03 Modelling a twin tube cavitating shock absorber M Alonso* and Á Comas Automóviles, Escuela Técnica de Ingenieros Industriales de Terrassa, Barcelona, Spain The manuscript was received on 3 December 2004 and was accepted after revision for publication on 0 January 2006. DOI: 0.243/09544070D2304 Abstract: An analytical method to quantify the damping force of a twin tube shock absorber is proposed. Fluid and chambers compressibility effects and fluid cavitation are included. A comparison of a calculated damping force against the ideal damping force (assuming that non-cavitation occurs) is presented. Keywords: Automobile, blow-off valve, cavitation, compressibility, elasticity, mass conservation, polytropy, shock absorber, vehicle, valves INTRODUCTION camera during the extension stroke, leaving it when the shock absorber works in compression. Within the The system of suspension within a vehicle must reserve camera there also exists a gas (generally air satisfy two requirements [ 4]. Because the force or nitrogen) that can be in direct contact with the oil between the tyre and the ground is responsible for and whose aim is to permit the oil volume variation the handling and security of the vehicle, the first and, at the same time, to ensure a minimum pressure requirement is not only to ensure contact between the in the reserve chamber. ground and the tyre, but also that transitory effects The single tube shock absorber absorbs the volume appearing in this contact when the vehicle is driven variation in another way, as a consequence of the irregularly or maneuvring are minimized, keeping rod s movement. This method uses a gas tank within the contact force as uniform and stable as possible. the compression chamber so that when the shock The second requirement of the suspension system is absorber is in the compression stroke, the gas to guarantee the comfort of passengers and/or loads, compresses by absorbing the additional volume protecting them from such irregularities. Nowadays, introduced by the rod. The opposite effect occurs sophisticated suspension systems that try to optimize when the shock absorber is working in traction. these two requirements already exist. The aim of this paper is to present a new Shock absorbers are part of the suspension system. mathematical model for a twin tube shock absorber Their primary target is the dissipation of energy that enables a better understanding of the physical absorbed by the spring and torsion bars. Additionally, phenomena that takes place when the shock in some cases, they are also used to reduce the absorber works. preload force of the spring [5]. In the automotive field, two different types of shock absorber are currently in use: the single and the twin tube. The main difference between these is 2 STATE OF THE ART in how the system s volume variation due to the rod Many authors have presented works to describe shock movement is compensated. Regards the twin tube absorber behaviour [6 9]. Most of them consider the shock absorber (Fig. ), there is a concentric chamber work of Segel and Lang [9] as the most realistic to to the working cylinder in which a certain amount date. Segel and Lang take into account the main of fluid remains and which enters the working physical phenomena affecting shock absorbers. This results in a proper correlation between model pre- * Corresponding author: Automóviles, Escuela Técnica de diction and experimental results. What makes their Ingenieros, Industriales de Terrassa, Musitu 22, Barcelona, 08023, work different is that they introduce a realistic model Spain. email: marcos7522@terra.es of cavitation based on semi-empirical data.
032 M Alonso and Á Comas Fig. Outline of a twin tube shock absorber Regards thermal analysis (heat transfer and temperature prediction) applied on shock absorbers, no complete and/or realistic works have yet been found, merely mention of how temperature affects damping force. 3 MODELLING THE VALVES Inside the twin tube shock absorber are found two sets of valves. The first set is located on the base valve and its main function is to control the pressure in the compression camera. The second set is placed on the piston and produces a differential pressure between the compression and the traction chamber, generating the damping force. The differential pressure that valves produce generates most of the damping force. Therefore, it is easy to understand that valves are an essential part of the shock absorber. In general, the valves of a shock absorber are not simple channels that link the working chambers because, in addition to these channels, there also exist the blow-off valves. Blow-off valves are used simply for low damping speeds and for the differential pressure between the working cameras when the working fluid circulating throughout a simple circular channel is enough. However, as soon as volume increases (which means higher damping speeds), the differential pressure generated produces an excessive damping force. Clearly, this situation is unacceptable. The easiest solution is to introduce a valve that, from a certain pressure, permits the circulation of the fluid. The result is that, from a certain pressure, the blow-off valve opens and the differential pressure grows less rapidly at the initial damping speed. Although there are more sophisticated works to describe valves [0], the proposed model of valve for this study is outlined in Fig. 2 and consists of an
