Jtam-A4.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 54, 2, pp. 321-332, Warsaw 2016 DOI: 10.15632/jtam-pl.54.2.321 VIBRATION ANALYSIS OF A HALF-CAR MODEL WITH SEMI-ACTIVE DAMPING Urszula Ferdek, Jan Łuczko Cracow University of Technology, Faculty of Mechanical Engineering, Krakow, Poland e-mail: uferdek@mech.pk.edu.pl; jluczko@mech.pk.edu.pl In this paper, analysis of a half-car model with linear and nonlinear semi-active dampers is performed. Using Matlab-Simulink software, a response of the system to a harmonic excitation of variable frequency and to an impulse excitation is found. The effect of both the distribution of spring-supported mass and the asymmetry of the support on the frequency characteristics of velocities and displacements at the mounting points of the dampers are analyzed. Additionally, characteristics of forces generated by the semi-active dampers and the response of the systemwhen crossing an obstacle are determined. Keywords: semi-active damper, vibration damping, car suspension, hysteresis Notations a0 – excitation amplitude cb – sum of damping parameters of suspension c1f,c2f,c1r,c2r – damping parameters of front and rear semi-active damper cwf,cwr – damping coefficients of front and rear wheel Ib – moment of inertia of body kbf,kbr,kb – stiffness of front and rear spring, sum of stiffness kwf,kwr – stiffness parameters of front and rear wheel lf, lr – distance of axles from center of bodymass l – distance between both axles mb,mwf,mwr – mass of body (spring-supportedmass), front and rear wheel (non-spring-supportedmass) pmin,pmax,pmean – minimum,maximum and the mean value of the parameter p t0 – delay time of kinematic excitation acting on rear wheel uf,ur – control force of front and rear semi-active dampers V0 – driving speed vbf,vbr – dimensionless velocity of front and rear body points wf,wr – kinematic excitation applied to front and rear wheel xbf,xbr,xwf,xwr – dimensionless displacements of body points, front and rear wheel yb – displacement of center of bodymass ybf,ybr,ywf,ywr – displacements of body points and front and rear wheel α,α0 – dimensionless and dimensional scaling factors of Bouc-Wen force β,γ,A,n – control shape parameters of hysteresis loops δ – dimensionless amplitude of parameters c1f,c2f,c1r,c2r α0f,α0r κc – ratio of stiffness of front spring to sum of stiffnesses η – dimensionless excitation frequency λ – mass distribution index of body 322 U. Ferdek, J. Łuczko φb – angle of rotation of body ω,ω0 – excitation and reference frequency 1. Introduction Theprimary cause of vibrations affecting the driver of a car are kinematic disturbances resulting from road surface irregularities. Elimination of these vibrations is essential in order to improve both the comfort and the safety of the passenger.When the vehicle is driving across a roadwith large irregularities (obstacles), itswheelsmight get separated fromthe surface of the road,which in turn decreases the efficiency of force transmission of the drive, braking and steering systems of the car. An improved driving dynamics and better road traction on curves and bumps can be achieved by using the so-called “hard suspension”.However, the cost is the reduction of comfort of thepassenger.The criteria for assessing thequality of shock absorbers should therefore include both theminimization of car body vibration and appropriate wheel-road adhesion (Łuczko and Ferdek, 2012). In order to perform dynamical analysis, either a quarter-car (Gopala Rao and Narayanan, 2009; Huang andChen, 2006) or a half-car (Ihsan et al., 2009; Sapiński andRosół, 2008) model can be used. The quarter-car model that consists of a non spring-supportedmass (a wheel with partial of suspension) and a spring-supportedmass (1/4 car body) is a two-degrees of freedom model and is usually used for testing of the performance of control algorithms. Thehalf-car four- -degrees of freedom model consist of two non-spring supported masses and a spring-supported one (1/2 car body). It additionally includes rotation angle of the body and allows analysis of the response to the excitation applied to both wheels of the vehicle. Dampers used in the suspension