matrix-valued functions. This computational technique is adept at swiftly acquiring recursive differential equations of motion for multibody systems. Thus, the expenditure of computational costs associated with the simulation was efficaciously minimized. The mechanisms of Andrew's squeezing and carpet scrapping were employed in conjunction with kinematic constraints to authenticate the validity of the proposed methodology. The findings suggest that, notwithstanding the fact that the root mean square error (RMSE) in the generalized coordinates between the two ODE and differential algebraic equation (DAE) techniques is small in the Andrew's squeezing mechanism (at approximately 0.02) and in the carpet scraping mechanism (at approximately 0.7), there are still discrepancies between the two methods. The degree of convergence in the aforementioned methodologies, despite the slight RMSE difference between them, shows a clear disparity. By way of illustration, when employing the conventional DAE technique, the duration required to resolve the Andrew's squeezing and carpet scraping mechanisms is 5 and 11.34 s, respectively. However, by utilizing the ODE methodology in the aforementioned mechanisms, the time required to determine a solution is truncated to 1.19 and 2.1 s, sequentially.

Определение параметров упругодемпфирующих устройств в механизмах карьерных экскаваторов

Introduction. The aim of the work is to develop a procedure for synthesizing the parameters of elastic-damping devices for shovel based on the representation of these devices in the form of additional feedbacks and optimization of the transfer functions of these feedbacks, which allows abstracting from specific methods of their implementation and making a comparative assessment and reasonable choice of device parameters. Research method. Optimization of the parameters of elastic-damping devices is carried out on the basis of using the root method for studying the characteristic polynomial of the transfer function of the additional feedback and setting the desired nature of the transient process, which is the solution of the original differential equation of motion with given roots. Results. A procedure for synthesizing the parameters of elastic-damping devices based on the representation of these devices in the form of additional feedbacks and optimization of the transfer functions of these feedbacks using root methods is proposed, which allows one to abstract from specific methods of their implementation and make a comparative assessment and reasonable choice of device parameters. Analytical dependencies are obtained, that establish a relationship between the parameters of elastic-damping devices and the quality of transient processes, making it possible to determine the effective range of these parameters. Numerical modeling of the electromechanical system of the digging mechanism of a mining excavator was carried out based on the use of its real characteristics, which confirmed the high efficiency of the proposed method for the synthesis of stiffness and damping parameters of elastic-damping devices. Currently, an experimental unit equipped with a prototype of an elastic-damping device is being created, on which it is expected to evaluate the performance and effectiveness of the proposed method for synthesizing parameters. After this, it is planned to contact potential consumers of such devices to determine the possibilities of practical application of both the devices themselves and the proposed method for synthesizing their parameters.

Dynamic analysis of the joint movement of the hoisting and slewing mechanisms of a boom crane

To increase the productivity of boom cranes, the operation of individual mechanisms is combined. At the same time, dynamic loads on structural elements, drive mechanisms and loads on a flexible suspension increase, which reduces the reliability of crane operation and increases energy losses. Therefore, the research aims to consider the problem of the dynamics of the joint movement of the mechanisms for load slewing and hoisting of a boom crane. To study the dynamics of the joint movement of the mechanisms, the boom system was represented by a mechanical system with 6DOF, where the basic movement of the mechanisms and the oscillatory movement of the structural links with elastic and dissipative properties, as well as the load on a flexible suspension in the plane of crane slewing and hoisting were considered. For such a mechanical system of a crane, the differential equations of the joint motion of the crane slewing and hoisting mechanisms were developed. The obtained equations are a system of the second order nonlinear differential equations, for solving which a numerical method in the form of a computer program was used. Using the developed program, the dynamics of the joint movement of the mechanisms of a jib crane with specific numerical parameters were calculated. Based on the calculations, a dynamic analysis of the joint movement of the mechanisms for slewing and hoisting the load of a jib crane with a hoisting boom was carried out, which revealed high-frequency vibrations of links with elastic and dissipative properties, as well as low-frequency oscillations of the load on a flexible suspension. The greatest impact of oscillations is observed during the start-up of mechanisms, where high-frequency oscillations dampen during the transient process, and low-frequency oscillations dampen over a fairly significant period. To improve the dynamic properties of the mechanisms for turning and hoisting a load during their joint movement, it is proposed to optimise the mode of movement in the areas of transient processes (start-up, braking). The research results can be used in the development and operation of cranes in mechanical engineering, construction, and other industries

