Development and optimization of an energy ...

Journal of Sound and Vibration 357 (2015) 16–34

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Development and optimization of an energy-regenerative suspension system under stochastic road excitation Bo Huang, Chen-Yu Hsieh, Farid Golnaraghi n, Mehrdad Moallem School of Mechatronic Systems Engineering, Simon Fraser University, Surrey, BC, Canada

a r t i c l e in f o

abstract

Article history: Received 6 October 2014 Received in revised form 3 May 2015 Accepted 3 July 2015 Handling Editor: H. Ouyang Available online 1 August 2015

In this paper a vehicle suspension system with energy harvesting capability is developed, and an analytical methodology for the optimal design of the system is proposed. The optimization technique provides design guidelines for determining the stiffness and damping coefficients aimed at the optimal performance in terms of ride comfort and energy regeneration. The corresponding performance metrics are selected as root-meansquare (RMS) of sprung mass acceleration and expectation of generated power. The actual road roughness is considered as the stochastic excitation defined by ISO 8608:1995 standard road profiles and used in deriving the optimization method. An electronic circuit is proposed to provide variable damping in the real-time based on the optimization rule. A test-bed is utilized and the experiments under different driving conditions are conducted to verify the effectiveness of the proposed method. The test results suggest that the analytical approach is credible in determining the optimality of system performance. & 2015 Elsevier Ltd. All rights reserved.

1. Introduction In the vehicle suspension system, the damper acts as a vibration isolator to suppress the disturbance from the road excitation by dissipating the energy into heat. Recently, with energy saving and fuel efficiency getting more and more concern, regenerative suspension systems are taken into consideration. Rather than wasting the energy through viscous friction, they are able to convert the mechanical energy from systems into the electrical energy and charge the battery directly. Different strategies of regenerative suspension systems have been investigated by a number of researchers. An electromagnetic (EM) damper was designed that consists of a DC motor and a ball screw mechanism as an active suspension and generator [1]. Kowal et al. [2] developed an electro-hydraulic self-powered suspension system that accumulated energy in a pneumohydraulic accumulator. Ebrahimi et al. [3,4] proposed a hybrid linear EM damper that acted as a viscous damper for ride comfort improvement, with self-powered capability. A linear motor based active suspension topology with energy regeneration capabilities was introduced by David and Bobrovsky [5]. Li et al. [6] proposed a rack-pinion mechanism connected to the DC motor for an EM shock absorber. Sabzehgar et al. [7] designed an algebraic screw mechanism paired a permanent-magnet synchronous generator that offered a high efficiency. Different from the conventional passive suspension system, the regenerative suspension systems are expected to capture energy within a wide range of excitation frequencies when energy saving is concerned. On the other hand, they also act as

n Correspondence to: Room 4388 Galleria, School of Mechatronic Systems Engineering, Simon Fraser University, 250-13450 102nd Avenue, Surrey, BC, Canada V3T 0A3. Tel.: þ 1 778 782 8054. E-mail address: [email protected] (F. Golnaraghi).

http://dx.doi.org/10.1016/j.jsv.2015.07.004 0022-460X/& 2015 Elsevier Ltd. All rights reserved.

B. Huang et al. / Journal of Sound and Vibration 357 (2015) 16–34

Nomenclature

Δiin

m cf fn ω x; x_ ; x€

k ζ ωn r y; y_ ; y€

z; z_ ; z€ a Svv P V emf Rload I kt Jm τb l d ηb n0 w Gq ðn0 Þ f nfc Zs iref Vc V DC ierr ΔI

sprung mass friction coefficient natural frequency excitation frequency displacement, velocity, acceleration of the sprung mass stroke displacement, velocity, acceleration of the shock absorber absolute acceleration of the sprung mass PSD of relative velocity between sprung mass and excitation instantaneous power generated by the damper back EMF voltage load resistance current through the load resistor torque constant inertia of the motor output torque of the shock absorber lead ratio of the ball screw coefficient of the ball screw efficiency of the ball screw reference spatial frequency ( ¼ 0:1 m  1 ) road index (¼2) road roughness coefficient temporal excitation frequency full cut-off spatial frequency internal impedance of DC motor reference current of SMR controlled voltage source average DC voltage error current between reference current and terminal current small error-band

_ θ€ θ; θ; Saa ωc EP ke Rint V load τi Jb Fb Jg kg ηg n V Gq ðnÞ W f fc La iin Vx kv f sw Rin

17

charging rate of terminal current of SMR circuit spring constant damping ratio radian natural frequency damping ratio (¼ω=ωn ) displacement, velocity, acceleration of base excitation angular displacement, velocity, acceleration of the motor gear shaft PSD of absolute acceleration of the sprung mass radian cut-off frequency in the ground–tire interface expectation of generated power by the damper back EMF coefficient internal resistance of DC motor voltage of the load resistor output torque of the motor inertia of the ball screw output force of the shock absorber inertia of the gearbox transmission ratio of the gearbox efficiency of the gearbox spatial frequency ( ¼ 0:01 m  1 10 m  1 ) vehicle speed road displacement PSD length of the tire footprint full cut-off temporal frequency physical inductor in SMR terminal current of SMR terminal voltage of DC motor modulation index switch frequency of PWM modulation synthesized variable resistor