system can be either passive, semi-active or active. Dyna- mical properties of the dampers are usually defined by models with hysteresis characteristics, such as Bingham (Prabakar et al., 2009), Bouc-Wen (Dominguez et al., 2008; Yao et al., 2002) or Spencer model (Spencer et al., 1996). Requirements set for the comfort and safety of driving can be fulfilled by using semi-active suspension systems, introduced by Crosby and Karnopp (1973). In comparison to passive ones, the semi-active systems allow the damping force to be adjusted depending on driving conditions. Additionally, they require less power than similar active systems. Several methods of control have been used, some of which can be found in the paper byAh- madian (2001). Liu et al. (2005) aswell asWuandGriffin (1997), when analyzing on-off control, assume that the damping force should by high if the product of relative and absolute velocity is more than zero. Fischer and Isermann (2004) analyzed the relation between parameters of the car suspension system and the driving comfort as well as the safety indexes. They defined the comfort index as the effective acceleration value while the safety index as the effective ratio of the dynamic and static response. In the study by Sapiński andMartynowicz (2005), the results were presented for the theoretical and experimental half-car model, in which the car suspension was controlled by two separate magneto-rheological dampers (MR damper). Some interesting options for control of a semi-active car suspensionwere presented byAhma- dian (2001). Themost commonmodel tobeanalyzedwas thequarter-car one. In the steady-state case, the response to the harmonic excitation was analyzed, while in the transient one (Ahma- dian and Vahdati, 2006), the response to the unit step. To ensure a compromise between the requirements for both comfort and safety, hybrid control with aMRdamper is used (Goncalves and Ahmadian, 2003) and a combination of sky-hook and ground-hook control. The damping control algorithm was changed by a step function (on-off control) in order to simulate the be- haviour of the damper between the constant reference point and a spring-supported (sky-hook – comfort) or non spring-supportedmass (ground-hook – safety). Vibration analysis of a half-car model with semi-active damping 323 In the paper by Łuczko and Ferdek (2012), the effectiveness of damping of vibration of a quarter-car model by both semi-active and passive dampers was compared. Several different algorithms were proposed for semi-active dampers. The effect of these algorithms on the factors corresponding to driving safety and comfort were analyzed. In this paper, analysis of a half-car model of a car by semi-active suspension is performed. The influence of parameters of themodel on the efficiency of spring-supportedmass damping is considered. 2. Half-car suspension model Figure 1 shows the analyzed half-car model of an automobile. Vibration of the system around the static equilibrium position can be written using the following differential equations mwf ÿwf =−cwf(ẏwf − ẇf)−kwf(ywf −wf)+kbf(ybf −ywf)−uf mwrÿwr =−cwr(ẏwr− ẇr)−kwr(ywr−wr)+kbr(ybr −ywr)−ur mbÿb =−kbf(ybf −ywf)−kbr(ybr −ywr)+uf +ur Ibφ̈b = lfkbf(ybf −ywf)− lrkbr(ybr−ywr)− lfuf + lrur (2.1) where ywf and ywr, are displacements of the front and rear suspension systems (i.e. the non- -spring-supported mass mwf and mwr), ybf and ybr are displacements of the points connecting the car body (spring-supportedmass mb of inertia Ib) with the suspension systems. Fig. 1. Half-car model Additionally, two parameters are introduced which are respectively: position yb of the mass center and rotation angle φb of the car body. They can be found using the following equations (on the assumption of small displacements) yb = lrybf + lfybr l φb = ybr−ybf l (2.2) The functionwf(t) andwr(t)=wf(t−t0) define the applied kinematic excitation, which corre- sponds to the profile of the road. The delay time t0 is related to driving velocity V0 and distance l = lf + lr between both vehicle axles. The parameters: kwf, kwr and cwf, cwr define stiffness 324 U. Ferdek, J. Łuczko anddamping parameters of the front