Imitation Model of a Dual-Mass Machine with Universal Joint Shaft Transmission

The research is devoted to developing a mathematical model describing the dynamics of a dual-mass machine unit with a cardan gear, taking into account the catalogue data of an induction motor. A reference simulation model of a machine unit of sequential structure driven by a given induction motor is obtained. The simulated machine unit consists of a reducer and two rotating masses with a cardan gear with a given torsional stiffness between them. The differential equations of motion of the machine unit were calculated. A mathematical description of mechanical characteristics on the basis of Klos formulae has been developed and the influence of machine unit parameters on its performance in start-up, steady-state and emergency modes of operation has been studied. The analysis of arising oscillations and dynamic loads is carried out and the results of numerical implementation of the obtained results are presented.

Analytical solution for size-dependent nonlinear behavior of AFM microcantilever with assembled probe in liquid environments

Using modified strain gradient theory, this study aims to examine the size-dependent nonlinear behavior of atomic force microscopy (AFM) with an assembled cantilever probe (ACP) in various liquid environments. The largest contribution of this theory lies in its consideration of the material length scale parameter (MLSP) to analyze the effect of size on physical characteristics. When studying ACP in liquids, the hydrodynamic force creates a different behavior compared to an ACP beam enclosed in air. In this study, Hamilton’s principle is used to derive the nonlinear partial differential equation (PDE) of motion and associated boundary conditions (BCs) of the ACP in liquid environments. To achieve this, a combination of Euler-Bernoulli’s theory and strain gradient concepts are employed. As the next step, a reduced order model of the system is developed using the Galerkin method. To assess the nonlinear frequency response of the ACP in various liquid environments, perturbation technique, including the method of multiple scales (MMS), is applied to the developed reduced order model. The study also calculates resonance frequencies, phase planes, stability, and the potential function of the system analytically. The results of this study indicate that the system becomes nonlinear as the ratio of beam thickness to MLSP decreases. Several factors, including the ratio of beam thickness to MLSP as well as the viscous damping coefficient, can influence the jump phenomenon. Furthermore, the results of the phase and amplitude of the ACP in liquid environments demonstrate that an increase in the (h/l) ratio results in a decrease in response amplitude and a leftward shift in the phase of the response diagram. This phenomenon is indicative of a softening behavior exhibited by the system. Lastly, the paper examines the validity and accuracy of the MMS solution by comparing it to a numerical solution.

ელექტროგამტარ სითხეში წრიული ფოროვანი ფირფიტის ბრუნვის სტაციონარული ამოცანა გაჟონვის სიჩქარის დიდი მნიშვნელობებისათვის თბოგადაცემით, მაგნიტური ველისა და ჯოულის სითბოს გათვალისწინებით

The stationary problem of rotation of a circular porous plate in an electrically conductive fluid is studied by the sequential approximation method (Green's function and small parameter method) for large values of the suction velocity with heat transfer, taking into account a weak magnetic field. The problem considers the case when the influence of dissipative members on heat transfer is very small, the energy equation takes into account the influence of Joule's heat on heat transfer, and it is assumed that the temperature along with the radius of the plate changes according to the square law. For solving the problem, the Navier-Stokes and energy partial differential equations of fluid motion in a magnetic field are reduced to ordinary nonlinear differential equations using generalized Karman embeddings, the solutions of which are sought in the form of infinite series for small values of the suction parameter. The solution of the problem by the means of Green's function is reduced to the solution of integral-differential equations. Recurrence formulas have been obtained, by the means of which it is possible to calculate the solution with any approximation. The first two approximations are clearly calculated. All kinematic characteristics of fluid flow are calculated. Images for heat transfer and pressure distribution on a circular plate are obtained. The moment of resistance to rotation of the plate and the heat transfer coefficient are also calculated.

