242_1

J. Alanoly Research Scientist.

S. Sankar Professor and Director.

CONCAVE Research Centre, Department of Mechanical Engineering,

Concordia University, Montreal, Canada H3G1M8

A New Concept in Semi-Active Vibration Isolation Semi-active suspensions can achieve performance close to that of active suspensions with much lower cost and complexity. They use an active damper in parallel with a passive spring. The forces in the damper are generated merely by the modulation of fluid-flow orifices based on a control scheme involving feedback variables. This paper presents an original control strategy employing only directly measurable variables in vehicle applications. The relative displacement and relative velocity across the suspension are the only feedback signals and the damper force can be continuously modulated (as opposed to on-off control). Vibration isolation performance of the new semi-active scheme is compared to semi-active sky-hook suspension, as well as passive and active suspensions.

1 Introduction

Suspensions are an integral part of any ground vehicle. They perform the important task of isolating the passenger and cargo from terrain-induced shock and vibration. A vehicle suspension unit usually consists of a spring and damper. Normally they are "passive units" because they do not require any external power. By using hydraulic or pneumatic power and sophisticated control devices, it is possible to make "active suspensions" which could give substantially improved performance over passive ones. But they are more complex, expensive, and less reliable than passive suspensions.

A compromise between the active and passive types is the "semi-active" suspension system. In this system, virtually no external power is required. Desired forces are generated in a damper by modulating orifice areas for fluid flow. However, like an active suspension system, this also requires an instrumentation package and control devices. It has been shown that a semi-active suspension can provide a performance superior to that of a passive one without the cost and complexity of a fully active suspension.

2 Passive and Active Isolators

In Fig. 1, a single-degree-of-freedom vehicle model is shown with (a) passive, (b) active, and (c) semi-active suspensions. The passive system using linear elements has the equation of motion

x+2fa„(x-y) +o>2„(x-y) = 0 (1) where

km u>l = k/m and f=c/2\ Its vibration isolation can be characterized by acceleration transmissibility (defined as the ratio of the steady-state peak response acceleration to peak input acceleration for sinusoidal

Contributed by the Design Automation Committee and presented at the Design Engineering Technical Conference, Columbus, Ohio, October 5-8,1986, of THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS. Manuscript received at

ASME Headquarters, July 9, 1986. Paper No. 86-DET-28.

X

)Active force generator

4-' (a) (b) <c>

Fig. 1 Schematic of (a) passive, (b) active, and (c) semi-active isolators

input). For the passive system of equation (1), the transmissibility, Tis given by

1 + T=-

W ['-£)]>

(2)

Figure 2 shows this transmissibility plotted for several values of damping ratio, f. This plot illustrates the fundamental performance characteristics of most passive suspensions, both linear and nonlinear.

Based on this transmissibility plot, it can be seen that lower damping gives good isolation at high frequencies but poor resonance characteristics. However, higher damping results in good resonance isolation at the expense of high frequency performance [1].

When active suspensions are used the suspension force can be generated based on any number of control strategies [2]. Using optimal control theory and a commonly used quadratic performance criterion it was shown [3] that an optimum single-degree-of-freedom isolator must generate suspension force as

Fs/m=-2fanx-o>Hx-y) (3) leading to a sprung mass equation of motion as

x + 2fr„x + u2„{x-y)=Q (4) The optimum value of damping ratio, f, is 1/V2. The transmissibility of this system is given by

242/Vol. 109, JUNE 1987 Transactions of the ASME

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1.0 -

0.1 -

0.001

=ML

-A 1

\ \

5-1.0 s

0 . 7 0 7 -

: . 0.5 —

0.25 -

0.1 - |

0.0 J~l\" • \ \

\ \

10.0

1.0 ai/co

10.0

1.0

0.1

0.01

! \ >— r

-5=0.1

- 0.25

- 0.5 - 0.707(

- 1.0

i

^

^ V-IC K Vtt, s

optimal)

s 0.1 1.0

u)/u„ 10.0

Fig. 2 Transmissibility of passive isolator Fig. 3 Transmissibility of active isolator

T=-

[•-£)H*3: (5)

and is shown plotted in Fig. 3. It should be noted from this figure that the optimum transmissibility has no resonance amplification and that its performance is much superior to any passive system throughout the frequency range. However, the implementation of this active isolator requires accurate measurement of sprung mass absolute velocity, as well as ser-voactuators and power supply.

