Jtam-A4.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 56, 3, pp. 615-629, Warsaw 2018 DOI: 10.15632/jtam-pl.56.3.615 NONLINEAR MODELING AND ANALYSIS OF A SHOCK ABSORBER WITH A BYPASS Urszula Ferdek, Jan Łuczko Cracow University of Technology, Faculty of Mechanical Engineering, Kraków, Poland e-mail: uferdek@mech.pk.edu.pl; jluczko@mech.pk.edu.pl The model of a mono-tube shock absorber with a bypass is proposed in this paper. It is shown that the application of an additional flow passage (bypass) causes changes to the damping force characteristics when the excitation amplitudes are large. In such cases, the damping force values increase, thereby improving safety of the ride. For small excitation amplitudes, the shock absorber behaves in a similar fashion as shock absorbers without a bypass, ensuring a high comfort level of the ride on roads with smooth surfaces. Keywords: shock absorber, hydraulic damper, vehicle suspension, nonlinear, vibrations 1. Introduction The problems with the modelling and analysis of hydraulic dampers are discussed in several papers. It results from the fact of their wide applications (especially in the automotive industry) as well as from frequent employement of new damper designs (Norgaard and Cimins, 2009; King, 2014;Marking, 2014). These new concepts, often intuitively introduced, require successive theoretical solutions. One of such ideas is the application of an additional flow passage,meaning bypasses. Such solutions are introduced in order to improve safety of the ride, mainly in cross- -country vehicles. When such vehicles run onto a large obstacle, the oil flow through a bypass becomes blocked off. As a result, the damping force increases suddenly, changing the damper characteristics. The constructional parameters of the damper with a bypass should be selected in such a way as to have – during travelling on smooth surfaces – the characteristic not worse than the characteristic of a hydraulic damper without a bypass. Good comfortable rides are provided by dampers of ‘soft’ characteristics while dampers of ‘hard’ characteristics increase the safety, assuring better control and higher braking forces. In order to bring together these contradictory requirements, semi-active systems (Ferdek and Łuczko, 2015, 2016) usually magneto-rheological (Sapiński and Rosół, 2007; Gołdasz, 2015) are often applied. Compared to passive systems, semi-active dampers provide the possibility of adjusting the damping force to specific conditions of the ride. However, they have more complicated construction and due to that, they are more expensive both in production and in operations. Currently used dampers have a flow passage which can be situated inside the piston rod (King, 2014) or outside the working cylinder (Norgaard and Cimins, 2009; Marking, 2014). Such bypasses can have an additional pressure valve controlling the oil flow. The application of the bypass changes the dampers characteristics. When a vehicle is going on relatively smooth surfaces, the displacement of the piston rod is small, and the oil flow between the chambers occurs both through the piston bleed orifices and through the bypass. This in turn creates a soft damper characteristic ensuring comfort of the ride.When the vehicle has to clear large obstacles, it means large piston strokes, the oil is not flowing through the bypass, and the damping force significantly increases (‘hard’ characteristic). 616 U. Ferdek, J. Łuczko The analysis of the carmodel requires the introduction of a relatively simple hydraulic dam- permodel properly describing its basic properties and allowing simultaneously the investigation of the influence of essential parameterswithin thewide range of their changes. Tests ofmodelling twin-tube dampers (Ramos et al., 2005; Alonso andComas, 2006),mono-tube dampers (Talbott and Starkey, 2002; Titurus et al., 2010; Farjoud et al., 2012) and others have been undertaken. Theymainly differ in the approach to describing the oil flow through valves. Alonso andComas (2006) investigated the twin-tube dampermodel taking into account the cavitation problemand the damper chambers elasticity. Talbott and Starkey (2002) investigated themono-tube damper modelling the influence of the shim stack by the preliminarily pressed spring. They assumed that the laminar oil flowwas a result of leakage in the piston-cylinder system, in contrast to the turbulent flow through the orifice system in the piston. Farjoud et al. (2012) investigated the influence of the shim stack properties on characteristics of the mono-tube damper. The authors compared the obtained results with the experimental ones. In the paper by Czop and Sławik (2011), the