Differential Equations Of Motion Research Articles

Periodic and torus motions of a two-degree-of-freedom dry friction vibration system

Vibration induced by dry friction is ubiquitous in various engineering fields. To explore the vibration characteristics for further studies and/or controls, it is of great theoretical and practical significances to investigate the non-linear dynamic behaviors of the friction systems. This study considers the slight vibration of a two-degree-of-freedom non-linear dry friction excitation system. The differential equations of system motion are established according to Newton’s law of motion. Moreover, the system’s non-linear dynamic is studied when the block velocity is always less than the friction surface velocity. The results indicate that the linearized matrix of the vibration system has a pair of purely imaginary eigenvalues for some critical values of the relevant parameters. The Poincaré-Birkhoff normal forms are utilized to simplify the motion equation under the non-resonant assumption to obtain a simplified equation with only the resonant terms. Furthermore, the truncated part of the simplified equation is analyzed in the case of only linear terms degeneration. Finally, numerical simulations reflect some qualitative conclusions about the system’s local dynamic properties, including equilibrium point, periodic motion, torus motion, and their stability.

Electromechanical model of exoskeleton with three mobile links

The article considers an electromechanical model of an exoskeleton with three links, for which a system of differential equations of motion is written and inverse and direct problems of dynamics are solved. The angles between the links that specify anthropoid motion are figured out analytically. The electric drive controlling torques are found as a result of solving the inverse dynamics problem. The found torques are approximated with stepwise piecewise-constant functions simulating the impulse control of the exoskeleton motion. The solution of the direct problem of dynamics is carried out and the link rotation angles as functions of time are found. The results of the numerical solution of the system of differential equations are compared with the initial motion of the links. It has been found that the results of simulation with impulse control are in good agreement with the original motion. The total energy expenditures have been calculated. The exoskeleton simulation, taking into account the electric drive impacts also has been carried out. For this purpose, a system of differential equations of motion has been compiled; a numerical solution of the Cauchy problem for the compiled system has been carried out. The significance of the electric drive impact on the dynamics of the mechanism has been confirmed.

Метод расчета нагрузок при наполнении парашютов с учетом податливости конструкции

The paper analyzes various methods for determining forces during the opening of parachutes, based on the solution of a system of differential equations of motion of a mechanical system as a material point. The parachute inflation is a complex process in which the dynamics of air currents is closely related to the dynamics of the movement of the entire mechanical system: the payload, the parachute, and the involved air masses. The main problem in calculating the process of its inflation is that it is required to determine the maximum value of the loads during inflation, taking into account unknown parameters of the air mass mition directly near the parachute system and considering the implementation features of the design of parachutes. For the first time, a synthesized method for calculating loads when inflating parachutes is presented, considering the compliance of the structure, for which two methods were combined: (1) the existing one, to determine the reduced speed of the system when inflating the parachute, and (2) the new one, to determine the coefficient of parachute dynamism under conditions of “infinite” mass.

Nonlinear Dynamical Analysis of Composite Boring Bar with Nano-carbon Materials

Preventing resonance in boring bar vibration is the key to achieve high quality and efficiency. In this paper, using the three-dimensional nonlinear dynamic model of boring process, the cutter bar is simplified as a rotating non planar bending axis, starting from the constitutive relationship and stress displacement relationship of composite materials. Based on Hamilton principle, the nonlinear motion differential equation of cutting process is established. The nonlinear partial differential equations of bending vibration are discretized by Galerkin method. Using the multi-scale method, the frequency response functions of the system in the primary resonance and super resonance are obtained. Using this model, the effects of the length diameter ratio of nano-carbon materials and the content of nano-carbon materials on the vibration frequency response of the boring bar are studied. The results show that the steady vibration amplitude of the cutter bar can be reduced by adding nano-carbon materials to the fiber reinforced polymer composite (FRP) beam.