vibration isolators to suppress the road disturbance induced vibrations for improving passengers' comfort. It is necessary to note that the damper will behave contradictorily under these two scenarios. As a result, in the design of the system parameters, it is impossible to achieve both objectives, simultaneously. In practical applications, most optimization problems are multi-objective in their nature. The best solution is usually subjective, taking many issues into consideration, and is subject to compromise. Therefore, developing a systematic optimization method becomes the key challenge in determining the proper system parameters for the energy harvesting – ride control scenarios. To date, there are various approaches to the optimization problems of the suspension system. Jazar et al. [8] proposed an optimization method for a one-degree-of-freedom (1-dof) quarter-car model by minimizing the cost functions which are the root-mean-square (RMS) of absolute acceleration and relative displacement. Arzanpour et al. [9] extended the RMS optimization method from the frequency domain to the time domain. However, only a few researchers consider the actual road excitation in the form of stochastic profile. Tamboli and Joshi [10] suggested an optimization method for minimizing the RMS of the acceleration response based on an exponential-type power spectral density (PSD) curve that approximated to the highway type of road condition. Gobbi and Mastinu [11] proposed an optimization method based on multi-objective programming (MOP) and monotonicity analysis, subjected to the excitation from actual road irregularities. Verros et al. [12] introduced a methodology for optimizing the suspension parameters based on the random road excitation simulated by a series of sinusoidal inputs. Jayachandran and Krishnapillai [13] analyzed a quarter-car model and used genetic algorithm to derive the optimized parameters of passive and semi-active suspension system. All previous studies have been theoretically proven and numerically verified. However, they show a lack of experimental verification, and may not be realizable in practice. This paper aims at introducing an energy-regenerative suspension system consisting of a translation-to-rotation mechanism and a low power switch-mode rectifier (SMR). A graphical optimization design method is presented by

18

B. Huang et al. / Journal of Sound and Vibration 357 (2015) 16–34

Sprung Mass m

x

Shock absorber

Ball screw

Fb

d

Load Resistor (Battery / Power electronics)

Physical spring

R load

k DC Motor/ Generator

y

Base Fig. 1. 1-dof model of the regenerative suspension system.

selecting the system parameters such as spring rate and damping coefficient. Adapted for the standard road profile defined by ISO 8608:1995 [14], the strategy provides analytical solutions for optimizing a vehicle's ride control or power regeneration performance. The SMR circuit is capable of adjusting terminal resistance which is used to provide a variable damping in real-time to change the suspension system's performance according to the optimization rule. Furthermore, a physical test-bed is developed capable of applying standard road profiles to the system. Experimental results obtained are aimed at verifying the optimization method and realization of variable damping. The contribution of this work is threefold: (1) An integrated regenerative suspension system is developed which is capable of harvesting energy while adjusting damping force, thus mitigating the tradeoff between conventional passive suspension with inferior control performance and active suspension with high power consumption. (2) This work presents the technical contribution in terms of practical realization of the proposed technique. The optimization results can be useful guidelines for the development of advanced regenerative suspension systems and easily applied in real-world industrial problem. (3) The proposed SMR circuit is able to accurately synthesize desired resistors by regulating converter current according to line voltage that is unknown apriori. The SMR is placed under ISO road profile for variable damping supply, showing its practicality of active damping tuning according to optimization outcomes.

2. System description and modeling The regenerative suspension system can be simplified as a 1-dof quarter-car model, which is shown in Fig. 1. The model comprises sprung mass, physical springs, and shock absorber consisting of DC motor/generator, planetary gearbox, and a ball screw mechanism to convert the vertical motion between sprung mass and base excitation to the rotary motion of the screw shaft. A load resistor is connected to the motor terminal that acts as a battery. The motor is able to provide different values of damping force by adjusting the value of load resistor. The output torque of the motor is expressed in terms of the current which goes through the load resistor as τ i ¼ kt I ¼

kt V emf Rint þ Rload

(1)

_ The where V emf is the back EMF voltage generated due to the rotation of the motor and can be defined as V emf ¼ ke kg ηg θ. dynamic equation of the shock absorber consisting of the DC motor with a gearbox and a ball screw mechanism is described as
 τb  J g þ J b θ€  τi ¼ J m kg ηg θ€ (2) kg ηg The output torque of the shock absorber is given by 2

τb ¼

ke kt kg η2g 2 θ_ þ ðJ m kg η2g þ J g þ J b Þθ€ Rint þRload

(3)

According to the properties of the ball screw mechanism, the relationship between torque and force, angular and stroke motion are given by   (4) τb ¼ dηb F b θ ¼ z= dηb

B. Huang et al. / Journal of Sound and Vibration 357 (2015) 16–34

19

Table 1 Equivalent parameters of the suspension model. Symbol

Description

meq

mþ

m0

2

J m kg η2g þ J g þ J a d2 η2b

2

J m kg η2g þ J g þ J a d2 η2b

c

2

ke kt kg η2g ðRint þ Rload Þd2 η2b

where d ¼ l=2π. Therefore, the equivalent output force of the shock absorber is obtained as 2

Fb ¼

2

ke kt kg η2g 2 ðRint þ Rload Þd η2b

z_ þ

ðJ m kg η2g þJ g þJ b Þ 2

d η2b

z€

Consequently, the dynamic equation of the suspension system can be derived as ! ! 2 2 J m kg η2g þ J g þ J a ke kt kg η2g mþ x€ þ cf þ x_ þ kx 2 2 d η2b ðRint þ Rload Þd η2b ! 2 2 J m kg η2g þ J g þ J a ke kt kg η2g € þ cf þ y y_ þ ky ¼ 2 2 d η2b ðRint þ Rload Þd η2b

(5)

(6)

or 2

mþ

J m kg η2g þ J g þ J a

Eqs. (6) and (7) can be simplified as

and

2

d η2b

! z€ þ cf þ

!

