Hybrid variable damping control: design, simulation

Microsyst Technol DOI 10.1007/s00542-014-2214-8

Technical Paper

Hybrid variable damping control: design, simulation, and optimization Roberto Ribeiro · Ehsan Asadi · Mir Behrad Khamesee · Amir Khajepour 

Received: 15 October 2013 / Accepted: 3 May 2014 © Springer-Verlag Berlin Heidelberg 2014

Abstract  This paper presents the design, formulation, and performance optimization of a new hybrid electromagnetic damper in response to the demand for a tunable, regenerative and fail-safe damping device for various applications. Damping in a multitude of engineering applications has a variable threshold requirement based on system excitation. Since system excitation is also variable; dampers are such that an adequate amount of damping is provided, opposed to an optimal amount as a function of excitation. In this research it was shown that, by implementing a hybrid damper design based on a bias component provided through a hydraulic medium and a variable component provided by electromagnetics, an optimal damping quantity can be obtained for a given excitation. The produced damping force and electrical power were formulated based on the structure’s geometry and input displacement. The presented design was optimized for a scooter scaled application and it was shown that the damping and regenerative characteristics can be adjusted for different requirements. Furthermore, it was illustrated that this design has the potential to be scaled for other applications as well.

R. Ribeiro · E. Asadi · M. B. Khamesee (*) · A. Khajepour  Mechanical and Mechatronics Engineering Department, University of Waterloo, 200 University Ave. West, Waterloo, ON N2L3G1, Canada e-mail: [email protected] R. Ribeiro e-mail: [email protected] E. Asadi e-mail: [email protected] A. Khajepour e-mail: [email protected]

1 Introduction System modeling in a multitude of engineering applications typically results in a simplified representation consisting of a series of masses, springs, and dampers. The damper causes the system to eventually settle to a point of zero motion given a discontinuous input. Damping in a system can be obtained by the structure itself, the addition of a viscous medium, through contact friction or by many other means; all yielding a time dependent decay in response. In all systems there is an optimal damping factor based on the input and the desired response. In most applications, the desired system response is known but the input varies with time. This variability results in the design engineer choosing a damping factor that will perform adequately at a wide range of inputs, due to the passive nature of most damping mechanisms. In order to achieve an optimal amount of damping given a variable input, the damping itself must dynamically fluctuate. The use of an active or semi-active damper could achieve this requisite functionality. Active damping is typically characterized by having an energy input for actuation, and a semi-active system is characterized by having a variable damping ratio modified in a closed loop configuration across a wide bandwidth (Savaresi et al. 2010). In the 1960s Citroen developed and commercially implemented the first adaptive suspension system with its automotive hydro-pneumatic suspension. Since then many derivatives of this concept have been developed by a wide range of manufacturers. The three principle technologies developed in recent years are as follows: the solenoid valve electrohydraulic damper (Sachs), the magnetorheological damper (Delphi), and the electroheological damper (Fludicon) (Savaresi et al. 2010). Of these technologies, the systems developed by Delphi and Fludicon are of interest

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due to the force and work densities available (Gomis-Bellmunt and Campanile 2010). Although all of the adaptive suspension technologies mentioned are functional and have been commercially implemented at some level, they all have significant shortcomings. All of these systems are actuated electronically resulting in a percentage of energy consumption, all energy absorbed by the damper is expelled into heat, and all systems have an overall lower level of reliability relative to an equivalent passive system. Research at a variety of institutions has resulted in many novel and interesting implementations of damping mechanisms, with the capacity to provide variable damping on the basis of electromagnetic theory. The two core topologies of these electromagnetic dampers are linear and rotary. The linear topology is typically configured in a manner similar to what is seen in linear motors. Gysen et al. (2011) developed an actuator which consisted of a brushless tubular permanent-magnet actuator with a coil spring in parallel. This system was configured to operate in an active manner to control the damping characteristics of a vehicle chassis. It also has the capacity to act as a generator during the down stroke of the actuator. The capacity of this system to act as a generator is greatly compromised by the eddy current component of this damper which acts as the bias for the system. By reducing the bias, higher efficiencies with respect to energy recovery can be obtained; but from a practical perspective this is not attainable due to the requisite fail-safe functionality. The rotary topology is typically characterized by a mechanism (normally a ball screw or rack and pinion) that converts linear motion to rotary motion and a linkage to a permanentmagnet generator. Zhongjie et al. (2013) developed a system using a rack and pinion mechanism with bevel gears and a planetary gear box connected to a permanent-magnet generator. In general the rotary topology has slower response compared to the linear topology due to backlash in the mechanical linkages. On the contrary the rotary the topology is typically capable of providing more damping force (relative to linear actuator) in a semi-active configuration. As an alternative to the topologies and configurations presented above, a hybrid design consisting of a passive and electromagnetic damper integrated together in a semi-active configuration is proposed. The remaining sections of this paper will outline the proposed design, theory related to the produced electromagnetic force, optimization of the geometry, simulation results, scalability of the design, the damper’s tuning mechanism, and possible applications for this design.

