SUPPRESSION OF HEAVY TRUCK DRIVER SEAT VIBRATION USING SLIDING MODE CONTROL & QUANTITATIVE FEEDBACK THEORY

Transcription

1 SUPPRESSION OF HEAVY TRUCK DRIVER SEAT VIBRATION USING SLIDING MODE CONTROL & QUANTITATIVE FEEDBACK THEORY Nuraj I. Rajapakse Gemunu S. Happawana* Yildirim Hurmuzlu Graduate student Assistant Profsor Profsor Department of Mecanical Engineering Soutern Metodist University, Dallas, TX 7575, USA Keywords: Nonlinear modeling, Sliding surface, Feedback control, Tracking error, Stability, Time varying surface, Boundary layer, Cattering, Error dynamics, First order filter, and Model uncertainty. Abstract Tis paper prents a robust control metod tat combin Sliding Mode Control (SMC) and Quantitative Feedback Teory (QFT) for digning a driver seat of a eavy veicle to reduce driver fatigue. A matematical model is considered to analyze tracking control caracteristics troug computer simulation in order to demonstrate te effectivens of te proposed control metodology. Te SMC is used to track te trajectory of te dired motion beavior of te seat. However, wen te system enters into sliding regime, cattering occurs due to switcing delays as well as veicle system vibrations. Te cattering is eliminated wit te introduction QFT inside te boundary layer to ensure smoot tracking. Furtermore, using SMC alone requir iger actuator forc for tracking tan using bot te control scem togeter, caus various problems in selecting ardware. Problems wit noise amplification, ronanc, prence of uncertainti and unmodeled ig frequency dynamics can largely be avoided wit te use of QFT over oter optimization metods. Te main contribution of tis paper is to provide te guidance in digning te controller to reduce eavy veicle seat vibration so tat te driver s sensation of comfort maintains a certain level at all tim. NOMENCLATURE A - Cross sectional area of te ydraulic actuator piston F af - Actuator force F - Combined nonlinear spring and damper force of te driver seat k - Stiffns of te spring between te seat and te sprung mass m - Mass of te driver and te seat m s - Sprung mass x - Vertical position coordinate of te driver seat - Vertical position coordinate of te sprung mass x s * Corrponding Autor 1. INTRODUCTION Relationsip between driver fatigue and seat vibration as been used in many literatur based only on anecdotal evidence witout doing appropriate rearc [1, ]. It is widely believed and proved in field tts, tat te lower vertical acceleration levels will increase comfort level of te driver [3-5]. Heavy veicle truck drivers wo usually experience vibration levels around 3 1

2 Hz wile driving are subjected to fatigue and drowsins [6]. Lack of concentration wile driving leads to road accidents tat caus due to fatigue and drowsins. Most of te early laboratorary and field rearc revealed and proved tis beavior for different road conditions [1-6]. Furtermore, it is strongly believed tat intermittent and random vibration can ave a stimulating or weakening effect on te driver. Body metabolism and cemistry can largely be affected due to tis vibration exposure and rult in fatigue [7]. Typical wole-body vibration exposure levels of eavy veicle drivers are in te range 0.4 m/s -.0 m/s wit a mean value of 0.7 m/s in te vertical axis [3-6]. Suspension system tat links te weels and te veicle body determin te ride comfort of te veicle and its caracteristics must be properly evaluated to dign a proper driver seat under various operating conditions. A properly digned driver seat caus ls driver fatigue and it maintains same vibration level against any external disturbance to provide te bt performanc in riding. It also improv veicle control, safety and stability witout canging te ride quality and addrs te areas of road olding, load carrying and passenger comfort. Furtermore, it provid directional control during andling maneuvers. Over te past decad, te application of sliding mode control as been focused in many directions suc as te use of sliding control in underwater veicl, automotive applications and robot manipulators [8-17]. Te combination of sliding controllers wit state observers was also developed and discussed for bot te linear and nonlinear cas [18, 19]. Nonlinear systems are difficult to model as linear systems since tere exist some parametric uncertainti and modeling inaccuraci tat can eventually ronate te system [9]. Te sliding mode control can be used for nonlinear stabilization problems in digning controllers. In tis paper, te sliding mode control teory is applied to track te motion beavior of a driver seat of a eavy veicle to a trajectory tat can reduce driver fatigue and drowsins. Te trajectory is not necsarily be fixed and can be varied accordingly wit rpect to te driver requirements. Tis control metodology can overcome most of te road disturbanc and provide predetermined seat motion pattern to avoid driver fatigue. However, due to parametric uncertainti and modeling inaccuraci cattering can be observed wic caus a major problem in applying SMC alone. In general, te cattering enanc te driver fatigue and also leads to premature failure of controllers. SMC wit QFT is a good approac for tis kind of problems to eliminate te cattering satisfactorily. Furtermore, it reduc te control effort necsary to maintain te dired motion of te seat and it is broadly discussed in tis paper.. MATHEMATICAL MODELING Figure 1 below sows a scematic of a driver seat of a eavy truck. Te model is consisting of an actuator, spring, damper and a motor sitting on te sprung mass. Te actuator provid actuation force by means of a ydraulic actuator to keep te seat motion witin a comfort level for any road disturbance, wile te motor maintains dired inclination angle of te driver seat wit rpect to te roll angle of te sprung mass. Driver seat mecanism is connected to te sprung mass by using a pivoted joint and it provid te flexibility to cange te roll angle. Te system is equipped wit sensors to measure te sprung mass vertical acceleration and roll angle. Hydraulic prsure drop and spool valve displacement are also used as feed back signals.

