JOURNAL OF THEORETICAL AND APPLIED MECHANICS 43, 3, pp. 655-674, Warsaw 2005 VIBRATION CONTROL IN A PITCH-PLANE SUSPENSION MODEL WITH MR SHOCK ABSORBERS Bogdan Sapiński Paweł Martynowicz Department of Process Control, University of Science and Technology, Cracow e-mail: deep@agh.edu.pl; pmartyn@agh.edu.pl The paper is concernedwith the experimental study of vibration control in a suspension equipped with independently controlledmagnetorheolo- gical shock absorbers (MRAs) in front and rear sections. For research purposes, the pitch-plane suspension model with bounce and pitch mo- tions was considered. This suspension model was tested in open-loop and feedback system configurations under harmonic and square excita- tions. The experiments were conducted on a specially designed experi- mental setupwithadataacquisitionandcontrol systemconfigured in the MATLAB/Simulink environment. The obtained results reveal effective- ness of MRAs for vibration suppression in the investigated suspension model. Key words: MR shock absorber, pitch-plane suspension model, LQ control 1. Introduction Typical vehicle suspension systems with passive shock absorbers are cha- racterised with unavoidable compromise between road roughness attenuation and drive stability of the vehicle. That is why active and semi-active vehic- le suspensions are used. In semi-active suspensions, conventional springs are retained but passive shock absorbers are replaced with controllable shock ab- sorbers (Nagai et al. 1996). Vibro-isolation properties of semi-active suspen- sions are close to those of active ones. Semi-active suspension systems use external power only to adjust damping levels and operate the controller and set of sensors, whereas active systems require significant amount of external energy to power the actuators. An example of semi-active suspension is the 656 B.Sapiński, P.Martynowicz Magnetic RideControl system that has been used inCadillac Seville STS cars since 2002. This system is equipped withMRAs made by Delphi Automotive Systems (MagneRide). Similar shock-absorbers are used in suspensions of Co- rvette racing cars. Such systems are durable and give very wide capabilities of adaptation to drive conditions and enhance vehicle-manoeuvring possibilities. The application ofMRAswas successfully investigated in vibration control of car systems (mainly concerning a quarter car) among others by (Pare, 1998; Song, 1998; McLellan, 1998; Kiduck and Doyoung, 1999; Simon, 2000) and others from a research group at Virginia Tech (Virginia Tech, 2002). Results of computer simulations for sucha scaled quarter carmodel provided inamong others (Virginia Tech, 2002; Yao et al., 2002) revealed that with semi-active damping control vibration of the suspension system is well controlled. The group fromVirginia Tech evaluated the performance of MR suspensions for a quarter car model test facility and for a heavy truck on a road under control schemes of skyhook, groundhook and hybrid semi-active control. That was also partially confirmed in the initial phase of experimental research described in (Sapiński et al., 2003). The suspensionmodel selected for the purpose of vibration analysis is lar- gely dependent on the objective of the analysis. The review of simple and crediblemodels that can be useful for fundamental vibration analysis in terms of resonant frequencies and forced vibration responses of sprung andunsprung masses provided in (Ahmed, 2002) shows that the vibration response of ve- hicles to different excitations can be investigated through analysis of various in-plane models. Because the wheelbase of the majority of ground vehicles is significantly larger than the trackwidth, rollmotions canbeconsiderednegligible compared tomagnitudes of vertical and pitchmotions. That is whywe focused on a two degree-of-freedom (2DOF) pitch-plane model of a suspension equipped with MRAs(Martynowicz, 2003).Thismodel enablesus to studyqualitative bounce and pitch motions of the sprung mass, assuming