2

ke kt kg η2g 2

ðRint þRload Þd η2b

z_ þ kz ¼ my€

(7)

    meq x€ þ cf þ c x_ þ kx ¼ m0 y€ þ cf þc y_ þky

(8)

  meq z€ þ cf þ c z_ þkz ¼ my€

(9)

The equivalent mass and damping coefficient are listed in Table 1. 3. Optimization method based on stochastic road excitation In this section, an image-based design optimization of the proposed quarter-car suspension model based on standard road excitation is examined. The optimization method is designed by deriving the analytical formulas of the performance indexes (i.e. ride comfort and power regeneration) over the whole frequency range. The analytical formulas represent a theoretical way for selecting optimum parameters such as stiffness and damping coefficients, and also create a design chart to visualize the optimal values of the suspension parameters when a number of constraints and conflicting requirements on system's performance have to be satisfied. 3.1. Standard road profile The road profiles are characterized in terms of displacement power spectral density (PSD) which are considered as typical stochastic inputs. According to ISO 8608 (1995), the displacement PSD in the spatial frequency can be described as  w n Gd ðnÞ ¼ Gd ðn0 Þ ¼ G0 n  w (10) n0 where G0 ¼ Gd ðn0 Þ=n0 w . Assuming a vehicle is driving at speed V, the temporal excitation frequency f and the spatial excitation frequency n have the relation as f ¼ n UV, and similarly we have ω ¼ 2πn UV. Therefore, the PSD of road excitation in spatial frequency domain can be transferred to that in temporal frequency domain by knowing Gd ðnÞdn ¼ Gd ðωÞdω and can be described as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2πG0 V 2πG0 V ¼ (11) G d ð ωÞ ¼ : jω ω2

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B. Huang et al. / Journal of Sound and Vibration 357 (2015) 16–34

3.2. Ground–tire interface The ground/tire interface motion into the vehicle is used as the measure of the terrain elevation [15], and the measurement is limited at the short wavelengths by the length of the tire footprint in contact with the terrain. Specifically, variations in the terrain elevation with a wavelength equal to or less than the length of the tire footprint cannot be measured, since the elevation seen by the tire at that wavelength is at the maximum attenuation. Hence, the tire acts as a low-pass filter of the terrain elevation, with a full cut-off spatial frequency of nfc ¼ W  1 m  1 , where W is the length of the tire footprint. Suppose the vehicle speed V, the full cut-off temporal frequency should be around f fc ¼ nfc V ¼ W  1 V Hz. Using a first-order low-pass filter to simulate the ground–tire interface, the cut-off frequency of the filter can be found at f c ¼ 0:4f fc , where  3 dB attenuation happens. Therefore, the low-pass filter is derived to approximate the effects of the tire by ωc H T ðjωÞ ¼ (12) jωþ ωc where ωc ¼ 0:8πVW  1 . 3.3. The ride comfort cost function The ride comfort cost function is defined as RMS of absolute acceleration of the sprung mass. The transfer function between the excitation and absolute acceleration is given by Eq. (8)   m0 ðjωÞ4 þ cf þ c ðjωÞ3 þkðjωÞ2 H a ðjωÞ ¼ (13)   meq ðjωÞ2 þ cf þc jω þ k According to the ISO 2631-1:1997 [16], a method of evaluation of the effect exposure to vibration on humans is designed by weighting the RMS acceleration with human vibration-sensitivity curves, which can be approximated by a second-order quasi-least-square filter whose transfer function is given by [17] H asen ðjωÞ ¼

86:51jω þ546:1 ðjωÞ2 þ 82:17jωþ 1892

Therefore, the PSD of absolute acceleration can be obtained as 2 2 2 Saa ¼ H a ðjωÞ U H asen ðjωÞ U H T ðjωÞ UGd ðωÞ And the expectation of absolute acceleration square is given by Z 1

 1 Saa dω E a2 ¼ 2π  1

(14)

(15)

(16)

The PSD of absolute acceleration, relative displacement and relative velocity all can be expressed as the fraction with polynomial of jω in both numerator and denominator. According to [18], the analytical solutions exist for E½a2  if Saa can be written as 2 (17) Saa ¼ H ðjωÞ where H ðjωÞ ¼

( ) B0 þ ðjωÞB1 þ ⋯ þ ðjωÞn  1 Bn  1 A0 þ ðjωÞA1 þ ⋯ þ ðjωÞn An

(18)

Similar approach has been utilized by a number of researchers in studies of optimization and energy harvesting from random vibrations [11,19–21]. The PSD of absolute acceleration based on Eq. (15) can be obtained as 2 ðjωÞB1 þ ðjωÞ2 B2 þ ðjωÞ3 B3 þ ðjωÞ4 B4 (19) Saa ¼ U2πG0 Vω2c A0 þ ðjωÞA1 þ ðjωÞ2 A2 þ ðjωÞ3 A3 þ ðjωÞ4 A4 þ ðjωÞ5 A5 where A0 ¼ 1892ωc k   A1 ¼ 1892ωc cf þ c þ ð82:17ωc þ 1892Þk   A2 ¼ 1892ωc meq þ ð82:17ωc þ 1892Þ cf þ c þ ðωc þ82:17Þk   A3 ¼ ð82:17ωc þ1892Þmeq þ ðωc þ 82:17Þ cf þ c þ k   A4 ¼ ðωc þ82:17Þmeq þ cf þc A5 ¼ meq B1 ¼ 546:1k   B2 ¼ 546:1 cf þ c þ86:51k   B3 ¼ 546:1m0 þ86:51 cf þc B4 ¼ 86:51m0

B. Huang et al. / Journal of Sound and Vibration 357 (2015) 16–34

21

 And the analytical solutions of E a2 can be derived and expressed as n



 E a2 ¼ πG0 Vω2c A0 B24 A0 A23 þ A21 A4  A0 A1 A5  A1 A2 A3 þ A0 A5 2B2 B4  B23 ðA1 A2  A0 A3 Þ

 o þ A0 A5  2B1 B3 þ B22 ðA0 A5 A1 A4 Þ þ A0 A5  B21 ðA3 A4 A2 A5 Þ
o n = A0 A5 A20 A25  2A0 A1 A4 A5  A0 A2 A3 A5 þA1 A22 A5 þA21 A24 þ A0 A23 A4  A1 A2 A3 A4

(20)

Therefore, the ride comfort cost function can be obtained as qffiffiffiffiffiffiffiffiffiffiffi

ffi RMSa ¼ E a2 :