and rearwheel, while kbf, kbr are stiffneses of the front and rear spring respectively. The definition of forces uf and ur which are generated by semi-active dampers and applied to the non-spring-supported and spring-supported masses, are defined in Section 2 of this paper. Equations (2.1) can be written in a matrix form. In order to do so, the third and the fourth equation of the system must be transformed (2.1), including additional relations (2.2). By choosing the displacements ywf, ywr, ybf and ybr for coordinates of the vector y, vibration of the system can be, after introduction of massM, dampingC and stiffness matrixK, presented in form of the second-order matrix equation Mÿ+Cẏ+Ky= B̃u+ F̃w(t) (2.3) where y = [ywf,ywr,ybf,ybr]T , u = [uf,ur] T and w = [wf,wr] T. The stiffness matrix is as shown below K=   kwf +kbf 0 −kbf 0 0 kwr+kbr 0 −kbr −kbf 0 kbf 0 0 −kbr 0 kbr   (2.4) The damping matrix, after inclusion of passive dampers present in the vibroisolation systems, has the identical structure as the stiffness matrix. As damping properties of the wheels are usually omitted (cwf = cwr = 0) and the effect of passive dampers is already included in the forces uf and ur, the matrix C is empty. Themass matrix can be presented in the form: M= [ Mw 0 0 Mb ] (2.5) where Mw = [ mwf 0 0 mwr ] Mb =    mbl 2 r + Ib l2 mblflr− Ib l2 mblflr− Ib l2 mbl 2 f + Ib l2    (2.6) From Eqs (2.5) and (2.6), it can be seen that when the condition mblflr − Ib = 0 is fulfilled, the matrix M becomes diagonal, and with the matrix K given in (2.4), decoupling of vertical vibration of the rear and front part of the vehicle, is possible. If the so-called “mass distribution index” λ= Ib/mblflr is close to 1, the excitation applied to one axle does not cause vibration of the other one. Thematrices B̃ and F̃ can be written as follows B̃=   −1 0 0 −1 1 0 0 1   F̃=   kwf 0 0 kwr 0 0 0 0   (2.7) In order to transform matrix equation of motion (2.3) in the first-order form suitable for performing the numerical simulations, a modified state vector that includes velocities, is intro- duced x= [ x1 x2 ] = [ y ẏ ] (2.8) Vibration analysis of a half-car model with semi-active damping 325 Motion of the system can be nowwritten using the equation ẋ=Ax+Bu+Fw(t) (2.9) The relation between the matrix A and the matrices present in equation (2.3) is A= [ 0(4×4) I(4×4) −M−1K −M−1C ] (2.10) where 0(4×4) and I(4×4) are, respectively, an empty and singular matrix of 4×4 size. The same holds for the matrices B= [ 0(4×2) M−1B̃ ] F= [ 0(4×2) M−1F̃ ] (2.11) where thematrix 0(4×2) is an emptymatrix of 4×2 size.Matrix equation (2.9) is well-suited for analysis of active systems in which the control vector u is treated as the sought optimal control vector. 3. Semi-active dampers The analysis presented below is limited to testing the effect of several selected half-car model parameters and two semi-active dampers on dynamical characteristics of the system. The forces generated by a simplifiedmodel of the semi-active damper (denoted as SA1) after introduction of the functions uLin(ẏ1, ẏ2)= { cmax(ẏ1− ẏ2) ẏ2(ẏ1− ẏ2)¬ 0 cmin(ẏ1− ẏ2) ẏ2(ẏ1− ẏ2)> 0 (3.1) can be calculated from the equations uf =u Lin(ẏwf, ẏbf) ur =u Lin(ẏwr, ẏbr) (3.2) In SA1 damper model, the forces are proportional to the relative velocity, with higher energy dissipation if the momentary power if less than zero (which means that energy is retrieved from the spring-supportedmass). For cmax = cmin, equations (3.1) and (3.2) define the passive damper (PS). The other type of a semi-active damper (SA2) is defined (Spencer et al., 1996) using the Spencermodel (Fig. 2).Themathematical description of the generated force ismore complicated in this case. Based on the study by Ferdek and Łuczko (2011), a concise force definition can be presented uf =u Spencer(ywf, ẏwf, ẏbf) ur =u Spencer(ywr, ẏwr, ẏbr) (3.3) where uSpencer(y1, ẏ1, ẏ2)= c2(ż1− ẏ2) (3.4) Additional parameters z1 and z2 can be obtained from the set of equations k1(y1−z1)+ c1(ẏ1− ż1)−α0z2 = c2(ż1− ẏ2) ż2 =Aż0{1− [γ+βsgn(z2ż0)]|z2| n} (3.5) 326 U. Ferdek, J. Łuczko Fig. 2. Model of the semi-active Spencer damper in which z0 = z1−y1 and β+γ =1(0<β< 