negligible contributions due to axle and tire assembly. In this case, the road input is taken to be the same as the wheels and is suitable for the estimation of bounce and pitch resonance frequencies.Thismodel is also considered applicable for studyof off- road vehicles without the sprung suspension, where the stiffness and damping elements relate to the properties of tires alone. The paper reports an experimental study of vibration control in a semi- active suspension equippedwithMRAs. For this purpose, we built an experi- mental setup of the 2DOFpitch-plane suspensionmodel.We employedMRAs of the RD-1005-3 series made by Lord Corporation (http://www.lord.com). Vibration control in a pitch-plane suspension model... 657 For the purpose of controller design, real-time data acquisition and control, MATLAB/Simulink environment was employed (The Mathworks, 2003). In the control system, we used solution to a linear-quadratic (LQ) problem for- mulated for this part of the system which does not contain MRAs and the hysteresis inverse model of theMRA (Sapiński andMartynowicz, 2004). 2. Pitch-plane suspension model For the purpose of the suspension analysis, we introduce a 2DOF pitch- plane model. Using this type of model, we can analyse pitch and vertical motions of the suspended body, which prove to be themost inconvenient (the lowest threshold of toleration) for a human body. This type of model is an introduction to analysis of the whole-vehicle behaviour. The model of suspension to be considered is shown in Fig.1. The vehicle body is simulatedbya rigid rectangle-intersection beamofmass m,moment of inertia I, total length L, width a, height b andcentre of gravity (c.o.g.) in Pg. The beam is supported in points Pf and Pr by two identical spring-MRAsets (hereinafter called suspension-sets), which are subject to bottom kinematic displacement excitations similar to these acting upon a conventional vehicle suspension. The distances from Pg to Pf and from Pg to Pr are denoted by lf and lr (respectively). The presented model possesses two degrees of freedom, which can be described as the vertical (bounce) displacement x and pitch displacement ϕ of the c.o.g. (Ahmed, 2002; Fuller, 1996; Sapiński and Martynowicz, 2004). Fig. 1. 2DOF pitch-plane suspensionmodel Excitations applied to the bottom of the front and rear suspension-sets are denoted by wf and wr, respectively, displacements of points Pf and Pr – by xf and xr, resultant forces exerted on the front and rear suspension-sets – 658 B.Sapiński, P.Martynowicz by Ff and Fr (respectively). Assuming a vehicle with a long base (lf + lr), therefore sinϕ∼=ϕ, yields xf =x+ lfϕ xr =x− lrϕ (2.1) Let us introduce formulas describing forces Ff and Fr Ff =FS,f +FMR,f Fr =FS,r+FMR,r (2.2) where FS,f and FS,r are respective spring forces FS,f =−kS,f(xf −wf) FS,r =−kS,r(xr−wr) (2.3) Designations FMR,f, FMR,r represent resistance forces of MRAs denoted by df and dr, respectively. To obtain the formula for FMR,f, a Spencermodel of the MRA is invoked (Fig.2), governed by the system of equations żf =−γ|(ẋf − ẇf − ẏf)zf |zf −β(ẋf − ẇf − ẏf)|zf| 2+A(ẋf − ẇf − ẏf) ẏf = 1 c0+ c1 [αzf + c0(ẋf − ẇf)+k0(xf −wf −yf)] (2.4) FMR,f =−c1ẏf −k1(x1−x0) whereparameters: α,β,γ,A, c0, c1,k0,k1,x0describenon-linear relationships which are inherent features of the MRA (Spencer, 1996). These parameters (varyingwithpistonvelocity of theMRA)are presented inTable 1 andTable 2 (Sapiński, 2004) in function of frequency and amplitude of the sine excitation. The equationsmodeling rearMRAbehaviour and force FMR,r can be defined in an analogous manner. Fig. 2. MRAmodel by Spencer Vibration control in a pitch-plane suspension model... 