3 Mass of te driver & Seat m x Spring Actuator Motor Sprung Mass, m s x s Fig. 1: Te driver seat on te sprung mass wit supporting devic (Actuator and a motor) 3. EQUATIONS OF MOTION Based on te matematical model developed above, te equation of motion in te vertical direction for te driver and te seat can be written as follows: ( 1/ ) ( 1/ ) ( 1) && x = m F + m F Were af F k d d C d& d& F 3 = ( + ) + (1 + ) af = AP L s 1i s d = ( x x ) a sinθ see te appendix for details In four-way valve-piston ydraulic actuator system, te rate of cange of prsure drop across te ydraulic actuator piston, P L,, is given by: VP& t L = Q CtpPL A( x& x& s ) 4βe Were Vt Total actuator volume βe Effective bulk modulus of te fluid Q Load flow Ctp Total piston leakage coefficient A Piston area 3

4 Te load flow of te actuator is given by: [ ] ω ρ Q = sgn Ps sgn( xv) PL Cd xv (1/ ) Ps sgn( xv) PL 3 Were Ps Hydraulic supply prsure ω Spool valve area gradient xv Displacement of te spool valve ρ Hydraulic fluid density C Discarge coefficient d Voltage or current can be fed to te servo-valve to control te spool valve displacement of te actuator for generating te force. Moreover, Stiction Model for ydraulic spool can be included to reduce te cattering even furter, but it is not discussed ere. 4. SLIDING MODE CONTROL PRELIMINARIES In sliding mode control a time varying surface of S(t) is defined wit te use of a dired vector, X d, and te name is given as te sliding surface. If te state vector, X, can remain on te surface, S(t) for all time, t>0, tracking can be acieved. In oter words, problem of tracking te state vector, X X d (n-dimensional dired vector) is solved. Scalar quantity, s, is te distance to te sliding surface and tis becom zero at te time of tracking. Tis replac te vector X d effectively by a first order stabilization problem in s. Te scalar s reprents a realistic measure of tracking performance since bounds on s and te tracking error vector are directly connected togeter. In digning te controller, a feedback control law U can be cosen appropriately to satisfy sliding conditions. Te control law across te sliding surface can be made discontinuous in order to facilitate for te prence of modeling imprecision and of disturbanc. Ten te discontinuous control law U is smooted accordingly using QFT to acieve an optimal trade-off between control bandwidt and tracking precision. Consider te second order single-input dynamic system [9]: && x = f( X) + b( X) U ( 4) Were X State vector, [ x T x& ] x f b Output of intert Nonlinear time varying or state dependent function Control gain U Control input torque Te control gain, b, can be time varying or state-dependent but is not completely known. In oter words, it is sufficient to know te bounding valu of b. 0< b b b 5 min max Te timated value of te control gain, b, can be found as [] 4