model of the twin-tube shock absorber was tested and experimentally verified. There is a separate group of research papers dealing with the cavitation problem being a result of sudden oil pressure changes in hydraulic dampers. In the papers by Cho et al. (2002), Van de Ven (2013) and by Manring (1997) various descriptions of the effective bulk modulus were given. Papers dealing with modelling of dampers with a bypass are relatively rare. Apart from patents (Norgaard andCimins, 2009;King, 2014; Marking, 2014), only in the paper by Lee and Moon (2006) the model of the displacement-sensitive shock absorber was discussed. Depending on the piston displacement, the flow control was realised by a proper configuration of the inner cylinder surface. The purpose of the hereby paper is to demonstrate that the introducing of an additional flow passage to a classical hydraulic damper causes a change of the damping force characteristics. In the case of high excitation amplitudes, an increased value of the damping force improves the safety of the ride. While in the case of small excitation amplitudes, the damper behaves in a similar fashion as the classical shock absorber, where an increased oil flow causes a decrease in the damping force providing higher riding comfort. To assure the proper functioning of the model within the wide range of amplitudes and excitation frequencies, the pressure influence on the oil compressibility modulus is taken into account. The proposedmodel of the damper is different than the presented in the paper by Lee andMoon (2006) and allows the investigation of the influence of amore number of constructional parameters of the shock absorberwithin the wide range of their changes. 2. Model of a variable damping shock absorber The scheme of the hydraulic shock absorber with a bypass as well as its model is presented in Fig. 1. Two chambers are in the main cylinder: chamber K1 above the piston (rebound chamber) and chamber K2 below the piston (compression chamber). Narrow orifices through which oil flows between both chambers are inside the piston. Some orifices are constantly open while the others are the most often covered by the shim stack. An additional external flow passage connects chambers K1 and K2, and distances h1 and h2 determine placements of the bypass orifices. The shock absorber is rigidly connected with a reserve cylinder, consisting of chamberK3 filledwith oil and chamberK4 filledwith gas under a high pressure of 2-3MPa.The floating piston of a relatively smallmass separates both chambers. Two phases of the piston rod motion are essential in the damper operations: the compression phase and rebound (expansion) phase. During the compression, the piston rod is moving down causing the pressure increase in Nonlinear modeling and analysis of a shock absorber with a bypass 617 chamberK2 and the oil flow into chambersK1 andK3. During the rebound process, due to the pressure increase in chamber K1, the oil returns to chamberK2. Fig. 1. The scheme and the model of the damper with a bypass In the case ofminor displacements andpressures, the oil flows only throughbleed orifices not covered by plates through leakages and, eventually, through the bypass if the pressure controlled valve is not installed. Along with the pressure increase, the valves in the piston and bypass are gradually opened. During the rebound phase, the oil flows through differently designed piston orifices (of adifferent cross-sectional area) than in the compressionphase,which–finally – causes the damper characteristic asymmetry. Asymmetrical characteristics of the shock absorber are desirable for comfort of the passengers (Silveira et al., 2014). When the piston exceeds the distance h1 (during compression) or h2 (during rebound), the proper entrance to the bypass becomes blocked, and the oil flows only through bleed orifices in the main piston. The oil flow from chamber K2 to K3 occurs through a relatively short and stiff conduit of a significant cross-section. Coordinate xp determines the piston motion, xc – motions of both cylinders, while xfp – the floating piston motion. The relative displacements of corresponding pistons are determined by coordinates x = xp − xc and y = xfp − xc. The motion of both pistons is measured from the static equilibrium position. Notation pi is used for pressures in chambers Ki (i = 1, . . . ,4), Ai – for surfaces of the main and floating pistons (A3 = A4) and Vi – for volumes of chambersKi. The resistance force depends mainly on the resultant pressure force acting on the piston, it corresponds to the oil pressures p1 and p2 in chambersK1 andK2. Taking into consideration the Coulomb friction force Ff1 (Lee