659 Table 1. Parameters of Spencer’s model: frequency 1Hz, amplitude 20 ·10−3m (peak to peak) Current Parameter value I α c0 c1 k0 A β ·10 4 γ ·104 k1 x0 ·10 −2 [A] [N/m] [Ns/m] [Ns/m] [N/m] [–] [1/m2] [1/m2] [N/m] [m] 0.0 29879.5 348.6 69870 1181.6 17.664 348.217 394.71 705.8 25.135 0.2 49069.8 1138.9 33713 1807.1 31.684 42.071 221.31 550.8 31.466 0.4 52501.5 2138.8 38551 674.2 36.483 14.834 42.21 633.0 27.011 Table 2. Parameters of Spencer’s model: frequency 4Hz, amplitude 3 ·10−3m (peak to peak) Current Parameter value I α c0 c1 k0 A β ·10 4 γ ·104 k1 x0 ·10 −2 [A] [N/m] [Ns/m] [Ns/m] [N/m] [–] [1/m2] [1/m2] [N/m] [m] 0.0 70082.3 1013.6 12414 2973.4 4.042 226.307 571.24 835.4 24.242 0.2 76315.9 1838.6 48180 2922.8 14.803 2.728 305.84 839.0 23.561 0.4 86070.1 4010.5 50239 2903.4 17.146 1.317 99.03 695.9 26.891 In the analysed system, we consider the initial deflection of springs and MRAs due to the suspended load (the beam) as a zero initial condition of displacements x and ϕ. This enables us to neglect the gravity forces of the sprungmass and todescribe thedynamics of the 2DOFpitch-plane suspension model by expressions (2.5). Equations (2.5) present balances of forces and moments of forces mẍ=Ff +Fr Iϕ̈=Fflf −Frlr (2.5) 3. Experimental setup For the purpose of the 2DOF pitch-plane suspension analysis, an experi- mental setup was devised comprising data acquisition and control equipment connected to PC-based communication and control software. As our analysis is limited to pitch-plane oscillations, the experimental se- tup should conform to appropriate construction demands, namely transverse rigidity. All motion components orthogonal to the pitch-plane or other than pitch and bounce should be eliminated. This implies introduction of appro- priate central guiding elementswith adequately small friction forces. All joints should have a high transverse rigidity. 660 B.Sapiński, P.Martynowicz Another demand is the excitations are stationarity. A limited output of the excitation sources available in laboratory conditions implies constraints to the total mass andmoment of inertia of the sprung system. The experimental set-up (Fig.3) consists of: steel beam with rectangular intersection (1) as a load element (vehicle body), two identical suspension-sets: spring (2) –MRA (3), central roller guiding (4) and two kinematic excitation sources. Each suspension-set is built as a parallel connection of a vertically mountedMRA inside and an outer screw-cylindrical reflex spring guided onto two thin-wall sleeves (5). The sleeves are guided one inside the other with a teflon slide ring between them.Both sleeves possess outer flanges as the spring support. Each suspension-set is connected at the top with the beam and at the bottomwith the shaker by means of pin joints. Fig. 3. Experimental setup Two types of kinematic shakers were available in the experiments. One of them was an electro-hydraulic cylinder (6) with the maximum force output 2.5·103Nandmaximum stroke 50·10−3m. Itwas anchored to the foundation withbolts.Theother shakerwas a 3.5·103Wasynchronous electricmotorwith a circular cam crankmechanism (7). Themotor was supplied by an electronic inverter enabling smooth control of rotation speed and excitation frequency in the range of (3.5,10)Hz. The beam’s c.o.g. was connected to the vertical guiding rail (4) by means of a roller bearing guaranteeing longitudinal and transversal rigidity of the system. Guiding bar (4) and shaker (7) were both mounted on the rigid cubicoid frame (welded section steel) (8). Vibration control in a pitch-plane suspension model... 661 Numerical values of parameters describing the mechanical elements are presented in the Table 3. Table 3.Technical specifications data Parameter Designation Value Distance (Pg,Pf) lf [m] 0.7 Distance (Pg,Pr) lr [m] 0.7 Total length of the beam L [m] 1.5 Width of the beam a [m] 0.124 Height of the beam b [m] 0.173 Mass of the beam m [kg] 253.3 Moment of inertia of the beam I [kgm2] 49.20 Elasticity factor of the kS,f (kS,r) 42016 front (rear spring) [N/m] Fig. 4. Diagram of data acquisition and control system All measurements were conducted by means of four PSz-20 transducers (two of them located on the beam– xf and xr, and the other two on the sha- kers – wf and wr) andamultipurpose I/Oboardof theRT-DAC4 seriesplaced in a standardPC (Fig.4). TheMATLAB/Simulink environment with RTWT (Real-TimeWindows Target) extension of RTW (Real-TimeWorkshop) tool- box (TheMathWorks, 2003) running on theWindows 2000 operating system 662 B.Sapiński, P.Martynowicz completed the set-up. On the basis of xf and xr measurements, the vertical displacement x and pitch displacement ϕ of the beam’s c.o.g. were received. MRAs currents if, ir calculated in the MATLAB/Simulink were output by means of RTWT/RTW andRT-DAC4, and then converted to PWM signals. 