(21)

3.4. The power generation cost function The power generated from the vibration has three parts: mechanical dissipation; electrical loss and electrical load. From the energy harvesting point of view, the amount delivered to the electrical load needs to be maximized. Therefore, the expectation of the generated power is chosen as the power generation cost function. The total current passing through the electrical load can be obtained as I¼

ke kg ηg z_ ðRint þ Rload Þdηb

The instantaneous electrical power generated by the damper can be expressed as ! 2 2 2 ke kg η2g Rload 2 c2 d η2b Rint _ ¼ c P ¼ I 2 Rload ¼ z U z_ 2 2 2 ðRint þ Rload Þ2 d η2b ke kt kg η2g Thus the expectation of the generated power can be obtained by ! 2 h i c2 d η2b Rint EP ¼ c  UE z_ 2 2 2 ke kt kg ηg

(22)

(23)

(24)

To obtain the analytical solutions of EP , the expectation of relative velocity square E½z_ 2  should be given firstly. The transfer function between the excitation and relative velocity can be derived based on Eq. (9) Hv ðjωÞ ¼

mðjωÞ3   meq ðjωÞ2 þ cf þ c jωþ k

Therefore, the PSD of relative velocity can be obtained as 2 2 Svv ¼ H v ðjωÞ U H T ðjωÞ UGd ðωÞ And the expectation of relative velocity square is given by Z 1 h i 1 E z_ 2 ¼ Svv dω 2π  1 Similarly, the PSD of relative velocity is obtained as 2 ðjωÞ2 D2 Svv ¼ U2πG0 Vω2c C 0 þ ðjωÞC 1 þ ðjωÞ2 C 2 þ ðjωÞ3 C 3

(25)

(26)

(27)

(28)

And the expectation of relative velocity square is given by

n o h i πG0 Vω2c  C 0 C 1 D22 2 E z_ ¼ C 0 C 3 ðC 0 C 3  C 1 C 2 Þ

where C 0 ¼ ωc k   C 1 ¼ ωc cf þ c þ k   C 2 ¼ ωc meq þ cf þ c C 3 ¼ meq D2 ¼ m

(29)

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B. Huang et al. / Journal of Sound and Vibration 357 (2015) 16–34

Therefore, the power generation cost function is

EP ¼



πG0 Vω2c C 0 C 1 D22

c2 d2 η2b Rint ke kt kg 2 η2g

 c

C 0 C 3 ðC 0 C 3  C 1 C 2 Þ

:

(30)

3.5. Optimization procedure 3.5.1. Optimization rule and design chart The following procedure is used to optimize the suspension stiffness and damping coefficients of the 1-dof quarter-car model for ride comfort and power generation. The objectives of this optimization are to minimize the RMS of absolute acceleration experienced by the sprung mass and maximize the expectation of the power generated by the damper. The constraints of the optimization rule include the physical limitation of the spring (k1 r k r k8 ) and the limited range of the adjustable damping that can be applied by the DC motor (c1 rc r c8 ). The values of the objective functions are calculated by Eqs. (21) and (30). The expression of the optimization rule is defined as min w:r:t: ðk; cÞ RMSa ðk; cÞ max w:r:t: ðk; cÞ Ep ðk; cÞ s:t:

klow rk r khigh clow rc r chigh

(31)

An optimization design chart is created by defining RMSa and EP as the Cartesian Coordinates, using k and c as parameters. As illustrated in Fig. 2, each solid curve indicates the values of (RMSa ; EP ) in terms of a fixed k and different c; each dashed curve indicates the values of (RMSa ; EP ) in terms of a fixed c and different k. By overlapping the solid and dashed curves, a grid figure is plotted where each intersection on the grid corresponds to a certain pair of (k; c), which exclusively determines a value of (RMSa ; EP ). Four different conditions are analyzed to deal with the practical situation in real life, which are shown in Table 2. By inspection of Fig. 2 it may be noted that (1) The global optimum points for ride comfort and power generation performance can be found where a red circle with the lowest value in x-axis and a green square with the highest value in y-axis. (2) For each stiffness denoted by a solid blue curve, the local optimum damping points can be found where a minima of absolute acceleration and a maxima of generated power exist. (3) For ride comfort optimization, low stiffness and low damping provides the best performance; with increase of stiffness, local optimal damping moves to a higher value. (4) For power generation optimization, high stiffness and high damping provides the best performance; local optimal damping does not depend significantly on the stiffness. (5) With low stiffness, the optimal ride comfort and power generation are in conflict; with high stiffness, both optimal damping points get closer and the optimal ride comfort and power generation become consistent, thus can be optimized simultaneously. (6) Under Highway conditions, the variation range of absolute acceleration is small. Therefore, optimal stiffness and damping coefficients can be selected to improve the power generation effectively while not to sacrifice the ride comfort significantly. (7) When the stiffness is pre-set, the damping coefficient can be changed between two local optimal points on the same blue curve in the real time by an electronic circuit proposed in this paper.