5 1/ b = ( bminbmax ) Bounds of te gain, b can be written in te form: b β b Were 1 β 1/ max β = b b min Te nonlinear function f can be timated ( f ) and te timation error on f is to be bounded by some function of te original stat of f. f f F ( 6) 7 In order to ave te system track on to a dired trajectory x(t) x d (t), a time-varying surface, S(t) in te state-space R by te scalar equation s(x;t) = s = 0 is defined. d s = + λ x = x& + λx dt Were T X = X Xd = [ x x& ] λ = Positive constant (First order filter bandwidt) Wen te state vector reac te sliding surface, S(t), te distance to te sliding surface, s, becom zero. Tis reprents te dynamics wile in sliding mode, suc tat s & = 0 ( 9) Wen te Eq. (9) is satisfied, te equivalent control input, U eq can be obtained as follows: b b bu U f f, Tis leads to U = f + && x λx&, and 10 d U is given by 1 U = U k x s b ( sgn) ( 8) Were 5

6 k(x) - Control discontinuity Te control discontinuity, k(x) is needed to satisfy sliding conditions wit te introduction of an timated equivalent control. However, tis control discontinuity is igly dependent on te parametric uncertainty of te system. In order to satisfy sliding conditions and te system trajectori to be remained on te sliding surface, te following must be satisfied. d s = ss & η s 1 dt ( 11) Were η is a strictly positive constant. Te control discontinuity can be found from te above inequality: s ( f bb f ) + (1 bb )( && xd + λ x& ) bb k( x)sgn( s) η s s ( f bb f ) + (1 bb )( && xd + λx& ) + η s bb k( x) s s kx bb f f + ( bb 1)( xd + λx) + bb η s && & For te bt tracking performance, k(x) must satisfy te inequality as follows: kx bb f f + ( bb 1)( && x + λ x& ) + bbη d As seen from te above inequality, te value for k(x) can be simplified furter by rearranging f as below: f = f +( f - f ) and by using f f F kx bb ( f f ) + ( bb 1)( f && xd + λx& ) + bb η kx bb ( f f ) + ( bb 1)( f && xd + λx& ) + bb η kx β( F+ η) + ( β 1) ( f && x + λx& ) kx β( F+ η) + ( β 1) U 1 d By coosing k(x) to be large enoug, sliding conditions can be guaranteed. Tis control discontinuity across te surface s = 0 increas wit te increase of uncertainty of te system parameters. It is important to mention tat te functions for f and F may be tougt of as any measured variabl external to te system and tey also may depend explicitly on time. 6

7 5. REARRANGEMENT OF THE SLIDING SURFACE Sliding condition as in te previous sections, s & = 0 do not necsarily provide smoot tracking performance across te sliding surface. In order to guarantee smoot tracking performance and to dign an improved controller, in spite of te control discontinuity, sliding condition can be redefined, i.e. s& = α s [8], so tat tracking of x x d would acieve an exponential convergence. Here te parameter α is a positive constant. Te value for α is determined by considering te tracking smootns of te unstable system. Tis condition modifi U as follows: U = f + && xd λx& αs and k( x) must satisfy te condition below. kx bb f f + ( bb 1)( && x + λx& ) + bbη α s d Furter kx can be simplified as kx β( F+ η) + ( β 1) U + ( β ) α s 13 Even toug te tracking condition is improved, cattering of te system on te sliding surface is remained as an inerent problem in SMC. Tis can indeed be removed by using QFT as explained in section QFT CONTROLLER DESIGN In te previous sections of sliding mode preliminari, digned control laws, wic satisfy sliding conditions, lead to perfect tracking even wit some model uncertainti. However, after reacing te boundary layer, te cattering of te controller is observed because of te discontinuity across te sliding surface. In practice tis situation extremely complicat digning ardware for te controller. Furtermore, even after digning te controller, a dirable performance could not be acieved because of te time lag of te ardware functionality. Also cattering excit undirable ig frequency dynamics of te system. By using QFT controller, te switcing control laws can be modified to eliminate cattering in te system since QFT controller works as a lowpass filter. In QFT, attractivens of te boundary layer can be maintained for all t >0 by varying te boundary layer tickns, φ, as follows: 1 d Wen s φ s ( & φ η) s [9] ( 14) dt It is evident from Eq. (14) tat te boundary layer attraction condition is igly guaranteed in te case of boundary layer contraction ( & φ < 0 ) tan te boundary layer expansion ( & φ > 0 ). Furter more Eq. (14) can be used to modify te control discontinuity gain, k(x), to smooten te performance by putting k( x)sat( s/ φ ) instead of kx sgn( s ). Relationsip between k( x) and k( x ) for te boundary layer attraction condition can be prented for bot te cas as follows: & φ > = & & φ < = & 0 k( x) k( x) φ/ β 15 0 k( x) k( x) φβ 16 Ten te control law, U, and s& become 7