Vibration isolation performance of a beam clamped with torsional quasi-zero-stiffness isolator

Elastic beam structures connected by hinges are a common basic component of many mechanical systems. In this study, the nonlinear transverse vibration of a beam with one end free and the other end simply supported with a torsion spring is investigated. A torsional quasi-zero-stiffness vibration isolator is employed at the fixed end of the beam for isolating the vibration excitation from the base. Using Hamilton’s principle, the differential equation of motion for the transverse vibration of the beam is obtained, and the natural characteristics of the beam are determined. The dynamic equation of the system is derived by using Galerkin truncation method and the harmonic balance method is adopted to obtain the nonlinear vibration response. Compared with the vibration transmissibility of the beam supported by the equivalent linear torsion spring, the quasi-zero-stiffness vibration isolation system shows superior vibration isolation performance. It is found that the beam fixed with a torsional quasi-zero-stiffness isolator can reduce the first resonant frequency and provide a wide effective isolation range at low frequency, which is rarely affected by the strong nonlinearity. The results would provide an innovative method of introducing the torsional quasi-zero-stiffness isolator into the passive vibration control of continuum system.

High-fidelity flow field reconstruction model for incompressible fluid with physical constraints

Research on the flow characteristics of incompressible fluids in the field of ship and ocean engineering is important. The direct numerical simulation method can obtain all information on the flow field by solving the full-scale Navier-Stokes equations. However, it has high computational cost and low computational efficiency. By contrast, the Reynold-averaged Navier-Stokes method has significantly better efficiency and is widely used in engineering, but it is not as accurate as the direct numerical simulation method (DNS). In recent years, deep learning has developed rapidly and provided a new method for the study of fluid flow characteristics. However, current models have many problems in three-dimensional cases, at high Reynolds numbers, and with inverse problem solving. To address these problems, we propose a high-fidelity flow field reconstruction model, namely, HFRIF for incompressible fluids. The method introduces reduced-order differential equation of viscous fluid motion and low-fidelity the Reynold-averaged Navier-Stokes method (RANS) simulation data as constraints in the loss function. We apply it to the reconstruction of two-dimensional and three-dimensional flow fields around a cylinder with Reynolds numbers of 100 and 10000, respectively, and compare it to the high-fidelity flow field data. Finally, super-resolution reconstruction of the flow field and the inversion of physical equation parameters are studied under the same model parameter settings. We find that even in three-dimensional cases and at high Reynolds numbers, the model only needs low-fidelity data to realize high-fidelity flow field reconstruction. Moreover, high-precision super-resolution flow field reconstruction and high-precision parameter inversion can be achieved without changing the model hyper-parameters.

Research on radiation field and driving based on super-low frequency mechanical antenna array

Mechanical antennas (MAs) directly use the mechanical motion of electric or magnetic charges to excite electromagnetic waves. The radiation distance of rotating magnetic dipole type mechanical antennas is related to the volume of the radiation source, so the volume of the radiation source is too large for long-distance communication. To solve the above problem, we first establish the magnetic field model and differential equations of motion of the antenna array. Then, we design the prototype of antenna array with operating frequency of 75-125Hz. Finally, we experimentally established the radiation intensity relationship between a single permanent magnet and an array of permanent magnets. The results indicate that our driving model reduces the tolerance of the signal by 47%. Through 2FSK communication experiments, this article verifies the feasibility of extending the communication distance in the form of an array, which provides an important reference for long-distance low-frequency communication.

Analytical propagation on relative motion of geostationary earth orbit satellites with orbital perturbations

AbstractAn accurate description of the relative motion of satellites is critical for space missions in the complex geostationary earth orbit space environment. Here, a new relative motion model is established considering major perturbations. To avoid the singularity of the classical orbital elements, a set of quasi‐non‐singular orbital elements are used to represent the relative motion states, which is further extended by considering the SRP coefficient difference. On the basis of existing classical model, an analytical model is derived by taking the Taylor expansion of the time derivative of the perturbation functions to the first order in the neighbourhood of the reference satellite, which contains the earth's oblateness perturbation, the third‐body perturbations and the solar radiation pressure perturbation. This method avoids solving the complex differential equations of relative motion directly and provides a general framework for including arbitrary perturbations into the model. The new model is verified in uncontrolled natural relative motion scenarios and position‐keeping control scenario, respectively. The results show that the proposed model has high accuracy and strong competitiveness.

