Torque Balance Equation Research Articles

Dynamic plastic damage analysis of bending-torsion coupling in cross stiffener systems under mass impact

ABSTRACT This study proposes a simpler theoretical model to predict the bending-torsion coupling dynamic plastic response of cross stiffener systems under mass impact. Traditional methods often rely on torque balance and momentum conservation equations, which involve complex and difficult calculations. To simplify this process, the study uses the law of energy conservation and applies the variational method to solve the dynamic plastic damage of the cross stiffener system. The plastic response of a single stiffener is first analyzed and compared with existing theories, showing consistent results for maximum displacement. The study then extends the model to more complex conditions, performing comparative analyses on stiffeners of different sizes through numerical simulation. The findings show that the maximum displacement at the load and node remains within a reasonable range, with a displacement difference of only 0.9 mm and an error margin of 4.79% in one of the test conditions.

Influence of gate cutoff effect on flow mode conversion and energy dissipation during power-off of prototype tubular pump system

Understanding the cutoff effect of gates is essential for enhancing the overall quality of the pump system's power-off process, minimizing energy losses, and reducing potential risks associated with hydraulic transients. In this study, both numerical simulations and experimental investigations were conducted on the power-off process of a tubular pump system, considering scenarios with and without gate functionality. The simulations utilized a dynamic mesh method to model gate movement, incorporated the torque balance equation to determine the real-time impeller speed, and applied the 3D-VOF method for free surface modeling in reservoirs. Power-off experiments were performed on a model pump system and a prototype pump system to validate the numerical simulation results. To elucidate the mechanism of the gate's cutoff effect, flow modes during the power-off process were categorized based on the four-quadrant static test results of the pump, revealing the deviations between transient and static characteristics. By comparing the transient flow structures of the pump system under different gate operating states, specific energy dissipation behaviors during gate cutoff were analyzed. The research findings enhance the understanding of hydraulic transients, which is essential for developing more sustainable and resilient energy systems.

Evaluation of polar alignment structure and surface anchoring energy in ferroelectric nematic liquid crystals by renormalized transmission spectroscopic ellipsometry

It is essential to estimate the surface polar anchoring energy in order to discuss the interfacial orientation of ferroelectric nematic liquid crystals. In this study, we accurately estimated the twist angle of a π -twist cell with an antiparallel rubbing manner, by means of renormalized transmission spectroscopic ellipsometry, which has a great deal of experience in measuring the twist angle of ordinary nematic liquid crystals. We also succeeded in estimating the reduced polar anchoring energy from the essential equation derived from the simplified torque balance equation. It is assumed that the reduced surface polar anchoring energy is on the order of 102.

Fast Assessment Method for Transient Voltage Stability of Photovoltaic Receiving‐End Grid

The incorporation of renewable energy on a broad scale into power grids increases the complexity to the issue of transient voltage stability in power systems. This research presents a rapid evaluation technique for assessing the stability of voltage fluctuations in power grids that receive electricity from photovoltaic sources. At first, a receiving‐end system model was developed, which includes photovoltaics, and an alternative circuit of the virtual induction motor (IM) is obtained by utilizing the Thevenin equivalent. Second, the torque balance equation and the Kirchhoff voltage equation are interconnected to determine the unstable slip of the IM following a malfunction. Finally, by merging the unstable slip with the conventional transient stability discriminant index, the effects of different photovoltaic outputs, IM ratios, and system contact impedances on transient voltage stability are investigated. The proposed method avoids the computational burden of solving the differential‐algebraic equations describing the complex transient processes of the IM. It also obviates the necessity of iteratively modifying the receiving load or fault clearance time in the simulation platform to achieve the constrained stability condition of the system’s transient voltage. The transient stabilization of voltage results is efficiently obtained by directly solving algebraic equations.