Dynamic finite element modeling of axially functionally graded Timoshenko microbeams under a moving mass

This paper presents a finite element model for dynamic analysis of axial functionally graded (AFG) microbeams subjected to amovingmass. Thematerial properties of themicrobeams are considered to vary in the axial direction by a power-law function, and they are evaluated byMori-Tanaka scheme. The first-order shear deformation theory is employed in conjunction with the modified couple stress theory to establish the differential equations of motion for the AFG microbeams. A two-node beam element with six degrees of freedom was derived and used in combination with Newmark method to solve the equations of motion and to compute the dynamic response of the microbeams. Numerical investigations are carried out on AFG microbeam with simply supported ends. The proposed method is validated by comparing with results in previous work. The effects of dimension-less material length scale parameters, the material distribution and the moving mass parameters on the dynamic characteristics of the AFG microbeams are studied and discussed in detail.

Development and investigation of the vibration-driven in-pipe robot

Vibratory machines are widely used for monitoring and cleaning various tubes, pipelines, intestines, vessels, etc. The problems of ensuring the prescribed dynamic characteristics of such equipment and simultaneous optimizing the power consumption are currently being solved by numerous researchers. The main purpose of this study is to investigate the locomotion characteristics of the novel vibration-driven design of the pipeline inspecting and cleaning robot actuated by an electromagnetic exciter and equipped with the size-adapting and self-locking mechanisms. The research methodology consists of four main stages: an overview of the enhanced robot design; constructing its dynamic diagram and deriving the differential equations of motion; performing the numerical modeling with the help of the Mathematica software and studying the robot’s kinematic characteristics; conducting virtual experiments by computer simulation of the robot motion in the SolidWorks software. The research results present the time dependencies of the robot’s displacement, speed, and acceleration at different working regimes (excitation forces, disturbing frequencies, etc.). The novelty of the performed investigations consists in substantiating the efficient locomotion conditions of the enhanced vibration-driven in-pipe robot. Further investigations can be focused on developing the full-scale laboratory prototype of the robot and conducting experimental studies. The obtained research results can be interesting for engineers and scientists who deal with similar vibration-driven pipeline robots.

The Hydrodynamics of Ammonoid Swimming: Equations of Motion and Rocking Resonances

This work explores the swimming of ammonoids, cephalopods related to living squids, octopuses, and nautilids and, like the latter, equipped with a coiled external shell. A mathematical model is introduced for theoretical ammonoid conchs. The two differential equations of motion (one for the centre of mass, including the drag force and the added mass coefficient, and one for the roll angle) are solved numerically for the theoretical conchs, and the results are analysed in terms of velocity and rocking angle. Destabilising resonances occur when the rocking motion is in phase with the propelling water jet. It is suggested that the ammonoids partly evolved avoiding the occurrence of such resonances in their construction.

Research on the dynamic characteristics of rotor-bearing-seal system with breathing cracks under steam flow excited force

In order to reveal the influence of breathing-type straight cracks on the motion characteristics and stability of a turbine rotor-bearing-seal system. The study examines the motion characteristic of a cracked rotor system under different loads with different crack depths, the coupled thermal and dynamic loads. The research derives motion differential equations for the cracked system, considering nonlinear dynamic properties such as nonlinear stiffness and damping. The equations incorporate the nonlinear steam flow excited force obtained from numerical simulations. The resulting motion equations are solved using the Runge-Kutta method, and the accuracy of the investigation of nonlinear dynamic characteristics is validated through experimental comparisons. Based on these findings, the motion characteristics and stability of the cracked rotor system under different conditions are analyzed. The results indicate that the motion characteristics of the rotor are progressively altered by the crack depth, and at a certain depth, the motion deviates from the established pattern. The influence of cracks on rotor stability is predominantly notable in the 34–58% THA region and under high load conditions. Moreover, the chaotic characteristics of rotor motion is more significant after applied the coupled thermal and dynamic loads, and the stability of the 34–58% THA region is significantly worse.