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integration into powered wheelchairs has numerous positive characteristics associated with it. This system is capable of better isolating the user from environmental input disturbances to increase overall comfort. It also has the potential to give the individual a better quality of life based on an increased level of functional capacity. This is achieved by integrating the system into the occupant’s seat, similar to what is seen in heavy equipment (i.e. earth movers and farm equipment). Since the system does not consume any power there would not be any reduction in the overall travel distance per charge; with the added benefit of the harvested energy. The recovered energy could be used to power local electronics such as onboard instruments and charge standalone systems (e.g. cell phones); and surplus energy could be stored in onboard batteries. On a similar scale this system can be used on motorcycles to achieve adaptive control of the steering damper attached to the handlebars. Energy harvesting at this scale has shown promising results, based on the work completed by Patel et al. (2012).

3 Design Based on the attribute of energy consumption associated with the previously mentioned commercial systems, the backlash in the rotary topology, and the reduction in efficiency for energy recovery due to the eddy current bias found in the system developed by Gysen et al. (2011), a hybrid design is proposed. The hybrid design is configured such that the passive (viscous) damping acts as the bias (or baseline) and the electromagnetic system accounts for the variable damping (Ebrahimi 2009). The electromagnetic component of this system functions by means of electromagnetic induction. An optimized configuration of iron poles and permanent magnets translates concentric to the axis of actuation through a stack of conductor coils. The relative motion of the iron poles and magnets through the conductor coils induces a voltage potential across the coils. Control is achieved by limiting the output current from the coils, resulting in a resistive force opposing the direction of actuation (Faraday–Lenz Law). A predefined ideal damping model in conjunction with inertial sensors enables the system to operate at an optimal level. In addition to the system’s augmentable damping capabilities, it can also recover energy. Refer to Fig. 1 for a schematic representation of a conceptual design for this system.

2 Applications 4 Hybrid damper force derivation This paper describes the design of a hybrid variable damping system, which has a wide variety of applications ranging from macro to micro scales of order. Most notably its

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The force produced by the hybrid damper is the result of a viscous and an electromagnetic damper working together.

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4.2 The variable damping via electromagnetic damping The electromagnetic component of the hybrid design is based on the Faraday–Lenz and Lorentz laws. Faraday’s law of induction states: the change of total flux in a closed circuit results in an induced electromotive force proportional to the rate of change of flux through the integral path (Zuza et al. 2012). According to Lenz’s Law, the induced magnetic field produced by the induced current will oppose the direction of the original magnetic field. The Faraday– Lenz law can be mathematically represented by the following expression:

Vemf = −

dψ d = −N dt dt

(2)

where Vemf represents the electromotive force in volts,  is the magnetic flux linkage, N represents the number of turns in a conductor coil and ψ represents the flux through each turn of the coil. Lorentz’s law quantifies the electromagnetic force FE experienced by the conductor coil and is obtained by:  FE = idl × B (3) Fig. 1  Conceptual hybrid damper design

In this section the hybrid damping force for these two components is analytically formulated. This formulation provides a basis for design and system dynamic modeling by predicting the thrust force and generated electrical energy. 4.1 The baseline damping via viscous damping Viscous dampers are the most widely used damping devices in the automotive industry. Their reliability, capability of handling large force and high speed, and their predictability made them a good candidate for producing a baseline damping force. Viscous dampers provide a resistive force against the motion which is linearly proportional to the velocity. This force is expressed as:

Fv = −Cv ·

dx dt

(1)

where Cv is the damping coefficient of the damper, and x is the displacement. This coefficient is dependent on the damper structure and the viscosity of the fluid. Viscous dampers, when correctly designed and manufactured, maintain a constant damping coefficient over the full range of velocities and displacements. In practice, this coefficient can be tuned to achieve optimum values based on the application. This tuning is typically achieved by implementing an adjustable valve in the fluid channel.

where i and l are magnitude and path of the induced current, B is the magnetic field flux density, and × denotes the cross product. In order to formulate the electromagnetic damper, the distribution of magnetic field needs to be obtained with respect to the given design parameters. This can be achieved by using any of the following three main techniques: 1-lumped equivalent magnetic circuit (PalomeraArias et al. 2008; Lu et al. 2005), 2-numerical solution (finite element method) (Basak et al. 1990; Zuo et al. 2010) and 3-analytical solution (separation of variables) (Wang et al. 2004; Tsai and Chiang 2010). When compared to the other two mentioned methods, the lumped equivalent magnetic circuit is straightforward and less computationally expensive. This has led to the lumped equivalent magnetic circuit method becoming the most common approach in establishing a relationship between design parameters and machine performance. This method aims to replace the spatially distributed electromagnetic system with a series of interconnected simple discrete elements. 4.3 Lumped magnetic circuit model Figure  2 shows a lumped magnetic circuit model of the electromagnetic damper. The analysis is conducted for a magnet-iron pair but the same concept holds for all the pairs. By neglecting the magnetic flux leakage in the center rod, Ampere’s law is applied to the contour C1 as follows:

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Microsyst Technol

Hm =

Bm − Brem µrec

(6)

where Brem and μrec are the remnant flux density and recoil permeability of the permanent magnets, respectively. These parameters are obtained from the demagnetization curve of the permanent magnets. Since the magnetic flux leakage is neglected, the continuity condition can be applied to determine the magnetic flux densities for the various volumes in terms of B m:

Am Bm = Apole Bpole = Ap tube Bp tube = Aair Bair = Acoil Bcoil

(7)

Solving Eq. 7 for magnetic flux densities yields:

Bpole = Bair =

  π r 2 −rs2 Bm  τi  , 2rπ 2  2 2 π rw −rs Bm τ  , 2rπ 2i

Bp tube = Bcoil =

  π rw2 −rs2 Bm   π r 2 −r 2  2 p 2 c π rw −rs Bm τ  2rπ 2i

(8)

Using Eq. 8, the elements of Eq. 5 can be obtained as follows:

Fig.  2  a Magnetic flux circuit. b Cross of electromagnetic component of damper with modeling nomenclature





H · dl =

C1

1(magnet)

H · dl

2&5&8(iron)



+



H · dl +



H · dl+

(4)

H · dl = 0

4&6(copper)

3&7(air)

H · dl = Hm τm + 2

rw

rs

C1

+2

ra

rw

Bair dr + 2 µ0

ra

(5)

Bcoil dr = 0 µ0 µcu

where μ0 is the permeability of free space and μFe and μcu are the relative permeability of iron and copper, respectively. In order to solve Eq. 5 and find the operation point of the machine, Hm, Bpole, Bptube, Bair, and Bcoil are written in terms of Bm. The permanent magnetic characteristic equation defines the relation between Hm and Bm as:

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   2 − r2 Bpole rw 1 rw s dr = − rs2 ln Bm = kpole Bm µ0 µFe µ0 µFe τi 2 rs

(9)

  2 − r 2 (τ + τ ) rw m i Bp tube s  Bm = kp tube Bm (τm + τi ) =  µ0 µFe rp2 − rc2 µ0 µFe

(10)

ra

r 2 − rs2 Bair dr = w ln µ0 µ0 τi

(11)

rc

r 2 − rs2 Bcoil dr = w ln µ0 µcu µ0 µcu τi

ra



 ra Bm = kair Bm rw



 rc Bm = kcoil Bm . ra

(12)

By substitution of Eqs. 9–12 and 6 into Eq. 5, the operation point of the electromagnetic damper is obtained as:

Bpole Bp tube dr + − (τm + τi ) µ0 µFe µ0 µFe rc

rs

rw

where H refers to magnetic field strength. By assuming that the magnetic field is axial inside the magnet and pressure tube and is radial inside the poles, air gaps and coils, Eq. 4 can be written as:



rw

Bm =

τm + µrec

Brem τm . 2kpole + kp tube + 2kair + 2kcoil



(13)

By substituting Eq. 13 into Eq. 8, the magnetic flux density at the different parts of electromagnetic damper can be evaluated. In the following sections, the obtained magnetic flux density in the coils is used to calculate the induced voltage across the coils. 4.4 Coil induced current The relative velocity of armature (magnet-pole arrays) relative to the stator coils induces voltage across the coils.

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Applying the Faraday–Lenz’s law for a single coil turn gives us:

dx Vemfsingle turn = −2π rBcoil  2 dt2  2π rw − rs Bm dx . Vemfsingle turn = − τi dt

(14)

−

FE = Ce

Rcoil +Rload Lcoil

t

kv2 Aω cos (ωt + θ) + (Rcoil + Rload )2 + (ωLcoil )2 (20)

Therefore the total electromagnetic induction across the coil is:   2π rw2 − rs2 Bm dx dx = −kv Vemf = −N (15) τi dt dt

where A and ω are the amplitude and angular frequency of the excitation and C is a constant determined based on the initial condition. As can be observed in Eq. 20, there is a phase shift between input displacement and output force and its value is given by:   ωLcoil . θ = tan−1 − (21) Rcoil + Rload

where N is the number of excited turns in the coil and

4.6 Hybrid damping force

  2π rw2 −rs2 Bm N τi

kv = is called the voltage constant of the coil. The relationship between the induced current (i) and the electromagnetic voltage (Vemf) in the coils is dependent on the how the coils are connected (star, delta, etc…) to the external circuit. For the case where each coil is connected to an external load Rload (coined as “standard configuration” for this paper), the induced current can be obtained from: Lcoil

dx di + (Rcoil + Rload )i = Vemf = −kv dt dt

4.5 Coil electromagnetic force For a single coil turn, the experienced electromagnetic force is obtained using Lorentz’s law as follows:   2π rw2 − rs2 Bm i = kf i. FEsingle turn = 2πrBcoil i = (17) τi Therefore the total electromagnetic damping force is:   2π rw2 − rs2 Bm (18) i = kf i FE = N τi   2π r 2 −r 2 B

m w s where kf = N is called the force constant of the τi coil and has the same value as kv. By multiplying Eq. 16 by kf, the force–velocity relationship of the electromagnetic damper is obtained as:

dx dFE + (Rcoil + Rload )FE = −kv2 . dt dt

Ft = Fv + PFE

(22)

where P is the number of magnet-iron pairs in armature, and Fv and FE are obtained from Eqs. 1 and 20, respectively. Unlike a conventional oil shock absorber, it is notable that the total force in the proposed damper is variable by changing the external load as seen in Eq. 20.

(16)

where Lcoil and Rcoil are the coil’s inductance and resistance, respectively.

Lcoil

The total hybrid damping force Ft is the summation of viscous and electromagnetic damping forces, which is:

(19)

For a sinusoidal armature displacement profile in the form of x = Asinωt, the electromagnetic damping force is given by:

5 Optimization of geometry The performance of a linear permanent magnet array, when used as an actuator with respect to regenerative capacity and damping force, is highly dependent on the geometry of the system. In a limited number of applications the geometric results yielded through optimization can be directly applied to a design, due to the availability of components such as permanent magnets. In addition, due to physical constraints with respect to integration with existing products and designs, it is ideal to limit the scope of the optimization. As found in other literature the goal of a new design in most cases should be to act as a drop in replacement for an original component with the added benefit of increased performance and return on investment. Based on the work done by Wang et al. (2004) one of the parameters of interest for optimization of an actuator is the ratio of the magnet thickness to the thickness of the iron pole. Figure 3 reflects the results obtained through optimization for this ratio. In Fig. 3, the horizontal axis reflects the ratio of the magnet thickness to the thickness of the iron poles(τm/τi), the vertical axis is the normalized damping force, and each of the curves reflects a different length for each stack of magnets and iron poles(τ = τm + τi). It can be seen from Fig. 3 that at approximately a ratio of 2.3 all curves become more

