- Email: [email protected]
Development of a thermal model for automotive twin-tube shock absorbers
Applied Thermal Engineering 25 (2005) 1836–1853 www.elsevier.com/locate/apthermeng
Development of a thermal model for automotive twin-tube shock absorbers J.C. Ramos a
a,*
, A. Rivas a, J. Biera b, G. Sacramento b, J.A. Sala
b
TECNUN—Escuela Superior de Ingenieros, Campus Tecnolo´gico de la Universidad de Navarra, Paseo de Manuel Lardiza´bal, 13. 20018 San Sebastia´n, Spain b AP Amortiguadores, S.A. Ctra. Irurzun, s/n. 31171 Ororbia, Navarra, Spain Received 12 November 2003; accepted 8 November 2004 Available online 16 December 2004
Abstract This paper presents a model to predict the thermal performance of automotive twin-tube shock absorbers simulating a thermal stability test. The objective of this test is to determine the stabilization temperatures of the components of the shock absorber when it is under a multifrequencial test. The shock absorber has been modelled by dividing it into various subsystems that correspond to each working chamber and main components. The conservation of energy equation is applied to each subsystem. These equations are solved to obtain the evolution of the temperatures of the subsystems during the test time. Four shock absorbers based on different designs have been put to the thermal stability test. The temperatures of some components have been measured to compare with the results of the model and to validate it. 2004 Elsevier Ltd. All rights reserved. Keywords: Shock absorber; Thermal model; Heat transfer; Automotive
1. Introduction Shock absorbers are one of the most important elements in the passive security of vehicles. Their function is to dissipate the energy introduced in the vehicle by road irregularities and by driving *
Corresponding author. Tel.: +34 943 219877; fax: +34 943 311442. E-mail address: [email protected] (J.C. Ramos).
1359-4311/$ - see front matter 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.applthermaleng.2004.11.005
J.C. Ramos et al. / Applied Thermal Engineering 25 (2005) 1836–1853
1837
Nomenclature Latin letters Aint internal area of the reserve tube, m2 Aext external area of the pressure tube, m2 Ap piston section area, m2 Ar rod section area, m2 c specific heat, J/kg K specific kinetic energy, J/kg ek specific potential energy, J/kg ep f1, f2 sign functions F shock absorber force, N h heat transfer coefficient, W/m2 K i specific enthalpy, J/kg k thermal conductivity, W/m K K, K 0 proportionality constants Lsubscript length of the control volume indicated by the subscript, m L_ subscript velocity of variation of the length of the control volume indicated by the subscript, m/s m mass, kg m_ oil mass flow rate, kg/s p pressure, N/m2 Pint internal perimeter of the pressure tube, m Q_ heat transfer, W R air constant, =287 J/kg K t time, s T temperature, K T_ velocity of variation of the temperature, K/s u specific internal energy, J/kg U_ velocity of variation of internal energy, W Vair reserve chamber air volume, m3 V_ air velocity of variation of reserve chamber air volume, m3/s W_ power, W x shock absorber displacement, m x_ shock absorber velocity, m/s Greek letter q density, kg/m3 Subscripts air reserve chamber air amb ambient air outside the shock absorber
1838
b bco def g1, g2 in out p po pt r rco reco rtout2 rtout3 r1, r2,
J.C. Ramos et al. / Applied Thermal Engineering 25 (2005) 1836–1853
bump chamber bump chamber oil deformation parts 1 and 2 of the rod guide inlet flow outlet flow piston orifices of the piston pressure tube rebound chamber rebound chamber oil reserve chamber oil part 2 of the lateral surface of the reserve tube part 3 of the lateral surface of the reserve tube r3: parts 1, 2 and 3 of piston rod
situations. This kinetic energy of the relative movement between the sprung mass and the unsprung mass of the vehicle is dissipated as heat by the shock absorber by means of the viscous heating of the inner oil as it flows through the calibrated orifices of the shock absorbers inner components. As shock absorbers are part of an automobiles suspension system, the main efforts in developing shock absorber models have been made in a dynamic direction. Dynamic simulators of shock absorbers focus on the prediction of forces as a function of speed and displacement, by modelling the characteristics of their inner components. These models usually contribute to more complex models of the suspension or of the whole car to predict its dynamic performance. AP Amortiguadores, a Spanish manufacturer of original equipment, has developed several hydraulic–dynamic models, capable of characterizing all of their designs, from twin-tube and mono-tube shock absorbers to variable shock absorbers for semi-active suspensions. In those models the shock absorber force is related to the pressure variation in the oil chambers. This pressure change takes into account the effects of head losses of the oil when passing through the orifices and valves. In these models, the only thermal effect considered is the variation of oil viscosity with temperature. Obviously, this variation has an influence on the shock absorber forces. In addition, AP Amortiguadores has integrated their hydraulic–dynamic models in their suspension dynamic model, SDV, so the effects of the shock absorber hydraulics are taken into account in the dynamics of the vehicle suspension [1]. Another dynamic model of interest is that developed by Reybrouck [2], who model the pressure variation in the chambers taking into account different parameters, some of which are very difficult to determine, depending on the design studied. In this model, again, the only thermal effect considered is the variation of oil viscosity with the temperature. Following on from this work, Duym and Reybrouck [3] and Duym [4] use the thermal effect of the variation of oil viscosity with temperature in the calculation of parameters characterizing shock absorber valves. The model developed by Lee [5] also introduces the effect of oil viscosity reduction by heating to predict the decrease of shock absorber forces.
J.C. Ramos et al. / Applied Thermal Engineering 25 (2005) 1836–1853
1839
A shock absorber model combining dynamic and thermal effects has been developed by Lion and Loose [6]. Those effects are related by means of the energy conservation equation formulated over time. The power produced in the system by the external forces is introduced as the product of shock absorber force and speed. The shock absorber forces are related to the frictional, viscous and elastic displacements of the shock absorber. The variation of oil viscosity with temperature is also taken into account in the viscous forces calculation. The internal energy variation consists of one term of elastic energy variation due to elastic displacements of the shock absorber and of another term of sensible energy variation related to oil temperature. The heat transfer is modelled as one-dimensional. The internal convection between the oil and the pressure tube, the conduction through the pressure tube and its convection with the ambient air are all considered in order to calculate experimentally the product of the overall heat transfer coefficient and the area. The differential equation obtained from the First Law of Thermodynamics is used together with the dynamic model equations to predict the forces and the oil temperature. In this model the dependence of the overall heat transfer coefficient with the temperature is modelled by means of an exponential function with coefficients that need to be obtained experimentally for each shock absorber. Finally, studies on heat transfer in magneto-rheological fluid shock absorbers by Gordaninejad and Breese [7] and Dogruoz et al. [8] and [9] are considered. The shock absorber is modelled as a lumped capacitance system, considering all the components at the same temperature and with one-dimensional heat transfer. In these models the external convective heat transfer coefficient is assumed to be constant and unaffected by the temperature increase of the damper. In the previously cited thermal models only the oil temperature is taken into account. This can be a good approximation to introduce them in a more general dynamic model [6] or to predict the decrease of the forces with the temperature [5]. But if the objective is to know the performance of the shock absorber components and the influence of the shock absorber design during a thermal stability test, the differences in the temperature of the components must be included. In this paper a purely thermal model for shock absorbers is developed. The model is predictive because the heat transfer coefficients do not need to be determined previously depending on the type of shock absorber tested and on the test conditions. The objective of the model is to obtain the variation of temperature of the shock absorber components with time during a thermal stability test. In this type of test the shock absorber is subjected to a displacement wave of fixed shape and duration to determine the maximum temperatures of the shock absorbers components and to see how critical they are for performance. The model is based on the application of the energy conservation equation to every subsystem or component into which the shock absorber has been divided, and in solving the equations system for the temperatures of the components. The temperature results of the model are contrasted with the experimental results obtained during the testing of four shock absorbers.
2. System description The shock absorber has been divided into the following subsystems: the oil within the rebound, bump and reserve chambers; the reserve chamber air; the pressure and the reserve tubes; the piston; the PTFE (PolyTetraFluoroEthylene) coating of the piston; the bottom valve; the rod guide, and the rod. This is shown in Fig. 1.
1840
J.C. Ramos et al. / Applied Thermal Engineering 25 (2005) 1836–1853 REBOUND CHAMBER BUMP CHAMBER RESERVE CHAMBER
RESERVE TUBE BOTTOM VALVE
ROD GUIDE
ROD RESERVE CHAMBER AIR
PTFE COATING
PISTON
PRESSURE TUBE RESERVE CHAMBER OIL
Fig. 1. Cross-section drawing of an automotive twin-tube shock absorber and its main components.
Additionally, some components have been subdivided into two or more parts to take into account the different boundary conditions that they can have because of their contact with other components of the shock absorber. For example, the reserve tube is divided into four parts. The first part corresponds to the base or bottom part of the shock absorber, the second and the third correspond to the lateral surface and the fourth to the top part of the lateral surface in contact with the rod guide. Thus, the shock absorber has been finally modelled into 17 subsystems or control volumes. Applying the First Law of Thermodynamics to each subsystem, 17 differential equations are obtained with temperature as dependent variable and time as independent variable. The following simplifying hypotheses have been made: • The oil flow between the rebound chamber and the reserve chamber, through the gap between the rod and the guide, is negligible. • The top cap of the shock absorber is only modelled using the rod guide. The seal, the top of the reserve tube, the cover or any other part in the top part of the shock absorber are not modelled. • The thermophysical properties of oil and air (density, viscosity, specific heat and thermal conductivity) are dependent on temperature and, indirectly through it, on time. • The thermophysical properties of the solid components of the shock absorber are assumed constant and independent of temperature. The system input consists of the displacement, x(t), and the velocity, x_ ðtÞ, of the piston with positive sign in the rebound movement. Length variations of the chambers are related to these variables through suitable geometrical relations.
3. Theoretical model The theoretical model of the shock absorbers thermal performance is created by applying the energy conservation equation to the different subsystems, into which the shock absorber has been divided.