Stability and Sommerfeld effect in a multi-resonant types vibrating system with isolated rigid frame driven by four exciters

The previous studies on the vibrating system with multiple exciters less considered the Sommerfeld effect, which were mainly focused on synchronization and stability problems of the system in the whole resonant region. A dynamical model is proposed in this paper to study the synchronization, stability and the Sommerfeld effect of four exciters in the multi-resonant types vibrating system (MRTVS). In order to improve the power of the system, the vibrating system with two rigid frames is driven by four especially distributed exciters. The motion differential equations of the system are given by Lagrange’s equations, and the dimensionless coupling equations and synchronization criterion are constructed as well by the average method. The theory condition for stability of four exciters in synchronous states is derived from the Hamilton principle. The coupling dynamic characteristics, stability and Sommerfeld effect of the system are discussed numerically. The whole resonant region of the system is divided into three segments by natural frequencies, and the synchronous and stable states in the different resonant types are analyzed respectively. The diversity phenomenon of the nonlinear system is observed, in other words, the system exists multiple stable equilibrium solutions at a certain particular resonant region. Then the Sommerfeld effect around the natural frequencies (NFs) is further revealed. The correctness of theoretical analyses is examined by simulation and the experiment. It’s shown that the reasonable working point in engineering should be selected in the sub-resonant region with respect to the natural frequency of the main vibrating system. The present work can provide theoretical guidance for designing some new types of vibrating machines with high driving power.

GPL-Reinforced composite piezoelectric microcantilever dynamics in atomic force microscope

Currently, Atomic Force Microscope (AFM) is utilized as an efficient tool for nanoscale measurements and nanomanipulation of particles. The central part of AFM is considered to be the microcantilever (MC) used in the device. AFM MC is generally made from silicon. Since functionally graded graphene-nanoplatelet-reinforced composite (GPLRC) materials are highly popular in industry and academic research, the present study aims to be the first attempt to investigate the possibility of utilizing these materials in AFM MC. Also, it seeks to compare the nonlinear vibrational behavior of the MC made from this type of material with that of the silicon MC and also to investigate the vibrational behavior of the GPLRC MC in interaction with the sample surface. To do so, the GPLRC MC vibrated with the piezoelectric layer on its top is utilized. In order to increase the sensitivity of the MC to the interaction force between the surface and the probe tip, the tip of the MC is considered narrower. The differential equation of the vibrational motion of the MC near the surface of the sample is derived by considering the Lennord-Jones model according to the Euler-Bernoulli theory and applying the Lagrange method. In order to solve the nonlinear differential equation of motion with partial derivatives in terms of the time and the position of each point on the MC, Galerkin variable-separation method and multiple time scale (MTS) method are applied. The results of the dynamic modeling near the surface of the sample indicate that the MC made of GPLRC is more sensitive to the nonlinearity of the interaction force as compared to the common silicon MC.

Performance analysis of nonlinear piezoelectric energy harvesting system under bidirectional excitations

In this paper, a cantilever beam piezoelectric energy harvester model of nonlinear partial differential equation based on Hamilton's Principle is established. A piezoelectric device using a bimorph cantilever beam with a fixed is subjected to horizontal and vertical excitation. Based on the Euler-Bernoulli thin beam model assumptions and inextensible deformation, the analysis further includes the effects of geometric nonlinearity and damping nonlinearity. The reduced-order nonlinear partial differential equations of motion with electro-mechanical coupling distributed parameter are derived by using Galerkin method. By using the method of multiple scales and solving the equations of motion, the energy harvester in its fundamental first-order resonance response is analyzed and the amplitudes of fundamental transverse deflection, output voltage and output power are determined. The major influence aspects of excitation amplitude, linear damping coefficient, nonlinear damping coefficient, impedance, nonlinear electro-mechanical coupling coefficient on transverse deflection, voltage and power are analyzed. The result shows that the linear damping coefficient and resistance affect the initial threshold of parametric excitation. The combination of parametric excitation and direct excitation gives full play to the advantages of parametric excitation and improve the energy conversion efficiency of the energy harvesting system.

Frequency stability analysis of a cantilever viscoelastic CNT conveying fluid on a viscoelastic Pasternak foundation and under axial load based on nonlocal elasticity theory

AbstractIn this paper, the frequency stability analysis of a fluid conveying cantilever viscoelastic carbon nanotube (CNT) in the Kelvin–Voigt material model with slip boundary condition (BC) regime on a viscoelastic Pasternak foundation and under axial load has been performed based on nonlocal Euler–Bernoulli thin beam theory. The governing partial differential equation of motion (EOM) and its associated BCs have been derived using the Hamilton principle. The governing EOM has been converted into an ordinary differential equation using mode summation technique and solved by applying the extended Galerkin method. Then, the frequency stability analysis has been carried out for the vibrational response of CNT in the state‐space form. The obtained results have been validated with the literature works. The effect of changes of various parameters like elastic stiffness, shear stiffness and damping coefficients of foundation, nonlocal scale‐effect parameter of the nanotube, fluid flow Knudsen number, mass ratio, structural damping of the nanotube and applied axial force have been investigated in terms of frequency to predict the occurrence of different instability modes for the system. It is concluded that flutter and divergence instabilities in the vibration of the viscoelastic CNT are significantly affected by changes of different above‐mentioned parameters.

