Showing papers on "Inertia published in 2012"

Set-Valued Force Laws: Dynamics of Non-Smooth Systems

Abstract: 1. Introduction.- 1.1 Friction Laws.- 1.2 Literature Survey.- 1.3 Subjects and Contents.- 2. Fundamental Concepts.- 2.1 Internal and External Forces.- 2.2 The Law of Interaction.- 2.3 The Dynamic Equilibrium.- 2.4 The Virtual Work of a Dynamic System.- 2.5 Resultant Force and Inertia Terms.- 3. Rigid Body Systems.- 3.1 Preliminaries on the Vector Product.- 3.2 Rigid Body Kinematics.- 3.3 Rigid Body Kinetics.- 3.4 The Dynamic Equilibrium of a Rigid Body.- 3.5 The Virtual Work of a Rigid Body System.- 3.6 Classical Bilateral Constraints.- 3.7 Generalized Coordinates.- 4. Motion and Discontinuity Events.- 4.1 Preliminaries on Integration of Functions.- 4.2 Displacements, Velocities, and Accelerations.- 4.3 Restriction to Finite Numbers of Discontinuities.- 5. Displacement and Velocity Potentials.- 5.1 Directional Newton-Euler Equations.- 5.2 Set-Valued Force Laws.- 5.3 Scalar Potential Functions.- 5.4 On the Modeling of Force Laws.- 6. Representation of Scalar Force Laws.- 6.1 Decomposition into Unilateral Primitives.- 6.2 Variational Formulations and Upper Subderivatives.- 6.3 The Convex Case: Conjugate Potentials and Duality.- 6.4 Force Elements in Engineering Dynamics.- 7. Force Laws on Different Kinematic Levels.- 7.1 Continuity Properties of the Trajectories.- 7.2 Displacement Force Laws on Acceleration Level.- 7.3 Velocity Force Laws on Acceleration Level.- 8. Index Sets and LCP-Formulation.- 8.1 Index Sets.- 8.2 Formulation on Different Kinematic Levels.- 8.3 The Linear Complementarity Problem.- 8.4 The Dual Principle of Least Constraints.- 9. Principles in Dynamics.- 9.1 The Principle of Least Constraints.- 9.2 The Principle of Gauss.- 9.3 The Principle of Jourdain.- 9.4 The Principle of d'Alembert/Lagrange.- 9.5 Remarks on d'Alembert/Lagrange's Principle.- 10. Spatial Coulomb Friction.- 10.1 Geometry of Surfaces.- 10.2 Contact Kinematics.- 10.3 Kinetics.- 10.4 Contact Laws.- 10.5 Sliding Contacts.- 10.6 Friction Pyramid for Rolling Contacts.- 10.7 Friction Cones and NCP Formulations.- 10.8 A Differentiable NCP for Rolling Contacts.- 10.9 Example and Remarks.- 11. Velocity Jumps due to C0-Constraints.- 11.1 On Impacts in Mechanical Systems.- 11.2 Mechanical Model and Problem.- 11.3 Bilaterally Constrained Motion.- 11.4 Velocity Jump by Time-Scaling.- 11.5 Velocity Jump by Reflection.- 11.6 Reflections and Collisions - Remarks.- 12. Electropneumatic Drilling Machine.- 12.1 Mechanical Model.- 12.2 Simulations.- 13. Percussion Drilling Machine.- 13.1 Mechanical Model of the Drilling Machine.- 13.2 Mathematical Model for Non-Contact.- 13.3 The Contact Model.- 13.4 State Transitions.- 13.5 Results.- 14. Turbine Blade Damper.- 14.1 The Damper Model and the Non-Contact Case.- 14.2 Contact Kinematics of the Damping Device.- 14.3 Numerical Results.- 15. Concluding Remarks.- References.

Inertia and Chiral Edge Modes of a Skyrmion Magnetic Bubble

Abstract: The dynamics of a vortex in a thin-film ferromagnet resembles the motion of a charged massless particle in a uniform magnetic field. Similar dynamics is expected for other magnetic textures with a nonzero Skyrmion number. However, recent numerical simulations reveal that Skyrmion magnetic bubbles show significant deviations from this model. We show that a Skyrmion bubble possesses inertia and derive its mass from the standard theory of a thin-film ferromagnet. In addition to center-of-mass motion, other low energy modes are waves on the edge of the bubble traveling with different speeds in opposite directions.