Rolling-Sliding Performance of Radial and Offset Roller Followers in Hydraulic Drivetrains for Large Scale Applications: A Comparative Study

Generally speaking, excessive side thrust and roller slippage are two different aspects affecting cam-roller mechanisms. In novel large-scale hydraulic drivetrains for offshore wind turbines, the highly dynamic nature of these mechanisms combined with the interplay of cyclic loads, frictional torques and inertia promote slippage at the cam-roller interface. At larger scales, the effects of roller inertia become much more pronounced, as the inertia escalates exponentially with the roller’s radius. This study presents a comparative analysis between radial and offset roller followers in novel large-scale hydraulic drivetrains, where offset followers are incorporated to minimize the side thrust. The framework encompasses a comprehensive kinematic and force analysis, to provide the inputs for two lubrication models integrated into the torque-balance equation, where the possibility of slippage is allowed. The findings reveal that the equivalent side thrust can be reduced by 51% with offset followers. Both configurations experience slippage during the low-load phase, but it rapidly diminishes during the high-load phase. This sudden transition in rolling conditions results in a sharp increase in surface temperature and traction force, emphasizing the importance of minimizing sliding at the interface. However, besides the substantial side thrust reduction, offset followers showed superior tribological performance, mitigating undesirable thermal and frictional effects.

Reduced-variance orientational distribution functions from torque sampling

We introduce a method to sample the orientational distribution function in computer simulations. The method is based on the exact torque balance equation for classical many-body systems of interacting anisotropic particles in equilibrium. Instead of the traditional counting of events, we reconstruct the orientational distribution function via an orientational integral of the torque acting on the particles. We test the torque sampling method in two- and three-dimensions, using both Langevin dynamics and overdamped Brownian dynamics, and with two interparticle interaction potentials. In all cases the torque sampling method produces profiles of the orientational distribution function with better accuracy than those obtained with the traditional counting method. The accuracy of the torque sampling method is independent of the bin size, and hence it is possible to resolve the orientational distribution function with arbitrarily small angular resolutions.

Robot Zero-Moment Control Algorithm Based on Parameter Identification of Low-Speed Dynamic Balance

This paper proposes a zero-moment control torque compensation technique. After compensating the gravity and friction of the robot, it must overcome a small inertial force to move in compliance with the external force. The principle of torque balance was used to realise the zero-moment dragging and teaching function of the lightweight collaborative robot. The robot parameter identification based on the least square method was used to accurately identify the robot torque sensitivity and friction parameters. When the robot joint rotates at a low speed, it can approximately satisfy the torque balance equation. The experiment uses the joint position and the current motor value collected during the whole moving process under the low-speed dynamic balance as the excitation signal to realise the parameter identification. After the robot was compensated for gravity and static friction, more precise torque control was realised. The zero-moment dragging and teaching function of the robot was more flexible, and the drag process was smoother.

Time constant analysis of micromotors for virtual reality helmets

This paper focuses on the study of an ultra-micro permanent magnet stepper motor with automatic adjustment of interpupillary distance for VR helmets. To precisely adjust the rotation of the motor, the control requirements for the motor are very important, which also makes the time constant of the micro permanent magnet stepper motor of great significance. Starting from the voltage, torque balance equations and thermal balance equations, the paper analyzes and compares the electrical time constant, mechanical time constant, electromechanical time constant and thermal time constant of the micro permanent magnet claw-pole stepper motor in basic theory. Different specifications of micro permanent magnet claw-pole stepper motors were used to obtain some relationships between time constant and motor performance, which provided reference for the design and control research of micro permanent-magnet claw-pole stepper motors.

Energy loss mechanisms of transition from pump mode to turbine mode of an axial-flow pump under bidirectional conditions

Pump-as-turbine stations with ultra-low head often face the deterioration risk during load rejection operations when the demands in water delivery and power generation are satisfied. In the course of load rejection, significant energy transfer and hydraulic instability may ensue from the pump-to-turbine transition mode, and vice versa. In order to investigate the variation of energy loss during the transition process of a bidirectional axial-flow pump, the entropy production method is herein utilized for visualization and quantitative analysis of high energy loss regions within key components. In the process of transient simulation, the torque balance equation is applied for calculating the rotational speed of impeller. Two-way transitions are considered with the bidirectional pump for comparing two distinct forms of pumps equipped with front and rear guide vanes. Results show that the simulated external characteristic curves and transition parameters of the pump are in good accordance with the experimental data. The transient process of the transition successively goes through four modes, i.e. pump, braking, turbine and runaway modes. In the process of the direction switch of the main flow and rotation, the different positions of guide vanes also deeply influence the inflow angle and flow pattern in impeller, and then change the distribution of the pressure coefficient, vorticity and entropy production rate. In pump mode, the entropy production rate and pressure coefficient at the impeller inlet edge under backward transition condition (BTC) is higher than that under forward transition condition (FTC), due to the smaller inflow angle caused by the front guide vanes under BTC. In runaway operation, the total entropy production (TEP) under FTC is obviously higher than that under BTC, because the energy loss is reduced by the broken shedding vortices caused by the obstruction of the guide vanes under BTC. This study's results can be referred to while seeking for the operation safety and stability of the pump stations consecutively used to pump and generate power.