3.5.2. Local optimization for energy harvesting When the spring constant ki is selected and its value is fixed, the local optimization for energy harvesting is determined by the damping coefficient chosen to satisfy the following condition: ∂EP ðki ; cÞ ¼0 ∂c

∂2 EP ðki ; cÞ o0 ∂c2

(32)

The complete closed-form expression of the optimum damping coefficient can be obtained by solving Eq. (32). It should be noted that the damping coefficient must be a positive real number. Therefore, only one analytical solution is found that

B. Huang et al. / Journal of Sound and Vibration 357 (2015) 16–34

12

10 9

Expectation of Generated Power (w)

Expectation of Generated Power (w)

c6

Optimum points for ride comfort

25

c7

Optimum points for power generation

11

c8

c5

8 7

c4

6 5

k8

k3

4 3

c3

2

k2 k 4

1

k1 6.8

k6

k5

k7

c2

Optimum points for power generation Optimum points for ride comfort c 6

20

7.2

7.4

7.6

7.8

8

8.2

15

c3

k2

5

10

10.5

c7

k8

c8 c4

10

5

k3 k7

c3

k1 k4

k5

k6 c2 c1

10.5

11

11.5

12

12.5

13

11

c1

11.5

12

12.5

14

c7

25

c4

k3

c8 k7

c3

20 15 10

k8

c5

30

11

14.5

c6

Optimum points for ride comfort

35

5 13.5

c2

2 RMS of Absolute Acceleration (m/s )

Expectation of Generated Power (w)

Expectation of Generated Power (w)

c5

k7

k6

k5

k1 k4

40

20

k2

k8

Optimum points for power generation

c6

25

15

c8

k3

10

Optimum points for power generation Optimum points for ride comfort

7

c4

RMS of Absolute Acceleration (m/s2)

30

c

c5

c1 7

23

k4

k2

k6

k5

c2 c1

k1 12

13

14

15

16

17

18

19

2

2

RMS of Absolute Acceleration (m/s )

RMS of Absolute Acceleration (m/s )

Fig. 2. Contour curves for functions RMSa and Ep in terms of ðk; cÞ and optimum points for ride comfort and power generation, based on different driving situations: (a) Highway I; (b) Highway II; (c) City; and (d) Off-road. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)

Table 2 Different driving situations. Driving situation

Road condition

Driving speed (km/h)

Highway I Highway II City Off-road

Grade-A Grade-B Grade-C Grade-D

120 90 50 30

maximizes the electrical power generation. The expression is copt ¼ 

β 1 ρ6 þ 4α 2

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ρ  4ρ26 2ρ3 þ 4 ρ6

(33)

24

B. Huang et al. / Journal of Sound and Vibration 357 (2015) 16–34

Expectation of Generated Power (w)

35

k8

30

k6

25

k4

20

k2

15

10

5

Graphic solution of optimum points Analytical solutions of optimum points

0

0

500

1000

1500

2000

2500

3000

3500

Damping coefficient (Ns/m) Fig. 3. Local optimization for energy harvesting under “City” driving situation when spring constant is selected as k2 ; k4 ; k6 ; k8 .

where 2

τ¼

c2 d η2b Rint 2

ke kt kg η2g

α ¼ ω2c τ

  β ¼ 2ωc τ ω2c meq þ2ωc cf þ k


    γ ¼ ω2c meq þ ωc cf þ k 4ωc cf τ  ωc þkτ þ ωc ωc c2f τ þ cf kτ þ k    δ ¼  2cf ω2c meq þ ωc cf þk ωc ωc cf τ kτ    φ ¼ cf ω2c meq þωc cf þ k ωc cf þ k ρ1 ¼ γ 2  3βδ þ 12αφ ρ2 ¼ 2γ 3  9βγδþ 27β2 φ þ27αδ2 72αγφ 8αγ  3β2 8α2

ρ3 ¼

β3  4αβγ þ 8α2 δ 8α3 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u 3 ρ þ ρ22  4ρ31 t 2

ρ4 ¼

ρ5 ¼

2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 1 ρ  ρ3 þ ðρ5 þ 1 Þ ρ6 ¼ 2 3 3α ρ5

In Fig. 3, local optimization curves for power generation under “City” driving situation are plotted. It can be seen that the analytical solutions match the maximum points on the curves. By decreasing the interval of the parameters selection, the optimum points from the design chart approach to the analytical solutions.

4. Principle and realization of variable damping in electrical domain For the present, the conventional passive suspension has fixed stiffness and damping characteristics that can only maintain a certain performance. The local optimization provides a guideline of tuning damping coefficient to satisfy different performance requirements when certain stiffness is selected. To realize the tunable damping in real time, the regenerative suspension system is integrated with a low-power SMR capable of synthesizing variable terminal resistors through a three-level double-band hysteresis current controller (DB-HCC). Through SMR the damping coefficient of the regenerative suspension can be changed to vary its dynamic response as affecting the ride comfort and power generation performance.

B. Huang et al. / Journal of Sound and Vibration 357 (2015) 16–34

25

iDC Z s i La in Vc Vemf

Q1'

Q3' VDC

Q2'

Q4'

Fig. 4. Switch-mode rectifier topology.

Zs

iref Rin

Vx

Zs

iin

Vx

La Vc

Fig. 5. Modeling of variable resistance synthesis.

4.1. Switch-mode rectifier Fig. 4 indicates the bi-directional power converter for synthesizing variable resistance. The power stage includes a physical power inductor and a single phase Voltage Source Inverter (VSI) for providing the desired V c through the corresponding PWM pulses driving the power MOSFETs. 4.2. Modeling of variable resistance synthesis Considering model of resistive load operation, shown in Fig. 5(a), which indicates the synthesis of a variable resistor Rin through the amplitude and phase relationships between terminal voltage V x , and current iref . Referring to Fig. 5(b), the variable resistor synthesizer consists of a physical impedance Z s and inductor La , as well as a controlled voltage source V c , which is generated by a single phase 3-level VSI. By equating the reference current iref and terminal current iin in Laplace domain, the expression of controller voltage source V c in Laplace domain can be obtained as V c ðsÞ ¼

Rin sLa V x ðsÞ Z s þRin

(34)

As shown in Eq. (34), the amplitude attenuations and phase shifts are determined by values of passive elements. Thus, controlling V c ðsÞ to follow the relationship with V x ðsÞ in (34) ensures the correct synthesis of Rin . 4.3. Controlling of SMR with DB-HCC In order to control V c for correct Rin synthesis the terminal current iin should be regulated to follow iref . In this work, a DB-HCC has been implemented to control iin . The modulating scheme is implemented for its wide bandwidth/fast response, unconditional stability [22–25]. The controller consists of hysteresis comparators with two different sizes of error-bands as shown in Fig. 6. With the desired gating signals for Q1–Q4, additional digital logic circuits were introduced for MOSFETs switching frequency (i.e. switching power loss) reduction and equalization, synonymous to carrier-based PWM modulation [26,27]. As a result, the average switching frequency approximates to f sw  V DC kv ð0:6 0:5kv ÞðLa ΔIÞ  1