Research on the Symmetry of the Hamiltonian System under Generalized Operators

Generalized operators have recently been proposed with great potential applications. Here, we present research carried out on Noether figury and perturbation to Noether symmetry for Hamiltonian systems within generalized operators. There are four parts, and each part contains two kinds of generalized operator. Firstly, Hamilton equations are established. Secondly, the Noether symmetry method is used for finding the solutions to the differential equations of motion, and conserved quantities are obtained. Thirdly, perturbation to Noether symmetry and adiabatic invariants are further explored. In the end, two examples are given to illustrate the methods and results.

Application of artificial intelligence technique in optimization and prediction of the stability of the walls against wind loads in building design

The present study is done to perform the optimal design of the structural components of the buildings against the unwanted wind load exerted on their outer face. To this end, the case study of the research is the outer wall made of a one-floor building modeled as a rectangular plate with only one free edge and three clamped ones. It is assumed that the wall is a sandwich plate whose core is made of auxetic material and dace-sheets are reinforced with nanoparticles of graphene platelets (GPL). Differential equations governing the system’s motion are obtained within the background of the plate’s shear-deformation theories. The stability analysis of the sandwich wall is performed based on the application of artificial intelligence (AI) methods optimized with an innovative optimization approach to gain a high level of accuracy. To determine the stability information of the system at the train points, the differential quadrature approach (DQA) is applied as the solver of differential equations of motion. The accuracy of the methods used in this paper is examined and verified by comparing the results with those acquired in the articles published previously. The results obtained in this study provide very useful information about the stability response of lightweight building components through AI-based solutions.

McKean-Vlasov stochastic differential equations driven by the time-changed Brownian motion

In this paper, we study a class of McKean-Vlasov stochastic differential equations driven by the time-changed Brownian motion and impulsive McKean-Vlasov stochastic differential equations driven by the time-changed Brownian motion. We will first discuss questions of existence and uniqueness of solutions under some appropriate conditions by some fixed point theorems. Subsequently, we consider various stability properties with respect to initial data and coefficients by some time-changed Gronwall-like inequalities and time-changed Gronwall-like inequalities with jumps, generalizing some known results.

A third transition in science?

Since Newton, classical and quantum physics depend upon the 'Newtonian paradigm'. The relevant variables of the system are identified. For example, we identify the position and momentum of classical particles. Laws of motion in differential form connecting the variables are formulated. An example is Newton's three laws of motion. The boundary conditions creating the phase space of all possible values of the variables are defined. Then, given any initial condition, the differential equations of motion are integrated to yield an entailed trajectory in the prestated phase space. It is fundamental to the Newtonian paradigm that the set of possibilities that constitute the phase space is always definable and fixed ahead of time. This fails for the diachronic evolution of ever-new adaptations in any biosphere. Living cells achieve constraint closure and construct themselves. Thus, living cells, evolving via heritable variation and natural selection, adaptively construct new-in-the-universe possibilities. We can neither define nor deduce the evolving phase space: we can use no mathematics based on set theory to do so. We cannot write or solve differential equations for the diachronic evolution of ever-new adaptations in a biosphere. Evolving biospheres are outside the Newtonian paradigm. There can be no theory of everything that entails all that comes to exist. We face a third major transition in science beyond the Pythagorean dream that 'all is number' echoed by Newtonian physics. However, we begin to understand the emergent creativity of an evolving biosphere: emergence is not engineering.

INTERNAL TIME OF A MECHANICAL SYSTEM

One of the most urgent unsolved problems of modern science is the extension of the second law of thermodynamics to conservative mechanical systems that are in a state far from thermodynamic equilibrium. How does the collective motion of the material points of the system, each of which is described by time-reversible equations, become irreversible? Thus, the increase in entropy of an ideal gas in a state close to equilibrium indicates the irreversible nature of the evolution of a conservative mechanical system, which is definitely an ideal gas. That is, entropy acts as internal time and sets the direction to the future or the "arrow of time". The paper deals with obtaining a characteristic of a conservative mechanical system that would have the properties of entropy. The goal is to extend the second law of thermodynamics to conservative mechanical systems that are in a state far from equilibrium. Analytical methods were used to analyze the differential equations of motion of a phase liquid particle as an image of a mechanical system in a multidimensional space. The evolution of the distribution of the probability of a particle of a phase fluid staying on a hypersurface of equal energy is considered. For a conservative mechanical system, a value is introduced whose properties allow us to apply the term "internal time" to it. Its growth determines the difference between future events and past events, which is an inherent property of subjective time. When approaching the state of equilibrium, internal time slows down, and physical time, accordingly, accelerates relative to internal time. Internal time is as universal as physical time, in the sense that it is determined for each system by a universal formula. The proposed approach made it possible to solve the fundamental problem of replacing the averaging of the probability density in the phase space by averaging in time along the trajectory of the point in the phase space. We used the fact that the equation of motion of a conservative system can be obtained as the equation of motion of a particle of a phase liquid, using the law of conservation of matter. The equations of motion of a particle were transferred to the motion of a point. In contrast, when using the traditional approach, the equation of motion of a point in phase space was taken as the basic law of nature. An understanding of internal time allows us understand the emergence of dissipative structures in the future.