Comparison of Surface Irrigation Simulation Models: Full Hydrodynamic, Zero Inertia, Kinematic Wave

Abstract: Some phenomena such as surface irrigation are so complex that it is very difficult to implement them in nature. For this purpose, simulation models are used. In this study, ability of full hydrodynamic, zero inertia, and kinematic wave models has been investigated in surface irrigation simulation. Using SIRMOD software, their performance has been compared. The results showed that full hydrodynamic and zero inertia models were very powerful in simulation process. For increasing of filed slope until amount of 0.01 full hydrodynamic and zero inertia models had not any difference but for more increasing of S0 due to the increasing of velocity, accuracy of zero inertia model dropped. In full hydrodynamic and zero inertia models for increase in Manning’s roughness coefficient amount of error was increased until n=0.15. After this amount, error remained constant thus n=0.15 determined as critical discharge. Accuracy of kinematic wave model reduced in clay and heavy clay soils, high discharges, high Manning’s roughness coefficient, and basin irrigation. However, in many situations all three models had the same answers and were capable tools to simulating of surface irrigation processes.

Sliding mode attitude tracking of rigid spacecraft with disturbances

Abstract: The attitude tracking control problem of a spacecraft nonlinear model with external disturbances and inertia uncertainties is addressed in this paper. First, a new sliding mode controller is designed to ensure the asymptotic convergence of the attitude and angular velocity tracking errors against external disturbances and inertia uncertainties by using a modified differentiator to estimate the total disturbances. Second, an adaptive algorithm is applied to compensating the disturbances, by which another sliding mode controller is successfully designed to achieve a high performance on the attitude tracking in the presence of the inertia uncertainties, external disturbances and actuator saturations. Finally, simulation results are presented to illustrate effectiveness of the control strategies.

Brownian motion at short time scales

Abstract: Brownian motion has played important roles in many different fields of science since its origin was first explained by Albert Einstein in 1905. Einstein's theory of Brownian motion, however, is only applicable at long time scales. At short time scales, Brownian motion of a suspended particle is not completely random, due to the inertia of the particle and the surrounding fluid. Moreover, the thermal force exerted on a particle suspended in a liquid is not a white noise, but is colored. Recent experimental developments in optical trapping and detection have made this new regime of Brownian motion accessible. This review summarizes related theories and recent experiments on Brownian motion at short time scales, with a focus on the measurement of the instantaneous velocity of a Brownian particle in a gas and the observation of the transition from ballistic to diffusive Brownian motion in a liquid.

Estimation of generator inertia available during a disturbance

Abstract: The inertia of a power system is a key factor in determining the initial frequency decline after a disturbance. Low system inertia can allow large frequency declines to occur that could lead to violation of frequency security limits, particularly in smaller power systems, or play a key role in allowing cascading outages to occur. Future developments in power systems will mean that the system inertia will become highly variable and take values that would traditionally have been considered very low. Presented is a method for the robust estimation of the generator inertia available in the system during a disturbance. This method has been validated using simulations of the IEEE 39-bus system in DIgSILENTTM PowerFactory®. Inertia estimates for a variety of disturbance types and noise conditions have been made, and found to have a median error of 1.53% with inter-quartile range of 6.6%.

Effects of nonlinear piezoelectric coupling on energy harvesters under direct excitation

Abstract: A nonlinear analysis of an energy harvester consisting of a multilayered cantilever beam with a tip mass is performed. The model takes into account geometric, inertia, and piezoelectric nonlinearities. A combination of the Galerkin technique, the extended Hamilton principle, and the Gauss law is used to derive a reduced-order model of the harvester. The method of multiple scales is used to determine analytical expressions for the tip deflection, output voltage, and harvested power near the first global natural frequency. The results show that one- or two-mode approximations are not sufficient to produce accurate estimates of the voltage and harvested power. A parametric study is performed to investigate the effects of the nonlinear piezoelectric coefficients and the excitation amplitude on the system response. The effective nonlinearity may be of the hardening or softening type, depending on the relative magnitudes of the different nonlinearities.