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Microsyst Technol Table 1  Nomenclature and chosen dimension for use in analysis; refer to Fig. 2b

Fig. 3  Normalized damping force vs. magnet to iron pole thickness ratio for varying values of τ (total thickness of iron poles and magnets)

or less linear, which shows that a minimum ratio of 2.3 is required to maximize the normalized damping force. This resulting ratio is consistent with the results published by Wang et al. (2004) after converting the ratio shown above into an equivalent ratio for comparison. Wang et al. (2004) also showed that although the normalized damping force is maximized at this ratio, an equivalent ratio between 2.3 and 3.1 will reduce the effects of force ripple on the actuator. Based on this result, a ratio of 3 for τm/τi and 8.47 mm will be chosen τ for calculations and modeling in subsequent sections. The remaining dimensions for the actuator will be chosen such that design will act as a drop replacement for an existing product or design.

6 Simulation results In order to model the electromagnetic aspect of the aforementioned design, a finite element package was employed. With the aid of COMSOL Multiphysics, a configuration consisting of four iron poles, nine neodymium N52 grade permanent magnets (where three magnets are equal to τm), and 15 multi-turn copper coils were modeled. This modeling yielded the magnetic flux distribution, the induced current, and the maximum force generated from the electromagnetics. Based on the geometry and nomenclature shown in Fig. 2b; Table 1 outlines the applied nomenclature and geometric dimensions used in this model. The ratio of the thickness of the magnet to the thickness of the iron pole was selected based on the previously shown optimization; the remaining dimensions were chosen based on a scale of actuators typically found on the scooters and

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Variable

Dimension (mm)

Description

rs

2.1158

Piston shaft radius

rw

6.3475

Magnet and iron pole outer radius

ra

6.7818

Coil inner radius

rc

8.0569

Coil outer radius

rp

8.4658

Pressure tube outer radius

τi

2.1158

Iron pole thickness

τm

6.3475

Magnet thickness

τc

2.7789

Coil thickness

motorcycle steering dampers. Subsequent sections will outline the scaling of this design as well as additional applications. Using the tabulated geometric data shown in Table 1 and the Magnetic Fields module within COMSOL; the magnetic flux distribution was obtained along with the two additional charts (illustrating the short circuited current per coil in phase b and the maximum damping force produced by the electromagnetics) for a given study period of 0.25 s and a maximum displacement of 4.2333 mm. From the simulation results, the magnetic flux density plot (Fig. 4) shows that the flux stream lines are being directed into the coils to minimize the leakage and maximize the output. If the iron poles were omitted from the design there would be a significant amount of flux leakage and consequently lower output. Alternatively, the use of radially magnetized permanent magnets in place of the poles would yield even better results with respect to flux leakage. In this case the design used the iron poles due to their accessibility, cost, and better availability of the required dimensions to meet the optimization requirements. Figure 5 shows the output current in amps for the study period in a short circuited configuration. With the linear topology; the maximum amount of force is generated when the system is configured in the short circuited configuration, and the maximum amount of energy is recoverable when the system’s internal resistance is matched with an external load. From Fig. 5 it is apparent that the output current has a periodic trend and that the excitation in the various coils varies in magnitude within a given phase and clear phase shift is present. It can also be seen that B1 and B5 are equal with a phase shift of 180°; similarly B2 and B4 are pairs. If these four coils are compared to B3, it can be seen that B3 experienced excitation during the entire study period and remaining coils experienced less excitation. Figure  6 shows the maximum amount of force that the simulated structure is able to produce solely based on the electromagnetics. It is anticipated that applying a phase shift to the output through electronics could further

Microsyst Technol Fig. 4  Magnetic flux density distribution

Fig. 5  Current in phase B coils in short circuited configuration

Fig. 6  Produced Force in short circuited configuration

increase the maximum attainable force. It is also important to note that this maximum force will be further prorated if the damper is used for energy harvesting. In the following sections the effect of scaling and energy recovery will be assessed.

simulations were done to determine the effects of scale on the damping coefficient. Figure 7 reflects a summary of these results. To produce the curve shown in Fig. 7; the design and parameters previously shown were all scaled up and down relative to the original design, which acted as the bias for the scale (scale 1). In order to accurately represent the scaling of this model; not only were the geometric parameters scaled, but also the model inputs such as the prescribed displacement. After completing the various simulations the peak force for each simulation was tabulated and the damping coefficient was determined based on the peak force from simulation and the peak velocity.

