Inertia Torque Research Articles

Corrected mathematical models for inertial torques generated by a spinning sphere

The recent publications about inertial torques acting on spinning objects describe their physics by mathematical models. The several inertial torques are generated by the rotating mass which produces centrifugal, Coriolis forces, and the change in the angular momentum. The values of the inertial torques depend on the geometries of the spinning objects. The volumetric geometries of the objects request complex analytical processing of the expressions for the inertial torques. The analysis of the known publication related to the spinning sphere shows the expressions of inertial torques derived from errors in the mathematical processing of integral equations. The wrong limits of the integral equations that give incorrect values for the inertial torques of the spinning sphere present these errors. This manuscript presents the corrected mathematical model for the inertial torques generated by the rotating mass of the spinning solid and hollow sphere.

Fundamental properties of the gyroscope oscillation

Despite partial solutions by famous scientists during the early Industrial Revolution, gyroscope problems remained unsolvable until the beginning of the twentieth century when several fundamental physical laws were finally formulated to describe them. Today, the principles of classical mechanics enable the formulation and description of the physical processes involved in the rotation of any object. Gyroscopic devices are objects that rotate and exhibit oscillation, which has been a challenging problem in engineering mechanics. The oscillation of a gyroscope is caused by the interaction between external and inertial torques. This is different from other examples of oscillation, such as pendulums and springs, which have been well documented. The main difference in the physics of gyroscopic oscillation is that the spinning rotors of the gyroscopic devices are supported on one side, with their axes perpendicular to the axis of oscillation. The oscillation of gyroscopic devices is interrelated with the potential and kinetic energy of their components. However, the physics of oscillation of such objects has not been fully described in publications until recently. The theory of gyroscopic effects for rotating objects has now been published and provides a solution to this problem. According to this theory, gyroscopic inertial torques represent the potential energy of the external torque and kinetic energy of the spinning rotor. This paper demonstrates the distribution of inertial torques about the axes of Cartesian coordinates, which enables the computation of gyroscope motion and oscillation.

Self-aligning active agents with inertia and active torque.

We extend the study of the inertial effects on the two-dimensional dynamics of active agents to the case where self-alignment is present. In contrast with the most common models of active particles, we find that self-alignment, which couples the rotational dynamics to the translational one, produces unexpected and nontrivial dynamics, already at the deterministic level. Examining first the motion of a free particle, we contrast the role of inertia depending on the sign of the self-aligning torque. When positive, inertia does not alter the steady-state linear motion of an a-chiral self-propelled particle. On the contrary, for a negative self-aligning torque, inertia leads to the destabilization of the linear motion into a spontaneously broken chiral symmetry orbiting dynamics. Adding an active torque, or bias, to the angular dynamics, the bifurcation becomes imperfect in favor of the chiral orientation selected by the bias. In the case of a positive self-alignment, the interplay of the active torque and inertia leads to the emergence, out of a saddle-node bifurcation, of solutions which coexist with the simply biased linear motion. In the context of a free particle, the rotational inertia leaves unchanged the families of steady-state solutions but sets their stability properties. The situation is radically different when considering the case of a collision with a wall, where a very singular oscillating dynamics takes place which can only be captured if both translational and rotational inertia are present.

Simultaneous Identification of Load Torque and Moment of Inertia of PMSM Based on Variable Structure Extended Sliding Mode Observer

Simultaneous Identification of Load Torque and Moment of Inertia of PMSM Based on Variable Structure Extended Sliding Mode Observer

Long-Axis Spinning of an Optically Levitated Particle: A Levitated Spinning Top.

An elongated object can be rotated around one of its short axes, like a propeller, or around its long axis, like a spinning top. Using optically levitated nanoparticles, short-axis rotation and libration have been systematically investigated in several recent studies. Notably, short-axis rotational degrees of freedom have been cooled to millikelvin temperatures and driven into gigahertz rotational speeds. However, controlled long-axis spinning has so far remained an unrealized goal. Here, we demonstrate controlled long-axis spinning of an optically levitated nanodumbbell with spinning rates exceeding 1GHz. We show that the damping rate in high vacuum can be as low as a few millihertz. Our results open up applications in inertial torque sensing and studies of rotational quantum interference.