Inertial particle relative velocity statistics in homogeneous isotropic turbulence

Abstract: In the present study, we investigate the scaling of relative velocity structure functions, of order two and higher, for inertial particles, both in the dissipation range and the inertial subrange using direct numerical simulations (DNS). Within the inertial subrange our findings show that contrary to the well-known attenuation in the tails of the one-point acceleration probability density function (p.d.f.) with increasing inertia (Bec et al., J. Fluid Mech., vol. 550, 2006, pp. 349–358), the opposite occurs with the velocity structure function at sufficiently large Stokes numbers. We observe reduced scaling exponents for the structure function when compared to those of the fluid, and correspondingly broader p.d.f.s, similar to what occurs with a passive scalar. DNS allows us to isolate the two effects of inertia, namely biased sampling of the velocity field, a result of preferential concentration, and filtering, i.e. the tendency for the inertial particle velocity to attenuate the velocity fluctuations in the fluid. By isolating these effects, we show that sampling is playing the dominant role for low-order moments of the structure function, whereas filtering accounts for most of the scaling behaviour observed with the higher-order structure functions in the inertial subrange. In the dissipation range, we see evidence of so-called ‘crossing trajectories’, the ‘sling effect’ or ‘caustics’, and find good agreement with the theory put forth by Wilkinson et al. (Phys. Rev. Lett., vol. 97, 2006, 048501) and Falkovich & Pumir (J. Atmos. Sci., vol. 64, 2007, 4497) for Stokes numbers greater than 0.5. We also look at the scaling exponents within the context of the model proposed by Bec et al. (J. Fluid Mech., vol. 646, 2010, pp. 527–536). Another interesting finding is that inertial particles at low Stokes numbers sample regions of higher kinetic energy than the fluid particle field, the converse occurring at high Stokes numbers. The trend at low Stokes numbers is predicted by the theory of Chun et al. (J. Fluid Mech., vol. 536, 2005, 219–251). This work is relevant to modelling the particle collision rate (Sundaram & Collins, J. Fluid Mech., vol. 335, 1997, pp. 75–109), and highlights the interesting array of phenomena induced by inertia.

Magnetization dynamics, gyromagnetic relation, and inertial effects

Abstract: The gyromagnetic relation—that is, the proportionality between the angular momentum L→ and the magnetization M→—is evidence of the intimate connections between the magnetic properties and the inertial properties of ferromagnetic bodies. However, inertia is absent from the dynamics of a magnetic dipole: The Landau–Lifshitz equation, the Gilbert equation, and the Bloch equation contain only the first derivative of the magnetization with respect to time. In order to investigate this paradoxical situation, the Lagrangian approach, proposed originally by Gilbert, is revisited keeping an arbitrary nonzero inertia tensor. The corresponding physical picture is a generalization to three dimensions of Ampere’s hypothesis of molecular currents. A dynamic equation generalized to the inertial regime is obtained. It is shown how both the usual gyromagnetic relation and the well-known Landau–Lifshitz–Gilbert equation are recovered in the kinetic limit, that is, for time scales longer than the relaxation time of the angu...

Unsteady swimming of small organisms

Abstract: Small planktonic organisms ubiquitously display unsteady or impulsive motion to attack a prey or escape a predator in natural environments. Despite this, the role of unsteady forces such as history and added mass forces on the low-Reynolds-number propulsion of small organisms, e.g. Paramecium, is poorly understood. In this paper, we derive the fundamental equation of motion for an organism swimming by means of the surface distortion in a non-uniform background flow field at a low-Reynolds-number regime. We show that the history and added mass forces are important as the product of Reynolds number and Strouhal number increases above unity. Our results for an unsteady squirmer show that unsteady inertial effects can lead to a non-zero mean velocity for the cases with zero streaming parameters, which have zero mean velocity in the absence of inertia.

Effects of the synthetic inertia from wind power on the total system inertia after a frequency disturbance

Abstract: The future power systems face several challenges; one of them is the use of high power converters that virtually decouple primary energy source from the AC power grid. An important consequence of this situation is their effect on total system inertia and the ability to overcome the system's frequency disturbances. The wind power industry has created a controller to enable inertial response on wind turbines generators: Synthetic Inertial. This paper evaluates the effects of the inertia emulation of wind turbines based on full-converters and their effect on total system inertia after frequency disturbances happen. The main contribution of this paper is to demonstrate (based on simulations) that during an under-frequency transients on future power systems, synthetic inertia does not completely avoid worse scenarios in terms of under-frequency load shedding. The extra power delivered from a wind turbine during frequency disturbances can increase “momentary” the total system inertia and substantially reduce the rate of change of frequency providing time for the active governors to respond. However, synthetic inertia might not completely avoid under-frequency load shedding.