1,0<γ < 1). Equation (3.5)1 can be interpreted as the equilibrium condition of forces acting on a massless middle element in the Spencer mo- del (Fig. 2). The dimensionless variable z2, which is the solution of differential equation (3.5)2 proposed in a similar form by Bouc-Wen (Spencer et al., 1996), determined the appearance of a hysteresis. Its shape depends on the parametersA, γ, β and n. The parameters β and γ have impact on the characteristic only when A is small. In such a case, an increase in β causes the width of the hysteresis to be slightly decreased. Most often, when forming the characteristic, the value n = 2 is chosen, in rare cases n = 1 is taken. The parameter A is the one that strongly influences the shape of the hysteresis. With a increase in the parameter A, the lower and upper limits of the Bouc-Wen model characteristic are symmetrical. In the numerical cal- culations, emphasis is put on the analysis of coefficients α0, c1 and c2 and their impact on the solution. The parameters of A = 50 m−1, γ = β = 0.5 and n = 2 have been taken from the literature. The slope of the force characteristic in the high velocity range depends on the value of a substitute damping coefficient cz = c1c2/(c1 + c2) with a relation close to linear. For lower velocities, the slope and inflection point of the characteristic are related to the parameter α0. In the semi-active damper (e.g. magneto-rheological one), it is assumed that the parameters α0, c1 and c2 of the Spencer model are linearly dependent on the control voltage. By taking a control algorithm analogical to (3.1), it can be assumed that ck = p(ẏ1, ẏ2,c max k ,c min k ), k=1,2 and α0 = p(ẏ1, ẏ2,α max 0 ,α min 0 ), with p defined using the formula p(ẏ1, ẏ2,p max,pmin)= { pmax pmin ẏ2(ẏ1− ẏ2)¬ 0 ẏ2(ẏ1− ẏ2)> 0 (3.6) 4. Results of numerical calculations In the numerical calculations, the emphasis is placed on analyzing the effect of a few selected parameters of the system, with other parameters assumed as follows: lf =0.94 m, lr =1.66 m, mb = 510 kg, mwf = mwr = 28 kg, Ib = λmblflr (variable λ), kwf = kwr = 180000 N/m, kb = 40000 N/m, k1 = 0.01kb, cwf = cwr = 0, A= 50 m −1 and β = γ = 0.5. When analyzing the effect of stiffness of the front and rear suspension system, it is assumed that kbf = κkkb, kbr =(1−κk)kb, (0<κk < 1), kb – is the sumof stiffness parameters. Similarly,whenconsidering the effect of energy dissipation, the parameters of PS damper are: cbf = κccb, cbr = (1−κc)cb, (0<κc < 1). Thevalue of cb =2260Ns/mhasbeen chosen such that the dimensionless damping factor, given by equation ζ = cb 2mbω0 = cbf + cbr 2mbω0 (4.1) Vibration analysis of a half-car model with semi-active damping 327 is ζ ≈ 0.25 – the recommended value for vehicle shock absorbers. The dimensionless angular velocity ω0 (close to the two highest vibration modes of the system) present in Eq. (4.1) is defined as follows ω0 = √ kb mb = √ kbf +kbr mb (4.2) When choosing the parameters of semi-active dampers, the same assumption is made re- garding the parameters of front and rear dampers as well as the energy dissipation level. For SA1 damper, it is assumed that the mean value of damping coefficients are: cmeanf = κccb, cmeanr =(1−κc)cb, while the extreme values can be calculated from pmax =(1+ δ)pmean pmin =(1− δ)pmean (4.3) where p= cf or p= cr and 0<δ< 1. SA2 damper has a higher number of significant parameters. Formulas (4.3) need to be used for the extreme parameters α0, c1 and c2 of the front and rear damper, while the mean values cmean1f ,c mean 2f ,c mean 1r and c mean 2r must be chosen such that the coefficient ζ has the desired value. From literature (Prabakar et al., 2009; Spencer et al., 1996), it can be seen that cmean1f ,c mean 1r values are an order lower from cmean2f , c mean 2r ones. These are chosen as follows: c1f = 1.1κccb, c2f = 11κccb, c1r = 1.1(1−κc)cb and c2r = 11(1−κc)cb. For the chosen values, the relation czf +czr = cb, where cz = c1c2/(c1+ c2) is true and coefficient (4.1) is equal to ζ =0.25. The values of varying parameters are presented with analysis of the results of numerical simulations. When presenting the results, the dimensionless parameters are