Image interpretation for kaolinite detachment from solid substrate: Type curves, stochastic model

Kaolinite clays are the most widespread natural fines attached to the rock surface; they can be mobilised by the flow with further migration in porous spaces, straining in thin pore throats and causing significant decline in the rock permeability. We conducted mathematical and laboratory studies of kaolinite detachment from solid substrates in visualisation cells. The mechanical equilibrium of the attached particles in the creeping flow was described by the torque balance. For uniform particles and substrates, this model presents fines detachment using only two rates that correspond to particles in primary and secondary energy minima, i.e., the attached concentration versus velocity is a piecewise-constant function. To observe this maximum retention function, we saturated a single-channel micromodel visualisation cell with a transparent top with kaolinite particles; it was then subjected to a flow with a piecewise-constant increasing rate. Images of the remaining attached fines were filmed after each rate increase. All the tests exhibit gradual fines detachment. To explain the phenomenon, we assumed a probabilistic distribution of the properties of the particles and the substrate. For two-parametric probability distribution functions, it adds the standard deviations to the list of model parameters. The continuous fines detachment versus velocity was highly matched by the torque balance equation with probabilistically distributed coefficients. The match allowed restoring probabilistic distributions of the selected model parameters from the measured maximum retention function. The sensitivity of the detachment rate to properties follows a decreasing order: semi-major axis, aspect ratio, lever arm ratio, and zeta potential. This work fundamentally advances lab-based mathematical modelling of colloidal detachment from solid surfaces by developing stochastic torque-balance equation, where standard deviations of the model coefficients are tuning / matching parameters along with their mean values. This approach allows determining the probabilistic distributions of the model coefficients from the image processing, and also calculating the attached concentration variation within six standard deviations of each parameter, permitting placing the model parameters in the order of their effect on particle detachment.

Torque for non-contact piezoelectric motor modulated by electromagnetic force

In this paper, the structural design and driving mechanism of a non-contact piezoelectric motor modulated by electromagnetic force are introduced. The equations of the electromagnetic force and torque of the electromagnetic modulation mechanism for the motor are deduced. Based on the piezoelectric constitutive equation and torque balance equations of the driving system, the equation of the piezoelectric driving torque is also obtained. Using these equations, the influences of the related parameters on the relationship between electromagnetic torque and angle difference between pole and rotor are studied. The influences of the main parameters and signal frequency on piezoelectric driving torques are analyzed. Based on them, the torque optimization of the motor is completed and the load-carrying ability of the motor is increased by about 50 times. A prototype of the motor is developed. Its output torque is tested and compared with calculated one which illustrates the theoretical model here.

Study on a horizontal axial flow pump during runaway process with bidirectional operating conditions

The ultra-low head pump stations often have bidirectional demand of water delivery, so there is a risk of runaway accident occurring in both conditions. To analyze the difference of the runaway process under forward runaway condition (FRC) and backward runaway condition (BRC), the whole flow system of a horizontal axial flow pump is considered. The Shear-Stress Transport (SST) k–ω model is adopted and the volume of fluid (VOF) model is applied to simulate the water surface in the reservoirs. Meanwhile, the torque balance equation is introduced to obtain the real time rotational speed, then the bidirectional runaway process of the pump with the same head is simulated. In addition, the vortex transport equation and swirl number are proposed to reveal the flow characteristics during the runaway process. The results show that the runaway process can be divided into five stages: the drop, braking, rising, convergence and runaway stages, according to the changing law of torque curve. In the rising stage, the pressure difference on the blade surface continues to increase, which contributes to the abnormal torque increase. In this stage, the flow hits the pressure surface (PS) at a faster speed enlarging the pressure on PS, and the flow separation takes place on the suction surface (SS) weakening the pressure on SS. During the convergence and runaway stage, the pulsation amplitude of torque and axial force under FRC is obviously larger than those under BRC. This is because the rotation frequency of the vortex rope is the same as main pressure fluctuation frequency in impeller under FRC, which enhances the pulsation amplitude. Whereas the vortices are broken due to the inhibitive effect from guide vanes under BRC.