8 1 U = U k x s b ( sat( / φ) ) 1 s& = bb ( k( x)sat( s/ φ) + αs) + g( x, xd ) 1 1 Were gxx (, d) = ( f bb f) + (1 bb )( && xd + λx& ) Since k( x ) and g are continuous in x, te system trajectori inside te boundary layer can be exprsed in terms of te variable s and te dired trajectory x d by te following relation: Inside te boundary layer, i.e., s φ sat( s/ φ) = s/ φ and x xd & Hence s = βd( k( xd)( s/ φ) + αs) + g( xd) 17 Were β d b ( x ) d max = b ( xd ) min 1/ Te dynamics inside te boundary layer can be written by combining Eq. (15) and Eq. (16) as follows: & φ > = & & φ < = & 0 k( xd) k( xd) φ/ βd 18 0 k( xd) k( xd) φβd 19 By taking te Laplace transform of Equation (17), It can be sown tat te variable s is given by te output of a first-order filter, wose dynamics entirely depends on te dired state x d. are te inputs to te first order filter, but tey are igly uncertain. g x d g( x d ) 1 + β k x φ+ α P d( ( d)/ ) s 1 P+ λ x φ selection s selection Fig. : Structure of te closed-loop error dynamics Were P is te Laplace variable. Tis sows tat cattering in te boundary layer due to perturbations or uncertainty of g( x d ) can satisfactorily be removed by a first order filtering as sown above in te diagram as long as ig-frequency unmodeled dynamics are not excited. Furter more te boundary layer tickns, φ, can be selected as te bandwidt of te first order filter of aving input perturbations same as tat of te first order filter from te definition of s, wic leads to tuning φ wit λ : 8

9 β [ k( x )/ φ+ α] = λ k x d d ( d) = ( λ/ βd α) φ 0 Combining Eq. (18) and Eq. (0) yields ( d) > φλ ( / βd α) and + d = dkxd kx & φ ( λ αβ ) φ β 1 Also by combining Eq. (19) and Eq. (0) rults ( d) < φλ ( / βd α) and + d d = k xd d k x & φ ( φ/ β )[( λ/ β ) α] / β Equations (15) and (1) & φ > = β β φ λ β α 0 kx kx ( d / ) [ kx ( d) ( / d )] 3 And combining Eq.(13) wit Eq. (19) & φ < = β β φ λ β α 0 kx kx ( / d) [ kx ( d) ( / d )] 4 In addition, initial value of te boundary layer tickns, φ (0), is given by substituting x d at t=0 in Eq. (0). k( xd (0)) φ(0) = ( λ / β ) α d Sections discussed above can be used for applications to track and stabilize igly nonlinear systems. Sliding mode control along wit QFT provid better system controllers and it leads to select ardware easier tan using SMC alone. Te application of tis teory to truck driver seat and its simulation are given in te following sections. 7. APPLICATIONS AND SIMULATIONS (MATLAB) Equation (1) can be reprented as, && x = f + bu ( 5) 9