and Moon, 2006; Farjoud et al., 2012; Gołdasz, 2015) between the piston rod and the main cylinder, the damping force can be described as F =(p1−p0)A1− (p2−p0)A2+Ff1 sgnẋ (2.1) where p0 is the nominal working pressure. In the simulations, the signum function is approxi- mated (Czop and Sławik, 2011) as follows: sgnẋ = tanh(ẋ/vref), where vref is the reference velocity value (in simulations vref =0.005m/s). In order to determine the shock absorber cha- racteristics, themost often a harmonic excitation is assumed in the form: x(t)= asinωt, where a and ω are the amplitude and frequency of the excitation. In order to determine pressures p1 and p2, the processes occurring in the chambers should be consideredwith a special attention directed to the proper description of the oil flows between 618 U. Ferdek, J. Łuczko the chambers. It will be assumed that the fluid is compressible, taking into account changes of the bulkmodulus, especially in the low pressures range. The equation dρi dpi = 1 βi ρi (2.2) describes the oil density change ρi in chamberKi caused by the pressure change pi (i=1,2,3). It is usually assumed that the bulk modulus value βi is constant (βi = β, where β is the bulk modulus for the given pressure value, e.g. for theworking pressure). This assumption is justified within the limited pressure changes, i.e. in a limited range of the amplitude and piston veloci- ty. For large displacements and velocities, the pressure in one chamber significantly increases while in the other decreases. The assumption of the constant value of the bulk modulus can lead to physically inadmissible solutions of the analysed equations, sometimes even to negative pressure values. In reality, the bulk modulus for large pressures insignificantly increases, and for very small pressures the cavitation effect occurs, during which – due to liquid evaporation – gas bubbles are formed. The accurate description of this phenomenon is more complex and depends on several other factors. The effect of cavitation is a sudden compressibility increase; meaning a bulk modulus decrease. Cho et al. (2002) proposed different descriptions of the bulk modulus depending on the pressure and the oil aeration degree. Comparisonswith experimental results were also presented. Van deVen (2013) provided selected equations for the effective bulk modulus βe. The simplest equation proposed by Merritt (1967) is of the following form βe =β 1 Rβ/κp+1 (2.3) whereR is the volume fraction of the air at the atmospheric pressure pa, whileκ is the adiabatic index. Hayward provides a slightly different equation βe =β R+pκ Rβ/κp+pκ (2.4) where p= p/pa. Another equation was proposed by Cho et al. (2002) βe =β R+pκexp[(pa−p)/β] Rβ/κp+pκexp[(pa−p)/β] (2.5) The shock absorber of properly selected parameters operates within the range of high pres- sures (∼ 2MPa), and then the compressibility modulus changes only insignificantly. However, in the designing process, the damper parameters can be changing in wide ranges. When the parameters are incorrectly selected, a malfunction of the shock absorber – manifested by large pressure changes – can occur. In such cases, there is a necessity of applying the proper equation for the effective bulkmodulus. The following one-parameter model is proposed in the hereby paper βe(p)=β tanh p ps (2.6) Along with decreasing of the parameter ps value (reference pressure) the bulk modulus faster obtains the limit value β. Figure 2 presents comparisons of the effective bulkmodulus diagrams obtained formodels: (2.3)-(2.6), for two relatively small values of the parameterR. For the given values of the parameter ps, the proposedmodel (2.6) indicates the best compatibilitywithmodel (2.4).Within the working pressures range (p0 =2MPa) the bulkmodulus is close to β, and for lowpressures it fastly approaches zero.Theadvantage of formula (2.6) constitutes the possibility Nonlinear modeling and analysis of a shock absorber with a bypass 619 Fig. 2. Bulk modulus versus pressure for different values ofR and ps of obtaining an analytical equation describing the oil density. After substituting formula (2.6) into equation (2.2), the following expression is obtained ρi = ρ0 [sinh(pi/ps) sinh(p0/ps) ]ps/β (2.7) where ρ0 is the oil density under the working pressure p0. In order to determine the oil pressure in chambersKi, equations of the general form can be used ρ̇iVi+ρiV̇i =Qi (2.8) where Qi = ṁi are mass flow rates. Volumes of chambers Ki (i= 1,2,3) can be calculated as follows V1 =A1(L1−x) V2 =A2(L2+x) V3 =A3(L3−y) (2.9) where distancesL1 andL2 are lengths of chambersK1 andK2 in theworking cylinder forx=0, while L3 and L4 are lengths of chambers K3 and K4 in the external