Modelling a twin tube cavitating shock absorber 033 Fig. 2 Valve modelling always-open channel and, in parallel, a blow-off valve. Here, channel a refers to an always-open channel and channel b to the blow off valve. The pressure loss that takes place on channel a can be modelled as [] set of equations Dp a =F(Q a ) Dp b =H(Q b ) Dp a =Dp b Dp a =p p 2 = 8rQ2 a p2d4 a A K a + f a L a D a B Q=Q a +Q b () where r is the fluid density, Q a is the volumetric flow, D is the channel diameter, K is the entry singular a a pressure loss constant, f is the Darcy Weisbach a friction coefficient and L is the channel length. a Alternatively, Blevins [2] approximates the blow-off valve behaviour as follows Dp b =p p 2 = 8rQ2 b p2d4 b C K b + f b L b D b +G A x2bd where Q is the volumetric flow, D is the blow-off b b valve inner diameter, K is the entry singular pressure b loss constant, f is the friction coefficient, L is the b b channel length, x is the distance between the disc and the seat valve and G(/x2) is a characteristic function valve. Generally, the disc lies on its seat owing to the existing spring, which normally has a certain preload. The only force able to move the disc by defeating the spring comes from the loss of pressure that takes place between the opposite sides of the disc. If the loss of pressure that takes place on the inner disc surface is linear, and if inertial forces are negligible, it leads to a third-degree equation with a unique positive real solution that corresponds to the true disc-to-seat distance. Once the individual valves are presented, the whole valve behaviour is described by the following where Q represents the volumetric flow that crosses the whole valve. 4 MODELLING THE CAVITATION Sometimes, during the working cycle of the shock absorber (generally at high speeds) local pressures inside the working cameras are below the vapour pressure. Under such conditions, there appear some oil vapour bubbles that implode quickly once the working pressure exceeds the vapour pressure. The implosion of the bubbles produces a set of waves of a very high magnitude that are highly destructive. This phenomenon is known as cavitation, and it causes abnormal damping forces during the shock absorber function (such as force delay or unexpected reductions of the damping force). Figure 3 represents the cavitation phenomenon, showing the oil vapour bubbles within the traction camera when the shock absorber works in compression. From the thermodynamic theory hypothesis, the presence of vapour bubbles is instantaneous. There is a delay as the bubble (which may be mixed with some gas dissolved in the oil) must evacuate a
034 M Alonso and Á Comas phase behaves as an ideal gas, the thermodynamic equations to achieve equilibrium between the liquid and vapour phases give V = m r l V V 0 v (Mp )/(RT ) r v v l where V is the vapour volume, m and V are the total v mass and total volume of the system (liquid plus vapour phases), r is the density of the liquid phase, l M is the molar mass of the vapour, p is the vapour v pressure, R is the universal gas constant and T is the system temperature. Deriving the above expression and taking into account that the system mass and/or volume may change (Fig. 4), temporal variation of the vapour volume is found to be dv v = (Mp )/(RT ) r v l A ṁ+r dv (2) l B where ṁ represents the mass flow that enters the volume V. 5 FLUID COMPRESSIBILITY Fluid mechanics define the isothermal compressibility Fig. 3 Twin tube shock absorber cavitation factor, which relates fluid density with pressure [] dr dp =b f r (3) certain volume of oil before it reaches its equilibrium volume. Therefore, delay is due to a mechanical where b is the isothermal compressibility factor. f effect. Because the pressure range at which the shock In accordance with the formulated Rayleigh absorber oil is exposed is not too wide, it is assumed theory [3], which is based on energetic concepts, that the compressibility factor is constant. Furtherthe collapse time (which is slightly lower than the more, assuming that density does not depend on bubble generation) is t=0.9r 0S r p 0 where R 0 is the initial bubble radius, r is the oil density and p 0 the hydrostatic pressure at which oil is exposed. For a fluid whose density is 880 kg/m3 and which allocates a bubble of 0. mm in radius, if the atmospheric pressure is applied (around 05 MPa), the collapsing time is t=4.3 0 3 s From this it can be assumed that the growth and extinction time of the bubble is instantaneous. Using this hypothesis, assuming that the system Fig. 4 Representation of the volumes for both the temperature remains constant and that the vapour liquid and the vapour phases