Dynamic Analysis and Seat Selection of Bus Driving Comfort under Different Road Conditions

The comfort of a bus running on different road conditions is a matter of public concern. In this paper, the differential equations of motion are established for a bus running on different road conditions and the whole driving process is mechanically analyzed. Firstly, the bump degree at different positions is quantitatively analyzed and it is found that the rear row is bumpier on different roads. Then, the relationship between the speed of the bus and the vertical displacement and acceleration is quantitatively described. Regardless of the speed, a similar displacement and acceleration will be eventually achieved, but the speed is higher, and the duration of maximum displacement and acceleration is longer. When the speed is 8 m/s, resonance occurs on the bus during road condition II. Finally, the change in vertical displacement and acceleration under the action of different spring stiffness coefficient ratios of the front and rear wheels is quantitatively analyzed. High stiffness ratios mean less displacement and acceleration. By establishing an actual excitation road surface, the differential equations and analytical solutions in this paper can be used to roughly analyze the mechanical response of a traveling bus. These results can provide some guidance for the design and driving of buses.

Research on Recoil Resistance Control Technology of Amphibious Vehicle Weapon

In order to improve the water shooting stability of amphibious vehicle weapon, the recoil motion differential equation of vehicle weapon is established by analyzing the force motion of amphibious vehicle during water shooting. the structure of valve-controlled recoil device and the optimal strategy of grey predictive fuzzy control are determined, and the recoil resistance characteristic curves with or without servo valve and different control voltages are analyzed numerically. And the water motion attitude of the amphibious vehicle under different recoil force. The simulation results show that the valve-controlled recoil device has a good recoil damping platform effect. By adjusting the liquid flow channel of the traditional anti-recoil device by using the servo valve, the recoil resistance of the vehicle weapon can be effectively controlled and the water firing stability of the amphibious vehicle weapon can be improved.

On the pressure–deflection relations and instability of carbon-based composite nonlinear pipes

In this paper, for the first time, the nonlinear buckling of curved pipes is investigated for different distributions of single-walled carbon nanotubes (SWCNTs) in the matrix. The curved pipe is subjected to uniformly distributed transverse pressure and experiences uniform temperature rise. Additionally, the pipe is in one-sided contact with an elastic nonlinear medium. Defined thickness-dependent and temperature-dependent material properties of SWCNTs are taken from the results of molecular dynamics (MD) simulations, well-established in the literature on the subject. Five technologically justified types of CNT distributions are considered. Shen’s rule of the mixture is applied to determine the effective properties of CNT-based composites. To avoid determining shear correction factor, the higher-order shear deformation theory is used. The nonlinear components of the Green–Lagrange strain tensor and the principle of virtual displacements are applied to derive the nonlinear system of partial differential equations of motion of the curved pipe. The solution to formulated boundary value problem is obtained by the perturbation technique for both simply-supported and clamped-clamped boundary conditions. The pressure–deflection curves of the curved pipe are obtained and discussed. The investigation of snap-buckling intensity and upper/lower limit load of the curved pipe is presented and new insights for its nonlinear stability are presented. For instance, it is shown that the V-type of CNT distribution profile affects on lower deflection of the curved pipe while the higher deflection exists for the Λ-type profile.