10 Were f = ( 1/ m) F b = 1/ m U = F af Te exprsion, f is a time varying function of x s and te state vector, x. Te time varying function, x s can be timated from te information of te sensor, attaced to te sprung mass and its limits of variation must be known. Te exprsion, f and te control gain, b are not required to be known exactly, but teir bounds sould be known in applying SMC and QFT. In order to perform te simulation, x s is assumed to vary between -0.3m to 0.3m and it can be approximated by te time varying function, Asin( ω t), were ω is te disturbance angular frequency of te road by wic te unsprung mass is oscillated. Te bounds of te parameters are given as follows: m m m x x x b b b min max smin s smax min max Estimated valu of m and x s : m = ( m m ) 1/ min max 1/ s = smin smax x x x Above bounds and te timated valu can rougly be obtained for some eavy trucks by utilizing field tt information [0-5]. Tey are as follows: m = 50 kg, m = 100 kg, x = 0.3 m, x = 0.3 m, ω = π (0.1 10) rad / s, A= 0.3 min max smin smax Te timated nonlinear function, f, and bounded timation error, F, are given by: ( / ) f = k m x x s F = f f b = β = max ( max min ) x = ( x ) *( x ) s s s 1/ Te sprung mass is oscillated due to road disturbanc and its canging pattern is given by te vertical angular frequency, ω = π( sin( πt) ). Tis function for ω is used in te simulation in order to vary te sprung mass frequency from 0.1 to 10 Hz. Nevertels, ω can 10

11 be measured by using te sensors in real time and be fed to te controller to timate te control force necsary to maintain te dired frequency limits of te driver seat. Expected trajectory for x is given by te function, xd = Bsinωdt, were ω d is te dired angular frequency of te driver to ave comfortable driving conditions to avoid driver fatigue in te long run. B and ωd are assumed to be.05 m and π *0.5rad/s during te simulation wic yields 0.5 Hz continuous vibration for te driver seat over te time. Furtermore, Mass of te driver and seat is considered as 70 kg trougout te simulation. Tis value cang from driver to driver and can be obtained by te attaced load cell to te driver seat to calculate te control force. Spring constant between te driver seat and te sprung mass is selected as 100 N/m in obtaining plots. It is important to mention tat, tis control sceme provid sufficient room to cange te veicle parameters of te system according to te driver requirements to acieve ride comfort. 8. USING SLIDING MODE ONLY In tis section te tracking is acieved by using SMC alone and te simulation rults are obtained as follows: Consider x = x(1)( m) and x& = x()( m/ s). Te Eq. (5) is reprented in state space form as follows: x& (1) = x() x& () = k / m x(1) x + bu s Combining Eq. (8), Eq. (10) and Eq. (5), timated control law becom, U = f + && xd λ[ x() x& d ] Figure 3 to Fig. 6 sows system trajectori, tracking error and control torque for te initial condition: [ x, x& ]=[0.1m, 1m/s.] using te control law. Figure 3 provid te tracked vertical displacement of te driver seat vs. time and perfect tracking beavior can be observed. Figure 4 exibits te tracking error and it is enlarged in Fig. 5 to sow it s cattering beavior after te tracking is acieved. Cattering is undirable for te controller tat mak impossible in selecting ardware and leads to premature failure of ardware. Te valu for λ andη in Eq. (8) and Eq. (11) are cosen as 0 and 0.1[9] to obtain te plots and to acieve satisfactory tracking performance. Te sampling rate of 1 khz is selected in te simulation. s & = 0 condition and te signum function are used. Te plot of control force vs. time is given in Fig. 6. It is very important to mention tat, te tracking is guaranteed only wit excsive control forc. Mass of te driver and driver seat, limits of it s operation, control bandwidt, initial conditions, sprung mass vibrations, cattering and system uncertainti are various factors tat cause to generate uge control forc. It is wortwile to mention tat tis selected example is governed only by te linear equations wit sine disturbance function, wic cause for te controller to generate periodic sinusoidal signals. In general, te road disturbance is sporadic and te smoot control action can never be expected. Tis will lead to cattering and QFT is needed to filter tem out. Moreover, applying SMC wit QFT can reduce excsive control forc and will ease te selection of ardware. In subsequent rults, te spring constant of te tir is selected as 100kN/m and te damping coefficient as 300kNs/m. Some of te trucks numerical parameters [8, 1] are used in obtaining plots and tey are listed below: m =100kg, m s =3300kg, m u =1000kg, k s = 00 kn/m, k = 1 kn/m, C s = 50 kns/m, C = 0.4 kns/m, J s = 3000 kgm, J u = 900 kgm, A i = 0.3 m, S i =0.9 m, and a 1i = 0.8 m. 11