J.C. Ramos et al. / Applied Thermal Engineering 25 (2005) 1836–1853
1841
Each of the 17 subsystems can then be grouped into one of three types with respect to the energy equation. These are: open systems (consisting of rebound chamber oil, bump chamber oil and reserve chamber oil); closed system (consisting of the reserve chamber air); and rigid body systems (consisting of the solid components). The following sections show the development of the energy equation of one system of each type: the rebound chamber oil, the reserve chamber air and the lower part of the rod. 3.1. Energy equation for the rebound chamber oil The oil chambers of the shock absorber are studied as control volumes where the oil content and the volume vary with piston displacement. It should be noted that the oil does not enter and leave a chamber simultaneously, but sometimes is entering and sometimes is leaving according to the alternative movement of the piston. This phenomenon is taken into account in the equations by means of functions that depend on the sign of the shock absorber speed: 1 if x_ ðtÞ P 0 1 þ sign½_xðtÞ 1 sign½_xðtÞ and f 2 ðtÞ ¼ with sign½_xðtÞ ¼ f1 ðtÞ ¼ 2 2 1 if x_ ðtÞ < 0 The general equation of energy conservation for the rebound chamber oil is: d ½m ðu þ ek þ ep Þrco ¼ ½m_ ði þ ek þ ep Þin ½m_ ði þ ek þ ep Þout þ Q_ rco W_ def–rco dt
ð1Þ
In the first term of the equation (the rate of change of the total energy of the oil in the rebound chamber), the kinetic and potential energies are neglected and the expression is such that: d d ½m ðu þ ek þ ep Þrco ¼ ½m urco dt dt d ¼ ½qoil ðT rco Þ ðAp Ar Þ Lrco ðtÞ coil ðT rco Þ T rco ðtÞ dt ¼ qoil ðT rco Þ ðAp Ar Þ Lrco ðtÞ coil ðT rco Þ T_ rco ðtÞ þ qoil ðT rco Þ ðAp Ar Þ L_ rco ðtÞ coil ðT rco Þ T rco ðtÞ
ð2Þ
This equation represents two components of the internal energy variation: the first, the internal energy variation due to changes of oil temperature with respect to a constant control volume; and the second, the internal energy variation due to the variation of the oil content as a consequence of shock absorber movement. The two first terms on the right hand side of Eq. (1) are the total energy terms of the inlet and outlet flows. Depending on the shock absorber movement, only one of these flows exists and this condition is introduced in the equation through the speed sign functions previously defined. The conditions and properties of these inlet and outlet flows are supposed to be the same of the oil of the bump chamber and of the oil of the rebound chamber, respectively, and therefore the kinetic and the potential energies are negligible and the expression is reduced to the enthalpy term: bm_ ði þ ek þ ep Þcin bm_ ði þ ek þ ep Þcout ¼ ½m_ iin ½m_ iout
ð3Þ
1842
J.C. Ramos et al. / Applied Thermal Engineering 25 (2005) 1836–1853
The enthalpy is developed into the internal energy term and the pressure term: p ðtÞ ½m_ iin ½m_ iout ¼ f1 ðtÞ qoil ðT rco Þ ðAp Ar Þ L_ rco ðtÞ coil ðT rco Þ T rco ðtÞ þ rco qoil ðT rco Þ pbco ðtÞ _ þ f2 ðtÞ qoil ðT bco Þ ðAp Ar Þ Lrco ðtÞ coil ðT bco Þ T bco ðtÞ þ qoil ðT bco Þ ð4Þ In this equation, for positive values of x_ ðtÞ the shock absorber is in rebound movement and sign½_xðtÞ ¼ 1, f1(t) = 1 and f2(t) = 0, so the oil leaves the rebound chamber with the properties (density, specific heat, temperature and pressure) of this chamber. For negative values of x_ ðtÞ the shock absorber is in bump movement and sign½_xðtÞ ¼ 1, f1(t) = 0 and f2(t) = 1, so the oil enters the rebound chamber with the properties of the bump chamber. The following term in the energy equation corresponds to the heat transfer from rebound chamber oil to the subsystems surrounding it; i.e., to the convection heat transferred to the pressure tube, to the three parts into which the rod has been divided, to the piston and to part 1 of the guide. These convective heat transfers are introduced according to Newtons law of cooling as the product of the convection coefficient, the transmission area and temperature difference. For example, heat transfer to the pressure tube is given by: Q_ rco–pt ðtÞ ¼ hrco–pt ðtÞ P int Lrco ðtÞ ðT rco ðtÞ T pt ðtÞÞ
ð5Þ
In Eq. (5), hrco–pt(t) is the convection heat transfer coefficient between the rebound chamber oil and the pressure tube. This heat transfer coefficient varies with time because it depends on shock absorber speed and on oil and tube temperatures by means of the convection correlation used to calculate it. In the same way, heat transfer to the other components is calculated. The last term of Eq. (1) is the power related to the work made on or by the oil of the rebound chamber by pressure forces due to the control volume deformation. In this case the only power acting corresponds to the work of deformation of the rebound chamber oil volume and it is equal to the product of the rebound chamber pressure by the rate of volume deformation. It is also necessary to introduce the speed sign functions to take into account that the oil volume is compressing in the rebound movement and is expanding in the bump movement: W_ def–rco ðtÞ ¼ f1 ðtÞ prco ðtÞ ðAp Ar Þ x_ ðtÞ f2 ðtÞ prco ðtÞ ðAp Ar Þ x_ ðtÞ
ð6Þ
The energy equation of the rebound chamber with all the previous terms taken into account is as follows: qoil ðAp Ar Þ Lrco coil T_ rco qoil ðAp Ar Þ L_ rco coil T rco þ f1 qoil ðAp Ar Þ L_ rco prco pbco _ þ f2 qoil ðAp Ar Þ Lrco coil T bco þ Q_ rco–pt Q_ rco–r1 Q_ rco–r2 coil T rco þ qoil qoil Q_ rco–r3 Q_ rco–p Q_ rco–g1 þ f1 prco ðAp Ar Þ x_ þ f2 prco ðAp Ar Þ x_ ¼ 0 ð7Þ The model under development aims at being a purely thermal model, that is, the only unknown quantities must be the temperatures of the components of the shock absorber. Thus, in the
J.C. Ramos et al. / Applied Thermal Engineering 25 (2005) 1836–1853
1843
previous equation the rebound chamber and the bump chamber pressures, prco and pbco, must be eliminated. The way to do this is to introduce, assuming some simplifications, the total energy equation of the flow inside the orifices of the piston connecting the rebound and the bump chambers: d ð8Þ ½m ðu þ ek þ ep Þpo ¼ ½m_ ði þ ek þ ep Þin ½m_ ði þ ek þ ep Þout þ Q_ po W_ po dt The left hand side of the above equation represents the rate of change of the total energy of the oil inside the orifices. This term has not been taken into account because the unsteady effects inside the orifices can be neglected. Moreover, the last two terms of the right hand side of Eq. (8) are also eliminated because there is no energy transfer as work of heat through the boundaries of the piston orifices. In the inlet and outlet flow energy terms the potential energy are neglected and also in the first one the kinetic energy is neglected because the oil conditions are assumed to be those of the inlet chamber, rebound or bump chamber depending on the movement of the shock absorber. With these assumptions the total energy equation (8) sets that the total enthalpy of the oil entering the orifices is maintained at the outlet: ð9Þ ½m_ iin ¼ ½m_ ði þ ek Þout ) iin ¼ ði þ ek Þout For clearness, only the terms corresponding to the rebound movement of this equation are going to be developed. So, for positive values of x_ ðtÞ the shock absorber is in rebound movement and the inlet and outlet chambers are, respectively, the rebound chamber and the bump chamber, and Eq. (9) adopts the following expression: p p 0 ¼ ðcoil T Þr–b þ þ ekðr–bÞ ð10Þ coil T þ qoil rco qoil bco By passing through the orifices of the piston the internal energy of the oil is going to be increased by the friction and other irreversibilities. This means that the temperature of the oil leaving the orifices is going to be higher than that of the oil at the chamber it is coming from (T 0 > Trco). As it is not desirable to introduce a new temperature in the model not corresponding to one of the components of the shock absorber, the effect of this heating has been introduced as a term proportional to the kinetic energy of the oil flowing from the rebound to the bump chamber, ek(r–b): ðcoil T 0 Þr–b ðcoil T Þrco ¼ K 0 ekðr–bÞ Introducing Eq. (11) into Eq. (10) and doing K = K 0 + 1: p p ¼ K ekðr–bÞ þ qoil rco qoil bco
ð11Þ
ð12Þ
A similar expression can be deduced for the case when the shock absorber is in bump movement, when the inlet chamber is the bump and the outlet is the rebound chamber: p p ¼ K ekðb–rÞ þ ð13Þ qoil bco qoil rco where ek(b–r) is the kinetic energy of the oil flowing from the bump to the rebound chamber.