introduced rela- ting the displacements with the excitation amplitude a0, velocities with the value ω0a0, while accelerations with ω20a0, and forces with kba0. One of less known parameters of the system is the delay time t0 between the functions describingmotion of the front and rearwheel of the vehicle. In order to estimate it, it is assumed that the angular frequency of the kinematic excitation ω is related for the given road profile linearly with the velocity V0, or using the formula: ω=µV0. For motion with constant velocity V0, the relation t0 = l/V0, where l is thedistancebetween bothaxles, is also true.Byadditionally assuming that the lowest vibrationmode of the approximate frequencyω0 is related to a known velocity Vr, the value of the parameter µ = ω0/Vr and the delay time t0 = ω0l/Vrω can be calculated, e.g. for given: Vr =20 km/h, l=2.6 m and ω0 =8.856 rd/s a value of t0 ≈ 4.15/ω is found. The frequency characteristics are ideal for the purpose of global evaluation of dynamical properties of the system. In the simulations, the kinematic excitation is usually defined by the harmonic function of a modulated angular frequency: (e.g. “Chirp Signal” in Simulink). If the simulation time is high enough, an approximate frequency characteristic can be acquired by graphing the maximal responses of the system. In order to illustrate the effect of car body mass distribution, characteristics of maximal dimensionless velocities vbf = ẋ3/ω0a0 andvbr = ẋ4/ω0a0 (in thepoints connecting thebodyand the suspension)with relation to the dimensionless angular frequency of the excitation η=ω/ω0 are shown in Fig. 3. The analysis is limited to vibration comparison of systems with identical shock absorbers (κk = κc = 0.5). The parameter defining the semi-active damper has been chosen as δ=0.5. The only modified value is the mass distribution ratio λ. The corresponding inertia of the spring-supportedmass is Ib = λmblflr ≈ 795.8λ. The actual value of λ should be close to one. Several exemplary values of this coefficient, calculated from the data presented in the literature, are equal: λ = 1.01 (Feng et al., 2003), λ = 1.09 (Sam et al., 2008), λ = 1.16 (Lozia et al., 2008), λ = 0.823 (Prabakar et al., 2009). However, in some cases this value is different, e.g. (Shamsi and Choupani, 2008) λ=0.544, which is far from one. 328 U. Ferdek, J. Łuczko Fig. 3. Effect of spring-supportedmass distribution on the frequency characteristics: (a) velocity vbf, (b) velocity vbr From the results presented in Fig. 3, it can be seen that the lower the coefficient λ becomes, the larger are the maximal velocities, especially within the resonance regimes. Although this model has four degrees of freedom, only two resonance regions can be seen on the presented frequency characteristics. This is caused by close proximity of the paired natural frequencies of the system. For example, when λ = 0.5, the natural frequency values of the linearized system related to ω0 are equal to: 0.88 (dominant displacement of the front part of the vehicle), 1.44 (rear part), 8.97 and 9.16 (wheel vibration), while for λ=2 these values are respectively: 0.67, 0.97, 9.14 and 9.21. A consequence of such a distribution of frequencies is dislocation of the lowest frequency region towards the lower frequency with an increase in λ and its slightly different disposition in the velocity characteristics vbf and vbr. The location of the “second” region is less vulnerable to a change in the parameter λ and is similar to Fig. 3a and Fig. 3b. Although the obtained results are only for SA1 damping system, the conclusions are more general, and essentially similar results are obtained for PS and SA2 systems. In the case of SA2 system described by the Spencer model, the effectiveness of the damper, depends on chosen values of cmean1f , c mean 2f , c mean 1r , c mean 2r and δ, but mostly on the parameter α0. Fig. 4 the shows the dimensionless displacement characteristics xwf = x1/a0 (non-spring- -supported mass) and xbf = x3/a0 (spring-supported mass) for several values of the parameter α=αmean0 /α mean 0 kba0. At κc =0.5 (symmetrical support), the relations are: c1f = c1r =0.55cb, c2f = c2r =5.5cb and the mean value of the coefficient ζ is close to 0.25. By analyzing Fig. 4b, one can see that the characteristic closest to the optimal is the curve obtained for α = 0.5. Too high values of α cause the forces to be much higher for low velocities, and also shift the inflation point location, which is undesirable especially in the range of high-frequency excitation. From the graphs shown in Fig. 4a, it can be concluded that within the range of high oscillation, which includes the third and fourth natural frequencies, the amplitudes of non-spring-supported Fig. 4. Effect of the parameterα on the frequency characteristics of SA2 system: (a) displacement xwf , (b) displacement xbf Vibration analysis of a half-car model with semi-active damping 329 masses (wheels) are significant. Additionally, within this range, the dynamic response values are also large, and so the indexes related to the driving safety and comfort are lower, which means that these semi-active dampers are not efficient. In order to compare SA1 and SA2 dampers, the dimensionless force characteristics Uf = uf/kba0 generated by both types of semi-active dampers are shown in Fig. 5. Only the front part of the car suspension is presented. As before, it is assumed that κk = κc = 0.5, (symmetrical model) with λ=1, α=0.5, ζ =0.25 and δ=0.5. Fig. 5. Damper SA1 and SA2 characteristics: (a) ω=1, (b) ω=4 The characteristic of SA1 damper is relatively simple. Two of its branches are straight lines of slopes equal to the given cminf and c max f values, while the lines linking the other two, should be, in theory, vertical. The reason for this deviation is due to the approximation used for the continuous switching step function (based on arctan). Such an approach is recommended for discontinuous functions due to its accuracy and the time of numerical computation required, and also in some cases for avoidance of undesirable effects caused by too frequent switching, e.g. chattering. Thecharacteristic of SA2damper in the rangeof highvelocities is similar to theonedescribed above. The differences are visible within the range of low velocities. The average slope of the characteristic is higher and depends primarily on the parameterα. The higher complexity of the graph is also due to the other parameters of the half-car model. Less complex characteristics can be obtained by analyzing a simplermodel, such as a quarter-carmodel (Łuczko andFerdek, 2012). Figure 6 shows the frequency response of dimensionless displacements xbf = x3/a0 and xbr =x4/a0 for passive PS and both semi-active SA1 and SA2 systems. Fig. 6. Effect of the parameters κk and κc on frequency characteristics PS, SA1 and SA2: (a) displacement xbf, (b) displacement xbr The case of symmetrical mounting κk = κc =0.5 (S systems, Fig. 6) and the asymmetrical one defined by parameters: κk = κc = lr/lf ≈ 0.638 (NS systems) is considered. For selected values of the parameter κk, the static deflections of the spring-supported mass under its own mass are identical in both locations of the connection with the suspension systems, while the 330 U. Ferdek, J. Łuczko center of stiffness overlaps the center of mass. The values λ=1, α=0.5, ζ =0.25 and δ=0.5 have been assumed, while the other parameters of the Spencermodel have been obtained using appropriate equations with known parameters κk and κc. When showing the results, only the region with the lowest natural mode (actually two lowest modes) are shown – the ones in which the car body vibration is dominant. As the parameters for models PS, SA1 and SA2 are chosen in such a way that the natu- ral frequencies are the same for the identical parameter κk, the dislocation of the resonance region is avoided. Such a dislocation is only visible between the curves obtained for κk = 0.5 (S systems) and κk = 0.638 (NS systems). The value of the parameter κk influences mostly the two lowest natural frequencies. The complex natural frequencies (related to ω0) are equal: η1 = −0.16 ± 0.83i, η2 = −0.29 ± 1.10i for S systems and: η1 = −0.195 ± 0.927i, η2 =−0.217±0.945i for NS ones. The location of the resonance regions, as shown in Figs. 6a and 6b is directly related to