7 Scalability of design As mentioned in previous sections, the geometry associated with the linear motor topology when used as an actuator is of great significance. To better understand the effects of scaling the previously presented design, an additional 10

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Fig. 7  Normalized damping coefficient vs. scaling factor

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Fig. 8  Damping force vs. time for varying external resistance in standard configuration

The results presented in Fig. 7 show that the scaling of the design is well-defined by a second order polynomial. Scaling of the original design by a factor of 1.5 would result in an increase in the normalized damping coefficient by a factor of approximately 3.5. The scaling of a design could become a metric of interest when it comes to determining the ratio of bias to electromagnetic force; also the ratio between the produced electromagnetic force and the regenerative capacity of the damper. The following section will detail the regenerative capacity ratio and its relationship to force.

8 Damping tuning mechanism As discussed in the previous sections this hybrid design has two core ratios; the ratio of electromagnetic force to the oil bias and the ratio of electromagnetic force to the regenerative capacity of the damper. The oil bias ratio in most applications is anticipated to be fixed within a safety tolerance specified by the application of the damper. As illustrated by Fig. 7, this ratio could very quickly be changed by simply scaling the design by a marginal factor. Similarly the force to regenerative capacity can be changed in a relatively trivial manner as well by adjusting the external resistance of the system. To quantify the effects of external electrical load to the capacity to produce electromagnetic damping force and recover energy, two configurations were investigated. The first was the “standard configuration” as previously detailed. To obtain the effects of electrical load on this configuration, a multitude of simulations were carried out to determine the effects on damping force and regenerative power. Figures 8 and 9 reflect the results of these simulations for damping and power respectively.

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Fig. 9  Regenerative power vs. time for varying external resistance in standard configuration

As seen in Figs. 8 and 9; the trend of having maximum electromagnetic force when the external load is zero (short circuited) and having maximum energy recovery capacity when the external resistance is matched with the internal resistance of the coil (in this case 0.423 Ohms), is maintained. The second configuration of interest is the “star configuration”. Similar to what was done for the “standard configuration”, simulations were run to obtain peak force and peak regenerative capacity for the “star configuration”. Figures 10 and 11 reflect a summary of the results obtained for both configurations and metrics of interest. From Figs. 10 and 11 it can be seen that the “standard configuration” produced a higher amount of damping force

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Fig. 10  Damping force and regenerative power vs. resistance (“Standard Configuration”)

bias damping force while the electromagnetic component provides a tunable damping force with the added benefit of regenerative capacity. The hybrid damping force was formulated using a lumped equivalent magnetic circuit to establish the relation between input displacement and output damping force. By targeting the scooters as the application, the electromagnetic component was optimized to produce the highest force density within the geometric restrictions. Finite element simulations were carried out for the optimized design to study damping variation in the electromagnetic damper. It was observed that the electromagnetic damping coefficient could be tuned to a desired value by adjusting the external resistance connected to the coils. This feature enables the driver to optimize characteristics of the hybrid damper for different road conditions. According to the results, the external resistance can be set to be equal to the internal resistance of the damper to maximize the regenerative capacity of the system. The proposed damper is designed for scooters and similar machines; however, the simulations for other scales are also included in this work to investigate the effect of scaling on the damping force. The obtained results show that scaling to smaller or larger sizes yields significant change in the range of damping force that can be achieved, which is of interest for other applications. Acknowledgments  The authors would like to acknowledge the financial support of “Auto 21” in this project.

References

Fig. 11  Damping force and regenerative power vs. resistance (“Star Configuration”)

and also had a higher capacity to recover energy, in comparison to the “star configuration”. Although the “standard configuration” outperformed the “star configuration”, there are many practical considerations that have not been taken into account. The results show that both configurations modify the damping force to regenerative capacity ratio, and choice of either configuration must also be weighted on the basis of implementation.

9 Conclusion In this work a hybrid damper consisting of a hydraulic and an electromagnetic damper was implemented into a single package. The hydraulic component produced a

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