Treating inertia in passive microbead rheology.

Abstract: The dynamic modulus G(*) of a viscoelastic medium is often measured by following the trajectory of a small bead subject to Brownian motion in a method called "passive microbead rheology." This equivalence between the positional autocorrelation function of the tracer bead and G(*) is assumed via the generalized Stokes-Einstein relation (GSER). However, inertia of both bead and medium are neglected in the GSER so that the analysis based on the GSER is not valid at high frequency where inertia is important. In this paper we show how to treat both contributions to inertia properly in one-bead passive microrheological analysis. A Maxwell fluid is studied as the simplest example of a viscoelastic fluid to resolve some apparent paradoxes of eliminating inertia. In the original GSER, the mean-square displacement (MSD) of the tracer bead does not satisfy the correct initial condition. If bead inertia is considered, the proper initial condition is realized, thereby indicating an importance of including inertia, but the MSD oscillates at a time regime smaller than the relaxation time of the fluid. This behavior is rather different from the original result of the GSER and what is observed. What is more, the discrepancy from the GSER result becomes worse with decreasing bead mass, and there is an anomalous gap between the MSD derived by naively taking the zero-mass limit in the equation of motion and the MSD for finite bead mass as indicated by McKinley et al. [J. Rheol. 53, 1487 (2009)]. In this paper we show what is necessary to take the zero-mass limit of the bead safely and correctly without causing either the inertial oscillation or the anomalous gap, while obtaining the proper initial condition. The presence of a very small purely viscous element can be used to eliminate bead inertia safely once included in the GSER. We also show that if the medium contains relaxation times outside the window where the single-mode Maxwell behavior is observed, the oscillation can be attenuated inside the window. This attenuation is realized even in the absence of a purely viscous element. Finally, fluid inertia also affects the bead autocorrelation through the Basset force and the fluid dragged around with the bead. We show that the Basset force plays the same role as the purely viscous element in high-frequency regime, and the oscillation of MSD is suppressed if fluid density and bead density are comparable.

Noise-Induced Drift in Stochastic Differential Equations with Arbitrary Friction and Diffusion in the Smoluchowski-Kramers Limit

Abstract: We consider the dynamics of systems with arbitrary friction and diffusion. These include, as a special case, systems for which friction and diffusion are connected by Einstein fluctuation-dissipation relation, e.g. Brownian motion. We study the limit where friction effects dominate the inertia, i.e. where the mass goes to zero (Smoluchowski-Kramers limit). Using the Ito stochastic integral convention, we show that the limiting effective Langevin equations has different drift fields depending on the relation between friction and diffusion. Alternatively, our results can be cast as different interpretations of stochastic integration in the limiting equation, which can be parametrized by α∈ℝ. Interestingly, in addition to the classical Ito (α=0), Stratonovich (α=0.5) and anti-Ito (α=1) integrals, we show that position-dependent α=α(x), and even stochastic integrals with α∉[0,1] arise. Our findings are supported by numerical simulations.

Effects of the synthetic inertia from wind power on the total system inertia: simulation study

Abstract: The future power systems face several challenges; one of them is the use of high power converters that virtually decouple primary energy source from the AC power grid. An important consequence of this modified the total system inertia and affecting its ability to overcome system frequency's disturbances. The wind power industry has created a controller to enable inertial response on wind turbines generators: Synthetic Inertial. This paper evaluates the effects of the synthetic inertia provided by wind turbines on the total system inertia after a system frequency disturbance. The main contribution of this paper is to demonstrate that during an under-frequency transients on future power systems, the synthetic inertia not completely avoid worse scenarios in terms of under-frequency load shedding. The extra power delivered from a wind turbine during frequency disturbances can increase “momentary” the total system inertia and substantially reduce the rate of change of frequency providing time for the active governors to respond. However, synthetic inertia might not completely avoid under-frequency load shedding.

