TECHNICAL NOTE: Properties of Air Spring as a Force Generator in Active Vibration Control Systems

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1 Vehicle System Dynamics /01/ $ , Vol. 35, No. 1, pp. 67±72 # Swets & Zeitlinger TECHNICAL NOTE: Properties of Air Spring as a Force Generator in Active Vibration Control Systems IGOR BALLO * SUMMARY In the active vibration control systems of electropneumatic type the air spring is often used not only as the elastic element of the system, but also as the compensation force generator, which caters for improved vibro-isolating effects. To provide this function of the air spring, it is necessary to know its characteristics in suf cient detail. In this paper the procedure of experimental estimation of these characteristics is described and the result is compared with the results of estimation based on theoretical considerations. The paper is completed by the graph of both types of characteristics. 1. INTRODUCTION As the actuator and elastic element of the active vibration control systems, an air spring is often used, into which air is fed or discharged from in such a way as to improve the vibration control effect. This air spring could be considered as a parallel combination of a passive spring and a compensation force generator, which caters for active vibro-isolating system effects whose aim is to improve the overall vibration control effect. To facilitate proper design of the control system, which governs the in ow and out ow of the air into and out from the air spring, it is necessary to know in suf cient detail the air spring characteristics in the function of compensation force generator. In some older papers (for example [2, 3]) these characteristics were derived from general aerodynamic arguments, not taking into account some speci c properties of real air springs. The result of this rst * Institute of Materials and Machine Mechanics, Slovak Academz of Sciences, RacÏianska str. 75, SK Bratislava 38, Slovakia.

2 68 I. BALLO approximation was, that the improvement in vibration control effect was manifest in a rather narrow frequency band only. The use of characteristics, estimated by a procedure described below, incurred substantial widening of the frequency band where the desired vibration control effect was obtained [1]. 2. AIR SPRING AS COMPENSATION FORCE GENERATOR The construction of air springs could be different. For the sake of this paper we will assume that their principal properties for use in vibration-isolation systems are solely determined by characteristics for which the following arguments are valid. A speci c type of air spring will be chosen as a typical example of all air springs. The convoluted type air spring, whose rubber body is reinforced by textile cord will be used throughout. The effective volume v x of such an air spring could be approximated by the following formula [2]: v x ˆv 0 v 1 : x v 2 : x 2 According to the same paper the effective cross section area s x of such an air spring could be expressed as: s x ˆs 0 s 1 : x s 2 : x 2 where the air spring compression is for positive x and dilatation for negative x. Other symbols in equations (1) and (2) denote the properties of a particular spring. In the state of static equilibrium the height of the air spring is H 0, the mass of air enclosed by the spring is m 0, it's absolute pressure is p 0 and overpressure is P p0. If, under isothermal conditions, air is admitted into the air spring or air is exhausted from the air spring into the surrounding atmosphere, an increase or decrease (i.e., change) in the internal pressure by p d is incurred. If r denotes the gas constant of air and T denotes the absolute temperature of the air enclosed then this process can be described in suf cient accuracy by the equation: p 0 p d v 0 v 1 x v 2 x 2 ˆ m 0 m r : rt 3 In the static equilibrium state following relations hold: 1 2 p 0 v 0 ˆ m 0 :rt v 0 ˆ s 0 H 0 4a; b

3 AIR SPRING AS A FORCE GENERATOR 69 After eliminating terms related to the static position and some re-arrangement of equation (3) following equation is obtained: p d s 0 ˆ mrrt p 0 p d v 1 x v 2 x 2 5 H 0 H 0 On the right hand side of equation (5) two terms are present. The rst one contains the amount of air in the air spring m r and is independent of air spring deformation x. In contrary, the second term depends on deformation x and is independent of the air mass m r. Hence, it is possible to assume, that the rst term is concerned with the air spring function as compensation force generator, whereas the second term describes predominantly the elastic properties of the air spring. The aim of this paper is to determine the characteristics of air spring as the compensation force generator. Therefore, the properties of the air spring around the equilibrium position, i.e., for x ˆ 0 will be further investigated. Under these circumstances, equation (5) has the form: s 0 p d Š xˆ0 ˆ mrrt 6 H 0 The in ow and out ow of the air into and out from the air spring is governed by electro-pneumatic sliding valve, whose action is controlled by electrical voltage u r. The slide function in a real system could be described in suf cient accuracy by equation: m r ˆ Dfu r g 7 where D is an integro-differential linear operator. After substitution and re-arrangement equation (6) has the form: 0 p d Š Dfu r gˆs0h xˆ0 8 rt Now Fourier transform is applied onto equation (8). Under this transform the linear operator D is transformed into the frequency domain as the Frequency Response Function (FRF) C! If the Fourier transforms of the dynamic variables are denoted by capital letters then for the description of FRF of the air spring as compensation force generator is following:! ˆ P dš xˆ0 : s 0H 0 U r rt 9