12 Fig. 3: Vertical displacement of driver seat vs. time Fig. 4: Tracking error vs. time Fig. 5: Zoomed in tracking error vs. time Fig. 6: Control force vs. time 9. USE OF QFT ON THE SLIDING SURFACE In order to lower te excsive control force and to furter smooten te control beavior wit a view of reducing cattering, QFT is introduced in side te boundary layer. Te following graps are plotted for te initial boundary layer tickns of 0.1 meters. 1

13 Fig. 7: Vertical displacement of driver seat vs. time Fig. 8: Tracking error vs. time Fig. 9: Zoomed in tracking error vs. time Fig. 10: Control force vs. time Fig. 11: Zoomed in control force vs. time Fig. 1: s-trajectory wit timevarying boundary layer vs. time 13

14 Figure 7 again sows tat te system is tracked to te trajectory of intert and it follows te dired trajectory of te seat motion over te time. Figure 9 provid zoomed in tracking error of Fig. 8 wic is very small and perfect tracking condition is acieved. Te control force needed to track te system is given in Fig. 10. Figure 11 provid control forc for bot te cas, i.e. SMC wit QFT and SMC alone. SMC wit QFT yields lower control force and tis can precisely be generated by using a ydraulic actuator. Increase of te parameter λ will decrease te tracking error wit te increase of initial control effort. Varying tickns of te boundary layer allows te better use of te available bandwidt, wic caus to reduce te control effort for tracking te system. Parameter uncertainti can effectively be addrsed and te control force can be smootened wit te use of bot te SMC and QFT rater tan applying SMC alone. Furtermore, a succsful application of QFT metodology requir selecting suitable function for F, since te cange in boundary layer tickns is dependent on te bounds of F. Increase of te bounds of F will increase te boundary layer tickns tat leads to overtimate te cange in boundary layer tickns and te control effort. Evolution of dynamic model uncertainty wit time is given by te cange of boundary layer tickns. Rigt selection of te parameters and teir bounds always rult in lower tracking errors and control forc, wic will ease coosing ardware for most of te applications. 10. CONCLUSIONS Tis paper provid adequate information in digning a road adaptive driver seat of a eavy truck via a combination of SMC and QFT. Based on te assumptions, te simulation sows tat te adaptive driver seat controllers provide superior driver comfort over a wide range of road disturbanc. However, Parameter uncertainty, te prence of unmodeled dynamics suc as structural ronant mod, neglected time-delays and finite sampling rate can largely cange te dynamics of te systems. SMC provid effective metodology to dign and tt te controllers in te performance trade-offs, tus te tracking is guaranteed witin te operating limits of te system. Combine use of SMC and QFT facilitat te controller to beave smootly and wit minimum cattering tat is an inerent obstacle of using SMC alone. Cattering reduction by te use of QFT supports in selecting ardware and also to reduce excsive control action. In tis paper simulation study is done for a linear system wit sinusoidal disturbance inputs. It is seen tat very ig control effort is needed due to fast switcing beavior in te case of using SMC alone. QFT smootens te switcing nature and te control effort can be minimized. Most of te controllers fail wen excsive cattering is prent and SMC wit QFT can be used effectively to smooten te control action. Furtermore, in tis example control gain is fixed and it is independent of te stat. Tis will ease te control manipulation. But, tis is not te case in many situations were cattering is most likely te problem. Tis developed teory can be used effectively in most of te control problems to reduce cattering and to lower te control effort. To mention one ting particularly is tat te acceleration feedback is not always needed for position control, but it depends mainly on te control metodology and te system employed. Here, we aven t discussed oter seat control metods and teir trade offs. In order to implement te control law, te road disturbance frequency, ω sould be measured at a rate more tan 1000Hz (just to comply wit te simulation requirements) to update te system, but iger te better. Te bandwidt of te 14