Modelling a twin tube cavitating shock absorber 035 temperature, the above equation can be integrated easily. The relation between density and pressure is then described as r=r 0 eb f (p p 0 ) where r 0 is the oil density when pressure is p 0. Deriving temporally dm i =r dv i i +V dr i i And, by relating the mass flow to the volumetric flow we have 6 DEFORMATION OF WORKING CYLINDER WITH THE PRESSURE Many authors define the apparent compressibility factor of the fluid by adding the effects of the working cylinder compressibility to the effects of the fluid compressibility. In this study, this concept will not be considered because it may lead to a false fluid density, which, if applied to the valves or mass conservation equations, will lead to error. For a cylinder whose inner radius is R and outer i radius is Re when the reference pressure is p, once 0 pressure becomes p its inner volume change is [4] DV=pLR2 i Dp 2 EA R2 i +R2 e R2 e R2 i +n B where DV is the volume change, L is its length, Dp is p p, E is its elasticity module and n is its Poisson 0 coefficient. The cylinder compressibility factor b is defined as c b c = 2 EA R2 i +R2 e R2 e R2 i +n B DV =V I b c Dp (4) 7 MASS CONSERVATION Hereon, properties referring to the compression chamber will be subscripted with, those referring to the traction chamber with 2, those referring to the reserve chamber with 3 and those referring to gas with g. Generically, the mass of an existing fluid in any camera is m i =r i V i Q = ṁ i [ Q = dv i i r i b f V dp i i i (5) where equation (3) has already been used. Assuming that the cavitation can only appear in the traction or in the compression chamber, the fluid volume inside the chambers can be found as follows V f =[V 0 (x x 0 )S c ](+b c Dp ) V v =V (+b Dp ) V I c v V =[V +(x x )(S S )](+b Dp ) V 2f 20 0 c v c 2 v2 =V 2I (+b c Dp 2 ) V v2 V 3f =V 30 +V g0 V g where V is the actual fluid volume, V is the initial if i0 fluid volume, Dp is the current pressure minus the i reference pressure, x is the current piston position, V is the non-deformed volume (i.e., the volume ii atzits reference pressure), x is the initial piston 0 position (time t=0), S and S the normal surfaces c v of the cylinder and rod respectively and V is the vi existing vapour volume (cavitating volume). The volume change is obtained by deriving the above expressions Therefore, if V is the inner volume (or non- I deformable volume) enclosed by the cylinder when the pressure is the reference pressure, then dv f = vs c (+b c Dp )+V I b dp c dv v dv 2f =v(s c S v )(+b c Dp 2 )+V 2I b dp 2 c dv v2 dv 3f = dv g (6) On the other hand, gas volume is supposed to behave as a polytropic gas. It is p g V n g =p g0 V n g0 [ dv g = V g np g dp g dv g = V g0 p/n g0 n dp g p(n+)/n (7)
036 M Alonso and Á Comas Introducing equation (7) within equations (6) and If V 0 v2 using equation (5) leads to the temporary pressure change dp 2 = Q 2 +v(s c S v )(+b c Dp 2 )+dv v2 / V b +b [V (+b Dp ) V ] dp 2I c f 2I c 2 v2 = Q +vs c (+b c Dp )+(dv v )/ 3. Mass conservation within the reservoir chamber V b +b [V (+b Dp ) V ] I c f I c v dp 3 = Q dp 3 2 [(V p/n g0 g0 )/n][/p(n+)/n ] = Q 2 v(s c S v )(+b c Dp 2 )+(dv v2 )/ V b +b [V (+b Dp ) V ] 3 2I c f 2I c 2 v2 +b {V +V V [( p )/p ]/n} f 30 g0 g0 g0 3 dp 3 4. Equation of continuity = Q 3 V p/n g0 g0 +b n p(n+)/n 3 fc V 30 +V g0 g0a V p g0 p 3 B /n D r r Q = Q 2 Q 3 2 r 3 r Within the above three equations, there exist 5. Vapour volume within the traction chamber eight unknown quantities; therefore, five additional equations must be introduced. The first two equations If p >p come from using equation (3) applied separately v for the compression and traction chambers. Two dv v more equations come from valve equations applied =0 separately for the piston and base valves. The fifth equation comes from mass conservation If p p v m +m 2 +m 3 =cte [ dm + dm 2 + dm 3 dv v = (Mp )/(RT ) r v A r Q +r dv B =0 [ ṁ +ṁ 2 +ṁ 3 =0 6. Vapour volume within the compression chamber which, converted into volumetric flow is Q = Q 2 r 2 r Q 3 r 3 r If p 2 >p v dv v2 =0 8 SUMMARY OF THE MODELLING EQUATIONS. Mass conservation within the compression chamber If V v >0 dp =0 If V v 0 dp = Q +vs c (+b c Dp )+dv v / V I b c +b f [V I (+b c Dp ) V v ] 2. Mass conservation within the traction chamber If V v2 >0 dp 2 =0 If p 2 p v dv v2 = (Mp )/(RT ) r v 2 A r 2 Q 2 +r dv 2 2 B 7. Loss of pressure through the piston valves If Dp Dp lim p p 2 = 8rQ2 a p2d4 a C K a + f a L a D a DK piston If Dp>Dp lim Dp a = 8rQ2 a p2d4 a C K a + f a L a D a DK piston Dp b = 8rQ2 b p2d4 b C K b + f b L b D b +G A x2bdk piston Dp =Dp a =Dp b Q =Q a +Q b