Composite synchronization on two pairs of vibrators in a far super-resonant vibrating system with the single rigid frame

In the present work, a new dynamical model with a single rigid frame driven by two pairs of vibrators, of which each pair of vibrators is engaged with each other by gear mechanism, is proposed to explore the composite synchronization of the system. The motion differential equations and vibration responses of the system are given first. The theory condition for achieving composite synchronization of the system is obtained, by using the average method to deduce the average torque balance equations of the two pairs of vibrators. According to the Hamilton’s theory, the system stability condition is presented, and it is mainly determined by the structural parameters of the system. The synchronous stable regions and stability ability versus the key parameters of the system are qualitatively discussed in numeric, and further quantitatively verified by simulations. It is shown that, in engineering, the reasonable working points of the system, should be selected in the region where the stable phase difference of the two pairs of vibrators is stabilized in the vicinity of zero. Only in this way, can the exciting forces of the two pairs of vibrators be positively superposed, and the linear motion of the system in the vertical direction be realized.

Stability and motion characteristics in a vibrating system with five rigid frames driven by two counter-rotating exciters

A new dynamical model with five rigid frames (RFs), driven by two counter-rotating exciters, is proposed to explore the synchronization, stability, and motion characteristics of the system in this paper. The motion differential equations and the corresponding responses of the system are given firstly. Using the average method, the average torque balance equations for the two exciters are deduced. According to the relationship between the difference of the dimensionless effective output electromagnetic torques for the two motors and the coupling torques of the system, the theory condition of realizing synchronization is obtained. Based on the Hamilton’s theory, the theory condition of stability of the system is deduced. The stability and motion characteristics of the system for different resonant regions are qualitatively discussed in numeric, including the stable phase difference of the two exciters, relative phase relationships among the five rigid frames, amplitude-frequency characteristics, stability coefficients, and the effective load torque between the two exciters. Simulations are carried out to further quantitatively validate the feasibility of the above theoretical and numerical qualitative results. It is shown that in engineering the reasonable working points of the system should be selected in Region II, only in this way, can the synchronous and stable relative linear motion of the system with the zero stable phase difference in vertical direction be realized, and in this case, the vibrations of the four inner rigid frames (IRFs) in the horizontal direction are compensated with each other, and the energy is also saved due to utilizing the resonant effect. Based on the present work, some new types of vibrating coolers/dryers or vibrating screening machines can be designed.

Modified mathematic model of a parallel two-dimensional piston flowmeter

The parallel two-dimensional piston (2D piston) flowmeter is a novel kind of positive displacement flowmeter that has been recently invented and proposed by our research group. Following an overview of the working principle of this flowmeter, a modified mathematic model was carefully established in this paper based on the analysis of the shortcomings of the presented primary model. The churning loss, as the first of two ignored parts in the previous research, is caused by the rollers rotation in oil, and it is considered a part of the torque balance equation in the pressure loss model. Besides, the flow rate of the transient leakage, which comes from the change in the leakage flow rate equation when the 2D piston rotates at a specific angle, was analytically modeled and added to the flow rate distribution equation. By comparing the experimental results from the last study, the modified mathematical model showed to be precise and became more suitable for predicting the flowmeter's performance and enhancing its measuring accuracy.

Energy-saving trajectory planning by using Fourier sine series and tracking control for a mechatronic elevator system

In this paper, Lagrangian method is proposed to formulate dynamic model for a mechatronic elevator system, which includes mechanical and electrical parts. Lagrange equations, i.e. the torque balance equation and voltage balance equation, are derived from the kinetic energy, potential energy and virtual work of the mechatronic system. From the dynamic equations, the energy balance equation is obtained including electromagnetic energy, electrical dissipation energy, mechanical kinetic energy, mechanical dissipation energy, and the mechanical potential energy. The minimum-energy trajectory by using Fourier sine series (FSS) and power series are found by real-coded genetic algorithm. From simulation results, it is found that FSS with less order can find the minimum input energy more quickly than the power series. The four-degree (4-D) FSS trajectory is given as a reference one for the adaptive tracking control. It is found that the proposed adaptive controller has good tracking performance for the 4-D FSS angular displacement and velocity.