1844
J.C. Ramos et al. / Applied Thermal Engineering 25 (2005) 1836–1853
Introducing Eq. (12) pre-multiplied by f1 and Eq. (13) pre-multiplied by f2 in the energy equation of the rebound chamber oil (7): qoil ðAp Ar Þ Lrco coil T_ rco qoil ðAp Ar Þ L_ rco coil T rco þ f1 qoil ðAp Ar Þ L_ rco coil T rco þ K f1 qoil ðAp Ar Þ L_ rco ekðr–bÞ p _ þ f1 qoil ðAp Ar Þ Lrco þ f2 qoil ðAp Ar Þ L_ rco coil T bco qoil bco p þ K f2 qoil ðAp Ar Þ L_ rco ekðb–rÞ þ f2 qoil ðAp Ar Þ L_ rco qoil rco Q_ rco–pt Q_ rco–r1 Q_ rco–r2 Q_ rco–r3 Q_ rco–p Q_ rco–g1 þ f1 prco ðAp Ar Þ x_ þ f2 prco ðAp Ar Þ x_ ¼ 0
ð14Þ
Assuming that the oil density difference between the two chambers in one time step is negligible, taking into account that L_ rco ðtÞ ¼ _xðtÞ and grouping the pressure terms, the previous equation is reduced to: qoil ðAp Ar Þ Lrco coil T_ rco qoil ðAp Ar Þ L_ rco coil T rco þ f1 qoil ðAp Ar Þ L_ rco coil T rco þ K f1 qoil ðAp Ar Þ L_ rco ekðr–bÞ þ f2 qoil ðAp Ar Þ L_ rco coil T bco þ K f2 qoil ðAp Ar Þ L_ rco ekðb–rÞ Q_ rco–pt Q_ rco–r1 Q_ rco–r2 Q_ rco–r3 Q_ rco–p Q_ rco–g1 þ f1 ðprco pbco Þ ðAp Ar Þ x_ ¼ 0 ð15Þ The product of the difference between the rebound and bump pressures by the area of the piston minus the area of the rod can be approximated to the force of the shock absorber, that is known experimentally: F(t) ffi [prco(t) pbco(t)] Æ (Ap Ar). In this approximation we are assuming that the total power of the shock absorber is used in the rebound chamber during the rebound movement, but a part of this power, proportional to pbco Æ Ar, is actually used in the reserve chamber to provoke the oil flow from this chamber to the bump chamber. Normally in the shock absorbers the restriction to the oil flow in the valves between the reserve and the bump chambers are very weak, so this part of the shock absorber power used in the reserve chamber can be assumed very small compared with the part exerted in the rebound chamber. During the bump movement of the shock absorber the total power is exerted in the bump chamber because the oil is only flowing out from it and no simplification is assumed. In Eq. (15) the constant, K, multiplying the kinetic energy of the oil inflow or of the oil outflow, depending on the movement, has to value more than 1 because it includes two effects, the kinetic energy of the flow and the increment of internal energy of the flow due to the mechanical energy losses when passing through the orifices of the piston. As the importance of this term is unknown, in a first approximation of the model the constant of proportionality is assumed to be 1, and the effect of its variation will be studied lately.
J.C. Ramos et al. / Applied Thermal Engineering 25 (2005) 1836–1853
1845
With this last assumption, the final expression of the energy equation of the rebound chamber oil is: qoil ðAp Ar Þ Lrco coil T_ rco qoil ðAp Ar Þ L_ rco coil T rco þ f1 qoil ðAp Ar Þ L_ rco coil T rco þ f1 qoil ðAp Ar Þ L_ rco ekðr–bÞ þ f2 qoil ðAp Ar Þ L_ rco coil T bco þ f2 qoil ðAp Ar Þ L_ rco ekðb–rÞ Q_ rco–pt Q_ rco–r1 Q_ rco–r2 Q_ rco–r3 Q_ rco–p Q_ rco–g1 þ f1 F x_ ¼ 0
ð16Þ
Similar expressions are obtained for the bump chamber oil and for the reserve chamber oil. In the latter, as it has been explained, the term for shock absorber power does not appear. 3.2. Energy equation for the reserve chamber air As this is a closed subsystem, the energy flow terms disappear from the energy equation: d ½m ðu þ ec þ ep Þair ¼ Q_ air W_ def–air dt
ð17Þ
In the reserve chamber air mass is constant and the increase or decrease of oil quantity with the movement of the shock absorber provokes variations of air pressure and volume. The ideal gas equation relates these pressure and volume changes to temperature variations: pair ðtÞ V air ðtÞ ¼ pair ðtÞ ðAint Aext Þ Lair ðtÞ ¼ mair R T air ðtÞ
ð18Þ
In Eq. (17) the kinetic and potential energy terms are negligible and the internal energy variation is only due to the change of the air temperature: d d d ½m ðu þ ec þ ep Þair ¼ ðm uÞair ¼ ½mair cair ðT air Þ T air ðtÞ dt dt dt ¼ mair cair ðT air Þ T_ air ðtÞ
ð19Þ
The heat transfer term of Eq. (17) includes the convection heat transfer from the air to the pressure tube, to the reserve chamber oil, to part 2 of the guide and to the reserve tube. The power term corresponds to the deformation work done to compress the air by the variation of the oil height with the movement of the shock absorber: L_ air ðtÞ W_ def–air ðtÞ ¼ pair ðtÞ V_ air ðtÞ ¼ pair ðtÞ ðAint Aext Þ L_ air ðtÞ ¼ mair R T air ðtÞ Lair ðtÞ
ð20Þ
The expression of the conservation of energy equation with all the terms is: mair cair ðT air Þ T_ air ðtÞ Q_ air–pt ðtÞ Q_ air–reco ðtÞ Q_ air–g2 ðtÞ Q_ air–rtout2 ðtÞ L_ air ðtÞ Q_ air–rtout3 ðtÞ þ mair R T air ðtÞ ¼0 Lair ðtÞ
ð21Þ
1846
J.C. Ramos et al. / Applied Thermal Engineering 25 (2005) 1836–1853
3.3. Energy equation for the lower part of the rod In the solid components of the shock absorber, the energy equation becomes the equivalence between the internal energy and the heat exchange: Q_ r1 ¼ U_ r1
ð22Þ
The internal energy variation of the lower part of the rod is given by its temperature variation: U_ r1 ðtÞ ¼ mr1 cr1 T_ r1 ðtÞ
ð23Þ
The heat transfer term includes the convection heat transfer to the rebound chamber oil and to the ambient air (if the lower part of the rod extends beyond the pressure tube during a rebound movement) and the conduction transfer to the piston, to part 2 (the intermediate part) of the rod and to the rod guide (assuming perfect contact when the rod goes through the guide). Conduction heat transfer is modelled, by considering the whole control volume at uniform temperature and by introducing the thermal resistance of each subsystem. For example, the equation for the conduction heat transfer between part 1 of the rod and the piston is: T r1 ðtÞ T p ðtÞ ð24Þ Q_ r1–p ðtÞ ¼ Lr1 =2 Lp =2 þ k r Ar k p Ar The final expression of the energy equation for this subsystem is: mr1 cr1 T_ r1 ðtÞ þ Q_ r1–rco þ Q_ r1–amb þ Q_ r1–p þ Q_ r1–r2 þ Q_ r1–g1 ¼ 0
ð25Þ
4. Model implementation Applying the First Law of Thermodynamics to the 17 subsystems of the shock absorber, as explained in the previous section, a system of 17 non-linear first order differential equations with time as independent variable and the temperatures as dependent variables is obtained. The non-linearity is a consequence of the dependence of the oil and the air properties on temperature and of the dependence of the convection coefficients on temperature, displacement and velocity. The differential equations are integrated by the Eulers explicit method, with a time step of 0.01 s and evaluating the non-linear coefficients at the preceding time. This integration method has been chosen after comparing the speed of the simulation and the results obtained with other high order methods (Runge-Kutta) [10]. The above mentioned numerical method has been implemented in Matlab. The known inputs of the system are the position and speed of the shock absorber, defined by the test wave being simulated. The model data are: the dimensions of the shock absorber and its components; the thermophysical properties of oil and air and their dependence on temperature; the force–speed curves of the shock absorber for different oil temperatures, and the relations between the shock absorber velocity and the oil velocity when flowing between the different chambers. These two last data are obtained from the hydraulic–dynamic models of AP Amortiguadores.
J.C. Ramos et al. / Applied Thermal Engineering 25 (2005) 1836–1853
1847
5. Convection correlations The convection correlations used in the model have been obtained from the bibliography: Incropera [11], Kakac¸ [12] and Kreith [13]. The correlations considered most appropriate for the flow conditions existing in the shock absorber have been used for each subsystem. For the heat transfer between the rebound chamber oil and the pressure tube, between the rebound chamber oil and the rod and between the bump chamber oil and the pressure tube a convection correlation for internal flow has been used. For the convection heat transfers between the reserve chamber oil and the pressure tube and between the reserve chamber oil and the reserve tube a similar correlation has been used. A correlation for flow perpendicular to a disc has been introduced to model the convective heat transfers between the oil of every chamber and the piston, the bottom valve, part 1 of the rod guide and the bottom part of the reserve tube, according to each case. To model the heat transfer from the base of the shock absorber and from the outside surface of the guide to the ambient air, a correlation for free convection on a horizontal isothermal surface has been introduced. This correlation has been also used for the heat transfers from the reserve chamber air to the inner surface of the guide and to the reserve chamber oil. The heat exchange between the outside lateral surface of the reserve tube and the ambient air is introduced into the model with a correlation for external flow on an isothermal surface. The Reynolds number has been evaluated at the instantaneous speed of the shock absorber. As in the test the rod is fixed (it is the reserve tube which moves), a correlation for free convection over a vertical cylinder has been used to calculate its heat transfer to the air surrounding the shock absorber. Finally, the correlation used for the heat transfer between the reserve chamber air and the pressure and the reserve tubes is that which corresponds to free convection in an enclosure. The convection correlations described above are for specific flow conditions not corresponding exactly to those existing in the shock absorber. Because of this, some coefficients of some correlations were adjusted to validate the thermal model, by comparing the temperatures obtained in the simulations of the model with the ones obtained in the experimental tests. Previous to this adjustment, the relative importance in the results of the correlations chosen was evaluated to determine the most influential ones. This sensibility analysis have been carried out comparing the results of the model obtained with the nominal expressions of the correlations as they appear in the bibliography with the results obtained when the correlations coefficients are increased by a factor of two. The study shows than the more influential correlations are the internal flow convection correlation used between the oil of the different chambers and the tubes of the shock absorber and the external flow convection correlation between the outside tube of the shock absorber and the ambient air. In the first case the relative difference in the incremental temperature from the initial values to the final values is of a 10% and in the second case is about 20%. For the rest of the correlations employed the relative difference in the incremental temperature between the two simulations of the model are lower than 3%. Taking into account these results, the coefficients of the oil internal flow correlation and the ambient air external flow correlation have been modified in order to adjust the results of the
1848
J.C. Ramos et al. / Applied Thermal Engineering 25 (2005) 1836–1853
model with the experimental ones. Both correlations are of the type Nu = C Æ Rem Æ Pr1/3, and the coefficients C and m are the ones who have been fitted.