Optimal position collaborative control of dual permanent magnet synchronous motor system

In order to improve the control accuracy and synchronous performance of the multi-motor control system, this paper proposes an optimal position collaborative control strategy for dual permanent magnet synchronous motor system. Firstly, based on the mathematical model of permanent magnet synchronous motor, an incremental predictive model is established to suppress constant and slow time-varying disturbances caused by changes in rotational inertia and load torque. Afterwards, the predicted compensation amount after multi-step prediction is obtained by minimizing the cost function which contains the synchronous error and tracking error simultaneously, and compensated to the given current loop in advance without changing the original dual motor system structure. Then the controllability and observability of the system are analyzed, and it can be proved that the system is asymptotically stable based on the optimal control theory. Secondly, considering the constraints such as maximum voltage and current limits, the optimal solution under the constraints is output to obtain the reference currents i q1 and i q2. Finally, it is demonstrated through the experiments that the proposed control strategy can achieve advance prediction compensation control of synchronous error, and the dynamic response speed is improved, while ensuring its steady-state performance remains unchanged.

Studying the rotational dynamics of a meter stick using the Arduino board and 3D printed parts

We present a low-cost experimental apparatus designed for studying the rotational dynamics of a meter stick. This apparatus uses everyday laboratory supplies, 3D printed parts, and an Arduino board for data acquisition. It consists of a common bearing fixed to a vertical axle, connected to the meter stick through a 3D printed spindle adapter. A thread attached to a hanging weight and winded around the spindle applies a constant external torque to the stick. The moment of inertia of the apparatus may be changed by incorporating weights into the apparatus through 3D printed adapters. An Arduino board records the time interval between consecutive half turns of the stick, allowing the calculation of the angular displacement over time. The data collected from experiments is compared with theoretical models. Among others, the effect of the applied torque, air resistance and moment of inertia of the stick is studied. This experimental apparatus is particularly suitable for undergraduate students enrolled in a classical mechanics course, offering them a practical, hand-on experience while remaining cost-effective.

Impact of Engine Inertia on P2 Mild HEV Fuel Consumption

The energy management system (EMS) of a hybrid electric vehicle (HEV) is an algorithm that determines the power split between the electrical and thermal paths. It defines the operating state of the power sources, i.e., the electric motor (EM) and the internal combustion engine (ICE). It is therefore one of the main factors that can significantly influence the fuel consumption and performance of hybrid vehicles. In the transmission path, the power generated by the ICE is in part employed to accelerate the rotating components of the powertrain, such as the crankshaft, flywheel, gears, and shafts. The main inertial components are the crankshaft and the flywheel. This additional power is significant during high-intensity acceleration. Therefore, the actual engine operation is different from that required by the power split unit. This study focuses on exploring the influence of engine inertia on HEV fuel consumption by developing a controller based on an equivalent consumption minimisation strategy (ECMS) that considers crankshaft and flywheel inertia. The optimal solution obtained by the ECMS controller is refined by incorporating the inertia effect of the main rotating components of the engine into the cost function. This reduces the engine operation during high inertial torque transient phases, resulting in a decrease in vehicle CO2 emissions by 2.34, 2.22, and 1.13 g/km for the UDDS, US06, and WLTC driving cycles, respectively.

Clustering of settling microswimmers in turbulence

Abstract. Clustering of plankton plays a vital role in several biological activities, including feeding, predation, and mating. Gyrotaxis is one of the mechanisms that induces clustering. A recent study (Candelier et al., 2022) reported a fluid inertial torque acting on a spherical microswimmer, which has the same effect as a gyrotactic torque. In this study, we model plankton cells as microswimmers that are subject to gravitational sedimentation as well as a fluid inertial torque. We use direct numerical simulations to obtain the trajectories of swimmers in homogeneous isotropic turbulence. We also investigate swimmers' clustering using Voronoï analysis. Our findings indicate that fluid inertial torque leads to notable clustering, with its intensity depending on the swimming and settling speeds of swimmers. Using Voronoï analysis, we demonstrate that swimmers preferentially sample downwelling regions where clustering is more prevalent.