4. Experiments In the next subsections, we present scenario and results of the experiments conducted in open-loop and feedback system configurations with the help of the experimental setup introduced above. 4.1. Open-loop system Toanalyse theopen-loop2DOFpitch-plane system,weobserved the centre of gravity Pg bounce andpitch responses (i.e. x and ϕ) to bottom excitations applied to the front suspension-set. One-side (front or rear) excitation is a ty- pical situation during driving the vehicle. Values of x and ϕ can be calculated as follows x= xf +xr 2 ϕ= xf −xr lf + lr (4.1) We considered two cases of the passive suspension (open-loop) system: in the first case (hereinafter calledOS1)we supplied the coils of bothMRAswith no current, in the second case (hereinafter called OS2) – with the current 0.1A. Let us introduce quality indexes to measure root-mean-square (RMS) ac- celerations of beam bounce (4.2)1 and pitch (4.2)2, and shaker (4.2)3 motions in the time range of (0, tfin) RMSẍ = √ √ √ √ √ tfin ∫ 0 [ẍ(t)]2 dt RMSϕ̈ = √ √ √ √ √ tfin ∫ 0 [ϕ̈(t)]2 dt (4.2) RMSẅf = √ √ √ √ √ tfin ∫ 0 [ẅf(t)] 2 dt To obtain RMS acceleration transmissibilites Tẍ (4.3)1 and Tϕ̈ (4.3)2, we measured the system response to front sine excitation series in the range of Vibration control in a pitch-plane suspension model... 663 (1,10)Hz, and then calculated RMSẍ, RMSϕ̈ and RMSẅf indexes for all analysed frequencies Tẍ = RMSẍ RMSẅf Tϕ̈ = lfRMSϕ̈ RMSẅf (4.3) The results are presented in Fig.5 and Fig.6. On the basis of these me- asurements, we calculated the frequency f-weightedRMSbounce acceleration transmissibility index TW (4.4), according to the weighting factors W(f) as in ISO 2631-1 (1997) – see Table 4 in Subsection 4.2 TW = √ ∑ f [W(f)Tẍ(f)]2 (4.4) Fig. 5. RMS bounce acceleration transmissibility Fig. 6. RMS pitch acceleration transmissibility Besides the resonance frequencies of OS1 are: fx ≈ 3.1Hz (bounce), fϕ ≈ 5.0Hz (pitch), the internal coupling between the two modes causes 664 B.Sapiński, P.Martynowicz the shift of the transmissibilities’ maximums for one-side excitations. Both characteristics reveal some embedded features of the experimental setup: at certain frequencies we observe high acceleration peaks. That means that at these frequencies there is an increased damping and rigidity of the analysed suspension-set due to the MRA nonlinearity. In further analysis, we assume that the front and rear suspension-sets are identical, i.e. possessing equal stiffness factors of the front and rear springs (kS,f = kS,r = k) and equal for bothMRAs values of parameters: α, β, γ, A, c0, c1, k0, k1, x0. 4.2. Feedback system As stated in the beginning of the study, one of the basic suspension pur- poses is the optimisation of drive comfort, thus minimisation of vibrations affecting humans (Kowal, 1996). Accordingly, we introduce a feedback system whose aim is to reduce RMS bounce and pitch acceleration transmissibility indexes Tẍ, Tϕ̈ and the frequency weighted RMS bounce acceleration trans- missibility index TW using the weighting factors as in ISO 2631-1 (1997). To accomplish it, we construct a semi-active cascade control system consisting of two stages (see Fig.7). In the first stage, the system determines values of for- ces F∗MR,f,F ∗ MR,r, whichminimise the assumed quality index J (4.5)1, where: qẍ, qϕ̈ are weighting factors of ẍ and ϕ̈, respectively (J depends on trans- missibility squares and RMSẅf – constant for the chosen excitation type, see equation (4.5)2). In the second stage, the system calculates values of currents if, ir, which cause MRAs to produce resistance forces FMR,f, FMR,r as close to F∗MR,f, F ∗ MR,r as possible to produce for instantaneous relative velocities of beam and shakers J = tfin ∫ 0 { qẍ[ẍ(t)] 2+qϕ̈[ϕ̈(t)] 2 } dt J = ( qẍT 2 ẍ + qϕ̈T 2 ϕ̈ l2 f ) RMSẅf (4.5) Fig. 7. Controller diagram Vibration control in a pitch-plane suspension model... 