Unsteady swimming of small organisms

Abstract: Small planktonic organisms ubiquitously display unsteady or impulsive motion to attack a prey or escape a predator in natural environments. Despite this, the role of unsteady forces such as history and added mass forces on the low-Reynolds-number propulsion of small organisms, e.g. Paramecium, is poorly understood. In this paper, we derive the fundamental equation of motion for an organism swimming by means of the surface distortion in a non-uniform background flow field at a low-Reynolds-number regime. We show that the history and added mass forces are important as the product of Reynolds number and Strouhal number increases above unity. Our results for an unsteady squirmer show that unsteady inertial effects can lead to a non-zero mean velocity for the cases with zero streaming parameters, which have zero mean velocity in the absence of inertia.

Parametric study of the dynamic response of thin rectangular plates traversed by a moving mass

Abstract: The governing differential equation of motion of a thin rectangular plate excited by a moving mass is considered. The moving mass is traversing on the plate’s surface at arbitrary trajectories. Eigenfunction expansion method is employed to solve the constitutive equation of motion for various boundary conditions. Approximate and exact expressions of the inertial effects are adopted for the problem formulation. In the approximate formulation, only the vertical acceleration component of the moving mass is considered while in the exact formulation all the convective acceleration components are included in the problem formulation as well. Parametric studies are carried out to investigate the effects of moving mass weight and velocity as well as its trajectory on the dynamic response of a simply supported plate. Rectilinear and orbiting paths are considered in the parametric studies as the two limiting cases for any possible moving mass trajectories. The obtained results demonstrate the importance of the moving mass inertia with respect to the moving load in most of the cases considered. In case of the rectilinear path, the approximate formulation underestimates the plate’s maximum response for mass velocities above certain limits. Furthermore, increasing the plate’s aspect ratio or the moving mass weight further reduces the range of velocities in which the approximate formulation can be used instead of the exact formulation. For the case of an orbiting path, the approximate formulation can capture the resonance excitation frequencies of the load reasonably well, even for large mass weight and radius of the orbiting mass. Considering small orbiting mass radii, the approximate formulation would provide an upper bound for the true response of the system for all orbiting frequencies as well as the mass weights. However, for larger radii, the maximum response values resulting from the approximate formulation are considerably lower than that of the exact one, especially for frequencies near to the resonance frequencies.

Virtual inertia for variable speed wind turbines

Abstract: Inertia provision for frequency control is among the ancillary services that different national grid codes will likely require to be provided by future wind turbines. The aim of this paper is analysing how the inertia response support from a variable speed wind turbine (VSWT) to the primary frequency control of a power system can be enhanced. Unlike fixed speed wind turbines, VSWTs do not inherently contribute to system inertia, as they are decoupled from the power system through electronic converters. Emphasis in this paper is on how to emulate VSWTs inertia using control of the power electronic converter and on its impact on the primary frequency response of a power system. An additional control for the power electronics is implemented to give VSWTs a virtual inertia, referring to the kinetic energy stored in the rotating masses, which can be released initially to support the system's inertia. A simple Matlab/Simulink model and control of a VSWT and of a generic power system are developed to analyse the primary frequency response following different generation losses in a system comprising VSWTs provided with virtual inertia. The possibility of substituting a 50% share of conventional power with wind is also assessed and investigated. The intrinsic problems related to the implementation of virtual inertia are illustrated, addressing their origin in the action of pitch and power control. A solution is proposed, which aims at obtaining the same response as for the system with only conventional generation. The range of wind speeds near the power limitation zone seems to be the most critical from a primary response point of view. The theoretical reasons behind this are elucidated in the paper. Copyright © 2012 John Wiley & Sons, Ltd.

The effect of slatted screens on waves

Abstract: A linearised model is proposed for the transmission of waves through thin vertical porous barriers, where both the inertial and dominant quadratic drag effects are included. A boundary-value problem is developed in which linear boundary conditions holding along the length of the screen are derived from a pair of canonical wave problems, one including an exact geometric description of a slatted screen to determine an inertia coefficient and the other using a quadratic drag law to determine an equivalent linear drag coefficient. The model is then applied to a range of wave scattering and sloshing problems involving thin vertical slatted screens in various settings. In each case results are verified by comparison to the solution of a direct non-linear calculation where the effects of drag have been isolated. We show that the solution to our canonical problem provides a good approximation to the solution of each of the model problems.