Asymptotic analysis of the angular motion of a projectile based on Lyapunov artificial-small-parameter perturbation

In the traditional analysis of the angular motion of a projectile, the coefficient freezing method is often used to simplify the differential equation of angular motion. However, this method gives a poor approximation for long time scales. In this paper, based on Lyapunov artificial-small-parameter perturbation theory, an asymptotic analytical solution for the angular motion of a projectile is derived by constructing artificial small parameters for the variable coefficient terms in the original equation. In an example, the angular motion of a single-channel rotating projectile with two-dimensional trajectory correction was analyzed. The method proposed in this paper was compared with the coefficient freezing method. The results show that the proposed method can give a good approximation over a long time scale. This study may be significant for the solution of differential equations with variable coefficients and may be useful for certain engineering applications.

Seismic analysis of a soil-liquid tank system using MATLAB Simulink

In this paper, a model is proposed for analyzing the long-term responses of a liquid tank system to earthquake loads while taking soil interaction into account. Three degrees of freedom (DOFs) in the sway-rocking form and the monkey tail DOF are used to model the soil-foundation system, while the fluid in the tank is modeled as a 2-DOF lumped parameter using the substructure method. By using the MATLAB Simulink tool to numerically integrate the system of differential equations of motion, the response of the DOFs, shear force, and moment at the tank bottom can be tracked over time. The El Centro 1940 earthquake's acceleration excited a vertical cylindrical tank with a radius of 10 m and a water mass of 8 m; the results of the analysis show that the horizontal displacement of the convective mass increased by 16.36%, and the shear force and moment significantly increased in comparison to the case of the tank rigidly linked to the ground.

Nonlinear Differential Equations of Flow Motion Considering Resistance Forces

For a stationary potential 2D planar open high-velocity water flow of the ideal liquid, we propose a closed system of nonlinear equations considering the resistance forces to the flow from the channel bottom. Tangential stresses on jet interfaces are ignored. The resistance force components are expressed in terms of velocity components. In this case, the flow equations can be solved through the method of characteristics, and the surface forces are reduced to equivalent volumetric forces. The system of non-linear equations is solved in the velocity hodograph plane; further, the transition to the physical plane takes place. Since the value of the hydrodynamic pressure decreases downstream of the flow, the friction forces to the flow in the first approximation can be considered by using the integral laws of resistance. At that, the form of the equations of motion in the plane of the velocity hodograph does not change. This fact is proved in the article. An example of calculating the water flow is provided. The kinecity, ordinates, and velocities of the flow along its extreme line are calculated without considering resistance forces. Validation of the model in the real flow is performed. Acceptable accuracy relative to experimental data is obtained.

Large amplitude free vibration analysis of circular arches with variable thickness

This study aims to investigate the large amplitude nonlinear vibration of variable thickness arches with different cross-sectional shapes under different boundary conditions. The governing equations of the arches are derived using Hamilton's principle by considering their geometric nonlinearities. Then, the differential quadrature method (DQM) is employed for the discretization to obtain the ordinary differential equations of motion for the circular arches with variable thickness. The theoretical model is validated by comparing the linear and nonlinear natural frequencies of arches. The nonlinear vibration characteristics of arches are solved using the direct iteration scheme. The effects of the slenderness ratio, modified slenderness ratio of the arch, and variable thickness parameters under different cross-sectional shapes on the nonlinear to linear frequency ratios of the arches with clamped and pin-ended boundary conditions are investigated. The results indicate that appropriate variable thickness parameters can improve the linear frequency and the nonlinear to linear frequency ratio of arches, which offers some suggestions for the design of variable thickness arches.