Reduced trunk movement control during motor dual-tasking in older adults

Older adults have a decreased trunk movement control which is linked to their higher fall risk. While motor/cognitive dual-tasking deteriorates balance and walking in older adults, there is limited understanding on how trunk kinematics and kinetics are affected by dual-tasking in scenarios where falls can occur. Therefore, the purpose of the study was to determine the impacts of a challenging motor dual-task, specifically obstacle avoidance during walking, on trunk and lower-body kinematics and kinetics of older adults compared to young adults. The study captured three-dimensional kinematic and kinetic data from 12 young adults and 10 older adults as they walked on a treadmill and stepped over an obstacle with both legs. The study analyzed trunk, hip, knee, and ankle angles and torques. Trunk torque was further broken down to trunk muscle torque, gravitational torque, and inertia torque. A linear mixed effects model was used to investigate the difference in each variable between the two groups. Older adults exhibited significantly increased trunk flexion angle and trunk extension muscle torque compared to young adults, with the trunk being the only segment/joint showing differences in both kinematics and kinetics. Trunk torque breakdown analysis revealed that larger trunk flexion led to a larger gravitational torque, which contributed to an increased compensatory trunk muscle torque. Moreover, older adults' less controlled trunk flexion during weight shifting from trail leg to the lead leg, necessitated a compensatory trunk deceleration during trail leg obstacle avoidance which was achieved by generating additional increase in trunk muscle torque. The study demonstrated that motor dual-tasking has the most negative effects on trunk control in older adults compared to young adults. This exposes older adults to a higher fall risk. Therefore, future work should focus on supporting trunk control during daily multi-tasking conditions where falls can occur.

Replicating the rolling-sliding dynamics of cam-roller contacts in large-scale hydraulic drivetrains: A small-scale approach

The rolling-sliding dynamics of large-scale cam-roller contacts are strongly influenced by the inertia of the roller, particularly when slippage occurs. Slippage can potentially impact the reliability of these rolling interfaces. This study introduces an approach to replicate the rolling-sliding dynamics of cam-roller contacts in a large-scale hydraulic drivetrain, on a small scale. For that, we have upgraded our two-roller tribometer to enable cyclic loading, allow the application of resisting torques, and generate inertia torques. These are three essential elements required to mimic the dynamics observed at large scales. A method has been proposed for scaling the roller inertia accordingly. Furthermore, we have implemented a modeling framework from previous work to make predictions under various dynamic conditions. The results show that our small-scale approach can replicate five key characteristics anticipated at a large scale, including those linked to slippage. Small increments in the resisting torque significantly increased the slide-to-roll ratio (SRR) and peak traction force, among others. The simulations also predicted these effects, capturing trends and producing reasonable predictions of the magnitude and relevant features of key parameters. The use of cyclic loading, extra inertia, and adjustable resisting torques, effectively generated repeatable and controllable dynamic rolling-sliding conditions. Our work is significant for the design and development of novel large-scale hydraulic drivetrains. Our findings highlight the importance of reducing slippage at low contact forces to prevent the brusque change in the rolling conditions during the high contact force phase. By doing so, surface damage and detrimental dynamic effects can be prevented.

Indicating the direction in conservation of angular momentum

This study developed a teaching material that makes the conservation of angular momentum, which students find very difficult, more concrete. The fact that angular momentum is a vector concept, the direction of the vector and the importance of this vector direction in angular momentum conservation are visualized. The teaching material consists of a table mounted on a freely rotating shaft, two electric motors mutually connected to the table, and a speed adjuster circuit used to change the rotation speeds of the motors. When electric motors begin to rotate, the angular momentum of the motors changes. How the angular momentum of the system is preserved thanks to the changing angular momentum of the parts interacting with each other, the directions of the vectors formed, the inertia torque, angular velocity, angular momentum relationship, and the relationship of the parts forming a system with each other have been tried to be explained in detail through the teaching material. When the motors were operated in different directions and speeds, the change in the rotation direction of the table was observed. This observation concretizes that angular momentum and angular momentum conservation are vector concepts. The right-hand rule, which describes the angular momentum vector direction with the material, can be demonstrated to students in practice. Video recordings showing how the teaching material works have been shared with readers.