(35)

where kv ¼ V emf =V DC is the modulation index, ΔI is the small error-band, and V DC is the inverter input voltage. According to controller state machine shown in Fig. 7, the current ripple (error) is primarily determined by small errorband hysteresis comparator. The large error-band hysteresis comparator is implemented to reverse the polarity of average DC voltage V DC , when the smaller error-band comparator is unable to regulate the current error ierr ¼ iin iref , defined as the difference between measured current and reference current. Under this condition, the larger error-band comparator will be activated to accomplish polarity reversal. Assuming the converter initially operates in positive line cycle, signðV in Þ 40, the smaller error-band comparator acts as a single-band hysteresis comparator, which switches on either Q1 or Q2 according to the error current ierr . When |ierr | oΔI, Q2/4 are switched on, connecting V in to La , which charges it at the rate of Δiin ¼ V x /La .

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B. Huang et al. / Journal of Sound and Vibration 357 (2015) 16–34

Q4=Q3 AND

J Q

larger band

XOR

K

iref Q1= Q2 it small band

Q

AND

OR

Q1' Q2'

AND

OR

Q3' Q4'

double-band HCC

Q

Fig. 6. Double-band 3-level HCC.

|iin-iref|< ∆I Mode 1 sign(Vin)> 0

Q2/4

start |iin-iref|< ∆I sign(Vin)< 0

Mode 3 Q1/3

|iin-iref|> ∆I Mode 2 |iin-iref|> ∆I

Q1/4

|iin-iref|< ∆I

|iin-iref|< ∆I |iin-iref|> ∆I Mode 4 |iin-iref|> ∆I

Small band lost regulation. Large band transition

Q2/3

Fig. 7. State diagram of DB-HCC.

Springs Ball screw Sprung mass

Accelerometer

Encoder Planetary gearhead DCmotor/ generator String potentiometer

Hydraulic shaker

Fig. 8. Experimental test setup for the regenerative suspension system.

When |ierr | 4ΔI, Q1/4 are switched on thus iin into battery at the rate of Δiin ¼ ðV x  V DC Þ=La . Assuming SMR initially operates in negative line cycle, signðV in Þ o0, Q3 will switch on with either Q1 or Q2 depending on the output of the smaller errorband comparator. The large error-band comparator switches on Q3 or Q4 for V DC polarity reversal, which takes place when the small error-band lost regulation.

5. Experimental tests with the regenerative suspension prototype The experimental system was developed to verify the proposed optimization method, as depicted in Fig. 8. The system includes a 1-dof quarter-car regenerative suspension prototype consisting of a mass plate on bearings and parallel springs.

B. Huang et al. / Journal of Sound and Vibration 357 (2015) 16–34

27

Table 3 Parameter values of the experimental setup. Parameter

Value

Sprung mass (m) Total spring constant (soft) (klow ) Total spring constant (hard) (khigh ) Rotor inertia of the DC motor (J m ) Torque constant of the DC motor (kt ) Back EMF coefficient of the DC motor (ke ) Internal resistance of the DC motor (Rint ) Rotor inertia of the gearhead (J g )

24 kg 10,795 N/m 73,320 N/m 120 g cm2 170 mN m/A 170 mV/rad/s 10.2 Ω 17.6 g cm2 12:1 83% 1248 g cm2 60 mm/r 95% 300 mm

Gear ratio of the gearhead (kg ) Efficiency of the gearhead (ηg ) Rotor inertia of the ball screw (J b ) Lear ratio of the ball screw (l) Efficiency of the ball screw (ηb ) Travel length of the ball screw

iref

1/Rload

Switching frequency reduction/ equalization digital logic

DB- HCC state flow

it

Vemf

Boot- strap gate drive

encoder Q1',2',3',4'

DC motor planetary gear

Bi- directional SMR

Mechanical

iDC VDC

Suspension Excitation input Fig. 9. Experimental configuration of the SMR prototype coupled to the regenerative suspension test system.

Bootstrap Gate drive

Bootstrap Gate drive

≈ 5 cm

Q1'-4'

V DC V emf

La ≈ 10 cm Fig. 10. SMR double-sided PCB prototype.

A Maxon DC motor (RE40-218011) was utilized to replace a traditional damper, attached with a planetary gearhead (GP52C223083). A THK precision ground ball screw (KX-10) was connected with the motor shaft to convert the vertical movement to the rotary motion. An MTS hydraulic shaker (Series 248) controlled by PC is used to provide the base excitation with pressure feedback. A dSPACE DS1103 system operates as the I/O interface to acquire data such as the voltage signal from the DC generator, the displacement signal measured by a string potentiometer, and the acceleration signal from an accelerometer. The parameters of the experimental setup are shown in Table 3. The experimental configuration of the SMR prototype coupled to the regenerative suspension test system is depicted in Fig. 9. And the SMR double-sided PCB prototype is shown in Fig. 10.

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B. Huang et al. / Journal of Sound and Vibration 357 (2015) 16–34

12

Amplitude of relative displacement (mm)

Damping coefficient (Ns/m)

2500

2000

1500

1000

500

0

0

50

100

150

200

250

300

350

400

450

R

=250 R

6

=120 R =100 =80

R R

4

=40 R

R

=30 R =20 R =10

2

3

4

5

6

7

8

9

7

8

9

Frequency (Hz)

25

R

20

Amplitude of generated power (W)

25

=10

15

10

R 5

R

=490

R

=490

=10

20

15

10

R

5

=490 R

0

=50

=60

2

Load resistor (Ohm)

Amplitude of absolute acceleration (m/s2)

=490

8

0

500

R

10

2

3

4

5

6

7

8

0

9

2

3

4

Frequency (Hz)

=10

5

6

Frequency (Hz)

Fig. 11. (a) Damping coefficient w.r.t. load resistor; frequency response of (b) relative displacement; (c) absolute acceleration; and (d) generated power.