665 The first stage of the control task utilises the optimal linear-quadratic (LQ) controller – a stabilising controller to be applied in linear dynamic sys- tems (Mitkowski, 1991). TheLQcontrol is the optimal control with respect to the assumed quadratic quality index JLQ (4.9), similar to J (4.5)1. In order that the linearity assumption and task conditions (minimised accelerations) be fulfilled, the model governed by equations (2.4)-(2.5) was modified accor- dingly, yielding a matrix state space equation, see (4.6), in which the state vector X, Eq (4.7)1, now includes velocities ẋ and ϕ̇ as well. Forces FMR,f and FMR,r (producedbyMRAs df and dr) definedby nonlinear relationships are transferred to the control vector U. In order that themodel canbewritten in a form of a linear state space equation, the control U should incorporate excitations wf, wr (4.7)2 Ẋ =AX+BU (4.6) where X = [x,ẋ,ϕ,ϕ̇]> U = [FMR,f,FMR,r,wf,wr] > (4.7) A=           0 1 0 0 − 2k m 0 − k(lf − lr) m 0 0 0 0 1 − k(lf − lr) I 0 − k(l2f + l 2 r) I 0           B=           0 0 0 0 1 m 1 m k m k m 0 0 0 0 lf I − lr I klf I − klr I           (4.8) Thepair (A,B) is stabilisable, hence thefirstnecessaryandsufficient condition for precisely one optimal control strategy is fulfilled (Mitkowski, 1999). Invo- king (4.7), the quality index in the LQ problem for the time horizon (0, tfin) is written in the form JLQ = tfin ∫ 0 { qx[x(t)] 2+qϕ[ϕ(t)] 2+qẋ[ẋ(t)] 2+qϕ̇[ϕ̇(t)] 2+qẍ[ẍ(t)] 2+ +qϕ̈[ϕ̈(t)] 2+rFMR,f [FMR,f(t)] 2+rFMR,r[FMR,r(t)] 2+ (4.9) +rwf [wf(t)] 2+rwr[wr(t)] 2+2rwfwrwf(t)wr(t) } dt where ẍ(t) and ϕ̈(t) are taken from the left-hand side of (4.6). After this substitution, we can write the quality index according to the LQ definition JLQ = tfin ∫ 0 { [X(t)]>QX(t)+ [U(t)]>RU(t)+2[X(t)]>NU(t) } dt (4.10) 666 B.Sapiński, P.Martynowicz where Q, R,N are described as follows Q=           qx+ 4k2 m2 qẍ 0 0 0 0 qẋ 0 0 0 0 qϕ+ 4k2l4 I2 qϕ̈ 0 0 0 0 qϕ̇           R=      rFMR,f +a1 a2 ka1 ka2 a2 rFMR,r +a1 ka2 ka1 ka1 ka2 rwf rwfwr ka2 ka1 rwfwr rwr      (4.11) a1 = 1 m2 qẍ+ l2 I2 qϕ̈ a2 = 1 m2 qẍ+ l2 I2 qϕ̈ N=           −2k m2 qẍ −2k m2 qẍ −2k2 m2 qẍ −2k2 m2 qẍ 0 0 0 0 −2kl3 I2 qϕ̈ 2kl3 I2 qϕ̈ −2k2l3 I2 qϕ̈ 2k2l3 I2 qϕ̈ 0 0 0 0           where lf = lr = l (full symmetryof the 2DOFpitch-plane suspensionsystem). The selection of the matrix elements Q, R was based on the principle that the cost of any variable must be inversely proportional to the squared allowable deflection from the nominal value (Brzzka, 2004). Besides, in order that quality index (4.9) approaches assumed form (4.5)1, the weighting fac- tors of the beam c.o.g. accelerations should be several orders of magnitude larger than theweighting factors of displacements qx, qϕ, velocities qẋ, qϕ̇ and maximal forces rFMR,f , rFMR,r. On account of the fact that the excitations present in the considered LQ problem are disturbances (they are not subject to control), strong constraints are imposed on wf and wr: rwf = rwr =10 11, rwfwr =10 10 and these components of the quality index are not taken into ac- count in the evaluation of system behaviour and ride comfort. Theweightings of the bounce and pitch displacements are taken as qx = qϕ =1.Additionally, it is assumed that: rFMR,f = rFMR,r = 16 · 10 −4 (weak constraints upon the maximum values of FMR,f and FMR,r due to reasonable MRA force output margins) and qẍ = qϕ̈ = 10 5 (large weightings of acceleration in the quali- ty index). To enhance the system behaviour at lower frequencies, we further Vibration control in a pitch-plane suspension model... 