6. Experimental tests Experimental tests have been undertaken in AP Amortiguadores in a hydraulic machine (Servosis, model no. 0119) with a maximum load capacity of 2000 da N, a maximum speed of 1.5 m/s, a maximum frequency of 25 Hz and a maximum stroke of 150 mm. Fig. 2 shows the arrangement of a shock absorber in the machine during the test. The thermal stability test consists of the repetition, a certain number of times, of a test cycle composed of a bi-frequency senoidal wave, consisting of the sum of a low-frequency, high-amplitude wave and a high-frequency, low-amplitude wave. The bi-frequency wave is followed by a single low-frequency wave. During the thermal stability tests, temperatures were measured by means of four thermocouples placed at three points on the lateral surface of the reserve tube and at an intermediate point of the rod. The three thermocouples were placed in the reserve tube towards the bottom part, at an intermediate point and at the top of the reserve tube, in contact with the rod guide, respectively. Fig. 3 shows the situation of the thermocouples on the shock absorber. Four shock absorbers with different valve settings and designs have been tested and the signals from the thermocouples recorded to compare with the results of the model, to adjust and to validate it.
Fig. 2. Experimental test machine with a shock absorber.
J.C. Ramos et al. / Applied Thermal Engineering 25 (2005) 1836–1853
1849
Fig. 3. Thermocouples on a tested shock absorber.
7. Theoretical–experimental agreement The theoretical model has been run for the conditions of the thermal stability test previously indicated and with the geometrical and functional characteristics of the four shock absorbers tested. Table 1 shows the results obtained in the tests and in the model simulations for the four temperatures measured in the four shock absorbers tested. The values correspond to the temperature achieved at the end of the thermal stability test, known as the stabilisation temperature.
Table 1 Comparison of stabilisation temperatures for the test and the model Shock absorber
A B C D
Experiment
Model
Relative difference T experiment T model T experiment T ini
Tini (C)
Ttop (C)
Tint (C)
Tlow (C)
Trod (C)
Ttop (C)
Tint (C)
Tlow (C)
Trod (C)
Ttop (%)
Tint (%)
Tlow (%)
Trod (%)
27 27 25 23
78 79 103 116
76 75 99 108
74 73 93 105
67 56 75 63
76 81 101 111
72 75 96 106
77 79 99 110
69 51 93 86
4 15 3 5
8 0 4 2
6 13 9 6
5 17 36 58
1850
J.C. Ramos et al. / Applied Thermal Engineering 25 (2005) 1836–1853
It can be observed that the relative differences of the temperature increments in the model results for the three temperatures measured in the reserve tube are smaller than 15%, giving a good correlation. For the rod temperature increments there is more dispersion in the values of the relative differences between the different shock absorbers tested. It is possible that several different effects not previously taken into account might explain this divergence. For example, shock absorbers C and D, which have the highest stabilisation temperatures, will have a large temperature gradient in the rod, between the internal part in contact with the oil and the external part in contact with the ambient air. In such cases, the standard subdivision of the rod into three parts may not be enough to simulate the temperature distribution, making it necessary to model the rod into more subdivisions. This indicates that when there is a large temperature gradient in the rod and it is modelled divided into more than three parts, the correspondence between the position of the thermocouples during the thermal stability test and the modelled parts of the rod must be checked to compare the results. Another possibility to explain the differences in the rod temperatures is that the conduction heat transfer through the clamping equipment of the rod to the test machine has not been modelled. The introduction of this heat transfer would lead to lower rod temperatures. Another difference between the model and the test results that can be observed in Table 1 is in the prediction of the stabilization temperatures of the intermediate and lower parts of the reserve tube. In the test the intermediate part has a higher temperature than the lower part, whilst in the model simulations the opposite occurs. It seems that, as in case of the rod, the main cause for the divergence in the temperature of the lower part of the external tube might be that the model does not take into account the conduction heat transfer through the clamping equipment of the reserve tube to the test machine. The introduction of this heat transfer would lead to a lower temperature of the bottom part and to a more suitable temperature gradient in the external tube. Figs. 4–7 compare the test and model results for the temperature evolution during the thermal stability tests of the four components measured in shock absorbers A and B. It can be seen that for both shock absorbers the prediction of the stabilisation temperatures of the model corresponds very well with the experimental results. Shock absorber B. T top 90
80
80
70
70
60 50 test model
40 30 20 10
Temperature (˚C)
Temperature (˚C)
Shock absorber A. T top 90
60 50 40
test model
30 20 10
0
0
0
1000 2000 3000 4000 5000 6000 time (s)
0
1000
2000
3000
4000
5000
time (s)
Fig. 4. Test and model temperatures of the top part of the reserve tube of shock absorbers A and B.
J.C. Ramos et al. / Applied Thermal Engineering 25 (2005) 1836–1853 Shock absorber B. T int
80
80
70
70
60 50 40
test model
30 20
Temperature (˚C)
Temperature (˚C)
Shock absorber A. T int
1851
10
60 50 40
test model
30 20 10 0
0 0
0
1000 2000 3000 4000 5000 6000
1000
2000
3000
4000
5000
time (s)
time (s)
Fig. 5. Test and model temperatures of the intermediate part of the reserve tube of shock absorbers A and B.
Shock absorber A. T low
90
Shock absorber B. T low 90 80
70 60 50 40
test model
30 20
Temperature (˚C)
Temperature (˚C)
80
10
70 60 50 test model
40 30 20 10
0
0 0
1000 2000 3000 4000 5000 6000
0
1000
2000
3000
4000
5000
time (s)
time (s)
Fig. 6. Test and model temperatures of the lower part of the reserve tube of shock absorbers A and B.
Shock absorber A. T rod
Shock absorber B. T rod
80
60 50
60 50 40
test model
30 20
Temperature (˚C)
Temperature (˚C)
70
40 30
test model
20 10
10 0
0
0
1000 2000 3000 4000 5000 6000 time (s)
0
1000
2000
3000
4000
5000
time (s)
Fig. 7. Test and model temperatures of the rod of shock absorbers A and B.
For shock absorber A the evolution of the four temperatures measured presents a good agreement between the test and the model results.
1852
J.C. Ramos et al. / Applied Thermal Engineering 25 (2005) 1836–1853
But for shock absorber B the reserve tube temperatures during the thermal stability test are, at some points, a 20% lower than the predicted by the model. This difference could be due to the fact that the adjustment of the coefficients of the convection correlations mentioned in point 5 has been done based only on the temperatures of the external tubes of the shock absorber but not on the temperatures of the internal components. In the experimental tests performed so far only the external temperatures of the shock absorber have been measured. In the sensibility analysis of the convection correlations, it was found that one of the more influential heat transfer coefficients affecting the behaviour of the whole system was the one associated to the heat transfer between the internal oil flow in the rebound and bump chambers and the pressure tube. As the adjustment of the coefficients of this correlation has been made without knowing the internal temperatures, the discrepancy in the transient temperatures can be attributed to this. It seems that a better adjustment would be obtained if the temperatures of the internal components of the shock absorber were measured.
8. Conclusions A thermal model for automotive twin tube shock absorbers has been developed to simulate the temperature evolution of their components during a thermal stability test. The thermal stability test has been carried out on four shock absorbers of different designs. During the tests the temperature evolution of certain components has been measured by means of thermocouples. The results have been recorded to compare with those obtained with the model in order to validate it. The relative differences of the reserve tube stabilisation temperature increments between the model and the test are less than 15% for the four shock absorbers. Relating the temperature evolution during the test, the maximum difference between the model and the experimental results is about a 20% for shock absorber B. To get a better correlation between the model and the experimental results, it is proposed that the temperatures of the internal components of the shock absorber are measured to adjust the coefficients of the internal convection correlations. It is also suggested that the conduction heat transfer to the test machine through the shock absorber clamping system is introduced. In the near future it is also proposed that the model be extended to accommodate other input signals, for example real road displacement signals.
References [1] G.J. Izaguirre, J.M. Jime´nez, J. Biera, J.A. Sala, SDV: A new multi-body environment for vehicle dynamics simulation, in: 6th US National Congress on Computational Mechanics, Dearborn, USA, 2001. [2] K. Reybrouck, A non linear parametric model of an automotive shock absorber, SAE paper 940869, SAE International Congress, SAE (eds.), Detroit, USA, 1994, pp. 79–86. [3] S. Duym, K. Reybrouck, Physical characterization of nonlinear shock absorber dynamics, European Journal of Mechanical and Environmental Engineering 43 (4) (1994) 181–188.
J.C. Ramos et al. / Applied Thermal Engineering 25 (2005) 1836–1853
1853
[4] S. Duym, Simulation tools, modelling and identification for an automotive shock absorber in the context of vehicle dynamics, Vehicle System Dynamics 33 (2000) 261–285. [5] K. Lee, Numerical modelling for the hydraulic performance prediction of automotive monotube dampers, Vehicle System Dynamics 28 (1997) 25–39. [6] A. Lion, S. Loose, A thermomechanically coupled model for automotive shock absorbers: theory, experiments and vehicle simulations on test tracks, Vehicle System Dynamics 37 (4) (1997) 241–261. [7] F. Gordaninejad, D. Breese, Heating of magneto-rheological fluid dampers, Journal of Intelligent Material System and Structures 10 (8) (1999) 634–645. [8] M. Dogruoz, F. Gordaninejad, E. Wang, A. Stipanovich, Heat transfer for magneto-rheological fluid dampers, in: S.C. Liu (Ed.), Smart Structures and Materials 2000: Smart Systems for Bridges, Structures and Highways, Proceedings of SPIE, Bellingham, USA, vol. 3988, 2000, pp. 84–93. [9] M. Dogruoz, F. Gordaninejad, E. Wang, A. Stipanovich, An experimental study on heat transfer from fail-safe magneto-rheological fluid dampers, in: D.J. Inman (Ed.), Smart Structures and Materials 2001: Damping and Isolation, Proceedings of SPIE, Newport Beach, USA, vol. 4331, 2001, pp. 355–369. [10] W.H. Press, S.A. Teulosky, W.T. Vetterling, B.P. Flannery, Numerical Recipes in C, second ed., Cambridge University Press, Cambridge, 1992. [11] F.P. Incropera, D.P. DeWitt, Fundamentals of Heat and Mass Transfer, fifth ed., John Wiley & Sons, New York, 2002. [12] S. Kakac¸, R.K. Shah, W. Aung, Handbook of Single-phase Convective Heat Transfer, first ed., John Wiley & Sons, New York, 1987. [13] F. Kreith, M.S. Bohn, Principios de transferencia de calor, sixth ed., Thomson-Paraninfo, Madrid, 2002.