100

Displacement of road profile (mm)

Displacement of road profile (mm)

15 10 5 0 -5 -10 -15 -20

80

60

40

20

0

-25 2

4

6

8

10

12

14

16

18

20

2

4

6

8

Time (sec) Fig. 12. Standard road profile for (a) Highway I and (b) City situations.

10

12

Time (sec)

14

16

18

20

B. Huang et al. / Journal of Sound and Vibration 357 (2015) 16–34

0

-2

10

10

Generated PSD Theoretical PSD

10

10

-6

-4

PSD of road profile

PSD of road profile

Generated PSD Theoretical PSD

-2

-4

10

-8

10

-10

10

-12

10

-6

10

-8

10

-10

10

10

-14

10

29

-12

-2

10

-1

0

10

1

10

2

10

10

3

10

10

-2

-1

10

0

10

10

Frequency (Hz)

1

2

10

10

3

10

Frequency (Hz)

0.6 0.4 0.2 0 -0.2 -0.4 -0.6

(b)

(c)

Rin= 10 Ω 0

(e)

(d) Rin= 30 Ω

5

10

Rin= 50 Ω

15

20

25

20 15 10 5 0 -5 -10 Rin= 100 Ω -15 -20 30

Vemf (V)

iin (A)

Fig. 13. PSD of road profile for (a) Highway I and (b) City situations.

2.4

2.6

2.8

3

iin (A)

0.6 0.4 0.2 0 -0.2 -0.4 -0.6 11

15.5

16

Time (sec)

iin (A)

15 10 5 0 -5 -10 -15 Rin= 50 Ω -20 16.5 17

0.6 0.4 0.2 0 -0.2 -0.4 -0.6

20 15 10 5 0 -5 -10 -15 Rin= 100 Ω -20 25.5 26 26.5 27 27.5 28 28.5

Vemf (V)

0.6 0.4 0.2 0 -0.2 -0.4 -0.6

11.5 12

Time (sec)

Vemf (V)

iin (A)

Time (sec)

20 15 10 5 0 -5 -10 -15 Rin= 30 Ω -20 12.5 13 13.5

Vemf (V)

20 15 10 5 0 -5 -10 -15 Rin= 10 Ω -20 3.2 3.4 20

0.6 0.4 0.2 0 -0.2 -0.4 -0.6

Vemf (V)

iin (A)

Time (sec)

Time (sec)

Fig. 14. (a) Variable resistance sweep from 10 to 100 Ω; (b) detailed corresponding line current iin and voltage V emf for synthesizing Rin ¼ 10 Ω, (c) Rin ¼ 30 Ω, (d) Rin ¼50 Ω; and (e) Rin ¼100 Ω under the stochastic excitation.

5.1. Frequency responses The performance of the experimental system was examined by producing the frequency response over frequencies of 2– 9 Hz, with base excitation amplitude of 10 mm. The variable damping coefficient was realized by adjusting the values of the load resistor according to Table 1. By changing the load resistance from 490 Ω to 10 Ω, the equivalent damping coefficient varies from 91.3 N s/m to 2281.9 N s/m. The relationship between load resistor and damping coefficient is shown in Fig. 11 (a). The frequency response of relative displacement between sprung mass and base excitation, sprung mass's absolute

30

B. Huang et al. / Journal of Sound and Vibration 357 (2015) 16–34

8.2

8.2

RMS of Absolute Acceleration (m/s2)

RMS of Absolute Acceleration (m/s2)

8.4

8 7.8 7.6

Experimental results 7.4 7.2

Analytical results

7 6.8 0

8

Experimental results

7.8

7.6 Optimum point for ride comfort

7.4

Analytical results 7.2

Optimum point for ride comfort 500

1000

1500

2000

2500

3000

7

0

3500

500

Damping coefficient (Ns/m)

2000

2500

3000

3500

14

13

2

RMS of Absolute Acceleration (m/s )

2

1500

Damping coefficient (Ns/m)

13.5

RMS of Absolute Acceleration (m/s )

1000

12.5

Experimental results 12 11.5 11

Analytical results

10.5

Experimental results

13.5

13

Optimum point for ride comfort 12.5

Analytical results

12

Optimum point for ride comfort 10

0

500

1000

1500

2000

2500

Damping coefficient (Ns/m)

3000

3500

11.5

0

500

1000

1500

2000

2500

3000

3500

Damping coefficient (Ns/m)

Fig. 15. Comparison of RMS of absolute acceleration between analytical solution and experimental test for (a) Highway I situation and soft spring setup, (b) Highway I situation and hard spring setup, (c) City situation and soft spring setup and (d) City situation and hard spring setup. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)

acceleration, and generated power are indicated in Fig. 11(b)–(d). From the results, it can be seen that the regenerative damper is able to provide any damping force as the traditional damper, by selecting the proper load resistance. 5.2. Road profile generation The road profile can be represented by a unit-intensity white noise signal passing through a first-order transfer function given by pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2πG0 V GðsÞ ¼ (36) s The numerical calculation for the road profile was carried out by Matlab/Simulink. Fig. 12 shows the generated road roughness in time domain for Highway I and City situations (see Table 2). Fast Fourier Transform (FFT) is used to get the PSD from the time domain to the frequency domain. Fig. 13 indicates that the transformed PSD curve fits the theoretical value well. 5.3. Variable resistance synthesis In order to examine how the proposed SMR can realize the variable resistance under the stochastic excitation, the synthesized resistors Rin is chosen to sweep from 10 Ω to 100 Ω in 30 s using standard road profile (Grade-A) and driving speed (30 km/h). The results are shown in Fig. 14, where (b)–(e) indicates the detailed instantaneous waveforms of V emf and iin . From figures it can be seen that the voltage and current are in-phase (i.e. high power factor) while maintaining the