3 Semi-Active Isolators

Semi-active suspensions represent a compromise between passive and active ones. The concept of semi-active suspension was first proposed by Crosby and Karnopp [4] in 1973. This scheme does not require hydraulic power. Forces are generated in a damper by modulating its fluid-flow orifices.

The damper force has the form

F.-F*- 2fa„x, x(x-y)>0

0, x(x-y)<0 (6)

This scheme was shown to give a response close to that of an active suspension [4, 5]. To implement this control logic, one needs a servovalve with a very large bandwidth as well as the measurement of absolute and relative velocities. However, the accurate measurement of the absolute velocity of a moving vehicle is a near impossible task. This is due to the fact that, in road vehicle suspension applications, the significant vibration levels occur below 10 Hz [6]. Since no inertial frame is available for a moving vehicle, absolute velocity measurement has to be obtained by integrating the acceleration signal. The measurement of very low frequency acceleration signals as well as the integration procedure are hampered by hardware limitations [7]. Therefore, this scheme has not been successfully implemented for vehicle suspensions to date. However, several theoretical results have been presented for vehicle applications including air-cushion vehicles, military tanks and

agricultural tractors [8-14]. The concept was also applied to the control of structural vibrations of vehicles and buildings [15-17]. In all these instances, it has been claimed that the semi-active isolator is superior to a passive system and that its performance is comparable to that of an active one. Experimental results also have been presented for laboratory prototypes [16-18].

The control scheme in equation (6) requires a continuous modulation of damper orifice area. A simpler on-off scheme was proposed and experimental results were presented [8, 13, 14, 21]. In this case, the damper force and the control logic were governed by

F1,-2fa„(x-y), x(x-y)>0

0, x(x-y)<0 (7)

The difference between equations (6) and (7) is that, in the latter, the force is proportional to the relative velocity across the damper. Thus this scheme can be implemented using an on-off damper. Experimental results reported indicate that the on-off semi-active isolator is superior to passive one.

Rakheja [22] and Rakheja and Sankar [23] have proposed an on-off semi-active damper based a different control scheme. The control requires the measurement of relative velocity and relative displacement, both of which are directly measurable even in vehicle applications. The scheme is

C2^u„{x-y)\x~y\, (x-y)(x-y)<0 F'd = \ (8)

\2$2u>n(x-y)\x-y\, (x-y)(x-y)>0 We will refer to the three semi-active control schemes,

represented by equations (6)-(8), as Type 1, Type 2, and Type 3 semi-active suspensions, respectively. In all these cases, the damper force is determined by one of two expressions depending on the sign of a certain function. This function will be called the "condition function." Types 1 and 2 are based on the same condition function. The former employs continuous control of damper forces while the latter uses on-off control. Type 3 is an on-off semi-active damper based on a different condition function [refer to equation (8)]. A continuous con-

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Fig. 4 Steady-state response of Type 4 system at (a) u/w„ = 0.5, (b) w/u„ = 1.0, and (c) u/u„ = 5.0; z = x - y

trol scheme based on this condition function is the new semi-active concept being proposed in this paper. This scheme will be referred to as a Type 4 system.

4 Type 4 Semi-Active Isolator

It is observed that the damping force in a passive damper tends to increase the acceleration of the sprung mass during part of a vibration cycle. This happens when the spring force and the damper force have the same direction. Therefore it is proposed that a semi-active damper be used which gives zero (or a very low) damping during this part of the cycle. When the spring force and the damper force are in opposite directions, the damper can generate a force with the same magnitude as the spring force but opposing it. This way the net force acting on the mass will be zero for this part of the vibration cycle. This idea is the basis of the Type 4 semi-active scheme.