the distribution of the natural frequencies of S and NS systems. As the low values correspond to the natural modes in which the displacement are dominant (and even more xbr), the resonance frequency of S systems is lower than that in NS ones for characteristics of the displacement xbf (Fig. 6a). The opposite effect is observed for the displacement xbr (Fig. 6b). The vibration reduction level is indeed related to the parameter κk (at least within the analyzed regimes). Using a more stiff front mounting, the comfort of the driver is only slightly decreased but seems to be important when considering safety of the driver (this is not a subject of analysis in this study). The systems with SA1 dampers reduce vibration by around 20% when compared with passive ones, while the semi-active SA2 dampers are proved to be even more effective, reducing the values of the displacement xbf and xbr twice in the fundamental resonance. Figure 7 shows the response of PS, SA1 and SA2 dampers (dimensionless displacements xbf and xbr, velocities vbf and accelerations abf in function of the dimensionless time τ = ω0t) to an impulse excitation defined by equation (Shekhar et al., 1999; Łuczko, 2011) wf(t)= e 4 a0 3∑ k=1 ωk(t− tk)] 2 exp[−ωk(t− tk)]H(t− tk) (4.4) whereH() is the unit step function. Function (4.4) is supposed to simulate the vehicle crossing the same obstacle at three different velocities V0 = θVr (where θ=1/2,1,4), which in equations (4.4) are represented by angular frequenciesω1 =0.5ω0,ω2 =ω0 andω3 =4ω0. The values of tk Fig. 7. Response of the system to an impulse excitation: (a) xbf , (b) xbr, (c) vbf , (d) abf Vibration analysis of a half-car model with semi-active damping 331 are chosen in such away that the vibration caused by the previous impulse would vanish before the next one is applied. When crossing the obstacle at a low speed (θ=0.5 and θ=1), themaximumdisplacements (Fig. 7a and Fig. 7b) are to a small extent related with the type of a damper, although the lowest ones are obtained for SA2 damper. The type of the used damper has a greater effect on the character of damping of vibration caused by an impulse excitation. For PS andSA1 systems, reduction of oscillatory vibration can be observed. In the case of SA2 systems, the damping is much faster and similar in shape to an exponential function. For high values of velocity, when crossing the obstacle (θ=4), the displacements are definitely lower. This is however at the cost of large acceleration values (Fig. 7d), for which themaximumvalues are practically independent of the type of the used vibroisolation system.With an increase in driving velocity, the time delay t0 is decreased while the displacements of both front and back parts of the body are similar. 5. Summary Based on the analysis of selected results of numerical calculations, several conclusions can be drawn: • The algorithms for control of a semi-active on-off damper inwhich the switching is related to the actual power, enable improvement the driving comfort, especially within the low- -frequency excitation range. • From the tested dampers, definitely the best one is SA2 damper with a nonlinear cha- racteristic. If the parameters for the damper are optimal, the vibration amplituda can be reduced twice as much within the range of fundamental resonance. • Semi-active SA2 system is also effective when subjected to an impulse excitation which simulates obstacles (bumps) in the road. • An improper mass distribution (low values of λ) might be an additional cause for an increased vibration level of the spring-supportedmass. • The introduction of an additional spring and damping elements to the front and back car suspension in which the center of stiffness overlaps the center of mass, does not cause a decrease in the indexes describing the driving comfort. It might be, however, beneficial when considering the safety. • Semi-active systemsare less efficient thanpassiveoneswhenthedrivingvelocity (excitation frequency) is high. References 1. Ahmadian M., 2001, Active control of vehicle vibrations, Encyclopedia of Vibration, Academic Press, London, 37-48 2. 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