667 assume that qẋ = qϕ̇ = 10 3. Such weighting factor values guarantee that all necessary and sufficient conditions for just one optimal control strategy U∗ be fulfilled (Q=Q> ­0,R=R> >0, pair (A,Q) is detectable) as given by formula (4.12) (Mitkowski, 1991) U ∗ = [ F∗MR,f,F ∗ MR,r,w ∗ f,w ∗ r ] > =−KX (4.12) where K is the feedback matrix calculated according to the LQ control law. In the second stage, the values of control currents if, ir are established such that for instantaneous relative velocities (ẋf−ẇf), (ẋr−ẇr), theMRAs produce resistance forces FMR,f, FMR,r equal to F ∗ MR,f, F ∗ MR,r (if possible), or of the same sign as F∗MR,f,F ∗ MR,r (if values F ∗ MR,f,F ∗ MR,r are impossible to produce for the instantaneous relative velocities of the beam and shakers), or equal to zero (if signs of F∗MR,f, F ∗ MR,r are impossible to be produced). This task utilises the hysteresis inverse model (HRM) of theMRA.On the basis of instantaneous values of the signals ẋ, ẍ, ϕ̇, ϕ̈, ẇf, ẇr, ẅf, ẅr and the MRA parameters α, β, γ,A, c0, c1, k0, k1, x0 at the given operating point (Table 1 and Table 2), the control currents if, ir are determined (Sapiński, 2004). Underlying the design of a nonlinear HRM is the velocity-force relationship for various current levels (Sapiński, 2004). Figure 8 presents a MATLAB/Simulink diagram of the developed HRM determining the control current of the front MRA. The main blocks of this diagram are: • Multi-port switch: HYSTERESIS modelling the hysteresis of the MRA; its output depends on (ẋf − ẇf) and (ẍf − ẅf) • Look-up table: Ff ∗ /(x ′ f −w ′ f) block for conversion of the quotient: F∗MR,f/(ẋf − ẇf) into the current if. The currents if and ir calculated in the described above manner are the output of the cascade controller. For the given parameters of the experimental setup described in Table 3, the elements of the feedback matrix are as folows K=      -41064 3615 29213 1001 -41072 3615 29242 -1000 0 0 0 0 0 0 0 0      Zero values of terms in the third and fourth row of thematrix K indicate that no controller action is required to regulate the excitations wf and wr such that the optimal value of the quality index is achieved. 668 B.Sapiński, P.Martynowicz Fig. 8. Simulink diagram of the hysteresis inversemodel Three variants of the cascade controller were studied: HRM-d, in which signals from transducers (xf,xr,wf,wr) are passed directly to the controller (only a primary low-pass filter is used); HRM with a 1st order inertial filter (time-constant 2 · 10−3 s) applied to xf, xr, wf, wr, and HRM-a with the same inertial filter (asHRM)andwith additional informationon thehysteresis width forwarded to the inverse model. Fig. 9. RMS bounce acceleration transmissibility Sine excitations in the range of (1,10)Hz were applied to the front suspension-set as it was done for the open-loop system. The effectiveness of the control algorithm described above was evaluated by performing analysis of the c.o.g. RMS bounce and pitch acceleration transmissibilities Tẍ and Tϕ̈ (see Fig.9 and Fig.10). Also, the frequencyweighted index TW was evaluated (see Table 4). Vibration control in a pitch-plane suspension model... 669 Fig. 10. RMS pitch acceleration transmissibility Table 4.RMS Acceleration transmissibility Type of Open-loop system Feedbacksystem–cascadecontroller excitation OS1 OS2 HRM HRM-a HRM-d wf if = ir =0A if = ir =0.1A Square Tẍ 0.1147 0.2214 0.1161 0.1039 0.1214 (experim.) Tϕ̈ 0.1373 0.2587 0.1376 0.1254 0.1394 Sine TW 2.0066·10 3 2.2612·103 1.9482·103 1.9306·103 1.9855·103 (experim.) Sine TW 2.3072·10 3 2.2176·103 2.0858·103 2.1046·103 – (simulat.) Time patterns of the acceleration ẍ, displacement xf and current if for the front sine excitation wf of the frequency of 6Hz for OS1 (open-loop) and HRM (feedback) systems are shown in Fig.11. Time patterns of ẍ, xf, if,wf for OS1 and HRM-d systems near the first resonance frequency (at 3.25Hz) are shown in the Fig.12. Fig. 11. Response to wf sine excitation at 6Hz 670 B.Sapiński, P.Martynowicz Fig. 12. Response to wf sine excitation at 3.25Hz Figure 13 presents time patterns of xf and if for the front square exci- tation of OS1 and HRM-a systems. Respective values of transmissibilities Tẍ and Tϕ̈ are presented in Table 4. Fig. 13. Response to wf square excitation To compare the experimental and simulation results we present, obta- ined theoretically, bounce andpitch acceleration transmissibilities (Fig.14 and Fig.15) and the frequency weighted transmissibility index TW (Table 4). 