4 70 I. BALLO 3. DETERMINATION OF THE FRF OF THE AIR SPRING AS COMPENSATION FORCE GENERATOR As seen, equation (9) supplies a general formula for determination of the sought air spring FRF C!. For determination of this dynamic characteristics the ratio of Fourier transforms of the pressure changes P d Š xˆ0 and control electrical voltage U r is decisive. This ratio could be determined best in experimental way [1, 4]. Under some speci c simplifying suppositions it would be possible to derive this ratio also by theoretical considerations [2, 3], however it is clear that in this way it will not be always possible to respect speci c properties of a given air spring, as it is possible by the experimental approach. The simplest, often used case of the electropneumatic sliding valve occurs, when a linear relation between the uid mass ow q [kg.s 1 ] and the control voltage u r, governing the slide operation, is stipulated [2, 3, 4]: q ˆ k 3 : u r 10 a The mass of air m r, transferred via the slide valve will be: or, after substitution: m r ˆ Z t 0 Z t m r D u r ˆk 3 u r dt q: dt 10 b 0 11 In same manner as before, applying the Fourier transform and some rearrangement the FRF C! is obtained in following form:! ˆk3 j! GRAPHICAL COMPARISON OF THE EXPERIMENTALLY DETERMINED FRF AND THE SIMPLIFIED, THEORETICALLY DERIVED FRF In the following Figure 1, two FRF courses are depicted. The rst curve, denoted A, represents the FRF determined according to equation (12), whereas

5 AIR SPRING AS A FORCE GENERATOR 71 Fig. 1. Curve A represents the simpli ed theoretical FRF, whereas curve B depicts experimentally determined characteristics. curve B represents the experimentally determined characteristics. Both axis in this gure are logarithmic. 5. CONCLUSION Based on simple theoretical reasoning the course of the frequency response curve of the air spring as compensation force generator for use in active vibroisolating systems has been derived. According to this expression the FRF of a speci c air spring was determined experimentally (curve B in Fig. 1). In the same gure the air spring FRF, derived by simpli ed aerodynamic reasoning is depicted too (curve A). Note the marked difference in the course of both curves, especially in the low frequency region, which is the most important one for practical application of active vibration control systems of electropneumatic type. REFERENCES 1. Ballo, I.: Improving the Ef ciency of Electropneumatic Vibration Control Systems by Means of the Force Generator Principle (in Slovak). StrojnõÂcky cïasopis 50(2) (1999), pp.82±91.

6 72 I. BALLO 2. GajarskyÂ, M.: Some Properties of Electropneumatic Active Vibration Control Systems (in slovak). StrojnõÂcky cïasopis 35(1±2) (1984), pp.51± Stein, J. and Ballo, I.: Active Vibration Control System for Driver's Seat for Off-Road Vehicles. Vehicle System Dynamics 20 (1991), pp.57± Ballo, I.: Improved model of active pneumatic vibroizolating system. Proc. of the conference Mechatronics and Robotics, Czech Institute of Technology, Brno, 1997, pp.125± Kowal, J.: Active and Semiactive Methods of Vibroisolation of Mechanical Systems. Scienti c Bulletins of Stanislaw Staszic Academy of Mining and Metallurgy, Cracow, No. 1310, Mechanics 23 (1990). 6. Engel, Z. and Niziol, J.: The Perspectives of the Development of the Active Methods of Vibration and Sound Control (in Polish). Proc. of the II. School `Active Methods of the Reduction of Vibrations and Sound', Academy of Mining and Metallurgy, Cracow- Zakopane, 1995, pp.11± Ballo, I.: Power Requirement of Active Vibration Control Systems. Vehicle Systems Dynamics 24 (1995), pp.683± Stein, G.J.: New Active Electro-Pneumatic Suspension for Driver's Seats. In: Proc. `Noise Control `98', Krynica, Poland, 1998, pp.525± Sharp, R.S. and Crolla, D.A.: Road Vehicle Suspension System Design - A Review. Vehicle System Dynamics 16 (1987), pp.167± Yi, K. and Hedrick, J.K.: Active and Semi-Active Heavy Truck Suspensions to Reduce Pavement Damage. SAE Technical Paper (1989). 11. Karnopp, D., Crosby, M.J. and Harwood, R.A.: Vibration Control Using Semi-Active Force Generators. Transactions of ASME, J. of Engineering for Industry 96 (1974), pp.619±626.

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