Development of a thermal model for automotive twin-tube shock absorbers J.C. Ramos a
a,*
, A. Rivas a, J. Biera b, G. Sacramento b, J.A. Sala
b
TECNUN—Escuela Superior de Ingenieros, Campus Tecnolo´gico de la Universidad de Navarra, Paseo de Manuel Lardiza´bal, 13. 20018 San Sebastia´n, Spain b AP Amortiguadores, S.A. Ctra. Irurzun, s/n. 31171 Ororbia, Navarra, Spain Received 12 November 2003; accepted 8 November 2004 Available online 16 December 2004
Abstract This paper presents a model to predict the thermal performance of automotive twin-tube shock absorbers simulating a thermal stability test. The objective of this test is to determine the stabilization temperatures of the components of the shock absorber when it is under a multifrequencial test. The shock absorber has been modelled by dividing it into various subsystems that correspond to each working chamber and main components. The conservation of energy equation is applied to each subsystem. These equations are solved to obtain the evolution of the temperatures of the subsystems during the test time. Four shock absorbers based on different designs have been put to the thermal stability test. The temperatures of some components have been measured to compare with the results of the model and to validate it. 2004 Elsevier Ltd. All rights reserved. Keywords: Shock absorber; Thermal model; Heat transfer; Automotive
1. Introduction Shock absorbers are one of the most important elements in the passive security of vehicles. Their function is to dissipate the energy introduced in the vehicle by road irregularities and by driving *
Corresponding author. Tel.: +34 943 219877; fax: +34 943 311442. E-mail address: [email protected] (J.C. Ramos).
1359-4311/$ - see front matter 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.applthermaleng.2004.11.005
J.C. Ramos et al. / Applied Thermal Engineering 25 (2005) 1836–1853
1837
Nomenclature Latin letters Aint internal area of the reserve tube, m2 Aext external area of the pressure tube, m2 Ap piston section area, m2 Ar rod section area, m2 c specific heat, J/kg K specific kinetic energy, J/kg ek specific potential energy, J/kg ep f1, f2 sign functions F shock absorber force, N h heat transfer coefficient, W/m2 K i specific enthalpy, J/kg k thermal conductivity, W/m K K, K 0 proportionality constants Lsubscript length of the control volume indicated by the subscript, m L_ subscript velocity of variation of the length of the control volume indicated by the subscript, m/s m mass, kg m_ oil mass flow rate, kg/s p pressure, N/m2 Pint internal perimeter of the pressure tube, m Q_ heat transfer, W R air constant, =287 J/kg K t time, s T temperature, K T_ velocity of variation of the temperature, K/s u specific internal energy, J/kg U_ velocity of variation of internal energy, W Vair reserve chamber air volume, m3 V_ air velocity of variation of reserve chamber air volume, m3/s W_ power, W x shock absorber displacement, m x_ shock absorber velocity, m/s Greek letter q density, kg/m3 Subscripts air reserve chamber air amb ambient air outside the shock absorber
1838
b bco def g1, g2 in out p po pt r rco reco rtout2 rtout3 r1, r2,
J.C. Ramos et al. / Applied Thermal Engineering 25 (2005) 1836–1853
bump chamber bump chamber oil deformation parts 1 and 2 of the rod guide inlet flow outlet flow piston orifices of the piston pressure tube rebound chamber rebound chamber oil reserve chamber oil part 2 of the lateral surface of the reserve tube part 3 of the lateral surface of the reserve tube r3: parts 1, 2 and 3 of piston rod
situations. This kinetic energy of the relative movement between the sprung mass and the unsprung mass of the vehicle is dissipated as heat by the shock absorber by means of the viscous heating of the inner oil as it flows through the calibrated orifices of the shock absorbers inner components. As shock absorbers are part of an automobiles suspension system, the main efforts in developing shock absorber models have been made in a dynamic direction. Dynamic simulators of shock absorbers focus on the prediction of forces as a function of speed and displacement, by modelling the characteristics of their inner components. These models usually contribute to more complex models of the suspension or of the whole car to predict its dynamic performance. AP Amortiguadores, a Spanish manufacturer of original equipment, has developed several hydraulic–dynamic models, capable of characterizing all of their designs, from twin-tube and mono-tube shock absorbers to variable shock absorbers for semi-active suspensions. In those models the shock absorber force is related to the pressure variation in the oil chambers. This pressure change takes into account the effects of head losses of the oil when passing through the orifices and valves. In these models, the only thermal effect considered is the variation of oil viscosity with temperature. Obviously, this variation has an influence on the shock absorber forces. In addition, AP Amortiguadores has integrated their hydraulic–dynamic models in their suspension dynamic model, SDV, so the effects of the shock absorber hydraulics are taken into account in the dynamics of the vehicle suspension [1]. Another dynamic model of interest is that developed by Reybrouck [2], who model the pressure variation in the chambers taking into account different parameters, some of which are very difficult to determine, depending on the design studied. In this model, again, the only thermal effect considered is the variation of oil viscosity with the temperature. Following on from this work, Duym and Reybrouck [3] and Duym [4] use the thermal effect of the variation of oil viscosity with temperature in the calculation of parameters characterizing shock absorber valves. The model developed by Lee [5] also introduces the effect of oil viscosity reduction by heating to predict the decrease of shock absorber forces.
J.C. Ramos et al. / Applied Thermal Engineering 25 (2005) 1836–1853
1839
A shock absorber model combining dynamic and thermal effects has been developed by Lion and Loose [6]. Those effects are related by means of the energy conservation equation formulated over time. The power produced in the system by the external forces is introduced as the product of shock absorber force and speed. The shock absorber forces are related to the frictional, viscous and elastic displacements of the shock absorber. The variation of oil viscosity with temperature is also taken into account in the viscous forces calculation. The internal energy variation consists of one term of elastic energy variation due to elastic displacements of the shock absorber and of another term of sensible energy variation related to oil temperature. The heat transfer is modelled as one-dimensional. The internal convection between the oil and the pressure tube, the conduction through the pressure tube and its convection with the ambient air are all considered in order to calculate experimentally the product of the overall heat transfer coefficient and the area. The differential equation obtained from the First Law of Thermodynamics is used together with the dynamic model equations to predict the forces and the oil temperature. In this model the dependence of the overall heat transfer coefficient with the temperature is modelled by means of an exponential function with coefficients that need to be obtained experimentally for each shock absorber. Finally, studies on heat transfer in magneto-rheological fluid shock absorbers by Gordaninejad and Breese [7] and Dogruoz et al. [8] and [9] are considered. The shock absorber is modelled as a lumped capacitance system, considering all the components at the same temperature and with one-dimensional heat transfer. In these models the external convective heat transfer coefficient is assumed to be constant and unaffected by the temperature increase of the damper. In the previously cited thermal models only the oil temperature is taken into account. This can be a good approximation to introduce them in a more general dynamic model [6] or to predict the decrease of the forces with the temperature [5]. But if the objective is to know the performance of the shock absorber components and the influence of the shock absorber design during a thermal stability test, the differences in the temperature of the components must be included. In this paper a purely thermal model for shock absorbers is developed. The model is predictive because the heat transfer coefficients do not need to be determined previously depending on the type of shock absorber tested and on the test conditions. The objective of the model is to obtain the variation of temperature of the shock absorber components with time during a thermal stability test. In this type of test the shock absorber is subjected to a displacement wave of fixed shape and duration to determine the maximum temperatures of the shock absorbers components and to see how critical they are for performance. The model is based on the application of the energy conservation equation to every subsystem or component into which the shock absorber has been divided, and in solving the equations system for the temperatures of the components. The temperature results of the model are contrasted with the experimental results obtained during the testing of four shock absorbers.
2. System description The shock absorber has been divided into the following subsystems: the oil within the rebound, bump and reserve chambers; the reserve chamber air; the pressure and the reserve tubes; the piston; the PTFE (PolyTetraFluoroEthylene) coating of the piston; the bottom valve; the rod guide, and the rod. This is shown in Fig. 1.
1840
J.C. Ramos et al. / Applied Thermal Engineering 25 (2005) 1836–1853 REBOUND CHAMBER BUMP CHAMBER RESERVE CHAMBER
RESERVE TUBE BOTTOM VALVE
ROD GUIDE
ROD RESERVE CHAMBER AIR
PTFE COATING
PISTON
PRESSURE TUBE RESERVE CHAMBER OIL
Fig. 1. Cross-section drawing of an automotive twin-tube shock absorber and its main components.
Additionally, some components have been subdivided into two or more parts to take into account the different boundary conditions that they can have because of their contact with other components of the shock absorber. For example, the reserve tube is divided into four parts. The first part corresponds to the base or bottom part of the shock absorber, the second and the third correspond to the lateral surface and the fourth to the top part of the lateral surface in contact with the rod guide. Thus, the shock absorber has been finally modelled into 17 subsystems or control volumes. Applying the First Law of Thermodynamics to each subsystem, 17 differential equations are obtained with temperature as dependent variable and time as independent variable. The following simplifying hypotheses have been made: • The oil flow between the rebound chamber and the reserve chamber, through the gap between the rod and the guide, is negligible. • The top cap of the shock absorber is only modelled using the rod guide. The seal, the top of the reserve tube, the cover or any other part in the top part of the shock absorber are not modelled. • The thermophysical properties of oil and air (density, viscosity, specific heat and thermal conductivity) are dependent on temperature and, indirectly through it, on time. • The thermophysical properties of the solid components of the shock absorber are assumed constant and independent of temperature. The system input consists of the displacement, x(t), and the velocity, x_ ðtÞ, of the piston with positive sign in the rebound movement. Length variations of the chambers are related to these variables through suitable geometrical relations.
3. Theoretical model The theoretical model of the shock absorbers thermal performance is created by applying the energy conservation equation to the different subsystems, into which the shock absorber has been divided.