Effect of slip-induced fluid inertial torque on the angular dynamics of spheroids in a linear shear flow

The angular dynamics of tiny spheroidal particles in shear flows have been widely investigated, but most of the studies mainly focus on the effect of strong shear, while the combined effect of both shear and slip velocity at the center of the particle has been less considered. Actually, the fluid inertial torque induced by the slip velocity between particle and fluid plays a significant role in spheroid angular dynamics. However, it is difficult to investigate these dynamics theoretically until the analytical expression of the fluid inertial torque at a small Reynolds number was derived by Dabade et al. [J. Fluid Mech. 778, 133–188 (2015)]. In this study, the effect of the fluid inertial torque on the particle rotations is considered in a linear shear flow with a small streamwise slip velocity at the center of the particle. We find that as the fluid inertial torque dominates, the prolate spheroids tend to logroll while oblate ones have a tendency to tumble or align to a direction with a relative angle to the streamwise direction. These results are opposite to the earlier results in the absence of the fluid inertial torque. Different ultimate rotation modes of spheroids are dependent on the relative importance between the fluid inertial torque and the particle inertia, as well as the initial orientations. This reflects a non-trivial effect of fluid inertial torque on the angular dynamics of inertial spheroidal particles.

Dynamic behavior of pin-free beam with torsional spring and hub at pinned end

This paper described the free vibration of a Euler-Bernoulli beam and discovered the first five dimensionless natural frequencies and mode shapes of motion for the pin-free beam model with a helical spring and rotating mass on the pinned side of the beam. The effect of changing the dimensionless torsional stiffness ratios was investigated, as was the effect of changing the dimensionless rotational inertia torque ratios Concerning the vibration properties of the beam. The mathematical equations that regulate the pin-free beam model were constructed using the exact analytical solution, establishing the Admissible Function and the natural frequency equation for the basic pin-free beam system. The approximate Rayleigh-Ritz approach was then used to determine the first five dimensionless natural frequencies, as well as the movement pattern for the pin-free beam model with a helical spring and a rotating mass on the side of the hinge. The results demonstrated that modifying the beam's natural frequencies is dependent on the ratios of the additional torsional spring parameters as well as the ratios rotational moment of inertia. The threshold system frequencies increase when the spring parameters are the major factor, and decrease when the inertia moment parameters are the main factor. Both play a role in influencing natural frequency to varied degrees. Unifying the final distribution of parameters might lead to optimizing the impact of parameters on the natural frequencies of the system.

Matrix projective synchronization and mechanical analysis of unified chaotic system

This article explores the mechanical analysis of a unified chaotic system and matrix projective synchronization (MPS). The sufficient conditions to achieve MPS of unified chaotic system have derived. The mechanics of unified chaotic system have been examined in contrast with Kolmogorov system, Euler equation, and Hamiltonian function. The Casimir energy function is also introduced to analyze the system dynamics. The unified chaotic system has been transformed into Kolmogorov type system and decomposed into four types of torques: inertial, internal, dissipation, and external torque. In order to view the mechanics behind the unified chaotic system, different particular cases have been discussed. Five scenarios are examined using combinations of various torques in order to identify the key elements in chaos creation and their physical significance. The bifurcation diagram, Lyapunov exponent and Lyapunov dimension validates the generation of chaos for different particular cases.

Study on the dynamic characteristics of planetary gear transmission mechanism of metal cold rolling mill

Using the parametric modelling feature in Solidworks software, a three-dimensional solid model of a planetary gearbox with linear reciprocating motion of the output shaft was constructed. By conducting theoretical calculations related to the kinematics of the transmission system and combining Adams virtual prototyping simulation technology, a dynamic simulation model of the planetary gear transmission mechanism was established to analyse the motion laws and dynamic characteristics. The simulation results were compared and analysed against the theoretical calculation results, and they showed good agreement and consistency. The study also investigated the impact of counterweights on the inertia forces and moments of the output shaft in the transmission mechanism. The results indicated that adding counterweights effectively reduced the inertia impact caused by inertia forces at the start and end positions of the stroke, as well as the inertia torque caused by changes in angular acceleration during the startup and stopping phases, thereby enhancing the smooth operation of the mechanism. Additionally, different thicknesses of counterweights had varying effects on balancing the inertia forces and moments of the output shaft. The study aimed to find the optimal thickness of counterweights to achieve the best balance effect. Furthermore, the study examined the influence of different speeds of the driving gear on the inertia forces of the output shaft. The results showed that at a driving gear speed of 1500 rpm, the horizontal thrust generated by the output shaft was 63566 N, which could provide the required thrust for the rolling mill operation, and the inertia forces of the output shaft tended to stabilize.