Modelling a twin tube cavitating shock absorber 037 8. Loss of pressure through the base valves In the simulation, the conditions of which are indicated in the figure legend, the cavitation If Dp Dp lim base phenomenon does not take place because the speed is not sufficiently elevated to make the pressure p p = 8rQ2 a 3 p2d4 a C K a + f a L a D a DK within the compression chamber decrease to the base vapour pressure (which is a few pascals). If Dp>Dp lim base Figure 6 compares the damping force (A area shown in Fig. 5) which is obtained for the same shock Dp = 8rQ2 a a base p2d4 a C K a + f a L a absorber and under the same cycle conditions but D a DK base allowing for a higher vapour pressure of the working fluid which helps the cavitation appearance. Dp b base = 8rQ2 b p2d4 b C K b + f b L b D b +G A x2bdk base When vapour generation starts on the traction chamber (point A), part of the volume evacuated by Dp =Dp =Dp base a base b base the piston movement is occupied by the vapour. Next, the volumetric flow that crosses the piston Q =Q +Q base a base b base valves from the compression to the traction chamber 9. Oil volume on the compression chamber is reduced in relation to the non-cavitating flow. Because piston movement is constant, the fluid V =[V (x x )S ](+b Dp ) V f 0 0 c c v volume that should move from the compression to =V (+b Dp ) V I c v the traction chamber moves to the reserve chamber. Consequently, the compression pressure increases. 0. Oil volume on the traction chamber The damping force can be calculated as follows V =[V +(x x )(S S )](+b Dp ) V 2f 20 0 c v c 2 v2 F =p S p (S S )=( p p )S +p S =V (+b Dp ) V a c 2 c v 2 c 2 v 2I c 2 v2 The higher the pressure on the traction chamber. Oil volume on the reserve chamber and the lower the pressure loss between working V =V +V V chambers, the lower the damping force. Consequently, 3f 30 g0 g when cavitation starts, damping force decreases. 2. Oil density Once cavitation is underway, the modelling equations r=r eb for each model are different, and therefore, damping 0 f (p p 0 ) force evolution can only be known by integration. 9 RESULTS 0 CONCLUSIONS A new shock absorber model that is able to predict damper force from the main shock absorber geo- metry (i.e. rod diameter, piston diameter, chambers Assuming a positive speed when the shock absorber works in compression, numerical integration of the presented equations for a sinusoidal cycle leads to the force speed characteristic shown in Fig. 5. Fig. 5 Force-speed characteristic for a sinusoidal cycle. Frequency 7 Hz, amplitude 4 cm
038 M Alonso and Á Comas Fig. 6 Force versus speed for the same shock absorber but with fluids with different vapour pressure length, etc.) and the main physical properties of REFERENCES its parts (i.e. fluid viscosity and density, chambers elasticity, etc.) has been presented. If it is assumed Dixon, J. J. The shock absorber handbook, 200 that cavitation bubbles appear or disappear instant- (Society of Automobile Engineers, UK). aneously, no empirical or semi-empirical data is 2 Yabuta, K., Hidaka, K., and Fukushima, N. Influence of suspension friction on riding comfort, required to describe cavitation phenomena. This is the dynamics of vehicles on roads and on tracks. In the innovation of the presented model. Proceedings of the 7th International Association for As cavitation on twin tube shock absorbers Vehicle System Dynamics (IAVSD) Symposium, 98 normally takes place in the traction chamber during (Swets & Zeitlinger, Cambridge). compression, the model shows that the compression 3 Milliken, W. F. and Milliken, D. L. Race car vehicle force is reduced when cavitation occurs. dynamics, 995 (Society of Automobile Engineers, UK). 4 Bastow, D. Car suspension and handling, 987, Ch. 4 FUTURE WORK (Pentech Press, London). 5 Warner, B. and Rakheja, S. An analytical and experimental investigation of friction and gas spring As oil temperature greatly affects its physical proper- characteristics of racing car suspension dampers, ties (i.e. viscosity, density, conductivity, etc.), tem- SAE paper 962548, 996, pp. 489 502. perature affects the damping force. In order to know 6 Surace, C., Worden, K., and Tomlinson, G. R. On the actual damping force when the shock absorber the non-linear characteristics of automotive shock absorbers, Proc. IMechE, Part D: J. Automobile has been operating for a certain period of time under Engineering, 992, pp. 3 6. known working conditions (i.e. shock absorber speed, 7 Audenino, A. L. and Belingardi, G. Modelling the ambient temperature, etc.), a thermal analysis will dynamic behaviour of a motorcycle