J.C. Ramos et al. / Applied Thermal Engineering 25 (2005) 1836–1853
1841
Each of the 17 subsystems can then be grouped into one of three types with respect to the energy equation. These are: open systems (consisting of rebound chamber oil, bump chamber oil and reserve chamber oil); closed system (consisting of the reserve chamber air); and rigid body systems (consisting of the solid components). The following sections show the development of the energy equation of one system of each type: the rebound chamber oil, the reserve chamber air and the lower part of the rod. 3.1. Energy equation for the rebound chamber oil The oil chambers of the shock absorber are studied as control volumes where the oil content and the volume vary with piston displacement. It should be noted that the oil does not enter and leave a chamber simultaneously, but sometimes is entering and sometimes is leaving according to the alternative movement of the piston. This phenomenon is taken into account in the equations by means of functions that depend on the sign of the shock absorber speed: 1 if x_ ðtÞ P 0 1 þ sign½_xðtÞ 1 sign½_xðtÞ and f 2 ðtÞ ¼ with sign½_xðtÞ ¼ f1 ðtÞ ¼ 2 2 1 if x_ ðtÞ < 0 The general equation of energy conservation for the rebound chamber oil is: d ½m ðu þ ek þ ep Þrco ¼ ½m_ ði þ ek þ ep Þin ½m_ ði þ ek þ ep Þout þ Q_ rco W_ def–rco dt
ð1Þ
In the first term of the equation (the rate of change of the total energy of the oil in the rebound chamber), the kinetic and potential energies are neglected and the expression is such that: d d ½m ðu þ ek þ ep Þrco ¼ ½m urco dt dt d ¼ ½qoil ðT rco Þ ðAp Ar Þ Lrco ðtÞ coil ðT rco Þ T rco ðtÞ dt ¼ qoil ðT rco Þ ðAp Ar Þ Lrco ðtÞ coil ðT rco Þ T_ rco ðtÞ þ qoil ðT rco Þ ðAp Ar Þ L_ rco ðtÞ coil ðT rco Þ T rco ðtÞ
ð2Þ
This equation represents two components of the internal energy variation: the first, the internal energy variation due to changes of oil temperature with respect to a constant control volume; and the second, the internal energy variation due to the variation of the oil content as a consequence of shock absorber movement. The two first terms on the right hand side of Eq. (1) are the total energy terms of the inlet and outlet flows. Depending on the shock absorber movement, only one of these flows exists and this condition is introduced in the equation through the speed sign functions previously defined. The conditions and properties of these inlet and outlet flows are supposed to be the same of the oil of the bump chamber and of the oil of the rebound chamber, respectively, and therefore the kinetic and the potential energies are negligible and the expression is reduced to the enthalpy term: bm_ ði þ ek þ ep Þcin bm_ ði þ ek þ ep Þcout ¼ ½m_ iin ½m_ iout
ð3Þ
1842
J.C. Ramos et al. / Applied Thermal Engineering 25 (2005) 1836–1853
The enthalpy is developed into the internal energy term and the pressure term: p ðtÞ ½m_ iin ½m_ iout ¼ f1 ðtÞ qoil ðT rco Þ ðAp Ar Þ L_ rco ðtÞ coil ðT rco Þ T rco ðtÞ þ rco qoil ðT rco Þ pbco ðtÞ _ þ f2 ðtÞ qoil ðT bco Þ ðAp Ar Þ Lrco ðtÞ coil ðT bco Þ T bco ðtÞ þ qoil ðT bco Þ ð4Þ In this equation, for positive values of x_ ðtÞ the shock absorber is in rebound movement and sign½_xðtÞ ¼ 1, f1(t) = 1 and f2(t) = 0, so the oil leaves the rebound chamber with the properties (density, specific heat, temperature and pressure) of this chamber. For negative values of x_ ðtÞ the shock absorber is in bump movement and sign½_xðtÞ ¼ 1, f1(t) = 0 and f2(t) = 1, so the oil enters the rebound chamber with the properties of the bump chamber. The following term in the energy equation corresponds to the heat transfer from rebound chamber oil to the subsystems surrounding it; i.e., to the convection heat transferred to the pressure tube, to the three parts into which the rod has been divided, to the piston and to part 1 of the guide. These convective heat transfers are introduced according to Newtons law of cooling as the product of the convection coefficient, the transmission area and temperature difference. For example, heat transfer to the pressure tube is given by: Q_ rco–pt ðtÞ ¼ hrco–pt ðtÞ P int Lrco ðtÞ ðT rco ðtÞ T pt ðtÞÞ
ð5Þ
In Eq. (5), hrco–pt(t) is the convection heat transfer coefficient between the rebound chamber oil and the pressure tube. This heat transfer coefficient varies with time because it depends on shock absorber speed and on oil and tube temperatures by means of the convection correlation used to calculate it. In the same way, heat transfer to the other components is calculated. The last term of Eq. (1) is the power related to the work made on or by the oil of the rebound chamber by pressure forces due to the control volume deformation. In this case the only power acting corresponds to the work of deformation of the rebound chamber oil volume and it is equal to the product of the rebound chamber pressure by the rate of volume deformation. It is also necessary to introduce the speed sign functions to take into account that the oil volume is compressing in the rebound movement and is expanding in the bump movement: W_ def–rco ðtÞ ¼ f1 ðtÞ prco ðtÞ ðAp Ar Þ x_ ðtÞ f2 ðtÞ prco ðtÞ ðAp Ar Þ x_ ðtÞ
ð6Þ
The energy equation of the rebound chamber with all the previous terms taken into account is as follows: qoil ðAp Ar Þ Lrco coil T_ rco qoil ðAp Ar Þ L_ rco coil T rco þ f1 qoil ðAp Ar Þ L_ rco prco pbco _ þ f2 qoil ðAp Ar Þ Lrco coil T bco þ Q_ rco–pt Q_ rco–r1 Q_ rco–r2 coil T rco þ qoil qoil Q_ rco–r3 Q_ rco–p Q_ rco–g1 þ f1 prco ðAp Ar Þ x_ þ f2 prco ðAp Ar Þ x_ ¼ 0 ð7Þ The model under development aims at being a purely thermal model, that is, the only unknown quantities must be the temperatures of the components of the shock absorber. Thus, in the
J.C. Ramos et al. / Applied Thermal Engineering 25 (2005) 1836–1853
1843
previous equation the rebound chamber and the bump chamber pressures, prco and pbco, must be eliminated. The way to do this is to introduce, assuming some simplifications, the total energy equation of the flow inside the orifices of the piston connecting the rebound and the bump chambers: d ð8Þ ½m ðu þ ek þ ep Þpo ¼ ½m_ ði þ ek þ ep Þin ½m_ ði þ ek þ ep Þout þ Q_ po W_ po dt The left hand side of the above equation represents the rate of change of the total energy of the oil inside the orifices. This term has not been taken into account because the unsteady effects inside the orifices can be neglected. Moreover, the last two terms of the right hand side of Eq. (8) are also eliminated because there is no energy transfer as work of heat through the boundaries of the piston orifices. In the inlet and outlet flow energy terms the potential energy are neglected and also in the first one the kinetic energy is neglected because the oil conditions are assumed to be those of the inlet chamber, rebound or bump chamber depending on the movement of the shock absorber. With these assumptions the total energy equation (8) sets that the total enthalpy of the oil entering the orifices is maintained at the outlet: ð9Þ ½m_ iin ¼ ½m_ ði þ ek Þout ) iin ¼ ði þ ek Þout For clearness, only the terms corresponding to the rebound movement of this equation are going to be developed. So, for positive values of x_ ðtÞ the shock absorber is in rebound movement and the inlet and outlet chambers are, respectively, the rebound chamber and the bump chamber, and Eq. (9) adopts the following expression: p p 0 ¼ ðcoil T Þr–b þ þ ekðr–bÞ ð10Þ coil T þ qoil rco qoil bco By passing through the orifices of the piston the internal energy of the oil is going to be increased by the friction and other irreversibilities. This means that the temperature of the oil leaving the orifices is going to be higher than that of the oil at the chamber it is coming from (T 0 > Trco). As it is not desirable to introduce a new temperature in the model not corresponding to one of the components of the shock absorber, the effect of this heating has been introduced as a term proportional to the kinetic energy of the oil flowing from the rebound to the bump chamber, ek(r–b): ðcoil T 0 Þr–b ðcoil T Þrco ¼ K 0 ekðr–bÞ Introducing Eq. (11) into Eq. (10) and doing K = K 0 + 1: p p ¼ K ekðr–bÞ þ qoil rco qoil bco
ð11Þ
ð12Þ
A similar expression can be deduced for the case when the shock absorber is in bump movement, when the inlet chamber is the bump and the outlet is the rebound chamber: p p ¼ K ekðb–rÞ þ ð13Þ qoil bco qoil rco where ek(b–r) is the kinetic energy of the oil flowing from the bump to the rebound chamber.
1844
J.C. Ramos et al. / Applied Thermal Engineering 25 (2005) 1836–1853
Introducing Eq. (12) pre-multiplied by f1 and Eq. (13) pre-multiplied by f2 in the energy equation of the rebound chamber oil (7): qoil ðAp Ar Þ Lrco coil T_ rco qoil ðAp Ar Þ L_ rco coil T rco þ f1 qoil ðAp Ar Þ L_ rco coil T rco þ K f1 qoil ðAp Ar Þ L_ rco ekðr–bÞ p _ þ f1 qoil ðAp Ar Þ Lrco þ f2 qoil ðAp Ar Þ L_ rco coil T bco qoil bco p þ K f2 qoil ðAp Ar Þ L_ rco ekðb–rÞ þ f2 qoil ðAp Ar Þ L_ rco qoil rco Q_ rco–pt Q_ rco–r1 Q_ rco–r2 Q_ rco–r3 Q_ rco–p Q_ rco–g1 þ f1 prco ðAp Ar Þ x_ þ f2 prco ðAp Ar Þ x_ ¼ 0
ð14Þ
Assuming that the oil density difference between the two chambers in one time step is negligible, taking into account that L_ rco ðtÞ ¼ _xðtÞ and grouping the pressure terms, the previous equation is reduced to: qoil ðAp Ar Þ Lrco coil T_ rco qoil ðAp Ar Þ L_ rco coil T rco þ f1 qoil ðAp Ar Þ L_ rco coil T rco þ K f1 qoil ðAp Ar Þ L_ rco ekðr–bÞ þ f2 qoil ðAp Ar Þ L_ rco coil T bco þ K f2 qoil ðAp Ar Þ L_ rco ekðb–rÞ Q_ rco–pt Q_ rco–r1 Q_ rco–r2 Q_ rco–r3 Q_ rco–p Q_ rco–g1 þ f1 ðprco pbco Þ ðAp Ar Þ x_ ¼ 0 ð15Þ The product of the difference between the rebound and bump pressures by the area of the piston minus the area of the rod can be approximated to the force of the shock absorber, that is known experimentally: F(t) ffi [prco(t) pbco(t)] Æ (Ap Ar). In this approximation we are assuming that the total power of the shock absorber is used in the rebound chamber during the rebound movement, but a part of this power, proportional to pbco Æ Ar, is actually used in the reserve chamber to provoke the oil flow from this chamber to the bump chamber. Normally in the shock absorbers the restriction to the oil flow in the valves between the reserve and the bump chambers are very weak, so this part of the shock absorber power used in the reserve chamber can be assumed very small compared with the part exerted in the rebound chamber. During the bump movement of the shock absorber the total power is exerted in the bump chamber because the oil is only flowing out from it and no simplification is assumed. In Eq. (15) the constant, K, multiplying the kinetic energy of the oil inflow or of the oil outflow, depending on the movement, has to value more than 1 because it includes two effects, the kinetic energy of the flow and the increment of internal energy of the flow due to the mechanical energy losses when passing through the orifices of the piston. As the importance of this term is unknown, in a first approximation of the model the constant of proportionality is assumed to be 1, and the effect of its variation will be studied lately.