Thus for a Type 4 system

f -aco2„(x-y), (x-y)(x-y)<0 F'd = \ (9)

L 0, (X-y)(X-y)>Q where a is the gain.

As in the case of the Type 1 system [4], the possibility of damper lockup exists when (x — y) (x—y) = 0. In this situation, two special cases arise. In the first case (x—y) = 0, in which case, the damping force F'd = 0. The second case occurs when (x — y) = 0 and (x—y) ^ 0. In this case, the system will lockup if the desired force Fd, (F'd = — aoi2, (x — y)) is greater than the lockup force, which is

Fd=-y-o>2„(x-y) (10) It is possible to implement this scheme using a servoactuator

to control the damper orifice. This would be similar to the implementation of a Type 1 semi-active damper reported by Boonchanta [20]. From the control point of view, the obvious

1 advantage of the present system is that it requires only the v measurement of relative displacement and relative velocity.

These quantities can be measured directly even for a moving vehicle suspension.

5 Solution Procedure and Performance Characterization

5.1 Simulation. The equations of motion of the semi-active suspension system described in equation (9) have step

1 discontinuities and hence are nonlinear. A direct and fast way of solving the system equations is by computer simulation.

1 This is the approach used by previous investigators [4, 22]. In simulation, the differential equations are solved as an initial-value problem for a given set of parameters. When a frequency domain performance measure is desired (such as

' transmissibility), the simulation is carried out at each discrete frequency. The simulation proceeds in time until a steady state is reached. The response variables are then evaluated and the process is repeated for the next frequency.

During numerical integration, the discontinuities in the equation require special treatment. Otherwise the integration algorithm will be very inefficient and will cause inaccurate results [24]. Several researchers have studied the problem of numerical integration of ODE's with discontinuities [24-27], Their basic approach is to incorporate a zero-finder in the integration procedure. At each integration step, a test is performed to detect the change in sign of the condition function. If no change occurs, the integration proceeds. Otherwise a zero-finder determines accurately the time, r*, when the condition function is zero. The integration then stops with the old set of system functions and restarts with the appropriate new set. In the present study, a fourth-order Runge-Kutta method was used for numerical integration in conjunction with a bisection algorithm as the zero-finder.

5.2 Bifurcation Analysis. Bifurcation, or branching of solutions, is a phenomenon that occurs in nonlinear systems. Some engineering problems where this occurs include nonlinear oscillators, such as, Duffing spring [28], impact oscillators [29, 30], buckling of structures, etc. [31, 32].

In the case of a Type 4 semi-active system, it was observed in simulation runs that the periodic solutions did not exist for certain values of parameters. This suggested a bifurcation problem. It is not easy to examine bifurcation and stability problems using simulation techniques. Therefore the study was done with the help of a bifurcation analysis software package called AUTO [33, 34]. AUTO solves the system as a boundary-layer problem and determines stability of periodic solutions under perturbations. Details regarding the use of this software for semi-active suspensions can be found in [35].

5.3 Performance Characterization. Since the semi-active system is nonlinear with step changes in damper force, the acceleration response will have discontinuities. Previous researchers [5] have used displacement transmissibility (ratio of the peak displacement response to the input amplitude) to characterize the suspension performance. Since the human body or a suspended mass is sensitive to inertial forces, the characterization in terms of acceleration would be more appropriate. In this study, the ratio of the root-mean-square (rms) value of the response acceleration to the rms-value of the input acceleration is taken as the transmissibility.