cylinder for y = 0. The relative displacement y of the floating piston can be determined from the differential equation mfpÿ=(p4−p3)A3−Ff2 sgn ẏ (2.10) wheremfp is the floating piston mass, and Ff2 is the friction force between this floating piston and the reserve cylinder. Gas pressure p4 is determined from the equation of the polytropic process: p4V n 4 = p0V n 40 (Farjoud et al., 2012; Ferdek and Łuczko, 2012), where: V4 =A4(L4+y) and V40 =A4L4. Hence, it follows p4 = p0 Ln4 (L4+y)n (2.11) After using equation (2.2) and transforming equations (2.8), the equations describing the oil pressures in the chambersKi (i=1,2,3) take the form ṗ1 = β1 V1 (Q1 ρ1 +A1ẋ ) ṗ2 = β2 V2 (Q2 ρ2 −A2ẋ ) ṗ3 = β3 V3 (Q3 ρ3 +A3ẏ ) (2.12) where the moduli βi = βe(pi) in the respective chambers are determined by formula (2.6) and the densities by (2.7). System (2.12) of non-linear differential equations of the first order and differential equation (2.10) of the second order constitute the base for the determination of characteristic (2.1) of the mono-tube hydraulic shock absorber with and without the bypass. 620 U. Ferdek, J. Łuczko Fig. 3. Diagram showing the flow paths One can denote Qj−i (j = 1, i= 2 or j = 2, i= 1) – the mass flow rate from chamber Kj (e.g. rebound for j =1) to chamber Ki (e.g. compression i=2). In the case of the flow in the reverse direction: Qj−i = 0 (then Qi−j 6= 0). Since the oil flow between these chambers occurs through orifices in the piston (Fig. 3) and through the bypass, the flow rate can be written as Qj−i =Q piston j−i +Q bypass j−i (2.13) where the flow rateQ piston j−i is the sum of three flow rates (Fig. 3) Q piston j−i =Q leakage j−i +Q orifice j−i +Q valve j−i (2.14) representing the flow rates resulting from leakage past piston, flow through bleed orifices and flow through valves in the piston. The oil flow in the reversed direction, from chamberKi toKj, determined by the mass flow rate Qi−j, causes a mass decrease in chamber Ki. Thus, the oil mass change in chamberKi can be written in the following form Qi =Qj−i−Qi−j (2.15) Since, from the law of mass conservation betweenmass flow rates the following relation occurs: Q1+Q2+Q3 =0. It is enough to determine the mass flow rates Q1 andQ3 determining mass changes in chambers K1 and K3. It results that Q2 = −Q1 −Q3. The negative value of Qi is related to the oil outflow from chamberKi. Assuming the laminar flow (Talbott and Starkey, 2002), themass flow rateQ2-1leakage from the compression chamber to the rebound chamber can be determined from the equation Q leakage 2−1 =πdp (b3pc(p2−p1) 12lpν − ρ2bpcẋ 2 ) (2.16) where bpc is clearance, dp – piston diameter, lp – piston length, ν – coefficient of kinematic viscosity. The first component of equation (2.16) determines the flow rate caused by the pressure difference while the second by the relative piston velocity. In the case when the piston moves up (for ẋ > 0), the flow rate caused by the pressure difference is decreasing. Equation (2.16) is correct only forQ leakage 2−1 > 0. Otherwise, the flow rate is determined from the following equation Q leakage 1−2 =πdp (b3pc(p1−p2) 12lpν + ρ1bpcẋ 2 ) (2.17) In order to determine the remaining component of equation (2.11) for the turbulent flow (Titurus et al., 2010), the equation of a general form is used Qj−i =CdAj−i √ 2ρj(pj −pi) (2.18) Nonlinear modeling and analysis of a shock absorber with a bypass 621 whereCd is the discharge coefficient whileAj−i is the effective cross-sectional area of the proper orifice throughwhich the oil flows fromchamberKj toKi. Equation (2.18) is correct for pj >pi. After introduction of the function ϑ(pj,pi,ρj)=CdH(pj −pi) √ 2ρj(pj −pi) (2.19) whereH(·) is the unit step function, equation (2.13) obtains the form Qj−i =Q leakage j−i +(A orifice j−i +A valve j−i +A bypass j−i )ϑ(pj,pi,ρj) (2.20) Out of parameters:A oriffice j−i ,A valve j−i ,A bypass j−i , determining the effective areas of respective orifices, only the parameter A oriffice j−i is of a constant value. After referring the orifice area to the area of the compression side of the piston, this area depends on the dimensionless parameter αj−i in the following way A orifice j−i =αj−iA2 (2.21) Values of the remaining areas depend on the oil pressure in the neighbouring chambers of the shock absorber and on the relative piston displacement. The flow through the compression intake or through the rebound intake is controlled by pressures p1 and p2. Bleed orifices are the most often covered by a stack of circular plates (Fig. 3) deflecting under the influence of the resultant pressure force, and gradually uncovering the orifices. The effective cross-sectional area dependsmainly on