Dynamic modelling and state observer-based robust control for a mechatronic system with temperature effect

The temperature effect is considered in dynamic modelling of a simple mechatronic system, which includes both mechanical and electrical parts. The total input energy is equal to heat dissipation, heat store, electromagnetic, and mechanical output energies. The complete model of a simple mechatronic system consists of the torque balance, voltage balance and heat balance equations. The responses of angular velocity, electric current and temperature are compared by using different values of specific heat capacities and convective coefficients on the responses. Then, a supplemental state observer-based sliding mode control (SSOBSMC) consisted of the supplemental state observer (SSOB) and sliding mode control (SMC) is successfully proposed to perform the robust tracking control perform for a simple mechatronic system with the unavailable temperature. The simulation results show that the proposed controller demonstrates robust states estimation and tracking control performances, simultaneously. The contributions of this paper are: (1) the temperature effect is considered in the dynamic modelling by using the fact that the dissipation energy of the mechanical damper and electrical resistance convert to heat dissipation and heat store energies in a simple mechatronic system; (2) the complete dynamic model of a simple mechatronic system is successfully formulated, and consists of the torque balance, voltage balance and heat balance equations; (3) a SSOBSMC is successfully proposed to perform well state estimation ability and robust tracking control performance, simultaneously.

Numerical Analysis of the Nanoparticle Dynamics in a Viscous Liquid: Deterministic Approach

We study the deterministic dynamics of single-domain ferromagnetic nanoparticles in a viscous liquid induced by the joint action of the gradient and uniform magnetic fields. It is assumed that the gradient field depends on time harmonically and the uniform field has two components, perpendicular and parallel to the gradient one. We also assume that the anisotropy magnetic field is so strong that the nanoparticle magnetization lies along the anisotropy axis, i.e., the magnetization vector is "frozen" into the particle body. With these assumptions and neglecting inertial effects we derive the torque and force balance equations that describe the rotational and translational motions of particles. We reduce these equations to a set of two coupled equations for the magnetization angle and particle coordinate, solve them numerically in a wide range of the system parameters and analyze the role of the parallel component of the uniform magnetic field. It is shown, in particular, that nanoparticles perform only periodic rotational and translational motions if the perpendicular component of the uniform magnetic field is absent. In contrast, the nanoparticle dynamics in the presence of this component becomes non-periodic, resulting in the drift motion (directed transport) of nanoparticles. By analyzing the short and long-time dependencies of the magnetization angle and particle coordinate we show that the increase in the parallel component of the uniform magnetic field decreases both the particle displacement for a fixed time and its average drift velocity on each period of the gradient magnetic field.

Liquid metal motor.

SummaryLiquid metal has demonstrated an enormous potential for developing soft functional devices and machines. However, current liquid metal enabled machines suffer from several issues, such as the requirement of a liquid environment, generation of weak actuating forces, and insufficient maneuverability. To overcome these restrictions, here, a motor is developed based on the electrical actuation of liquid metal droplets without the need for conventional electromagnets. The approach is distinguished by (1) the encapsulation of electrolyte and multiple liquid metal droplets within an enclosed system, and (2) the creation of stable and continuous torque outside a liquid environment. In addition, a liquid metal electrical brush is introduced to operate the motor with low friction and low wear. The unique driving mechanism endows the motor with several advantages, including low friction, no sparking, low noise, versatile working environment, and being built from soft materials that could offer new opportunities for developing soft robotics.

Analysis of Dynamic Engaged Characteristics of Wet Clutch in Variable Speed Transmission of a Helicopter

The working principle and motion process of an aviation wet clutch are analyzed. The initial velocity before the friction pair engaged is solved. The transient Reynolds equation is modified, and an oil film bearing capacity model and a micro-convex bearing capacity model are derived. The film thickness equation between N friction pairs and a pressure-plate is derived. A dynamic engaged model of springs, pistons, friction pairs, and pressure plates are established. The torque balance equation is established of two pairs of friction pairs. The friction torque, rate of change in the oil film, and law of relative change in speed are obtained. The results demonstrate that the spring preload and the viscosity of the lubricating oil have a significant influence on the engagement characteristics. Increasing the quality of the friction plate will reduce the time of engagement, whereas the quality of the friction plate has slight effect on the friction torque characteristics and oil film thickness. The initial speed generated by the collision process will reduce the output speed, sharply increase the torque peak at the lock, and increase the shift shock.