6 Results and Discussion

Figure 4 shows the steady-state response of a Type 4 system at 3 frequencies for a = 1. At oi/o>„ = 0.5, Fig. 4(a), the damper operates in three phases: (a) exactly opposing the spring force, (b) zero force and (c) lockup force. The lockup condition occurs only at low frequencies. This type of system



10.0

1.0

0.1

0.01

• Displacement-

-]?* r

.— Acct

r- Vel(

\ \ \

"A >\

leration

ici t j

Table 1 Upper bound on a for stable solutions

0.1 1.0 m/m 10.0

Fig. 5 The rms transmissibility ot Type 4 system for a = 1.0

10.0

1.0 - ~

1.0

0.01

Act ive, e=0.

y>

707 -P

t\ L \_

t rP a s

\ VP a s

\ 4 s-' x . • ^ A s >-V v

/ A v-

\J

1 i

s ive , c=0.1 ' i i I 1

s ive, c=1.0

1- Type 4,a=l .0

N s.

i

V" r ^ —\

^

\

-\-l \

\ \

\

\ —v

* ± x ~S; v^-I V

\ \

\ \ \ 0.1 1.0 to/to 10.0

Fig. 6 Comparison of rms acceleration transmissibilities of Type 4 semi-active system with active and passive systems

has a unique behavior in that for nearly half of the time the mass acceleration is zero. This, in turn, leads to good isolation characteristics.

Figure 5 shows the rms transmissibility of a Type 4 system. The acceleration transmissibility is higher than the displacement and velocity transmissibilities. This will in fact be the case for most semi-active systems because of the discon-

Frequency Ratio, OJ/V

0.5

1.02

2.55

3.96

6.384

10.0

a

1.366

1.474

1.857

1.86

1.88

2.00

10.0

1.0

0.1

0.01

_ -s H>?

/^Acceleration

' "^ . -Disp

< s

> ^ \ \ \

\ \ \ \ \\ \

, V e l .

\ \ \s \ -

• ^

^

\ 0.1 1.0 w/w

Fig. 7 The rms transmissibility of Type 4 system for a

10.0

= 1.3

tinuities in the damper force. The rms transmissibility shows the excellent isolation characteristics of this system. This acceleration transmissibility has a peak value of 1.84 at «/co„ = 0.5 and reduces to 0.007 at co/a>„ = 10.

Comparison of this system to active and passive isolators is shown in Fig. 6. Compared to the passive isolator, a Type 4 system has a far superior performance for co/o>„ greater than 0.95. At wAo„ = 10, the isolation is almost 1.5 times better than that of an undamped passive isolator. However, there is a price to be paid in terms of inferior low frequency performance. Figure 6 also shows that a Type 4 system is superior to the optimum active isolator for o>/o>„ greater than 1.24. This result is extremely encouraging because it is achieved through semi-active control.

The damper force gain, a, is an important parameter affecting the system performance. When looking at the steady-state time plots of the system, it was felt that a larger than uni-



—

/

/ /

f\ i » _ _

,

L \J>

S,^5 (

-L-NNA—

" " T r V X1. X

\

1 i . L

.. 1 1 1

a=0.

1.0"

1.3

L \

V"T 3,- — r \ U-V

V 0.1 1.0 to/id 10.0

Fig. 8 Type 4 sys'tem rms acceleration transmissibility for variation in gain, a

10.0

==s^

C = l .

v- 'v7^~ A »-^

o ~S\\ \ \ N

_ i _.

- C=0.25

. 0.5

. 0.707

\

L

V . .-

t -"~ ^ VT X

0-1 1.0 u/ai 10.0

Fig. 9 Type 1 system rms acceleration transmissibility for variation in damping ratio, f

ty value of a will give better isolation. To test this, a was set to 2.0 and simulation was carried out, but the computer run indicated that the system failed to achieve periodic oscillations.

It was recognized that the periodic solutions may be unstable at certain values of gain. To study this in more detail, the software package AUTO [33, 34] was used. From this analysis, it was found that at each frequency, there is an upper

bound on the parameter, a, for stable periodic solutions to exist. For higher values of a, the solutions would not persist under small perturbations. Table 1 shows the upper bound on a for some frequencies.

The system is stable throughout the frequency range for a = 1.3. Simulation was carried out with a = 1.3. Figure 7 shows the transmissibility plot. It is obvious that this value of a gives a better performance than the unity gain system, Fig. 5. The rms acceleration transmissibility has a peak of 1.19, but outperforms even active suspensions at high frequencies. Figure 8 shows the transmissibility for various values of the gain a. As a is increased, the resonant peak decreases and the isolation performance improves. However, there is a limit on maximum a for stability as established earlier.