the pressure difference p1−p2 in the shock absorber chambers as well as on geometrical and physical parameters of the plates. Theaccurate determination of the area change law requires the accuratemodelling of the specific technical solution.Disregarding inertia of theplates, thevalve canbemodelledbymeansof a stiff plate preliminarily pressed down by a spring of a progressive characteristic. Forminute pressure differences and until the resultant pressure force is lower than the preload force, the bleed orifice remains closed. Only after exceeding the preload force, the orifice is gradually uncovered. The parameter Avalvej−i (effective area) decides which flow depends on the spring deflection. It can not, however, exceed the total area of the orifice cross-section. A function θ1 will be used for description of a change in the area. This function is defined as follows θ1(pj −pi,σ,k) =H(pj −pi−σ)tanh pj −pi−σ k (2.22) where the parameter σ determines the pressure difference value above which the plates start to deflect uncovering the bleed orifice joining the neighbouring chambers (θ1 6= 0 only for pj − pi > σ), while the parameter k characterises elastic properties of the shim stack. Using the hyperbolic tangent function in equation (2.22) ensures that the effective area Avalvej−i will not exceed the area of the orifice cross-section. The efficient (variable) valve area, it means the parameter Avalvej−i , is determined by Avalvej−i = δj−iA2θ1(pj −pi,σ valve,kvalve) (2.23) where the dimensionless parameter δj−i is the ratio of the orifice cross-sectional area to the area of the compression side of the piston. Values of dimensionless coefficients δ2−1 and δ1−2 (also α2−1 and α1−2), deciding respectively on flows in the compression and rebound process, can be different (the most often δ12 < δ21) for ensuring a higher resistance force during rebound (for ẋ > 0) in relation to the force created during the compression process (for ẋ < 0). The opening or closing of the bypass is controlled by the relative piston displacement x. The bypass whose openings are of a round cross-section of radius r is gradually closed within 622 U. Ferdek, J. Łuczko the range (h2−r,h2+r) in the rebound phase and within the range (−h1−r,−h1+r) in the compression process. To describe the variable areaA bypass j−i , the function definedbelow is suitable θ2(x,h1,h2,r)=                  0 for x­h2+r θ0[(x−h2)/r] for h2−r 0) than in the compression one. Fig. 8. Influence of the parameters δ and Sv on the force characteristics (f =1.4Hz, a=5cm, h=2cm, α=0.004, γ=0.012, kp =2, kb =0.5) The dimensionless parameter δ (δ2−1 = δ, δ1−2 = 0.8δ) determines the maximum areas of the orifices controlled by the pressure difference. It influences the characteristic in the higher velocities range (Fig. 8).When the piston velocity increases, the pressure difference in chambers K1 and K2 also increases and, in consequence, the valves in the piston are opening and the damping force decreases. For larger values of the parameter δ, in the compression as well as in the rebound process, the inclination of curves determining the damping force dependence on Nonlinear modeling and analysis of a shock absorber with a bypass 627 velocity decreases. For high piston velocity near the piston zero position, the damping force of the shock absorber abruptly changes. The reason of this sudden change constitutes unblocking (or blocking) of flows through the bypass, which entails a decrease (or increase) in the pressu- re difference in the neighbouring chambers. The dimensionless parameter Sv characterises the preload force and decides about the location of the characteristic inflection point. This means that the point inwhich the inclination angle of curves changes (Fig. 8).When the parameter Sv changes, the characteristic shape related to opening of the orifices changes for higher velocities ẋ. Simultaneously, the maximum values of the damping force increase. The damper performance within the range of large relative displacements of the pi- ston depends additionally on parameters characterising the bypass, i.e. on the parameter γ (γ2−1 = γ1−2 = γ) (determining the orifice cross-section area), on distance h (it is assumed that h1 = h2 = h) determining placement of bypass openings in the working cylinder, and on the parameter kb (characterising elastic properties of the valve). Fig. 9. Influence of the parameters h and kb on the force characteristics (f =1.4Hz, a=4cm, β=0.004, β=0.04, kp =2) Figure 9 shows the influences of the parameters h and kb on the damping force as a function of piston displacements. The sudden change of the damping force occurs in two ranges of the displacements:−h−r