Stability and coupling dynamic characteristics of a vibrating system with double rigid body driven by two motors considering energy balance

The present work mainly focuses the coupling dynamic characteristics and energy transfer relation of the system of double rigid bodies (RBs) driven by two motors, and then explores the ideal working state of the system. The motion differential equations of the system are established by Lagrange equation. Then the energy balance method and Hamilton's principle are employed to deduce the synchronization condition and stability criterion of the system. Then the numeric analyses are introduced to investigate the influence of excitation frequency on the amplitude, stable phase difference and energy of the system. Besides, the accuracy of theoretical research is demonstrated by the computer simulation and experiment. The research results indicate that, the anti-resonance state is the ideal working state of the proposed system; the elastic potential energy between two RBs is suddenly increased owing to the decrement of absolute value of stable phase difference of the system around anti-resonance state, which is used to magnify the relative motion between two RBs and weaken the vibration of the RB2. The results can help us understand the relationship between coupling dynamic characteristics and synchronous behaviors of the system with double RBs from the view of energy transfer.

Moment Lyapunov exponents and stochastic nonstability of a circular cylindrical shells

The paper investigates the stochastic stability conditions of circular cylindrical shells, compressed by time-dependent stochastic membrane forces. The main contribution of the paper is the determination of the moment Lyapunov exponents of the linear cylindrical shell for the first time. The moment and almost-sure stochastic stability of a three-degree-of-freedom coupled continuous system, under parametric excitation of white noise, are investigated. It is assumed that the system possesses a small damping. This type of a shell structure model requires a specific scheme for determining the moment Lyapunov exponents. The system of stochastic differential equations of motion is decoupled by using the contact transformation method with the obtained appropriate symplectic matrix of the system. Then a scheme for determining the moment Lyapunov exponents is presented which contains an analytical procedure. The largest Lyapunov exponent is calculated through its relation with the moment Lyapunov exponent. The moment and almost-sure stability boundaries are obtained analytically. These results are important and useful in engineering applications as an example of a procedure for dynamic analysis of mechanical systems with coupled stochastic oscillators.

Nonlinear dynamics analysis of a graphene laminated composite plate based on an extended Rayleigh–Ritz method

Linear and nonlinear vibration characteristics of the graphene platelet-reinforced composite (GPLRC) laminated plates are investigated. Based on the first-order shear theory and von Karman geometric nonlinearity, the energy expressions of the GPLRC laminates are established. The boundary elastic potential energy is established by penalty function method to simulate different boundary conditions. The linear and nonlinear frequencies of the GPLRC laminated plate are calculated by introducing boundary potential energy into Rayleigh–Ritz method. The convergence and accuracy of the method are verified by numerical examples, and the effects of different parameters on frequency are analyzed. Considering the cantilever boundary conditions, the nonlinear motion governing equations of the GPLRC laminated plate are obtained by Hamilton principle. The two-degree-freedom ordinary differential motion equations of the laminates are derived by Galerkin method. Considering the fundamental parameter resonance and 1:1 internal resonance, the amplitude–frequency response curves of the structure under transverse excitation are obtained. The effects of transverse excitation and damping coefficient on nonlinear vibration characteristics of the GPLRC laminated plates are investigated by numerical simulation.

Гидравлические удары в напорных трубопроводах при надземной прокладке

To determine the factors having the greatest impact on the risk of occurrence of water hammers in above-ground installation of pipelines, typical of permafrost areas. Determine the velocities of shock wave front propagation in steel pipes for different types of steel pipes. To calculate parameters of possible water hammer depending on pipeline wall thickness. Methods: Method of characteristics used for solution of differential equations of unsteady motion of liquid is the basis of calculation model for determination of flow parameters at occurrence of water hammer. Results: On the basis of the calculations carried out in the article the diagrams of pressure changes in the pressure pipeline made of steel pipes are analysed. The influence of speed of propagation of water hammer wave and speed of steady movement of a liquid flow on character of flow of an unsteady process and size of pressure at water hammer in the pressure pipeline in above-ground laying is estimated. The use of measures to prevent water hammers, including combined ones, with the purpose of increasing their reliability is much cheaper than the replacement of damaged pipes and equipment, elimination of washouts by leaking water and loss of drinking water. Practical significance: In permafrost conditions, many factors complicate construction and operation of pressure hydrotransport systems. Elimination of consequences of emergency situations accompanied by water leakage is difficult in harsh climatic conditions, and thermal impact on soils is not always avoidable. Therefore when designing new pressure systems and reconstructing existing ones it is necessary to take into account possible unsteady processes in pressure pipes.