Figure 9 shows the rms acceleration transmissibility of a Type 1 system for various values of the damping ratio, £*. It can be seen that higher values of f improves isolation at high frequencies. But this also leads to deterioration at the very low frequency end due to the sharper discontinuities in the damper force. It was noted in [5] that as f -~ oo, the high frequency performance approaches that of an active isolator with u„ = 0.6 -4kJrn and f = 1.0. This performance would be superior to that of a Type 4 isolator with a =1.3 . However, for more reasonable, finite values of f, a Type 4 system performs better than a Type 1 system. It is also important to point out that the main advantage of a Type 4 system over Type 1, or its fully active counterpart, is that Type 4 control logic only involves directly measurable variables such as relative velocity and relative displacement.

7 Conclusion

Active suspensions can achieve the kind of performance not possible with passive ones. However, their cost and complexity, together with requirements on power and increased weight, limits their applications. This is especially true in the area of vehicle suspensions. Semi-active suspensions have been proposed as a compromise between active and passive suspensions. They do not require much power and are less complex than active suspensions. The new concept in semi-active vibration control proposed in this paper is shown to give performance comparable to that of a fully active isolator. The control scheme's main attraction is that it uses only relative velocity and relative displacement measurements, both of which are easily measured in vehicle suspensions.

References

1 Thomson, W. T., Theory of Vibration With Application, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1972.

2 Goodall, R. M., and Kortum, W., "Active Controls in Ground Transpor-ta t ion-A Review of the State-of-the-Art and Future Potential," Vehicle System Dynamics, 12(4-5), August 1983, pp. 225-257.

3 Bender, E. K., "Optimum Linear Preview Control With Application to Vehicle Suspension," ASME Journal of Basic Engineering, 90(2), June 1968, pp. 213-221.

4 Crosby, M. J., and Karnopp, D. C , "The Active Damper-A New Concept for Shock and Vibration Control," The Shock and Vibration Bulletin, 43(4), June 1973, pp. 119-133.

5 Karnopp, D. C , Crosby, M. J., and Harwood, R. A., "Vibration Control Using Semi-Active Force Generators," ASME Journal of Engineering for Industry, 96(2), May 1974, pp. 619-626.

6 Healey, A. J., Nathan, E., and Smith, C. C , "An Analytical and Experimental Study of Automobile Dynamics With Random Roadway Inputs," ASME Journal of Dynamic Systems, Measurement and Control, December 1977, pp. 284-292.

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8 Margolis, D. L., Tylee, J. L., and Hrovat, D., "Heave Mode Dynamics of Tracked ACV With Semi-Active Airbag Secondary Suspension," ASME Journal of Dynamic Systems, Measurement and Control, 97(4), December 1975, pp. 399-407.

9 Margolis, D. L., "Semi-Active Suspensions for Military Ground Vehicles Under Off-Road Conditions," presented at the 52nd Symposium on Shock and Vibration, New Orleans, October 1981.



10 Margolis, D. L., "Semi-Active Heave and Pitch Control for Ground Vehicles," Vehicle System Dynamics, 11(1), February 1982, pp. 31-42.

11 Margolis, D. L., "Semi-Active Control of Wheel Hop in Ground Vehicles," Vehicle System Dynamics, 12(6), December 1983, pp. 317-330.

12 Kim, K., "Ride Simulation of Passive Active and Semi-Active Seat Suspensions for Off-Road Vehicles," PhD Thesis, University of Illinois at Urbana-Champaign, 1981.

13 Roley, D. G., "Tractor Cab Suspension Performance Modeling," PhD Thesis, University of California, Davis, CA, June 1975.

14 Roley, D. G., "Performance Characteristics of Cab Suspension Models," presented at the 1975 Winter Meeting of American Society of Agricultural Engineers, Paper No. 75-1517, December 1975.