5. Discussion of results The results of experiments evidence advantages of the feedback semi-active suspension systemwith a cascade controller presented in Section 4.2 over pas- sive (open-loop) systems OS1, OS2. As we observe in Fig.9 and Fig.10, the Vibration control in a pitch-plane suspension model... 671 Fig. 14. RMS bounce acceleration transmissibility (obtained theoretically) Fig. 15. RMS pitch acceleration transmissibility (obtained theoretically) RMS acceleration frequency transmissibilities of systems HRM, HRM-a both lie below the respective characteristics of OS1 and OS2 in most of the frequ- ency ranges. HRM, HRM-d and especially HRM-a systems are characterised with lowered acceleration peakswith respect toOS1, and lowered accelerations at higher frequencies in comparison with OS2. Figure 11a presents reduction of the acceleration ẍ due to controller operation for the 6Hz front sine excita- tion. Figure 11b shows the reduction of the displacement xf due to controller operation, and also the way in which this result is accomplished: the MRA is stiffened (due to current if) when the time changes of xf and wf are op- posite – this implies the reduction of xf and ẍ amplitudes. Therefore, the controller action is based on the phase difference between xf and wf; a si- gnificant phase difference guarantees that the amplitude of xf is smaller than the amplitude of wf. Figures 12a,b show that the reduction of xf and ẍ, and also the controller action at 3.25Hz, are not so evident as at 6Hz (see Fig.9). Controller operation at 3.25Hz (Fig.12b) is visible in the time ran- 672 B.Sapiński, P.Martynowicz ge of (0.06,0.09) s and also (0.24,0.31) s. The first action is due to very small phase difference between xf and wf, thus very small time range of the op- posite changes of xf and wf. The second action cause theMRA stiffening to guarantee that the amplitude of xf is not larger than wf near the resonance (too small phase difference between xf and wf to enable further reduction of xf amplitude). The time pattern of current if for the square excitation also shows the correctness of controller (HRM-a) operation. At the first stage of the response, during and just after the step change of the input, the MRA is soft – this guarantees reduction of the initial shock.After that, when the input is constant, theMRA is rigid (peak of if current) to accomplish reduction of over-steer. One can observe that in the lower frequency range of (1,5)Hz (including reesonance peaks), theHRM-d controller proves to be themost effective due to the smaller time lag causedby the inputfilters.However, at higher frequencies, HRM-d does not confirm its advantage over OS1 – the signal filtering seems insufficient and the controller is too sensitive to environmental disturbances, especially during signal differentiation necessary for theMRA inverse model. Theabove considerations are borneoutbyvalues of the frequencyweighted bounce acceleration transmissibility index TW derived on the basis of the ISO 2631-1 (1997) standard. Theoretical values of RMSbounce andpitch acceleration transmissibilities ofOS1,OS2,HRMandHRM-a systems (Fig.14 andFig.15) and relevant TW indexes (Table 4) confirm the advantage of the feedback system as well. 6. Conclusions Thepaperdealswith the experimental studyof vibrationcontrol inapitch- plane suspensionmodel equippedwithMRAs. Experimental results prove the necessity of implementation and the efficiency of MRAs in vibro-isolation of the investigated suspension system. As the feedback system with additional information on the hysteresis width exhibits the best vibro-isolation features in general, more efforts will be undertaken to develop amore accurate inverse model of theMRA. Further research will be broadened into a 3DOF pitch-plane model that will take into account also a driver seatwhose suspension is equippedwith the MRA. Vibration control in a pitch-plane suspension model... 