B. Huang et al. / Journal of Sound and Vibration 357 (2015) 16–34

12

31

12

10

8

Optimum point for power generation

6

Experimental results 4

2

0

0

500

1000

1500

2000

2500

3000

Expectation of Generated Power (w)

Expectation of Generated Power (w)

Analytical results 10

8

Optimum point for power generation 6

Experimental results 4

2

0

3500

Analytical results

0

500

2000

2500

3000

3500

3000

3500

Analytical results

Expectation of Generated Power (w)

Analytical results

Expectation of Generated Power (w)

1500

30

25

20

Optimum point for power generation 15

10

Experimental results

5

0

1000

Damping coefficient (Ns/m)

Damping coefficient (Ns/m)

0

500

1000

1500

2000

2500

3000

3500

Damping coefficient (Ns/m)

25

20

Optimum point for power generation 15

10

Experimental results

5

0

0

500

1000

1500

2000

2500

Damping coefficient (Ns/m)

Fig. 16. Comparison of expectation of generated power between analytical solution and experimental test for (a) Highway I situation and soft spring setup, (b) Highway I situation and hard spring setup, (c) City situation and soft spring setup, and (d) City situation and hard spring setup. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)

corresponding amplitude ratio. The results show that the SMR is able to provide real-time variations of desired load resistance in the electrical domain.

5.4. System performance of ride comfort and power generation In this experiment, the suspension was excited by the standard road profile as depicted in Fig. 12. The responses of the system such as absolute acceleration and instantaneous power were acquired, and RMS of absolute acceleration and expectation of generated power were calculated over the road range. Two different sets of springs were selected with soft stiffness and hard stiffness, respectively. The variable damping coefficient ranges from 70.2 N s/m to 3042.5 N s/m by adjusting the value of the load resistor from 640 Ω to 5 Ω. The absolute acceleration of the sprung mass (x€ ) and voltage of the load resistor (V load ) were measured, and the instantaneous electrical power was calculated based on P ¼ V 2load =Rload . To verify the proposed optimization method, the curves of RMS of absolute acceleration and expectation of generated power were built and compared with the optimization curves depicted in Fig. 2. For a better view of the comparison, the plot such as in Fig. 2 was divided into two equivalent plots which show ride comfort and power generation performance individually. Referring to Figs. 15 and 16, the optimum points derived from analytical solution (red solid circle and green solid square) and experimental test (red hollow circle and green hollow square) match well under different conditions, although some value differences between the curves exist due to the uncertainties in the experimental tests that cannot be estimated in analytical solution. Fig. 17 shows the absolute acceleration and generated power in time domain when damping is selected at ride comfort optimal point or power generation optimal point under Highway I situation; while Fig. 18 shows the results under City situation. Comparing the responses between different performance optimizations (ride comfort/

32

B. Huang et al. / Journal of Sound and Vibration 357 (2015) 16–34

10

30

8

Generated power (w)

2

Absolute acceleration (m/s )

9 20

10

0

-10

7 6 5 4 3 2

-20

1 -30

0

2

4

6

8

10

12

14

16

18

0

20

0

2

4

6

8

20

14

16

18

20

14

16

18

20

100

10

Generated power (w)

Absolute acceleration (m/s2)

12

120

30

0

-10

-20

80

60

40

20

-30

-40

10

Time (sec)

Time (sec)

0

2

4

6

8

10

12

Time (sec)

14

16

18

20

0

0

2

4

6

8

10

12

Time (sec)

Fig. 17. System responses under Highway I situation with optimal damping for best ride comfort performance: (a) absolute acceleration; (b) generated power; and for best power generation performance: (c) absolute acceleration; and (d) generated power.

power generation) in both Figs. 17 and 18, we can see that changing damping and stiffness values significantly influence suspension system's performances.

6. Conclusions In this paper, an energy regenerative suspension system was developed for capturing energy from road irregularities. A systematic optimization methodology was presented and closed-form solutions of the performance metrics were derived, which can be utilized to design an optimal regenerative suspension with specific stiffness and damping coefficients. The performance of the suspension system and optimization method was evaluated experimentally by a 1-dof quarter-car test bed under stochastic excitation which is defined by the standard road profile. The results demonstrated that the suspension system performance was determined by the road condition, driving speed, as well as stiffness and damping coefficients. The proposed optimization method well predicted the system performance under different road and driving conditions and was effective when used as the guideline for optimal design of the regenerative suspension system. A switch-mode rectifier capable of synthesizing variable load resistance was designed and fabricated. The proposed circuit provided desired equivalent damping to the connected regenerative suspension mechanism, thus solving the problem of the conventional passive suspension with fixed parameters and performance.

B. Huang et al. / Journal of Sound and Vibration 357 (2015) 16–34

20

50

18 16

30

Generated power (w)

2

Absolute acceleration (m/s )

40

20 10 0 -10

14 12 10 8 6

-20

4

-30

2

-40

33

0

2

4

6

8

10

12

14

16

18

0

20

0

2

4

6

8

10

12

14

16

18

20

14

16

18

20

Time (sec)

Time (sec) 250

50

200

30

Generated power (w)

2

Absolute acceleration (m/s )

40

20 10 0 -10 -20

150

100

50

-30 -40 -50

0

2

4

6

8

10

12

Time (sec)

14

16

18

20

0

0

2

4

6

8

10

12

Time (sec)

Fig. 18. System responses under City situation with optimal damping for best ride comfort performance: (a) absolute acceleration; (b) generated power; and for best power generation performance: (c) absolute acceleration; and (d) generated power.

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