15 Allan, R. R., and Karnopp, D. C , "Semi-Active Control of Ground Vehicle Structure Dynamics," AIAA Paper No. 75-821, presented at the 16th Structural Dynamics and Material Conference, Denver, CO, May 1975.

16 Karnopp, D. C , and Allan, R. R., "Semi-Active Control of Multimode Vibratory Systems Using ILSM Concept," ASME Journal of Engineering for Industry, August 1976, pp. 914-918.

17 Hrovat, D., Barak, P., and Rabins, M., "Semi-Active Versus Passive or Active Tuned, Mass Dampers for Structural Control," ASCE Journal of Engineering Mechanics, 109(3), June 1983, pp. 691-705.

18 Krasnicki, E. J., "Comparison of Analytical and Experimental Results of Semi-Active Vibration Isolator," The Shock and Vibration Bulletin, 50(4), September 1980, pp. 69-76.

19 Hrovat, D., and Margolis, D. L., "An Experimental Comparison Between Semi-Active and Passive Suspensions for Air-Cushion Vehicles," International Journal of Vehicle Design, 2(3), August 1981, pp. 308-321.

20 Boonchanta, P., "Comparison of Active Passive and Semi-Active Suspensions for Ground Vehicles," PhD Thesis, University of California, Davis, CA, 1982.

21 Krasnicki, E. J., "The Experimental Performance of an ON-OFF Active Damper," The Shock and Vibration Bulletin, 51(1), May 1981, pp. 125-131.

22 Rakheja, S., "Computer-Aided Dynamic Analysis and Optimal Design of Suspension Systems for Off-Road Tractors," PhD Thesis, Concordia University, Montreal, Quebec, Canada, 1983.

23 Rakheja, S., and Sankar, S., "Vibration and Shock Isolation Perfor

mance of a Semi-Active 'On-Off Damper," ASME Paper No. 85-DET-15, presented at the ASME Design Engineering Technical Conference, Cincinnati, OH, September 10-13, 1985.

24 Carver, M. B., "Efficient Integration Over Discontinuities in Ordinary Differential Equation Simulations," Mathematics and Computers in Simulation, XX, 1978, pp. 190-196.

25 Ellison, D., "Efficient Automatic Integration of Ordinary Differential Equations With Discontinuities," Mathematics and Computers in Simulation, XXIII, 1981, pp. 12-20.

26 Li, Bo-Hu, Yang, W. H., and Ye, Xin-An, "An Effective Method for a System of Ordinary Differential Equations With Right-Hand Side Containing Discontinuities," Proceedings of the 10th /MACS World Congress, Concordia University, Montreal, August 1982, Vol. 1, pp. 6-9.

27 Borthwick, W. K. D., "The Numerical Solution of Discontinuous Structural Systems," Proceedings of the 2nd International Conference on Recent Advances in Structural Dynamics, University of Southampton, England, April 1984, Vol. 1, pp. 307-316.

28 Guckenheimer, J., and Holmes, P. J., Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer-Verlag, New York, 1983.

29 Shaw, S. W., and Holmes, P. J., "A Periodically Forced Piecewise Linear Oscillator," Journal of Sound and Vibration, 90(1), 1983, pp. 129-155.

30 Bapat, C. N., "A Study of Vibroimpact Systems," PhD Thesis, University of Manitoba, 1981.

31 Thompson, J. M. T., Instabilities and Catastrophes in Science and Engineering, Wiley, New York, 1982.

32 Huseyin, K., "Vibration and Stability of Mechanical Systems: III ," Shock and Vibration Digest, 16(7), July 1984, pp. 15-22.

33 Doedel, E. J., "AUTO: A Program for the Automatic Bifurcation Analysis of Autonomous Systems," Proceedings of 10th Manitoba Conference on Numerical Methods and Computation, Congress Number 30, 1981, pp. 265-284.

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35 Alanoly, J., "Vibration Isolation Characteristics of a Class of Semi-Active Suspensions," M. Eng. Thesis, Concordia University, Montreal, Canada, December 1985.



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