J.C. Ramos et al. / Applied Thermal Engineering 25 (2005) 1836–1853
1845
With this last assumption, the final expression of the energy equation of the rebound chamber oil is: qoil ðAp Ar Þ Lrco coil T_ rco qoil ðAp Ar Þ L_ rco coil T rco þ f1 qoil ðAp Ar Þ L_ rco coil T rco þ f1 qoil ðAp Ar Þ L_ rco ekðr–bÞ þ f2 qoil ðAp Ar Þ L_ rco coil T bco þ f2 qoil ðAp Ar Þ L_ rco ekðb–rÞ Q_ rco–pt Q_ rco–r1 Q_ rco–r2 Q_ rco–r3 Q_ rco–p Q_ rco–g1 þ f1 F x_ ¼ 0
ð16Þ
Similar expressions are obtained for the bump chamber oil and for the reserve chamber oil. In the latter, as it has been explained, the term for shock absorber power does not appear. 3.2. Energy equation for the reserve chamber air As this is a closed subsystem, the energy flow terms disappear from the energy equation: d ½m ðu þ ec þ ep Þair ¼ Q_ air W_ def–air dt
ð17Þ
In the reserve chamber air mass is constant and the increase or decrease of oil quantity with the movement of the shock absorber provokes variations of air pressure and volume. The ideal gas equation relates these pressure and volume changes to temperature variations: pair ðtÞ V air ðtÞ ¼ pair ðtÞ ðAint Aext Þ Lair ðtÞ ¼ mair R T air ðtÞ
ð18Þ
In Eq. (17) the kinetic and potential energy terms are negligible and the internal energy variation is only due to the change of the air temperature: d d d ½m ðu þ ec þ ep Þair ¼ ðm uÞair ¼ ½mair cair ðT air Þ T air ðtÞ dt dt dt ¼ mair cair ðT air Þ T_ air ðtÞ
ð19Þ
The heat transfer term of Eq. (17) includes the convection heat transfer from the air to the pressure tube, to the reserve chamber oil, to part 2 of the guide and to the reserve tube. The power term corresponds to the deformation work done to compress the air by the variation of the oil height with the movement of the shock absorber: L_ air ðtÞ W_ def–air ðtÞ ¼ pair ðtÞ V_ air ðtÞ ¼ pair ðtÞ ðAint Aext Þ L_ air ðtÞ ¼ mair R T air ðtÞ Lair ðtÞ
ð20Þ
The expression of the conservation of energy equation with all the terms is: mair cair ðT air Þ T_ air ðtÞ Q_ air–pt ðtÞ Q_ air–reco ðtÞ Q_ air–g2 ðtÞ Q_ air–rtout2 ðtÞ L_ air ðtÞ Q_ air–rtout3 ðtÞ þ mair R T air ðtÞ ¼0 Lair ðtÞ
ð21Þ
1846
J.C. Ramos et al. / Applied Thermal Engineering 25 (2005) 1836–1853
3.3. Energy equation for the lower part of the rod In the solid components of the shock absorber, the energy equation becomes the equivalence between the internal energy and the heat exchange: Q_ r1 ¼ U_ r1
ð22Þ
The internal energy variation of the lower part of the rod is given by its temperature variation: U_ r1 ðtÞ ¼ mr1 cr1 T_ r1 ðtÞ
ð23Þ
The heat transfer term includes the convection heat transfer to the rebound chamber oil and to the ambient air (if the lower part of the rod extends beyond the pressure tube during a rebound movement) and the conduction transfer to the piston, to part 2 (the intermediate part) of the rod and to the rod guide (assuming perfect contact when the rod goes through the guide). Conduction heat transfer is modelled, by considering the whole control volume at uniform temperature and by introducing the thermal resistance of each subsystem. For example, the equation for the conduction heat transfer between part 1 of the rod and the piston is: T r1 ðtÞ T p ðtÞ ð24Þ Q_ r1–p ðtÞ ¼ Lr1 =2 Lp =2 þ k r Ar k p Ar The final expression of the energy equation for this subsystem is: mr1 cr1 T_ r1 ðtÞ þ Q_ r1–rco þ Q_ r1–amb þ Q_ r1–p þ Q_ r1–r2 þ Q_ r1–g1 ¼ 0
ð25Þ
4. Model implementation Applying the First Law of Thermodynamics to the 17 subsystems of the shock absorber, as explained in the previous section, a system of 17 non-linear first order differential equations with time as independent variable and the temperatures as dependent variables is obtained. The non-linearity is a consequence of the dependence of the oil and the air properties on temperature and of the dependence of the convection coefficients on temperature, displacement and velocity. The differential equations are integrated by the Eulers explicit method, with a time step of 0.01 s and evaluating the non-linear coefficients at the preceding time. This integration method has been chosen after comparing the speed of the simulation and the results obtained with other high order methods (Runge-Kutta) [10]. The above mentioned numerical method has been implemented in Matlab. The known inputs of the system are the position and speed of the shock absorber, defined by the test wave being simulated. The model data are: the dimensions of the shock absorber and its components; the thermophysical properties of oil and air and their dependence on temperature; the force–speed curves of the shock absorber for different oil temperatures, and the relations between the shock absorber velocity and the oil velocity when flowing between the different chambers. These two last data are obtained from the hydraulic–dynamic models of AP Amortiguadores.
J.C. Ramos et al. / Applied Thermal Engineering 25 (2005) 1836–1853
1847
5. Convection correlations The convection correlations used in the model have been obtained from the bibliography: Incropera [11], Kakac¸ [12] and Kreith [13]. The correlations considered most appropriate for the flow conditions existing in the shock absorber have been used for each subsystem. For the heat transfer between the rebound chamber oil and the pressure tube, between the rebound chamber oil and the rod and between the bump chamber oil and the pressure tube a convection correlation for internal flow has been used. For the convection heat transfers between the reserve chamber oil and the pressure tube and between the reserve chamber oil and the reserve tube a similar correlation has been used. A correlation for flow perpendicular to a disc has been introduced to model the convective heat transfers between the oil of every chamber and the piston, the bottom valve, part 1 of the rod guide and the bottom part of the reserve tube, according to each case. To model the heat transfer from the base of the shock absorber and from the outside surface of the guide to the ambient air, a correlation for free convection on a horizontal isothermal surface has been introduced. This correlation has been also used for the heat transfers from the reserve chamber air to the inner surface of the guide and to the reserve chamber oil. The heat exchange between the outside lateral surface of the reserve tube and the ambient air is introduced into the model with a correlation for external flow on an isothermal surface. The Reynolds number has been evaluated at the instantaneous speed of the shock absorber. As in the test the rod is fixed (it is the reserve tube which moves), a correlation for free convection over a vertical cylinder has been used to calculate its heat transfer to the air surrounding the shock absorber. Finally, the correlation used for the heat transfer between the reserve chamber air and the pressure and the reserve tubes is that which corresponds to free convection in an enclosure. The convection correlations described above are for specific flow conditions not corresponding exactly to those existing in the shock absorber. Because of this, some coefficients of some correlations were adjusted to validate the thermal model, by comparing the temperatures obtained in the simulations of the model with the ones obtained in the experimental tests. Previous to this adjustment, the relative importance in the results of the correlations chosen was evaluated to determine the most influential ones. This sensibility analysis have been carried out comparing the results of the model obtained with the nominal expressions of the correlations as they appear in the bibliography with the results obtained when the correlations coefficients are increased by a factor of two. The study shows than the more influential correlations are the internal flow convection correlation used between the oil of the different chambers and the tubes of the shock absorber and the external flow convection correlation between the outside tube of the shock absorber and the ambient air. In the first case the relative difference in the incremental temperature from the initial values to the final values is of a 10% and in the second case is about 20%. For the rest of the correlations employed the relative difference in the incremental temperature between the two simulations of the model are lower than 3%. Taking into account these results, the coefficients of the oil internal flow correlation and the ambient air external flow correlation have been modified in order to adjust the results of the
1848
J.C. Ramos et al. / Applied Thermal Engineering 25 (2005) 1836–1853
model with the experimental ones. Both correlations are of the type Nu = C Æ Rem Æ Pr1/3, and the coefficients C and m are the ones who have been fitted.