Review of Methods for Determining the Moment of Inertia and Friction Torque of Electric Motors

Review of Methods for Determining the Moment of Inertia and Friction Torque of Electric Motors

Effect of fluid inertial torque on the rotational and orientational dynamics of tiny spheroidal particles in turbulent channel flow

Rotation and orientation of non-spherical particles in a fluid flow depend on the hydrodynamic torque they experience. However, little is known about the effect of the fluid inertial torque on the dynamics of tiny inertial spheroids in turbulent channel flows, as only Jeffery torque has been considered in previous studies by point-particle direct numerical simulations. In this study, we investigate the rotation and orientation of tiny spheroids with both fluid inertial torque and Jeffery torque in a turbulent channel flow. By comparing with the case in the absence of fluid inertial torque, we find that the rotational and orientational dynamics of spheroids is significantly affected by the fluid inertial torque when the Stokes number, which is non-dimensionalized by fluid viscous time scale, is larger than the critical value $St_c\approx 2$ , indicating that the fluid inertial torque is non-negligible for most particle cases considered in earlier studies. In contrast to the earlier findings considering only Jeffery torque (Challabotla et al., J. Fluid Mech., vol. 776, 2015, p. R2), we find that prolate (oblate) spheroids with a large Stokes number tend to tumble (spin) in the streamwise–wall-normal plane in a thinner region near the wall due to the presence of the fluid inertial torque. Approaching the channel centre, the flow shear gradually vanishes, but the velocity difference between local fluid and particles is still pronounced and increasing as particle inertia grows. As a result, in the core region, fluid inertial torque is dominant and drives the particles to align with its broad side normal to the streamwise direction rather than a random orientation observed in earlier studies without fluid inertial torque. Meanwhile, the presence of fluid inertial torque enhances the tumbling rates of spheroids in the core region. In addition, the effect of fluid inertial force on the dynamics of spheroids is also examined in this study, but the results indicate the effect of fluid inertial force is weak. Our findings imply the importance of fluid inertial torque in modelling the dynamics of inertial non-spherical particles in turbulent channel flows.

Cascaded 2D Micromirror with Application to LiDAR.

This paper introduced a novel approach to enhance the vertical scanning angle of a large aperture 2D electromagnetic micromirror through the utilization of a cascaded torsional beam design. The primary objective was to increase the vertical scanning angle without compromising the robustness, which was achieved by optimizing the trade-off between the rotation angle and the first mode of resonant frequency. The cascaded design provides flexibility to either increase the outer frame's rotation angle without sacrificing torsional stiffness or enhance the torsion beam's stiffness while maintaining the same rotation angle, thus elevating the first-mode resonant frequency and overall robustness. The effectiveness of the cascaded design was demonstrated through a comparative study with a non-cascaded 2D micromirror possessing the same aperture size, torque, and mass moment of inertia. Theoretical analysis and finite-element simulation are employed to determine critical parameters such as the stiffness ratio between the cascaded torsion beams, and to predict improvements in the scanning angle and primary resonant frequency brought by the cascaded design. Prototypes of both cascaded and non-cascaded designs are fabricated using a flexible printed circuit board combined with Computer numerical control (CNC) machining of a Ti-alloy thin film, confirming the superior performance of the cascaded 2D micromirror. The cascaded design achieved vertical scanning angles up to 26% higher than the traditional design when both were actuated at close resonance frequencies. Additionally, the micromirror was successfully integrated into a 3D LiDAR system. The light detection and ranging (LiDAR) system was modelled in Zemax OpticStudio to find the optimized design and assembly positions.

Active torsional vibration suppression of hybrid electric vehicle drive system based on optimal harmonic current instruction calculation method

Torsional vibrations in the drive system of a hybrid electric vehicle powered by an engine are generated by cyclic torque fluctuations. These fluctuations stem from the combustion torque and the reciprocating inertial torque of the cylinder in the reciprocating piston engine. To address this issue, a real-time engine transient torque model that considers the thermodynamic characteristics of each component is established. A method was designed to identify the torsional vibration of a powertrain based on the engine torque, electric motor torque, and powertrain model. Building on this foundation, a method was proposed to calculate the optimal harmonic current instruction of the electric motor to mitigate torsional vibration in the Hybrid electric vehicles (HEV) powertrain. Furthermore, a proportional integral-resonance (PIR) controller was designed for the injection of an electric motor reference harmonic current. Finally, a method was proposed for suppressing the torsional vibration in HEV drive systems that incorporate active harmonic current injection. The simulation and test results indicate that the proposed method can effectively diminish the amplitudes of the drive system's 2nd, 4th, 6th, and 8th harmonic torque, ensuring stable operation of the powertrain.