damper, Proc. be implemented. Heat transfer analysis based on a IMechE, Part D: J. Automobile Engineering, 995, calculated local map of temperatures will be con- pp. 249 262. ducted. Then, an energy conservation equation will 8 Duym, W. R. Simulation tools, modelling and be applied to all shock absorber elements (solid parts identification, for an automotive shock absorber, in the context of vehicle dynamics, Monroe European will be divided into small elements and each fluid Center, Vol. 33, Number 4. on each chamber will be treated as a thermodynamic 9 Segel, L. and Lang, H. H. The mechanics of system), leading to the temporal evolution of the automotive hydraulic dampers at high stroking local map of temperatures. frequencies, Veh. Syst. Dynamics, 98, pp. 79 83. Temperature evolution permits the adjustment of 0 Böswirth, L. A model for valve flow taking non steady the physical properties of the concerned parts, affect- flow into account, 998 (Eigenverlag, Wien). ing damper force. Therefore, the implementation of White, F. M. Mecánica de fluidos, 2004, pp. 842 848 (McGraw-Hill, London). thermal analysis will reveal not only the damper 2 Blevins, D. Applied fluid dynamics handbook, 2000, force as a function of the temperature but also local pp. 38 89 (Krieger Publishing Company, Florida). temperatures. A comparison of model results with 3 Young, F. R. Cavitation, 999, Ch. 2 (McGraw-Hill, experimental data will also be presented. London).
Modelling a twin tube cavitating shock absorber 039 4 Shigley, J. E. Diseño en ingeniería mecánica, 990, Q volumetric flow which enters a Ch. 3 (McGraw-Hill, London). compression chamber Q volumetric flow which enters a traction 2 chamber Q volumetric flow through a channel APPENDIX a Q volumetric flow through b channel b R gas universal constant Notation R outer radius of the working chamber e D a channel diameter R inner radius of the working chamber a i D b channel diameter R initial radius of the cavity b 0 D piston diameter S normal surface of the working chamber c c D blow-off valve disc diameter S rod surface d v D external shock absorber diameter t time ext D reserve chamber diameter (external) T system temperature re D reserve chamber diameter (internal) v piston speed ri D rod diameter V total volume of the system v E elasticity module of working chambers V gas volume g f a channel Darcy Weisbach friction V initial gas volume a g0 coefficient V control volume i f b channel Darcy Weisbach friction V non-deformed volume of the working b I coefficient chamber F function V volume of fluid (liquid phase) l F damping force V vapour volume a v G blow-off valve characteristic function V vapour volume of the compression v H function chamber K a channel singular loss of pressure V vapour volume of the traction chamber a v2 constant V volume of the compression chamber K b channel singular loss of pressure V fluid volume on the compression b f constant chamber L working chamber length V non-deformed volume of the compression I L a channel length chamber a L b channel length V initial fluid volume on the compression b 0 m total system mass (liquid plus vapour) chamber m total mass inside the control volume V volume of the traction chamber i 2 m fluid (liquid phase) mass V fluid volume of the traction chamber l 2f m vapour mass V non-deformed volume of the traction v 2I m fluid mass inside the compression chamber chamber V initial fluid volume of the traction 20 m fluid mass inside the traction chamber chamber 2 m fluid mass inside the reserve chamber V fluid volume of the reserve chamber 3 3f M vapour molar mass V volume of the reserve chamber 3 n polytropy gas index V initial fluid volume of the reserve 30 p pressure chamber p gas pressure x distance between the disc of the g p initial gas pressure blow-off valve and its seat or piston g0 p vapour pressure position v p reference pressure to obtain R and R or x piston initial position 0 i e 0 p p 2 p 3 Q Q i to obtain r 0 pressure in the compression chamber b equivalent chamber compressibility c pressure in the traction chamber factor pressure in the traction chamber b isothermal compressibility factor f volumetric flow Dp increase of pressure volumetric flow which enters a working Dp loss of pressure on the a channel a chamber Dp loss of pressure on the b channel b
040 M Alonso and Á Comas Dp minimum pressure to open the blow-off n Poisson s coefficient lim valve r density Dp pressure on the compression chamber r generic density i minus reference pressure ( p p ) r liquid density 0 l Dp pressure on the traction chamber minus r reference liquid density 2 0 reference pressure ( p p ) r fluid density in the compression chamber 2 0 DV increase of volume on the working r fluid density in the traction chamber 2 chamber r fluid density in the reserve chamber 3 L working chamber length t extinction time of the cavity