6. Experimental tests Experimental tests have been undertaken in AP Amortiguadores in a hydraulic machine (Servosis, model no. 0119) with a maximum load capacity of 2000 da N, a maximum speed of 1.5 m/s, a maximum frequency of 25 Hz and a maximum stroke of 150 mm. Fig. 2 shows the arrangement of a shock absorber in the machine during the test. The thermal stability test consists of the repetition, a certain number of times, of a test cycle composed of a bi-frequency senoidal wave, consisting of the sum of a low-frequency, high-amplitude wave and a high-frequency, low-amplitude wave. The bi-frequency wave is followed by a single low-frequency wave. During the thermal stability tests, temperatures were measured by means of four thermocouples placed at three points on the lateral surface of the reserve tube and at an intermediate point of the rod. The three thermocouples were placed in the reserve tube towards the bottom part, at an intermediate point and at the top of the reserve tube, in contact with the rod guide, respectively. Fig. 3 shows the situation of the thermocouples on the shock absorber. Four shock absorbers with different valve settings and designs have been tested and the signals from the thermocouples recorded to compare with the results of the model, to adjust and to validate it.
Fig. 2. Experimental test machine with a shock absorber.
J.C. Ramos et al. / Applied Thermal Engineering 25 (2005) 1836–1853
1849
Fig. 3. Thermocouples on a tested shock absorber.
7. Theoretical–experimental agreement The theoretical model has been run for the conditions of the thermal stability test previously indicated and with the geometrical and functional characteristics of the four shock absorbers tested. Table 1 shows the results obtained in the tests and in the model simulations for the four temperatures measured in the four shock absorbers tested. The values correspond to the temperature achieved at the end of the thermal stability test, known as the stabilisation temperature.
Table 1 Comparison of stabilisation temperatures for the test and the model Shock absorber
A B C D
Experiment
Model
Relative difference T experiment T model T experiment T ini
Tini (C)
Ttop (C)
Tint (C)
Tlow (C)
Trod (C)
Ttop (C)
Tint (C)
Tlow (C)
Trod (C)
Ttop (%)
Tint (%)
Tlow (%)
Trod (%)
27 27 25 23
78 79 103 116
76 75 99 108
74 73 93 105
67 56 75 63
76 81 101 111
72 75 96 106
77 79 99 110
69 51 93 86
4 15 3 5
8 0 4 2
6 13 9 6
5 17 36 58
1850
J.C. Ramos et al. / Applied Thermal Engineering 25 (2005) 1836–1853
It can be observed that the relative differences of the temperature increments in the model results for the three temperatures measured in the reserve tube are smaller than 15%, giving a good correlation. For the rod temperature increments there is more dispersion in the values of the relative differences between the different shock absorbers tested. It is possible that several different effects not previously taken into account might explain this divergence. For example, shock absorbers C and D, which have the highest stabilisation temperatures, will have a large temperature gradient in the rod, between the internal part in contact with the oil and the external part in contact with the ambient air. In such cases, the standard subdivision of the rod into three parts may not be enough to simulate the temperature distribution, making it necessary to model the rod into more subdivisions. This indicates that when there is a large temperature gradient in the rod and it is modelled divided into more than three parts, the correspondence between the position of the thermocouples during the thermal stability test and the modelled parts of the rod must be checked to compare the results. Another possibility to explain the differences in the rod temperatures is that the conduction heat transfer through the clamping equipment of the rod to the test machine has not been modelled. The introduction of this heat transfer would lead to lower rod temperatures. Another difference between the model and the test results that can be observed in Table 1 is in the prediction of the stabilization temperatures of the intermediate and lower parts of the reserve tube. In the test the intermediate part has a higher temperature than the lower part, whilst in the model simulations the opposite occurs. It seems that, as in case of the rod, the main cause for the divergence in the temperature of the lower part of the external tube might be that the model does not take into account the conduction heat transfer through the clamping equipment of the reserve tube to the test machine. The introduction of this heat transfer would lead to a lower temperature of the bottom part and to a more suitable temperature gradient in the external tube. Figs. 4–7 compare the test and model results for the temperature evolution during the thermal stability tests of the four components measured in shock absorbers A and B. It can be seen that for both shock absorbers the prediction of the stabilisation temperatures of the model corresponds very well with the experimental results. Shock absorber B. T top 90
80
80
70
70
60 50 test model
40 30 20 10
Temperature (˚C)
Temperature (˚C)
Shock absorber A. T top 90
60 50 40
test model
30 20 10
0
0
0
1000 2000 3000 4000 5000 6000 time (s)
0
1000
2000
3000
4000
5000
time (s)
Fig. 4. Test and model temperatures of the top part of the reserve tube of shock absorbers A and B.
J.C. Ramos et al. / Applied Thermal Engineering 25 (2005) 1836–1853 Shock absorber B. T int
80
80
70
70
60 50 40
test model
30 20
Temperature (˚C)
Temperature (˚C)
Shock absorber A. T int
1851
10
60 50 40
test model
30 20 10 0
0 0
0
1000 2000 3000 4000 5000 6000
1000
2000
3000
4000
5000
time (s)
time (s)
Fig. 5. Test and model temperatures of the intermediate part of the reserve tube of shock absorbers A and B.
Shock absorber A. T low
90
Shock absorber B. T low 90 80
70 60 50 40
test model
30 20
Temperature (˚C)
Temperature (˚C)
80
10
70 60 50 test model
40 30 20 10
0
0 0
1000 2000 3000 4000 5000 6000
0
1000
2000
3000
4000
5000
time (s)
time (s)
Fig. 6. Test and model temperatures of the lower part of the reserve tube of shock absorbers A and B.
Shock absorber A. T rod
Shock absorber B. T rod
80
60 50
60 50 40
test model
30 20
Temperature (˚C)
Temperature (˚C)
70
40 30
test model
20 10
10 0
0
0
1000 2000 3000 4000 5000 6000 time (s)
0
1000
2000
3000
4000
5000
time (s)
Fig. 7. Test and model temperatures of the rod of shock absorbers A and B.
For shock absorber A the evolution of the four temperatures measured presents a good agreement between the test and the model results.
1852
J.C. Ramos et al. / Applied Thermal Engineering 25 (2005) 1836–1853
But for shock absorber B the reserve tube temperatures during the thermal stability test are, at some points, a 20% lower than the predicted by the model. This difference could be due to the fact that the adjustment of the coefficients of the convection correlations mentioned in point 5 has been done based only on the temperatures of the external tubes of the shock absorber but not on the temperatures of the internal components. In the experimental tests performed so far only the external temperatures of the shock absorber have been measured. In the sensibility analysis of the convection correlations, it was found that one of the more influential heat transfer coefficients affecting the behaviour of the whole system was the one associated to the heat transfer between the internal oil flow in the rebound and bump chambers and the pressure tube. As the adjustment of the coefficients of this correlation has been made without knowing the internal temperatures, the discrepancy in the transient temperatures can be attributed to this. It seems that a better adjustment would be obtained if the temperatures of the internal components of the shock absorber were measured.
8. Conclusions A thermal model for automotive twin tube shock absorbers has been developed to simulate the temperature evolution of their components during a thermal stability test. The thermal stability test has been carried out on four shock absorbers of different designs. During the tests the temperature evolution of certain components has been measured by means of thermocouples. The results have been recorded to compare with those obtained with the model in order to validate it. The relative differences of the reserve tube stabilisation temperature increments between the model and the test are less than 15% for the four shock absorbers. Relating the temperature evolution during the test, the maximum difference between the model and the experimental results is about a 20% for shock absorber B. To get a better correlation between the model and the experimental results, it is proposed that the temperatures of the internal components of the shock absorber are measured to adjust the coefficients of the internal convection correlations. It is also suggested that the conduction heat transfer to the test machine through the shock absorber clamping system is introduced. In the near future it is also proposed that the model be extended to accommodate other input signals, for example real road displacement signals.
References [1] G.J. Izaguirre, J.M. Jime´nez, J. Biera, J.A. Sala, SDV: A new multi-body environment for vehicle dynamics simulation, in: 6th US National Congress on Computational Mechanics, Dearborn, USA, 2001. [2] K. Reybrouck, A non linear parametric model of an automotive shock absorber, SAE paper 940869, SAE International Congress, SAE (eds.), Detroit, USA, 1994, pp. 79–86. [3] S. Duym, K. Reybrouck, Physical characterization of nonlinear shock absorber dynamics, European Journal of Mechanical and Environmental Engineering 43 (4) (1994) 181–188.
J.C. Ramos et al. / Applied Thermal Engineering 25 (2005) 1836–1853
1853
[4] S. Duym, Simulation tools, modelling and identification for an automotive shock absorber in the context of vehicle dynamics, Vehicle System Dynamics 33 (2000) 261–285. [5] K. Lee, Numerical modelling for the hydraulic performance prediction of automotive monotube dampers, Vehicle System Dynamics 28 (1997) 25–39. [6] A. Lion, S. Loose, A thermomechanically coupled model for automotive shock absorbers: theory, experiments and vehicle simulations on test tracks, Vehicle System Dynamics 37 (4) (1997) 241–261. [7] F. Gordaninejad, D. Breese, Heating of magneto-rheological fluid dampers, Journal of Intelligent Material System and Structures 10 (8) (1999) 634–645. [8] M. Dogruoz, F. Gordaninejad, E. Wang, A. Stipanovich, Heat transfer for magneto-rheological fluid dampers, in: S.C. Liu (Ed.), Smart Structures and Materials 2000: Smart Systems for Bridges, Structures and Highways, Proceedings of SPIE, Bellingham, USA, vol. 3988, 2000, pp. 84–93. [9] M. Dogruoz, F. Gordaninejad, E. Wang, A. Stipanovich, An experimental study on heat transfer from fail-safe magneto-rheological fluid dampers, in: D.J. Inman (Ed.), Smart Structures and Materials 2001: Damping and Isolation, Proceedings of SPIE, Newport Beach, USA, vol. 4331, 2001, pp. 355–369. [10] W.H. Press, S.A. Teulosky, W.T. Vetterling, B.P. Flannery, Numerical Recipes in C, second ed., Cambridge University Press, Cambridge, 1992. [11] F.P. Incropera, D.P. DeWitt, Fundamentals of Heat and Mass Transfer, fifth ed., John Wiley & Sons, New York, 2002. [12] S. Kakac¸, R.K. Shah, W. Aung, Handbook of Single-phase Convective Heat Transfer, first ed., John Wiley & Sons, New York, 1987. [13] F. Kreith, M.S. Bohn, Principios de transferencia de calor, sixth ed., Thomson-Paraninfo, Madrid, 2002.