Research on the inertia matching of the Stewart parallel manipulator

Abstract: With the development of the parallel manipulator, inertia matching as an essential factor to realize good potentials of the parallel manipulator is taken serious gradually. However, neither definite inertia index nor inertia matching method has been proposed so far. In this paper, the above issues are discussed by taking the Stewart parallel manipulator as a study object. Firstly, adopting limb Jacobian matrices, the concise algebraic expression of the joint-space inertia matrix of the Stewart parallel manipulator is deduced, based on the dynamic modeling. Next, on the basis of the coupling analysis of the joint-space inertia matrix, the inertia index of the parallel manipulator, the Joint-Reflected Inertia, is proposed. Then, the practical inertia matching principles of the Stewart parallel manipulator are concluded on the basis of simulations, considering multiple factors, such as mechanical resonance frequency, acceleration torque and dynamic performance. Finally, the available range of the motor inertia is deduced, and the inertia matching of the Stewart parallel manipulator is finished as the case study. The inertia index and inertia matching method suggested in this paper can be further used in other parallel manipulators for dynamic analysis and motion system design.

Brownian transport in corrugated channels with inertia.

Abstract: Transport of suspended Brownian particles dc driven along corrugated narrow channels is numerically investigated in the regime of finite damping. We show that inertial corrections cannot be neglected as long as the width of the channel bottlenecks is smaller than an appropriate particle diffusion length, which depends on the the channel corrugation and the drive intensity. With such a diffusion length being inversely proportional to the damping constant, transport through sufficiently narrow obstructions turns out to be always sensitive to the viscosity of the suspension fluid. The inertia corrections to the transport quantifiers, mobility, and diffusivity markedly differ for smoothly and sharply corrugated channels.

Shaft Positioning for Six-Phase Induction Machines With Open Phases Using Variable Structure Control

Abstract: In this paper, a new variable structure control law is proposed in order to obtain the shaft positioning of a symmetrical six-phase induction machine when phases are lost and when large variations of inertia occur. The algorithm based on a time-varying switching line guarantees the existence of a sliding mode since the beginning of the shaft motion. Indeed, the surface is first designed to pass through the initial representative point and, subsequently, to move toward a predetermined desired surface via shifting. With this algorithm, the induction machine behavior is exactly the same in healthy or in faulted modes when one or more stator phases are lost and whatever the mechanical shaft configuration is. The capacities of this control technique with a mechanical sensor have been tested in simulation and then validated experimentally on a specific setup.

Impact of synthetic inertia from wind power on the protection/control schemes of future power systems: Simulation study

Abstract: Future power systems face several challenges; one of them is the use of high power converters that decouple new energy sources from the AC power grid. This decreases the total system inertia affecting its ability to overcome system frequency's disturbances. The wind power industry has created a controller to enable inertial response on wind turbines generators: Artificial, Emulated, Simulated, or Synthetic Inertial. This paper deals with issues related to the synthetic inertia of wind turbines based on full-converters and their effect on the frequency protection/control schemes during the recovery period after system frequency disturbances happen. The main contribution of this paper is to demonstrate (based on simulations) the recovery period of under-frequency transients on future power systems which integrate wind turbines with synthetic inertia capability not completely avoid worse scenarios in terms of under-frequency load shedding. The extra power delivered from a wind turbine during frequency disturbances can substantially reduce the rate of change of frequency providing time for the active governors to respond. However, synthetic inertia might not completely avoid under-frequency load shedding. (6 pages)

Equivalent vehicle rotational inertia used for electric vehicle test bench dynamic studies

Abstract: This paper provides an easy yet accurate approach for estimating vehicle inertia for the purpose of simulating the effect of vehicle inertia using a large flywheel. The method used to properly map the linear inertia of a vehicle to an equivalent rotational inertia is described in detail and an expression for equivalent rotational inertia of a vehicle is derived analytically using two different approaches considering kinetic energy of a moving mass in the linear and rotational context and vehicle dynamic equations. A MATLAB/Simulink model of a test bench consisting of a torque actuator connected to a large flywheel is used to emulate the effect of vehicle inertia. Using this model, the dynamic inertia effect of an electric vehicle on its traction motor is illustrated for two standard drive cycles. The results obtained from simulation are validated by ADVISOR to confirm the effectiveness of the proposed method in estimating vehicle inertia and it is shown that the use of a flywheel with a calculated rotational inertia using the proposed method can be sufficient in emulating vehicle inertia effect on a test bench.