673 Acknowledgement The research work has been supported by the state Committee for Scientific Re- search as a part of the research programNo. KBN 4T07B01626. References 1. AhmedA.K.W., 2002,Encyclopedia of Vibration, GroundTransportation Sys- tems, Academic Press 2. Brzózka J., 2004,Regulatory i ukady automatyki, Warszawa 3. FullerC.R., 1996,Acive Control of Vibration, Londyn, Academic Press Ltd. 4. ISO 2631-1, 1997,Mechanical vibration and shock – Evaluation of human expo- sure to whole-body vibration, International Organisation for Standardisation 5. KiduckK.,Doyoung J., 1999,Vibration suppression in aMRFluidDamper Suspension System, Journal of IntelligentMaterial Systems and Structures,10, 779-786 6. Kowal J., 1996, Sterowanie drganiami, Kraków, Gutenberg 7. LORDCORPORATION, 2003, http://www.lord.com 8. Martynowicz P., 2003, Simulation of a vehicle suspensionwithmagnetorhe- ological dampers,Kwartalnik AGH, Mechanika, 22, 355-361 9. McLellan N.S., 1998, On the Development of Real-Time Embeded Digital Controller forHeavyTruck Semiactive Suspensions,VirginiaTech,MasterThe- sis 10. Mitkowski W., 1991, Stabilizacja systemów dynamicznych, Cracow, AGH 11. NagaiM., OndaM., HasegawaT., YoshidaH., 1996, Semi-active control of vehicle vibration using continuously variable damper, Third International Conference on Motion and Vibration Control, Chiba 12. PareC.A., 1998,Experimental Evaluation of Semiactive Magneto-Rheological Suspensions for Passenger Vehicles, Virginia Tech, Master Thesis 13. Sapiński B., Piłat A., Rosół M., 2003, Modelling of a quarter-car semi active suspension withMR damper,Machine Dynamics Problems, 27 14. Sapiński B., 2004, Linear magnetorheological fluid dampers for vibrationmi- tigation: modelling, control and experimental testing, Rozprawy, Monografie AGH, 128 674 B.Sapiński, P.Martynowicz 15. Sapiński B., Martynowicz P., 2004, Sterowanie liniowo-kwadratowe drga- niamiw ukadzie zawieszeniamagnetoreologicznego (Linear-quadratic vibration control for MR suspension system), Czasopismo Techniczne Politechniki Kra- kowskiej, 5, 335-343 16. SimonD.E., 2000,Experimental Evaluation of SemiactiveMagneto-Rheological Primary Suspensions for Heavy Truck Applications, VirginiaTech, PhDThesis 17. Song X., 1998, Design of Adaptive Control Systems with Applications to Magneto-Rheological Dampers, Virginia Tech, PhDThesis 18. Spencer B., 1996, Phenomenological model of a magnetorheological damper, Journal of Engineering Mechanics 19. THEMATHWORKS, 2003, Real TimeWorkshopUser’s Guide 20. VIRGINIA TECH, 2002, http://www.avdl.me.vt.edu 21. Yao G.Z., Yap F.F., Chen G., Li W.H., Yeo S.H., 2002,MR damper and its application for semi-active control of vehicle suspension system, Technical Note, Mechatronics, 12, 963-973 Sterowanie drganiami w płaskim modelu zawieszenia z amortyzatorami magnetoreologicznymi Streszczenie W artykule przedstawiono analizę eksperymentalną sterowania drganiami za- wieszenia z dwoma niezależnymi amortyzatorami magnetoreologicznymi. Do analizy wykorzystano model zawieszenia w płaszczyźnie przechyłów wzdłużnych, posiadają- cy dwa stopnie swobody (ruch pionowy i przechył wzdłużny). Model ten zbadano w układach otwartym i zamkniętym przy harmonicznych oraz prostokątnych wymu- szeniach kinetycznych. Eksperymenty przeprowadzono na wykonanym według wła- snegoprojektu stanowiskubadawczymprzyużyciu środowiskapomiarowo-sterującego MATLAB/Simulink.Wyniki pomiarówpotwierdziły skuteczność sterowania drgania- mi za pomocą amortyzatorówmagnetoreologicznych w rozważanym układzie zawie- szenia. Manuscript received December 20, 2004; zccepted for print April 18, 2005