15 actuator depends upon several factors, i.e. ow quickly te actuator can generate te force needed, road profile, rponse time, and signal delay etc. Driver fatigue is a contributing factor in between 4 to 30 percent of road cras and te low frequency vibrations in eavy veicl are mostly caused by te long wavelengts of te road surface. Te wavelengts occur on roads creating an undulating effect. Valuable scientific data on Pysiological and psycological effects are needed to develop guidelin for better truck seat digns. Tis approac can easily be adopted for most of te digns and provide satisfactory rults to reduce driver fatigue and drowsins. 11. REFERENCES [1] L. J. Wilson and T. W. Horner, Data Analysis of Tractor-Trailer Drivers to Asss Drivers Perception of Heavy Duty Truck Ride Quality, Report DOT-HS , National Tecnical Information Service, Springfield, VA, USA, [] J. M. Randall, Human subjective rponse to lorry vibration: implications for farm animal transport, J. Agric. Engng, R, Vol 5, pp , 199. [3] U. Landstrom and R. Landstrom, Cang in wakefulns during exposure to wole body vibration, Electroencepal, Clin, Neuropysiol, Vol 61, pp , [4] A.O. Altunel, Te effect of low-tire prsure on te performance of fort products transportation veicl, Master s tis, Louisiana State University, Scool of Fortry, Wildlife and Fiseri, [5] A.O. Altunel and C. F. de Hoop, Te Effect of Lowered Tire Prsure on a Log Truck Driver Seat, Louisiana State University Agric. Center, Baton Rouge, USA, Vol. 9, no., [6] N. Mabbott, G. foster and B. Mcpee, Heavy Veicle Seat Vibration and Driver Fatigue, Australian Transport Safety Bureau, Report No. CR 03, pp. 35, 001. [7] Y. Kamenskii and I. M. Nosova, Effect of wole body vibration on certain indicators of neuro-endocrine procs, Noise and Vibration Bulletin, pp , [8] E. Z. Taa, G. S. Happawana and Y. Hurmuzlu, Quantitative feedback teory (QFT) for cattering reduction and improved tracking in sliding mode control (SMC), ASME J. of Dynamic Systems, Measurement, and Control, Vol. 15, pp , 003. [9] E. S. Jean-Jacqu and L. Weiping, Applied Nonlinear Control, Prentice-Hall, Inc., Englewood Cliffs, New Jersey 0763, [10] J. K. Roberge, Te mecanical seal, Bacelor s tis, Massacusetts Institute of Tecnology, [11] R. C. Dorf, Modern Control Systems, Addison-Wley, Reading, Massacusetts, pp ,

16 [1] K. Ogata, Modern Control Engineering, Prentice-Hall, Englewood Cliffs, New Jersey, pp , [13] D. T. Higdon and R. H. Cannon, ASME J. of te Control of Unstable Multiple-Output Mecanical Systems, ASME Publication 63-WA-148, New York, [14] J. G. Truxal, State Models, Transfer Functions, and Simulation, Monograp 8, Discrete Systems Concept Project, [15] K. H. Lundberg and J. K. Roberge, Classical dual-inverted-pendulum control In Proceeding of te IEEE CDC 003, Maui, Hawaii, pp , 003. [16] L. C. Pillips, Control of a dual inverted pendulum system using linear-quadratic and H- infinity metods, Master s tis, Massacusetts Institute of Tecnology, [17] W. McC. Siebert, Circuits, Signals, and Systems, MIT Prs, Cambridge, Massacusetts, [18] J. K. Hedrick and S. Gopalswamy, Nonlinear Fligt Control Dign via Sliding Metod, Dept. of Mecanical Engineering, Univ. of California, Berkely, [19] A. G. Bondarev, S. A. Bondarev, N. Y. Kostylyova and V. I. Utkin, Sliding Mod in Systems wit Asymptotic State Observers, Autom. Remote Contr., 6, [0] B. Tabarrok, X. Tong, Directional Stability Analysis of Logging Trucks by a Yaw Roll Model, Tecnical Reports, University of Victoria, Mecanical Engineering Department, pp. 57-6, 1993 [1] T. D. Gillpie, Fundamentals of Veicle Dynamics, SAE, Inc. Warrendale, PA, 199. [] E. Esmailzade, L. Tong. and B. Tabarrok, Road Veicle Dynamics of Log Hauling Combination Trucks, SAE Tecnical Paper Seri 91670, pp , [3] J.Y. Wong, Teory of Ground Veicl, New York, N.Y., Jon Wiley and Sons, [4] B. Tabarrok, and L. Tong, Te Directional Stability Analysis of Log Hauling Truck Double Doglogger, Tecnical Reports, University of Victoria, Mecanical Engineering Department, DSC-Vol. 44, pp , 199. [5] P.V. Aksionov, Law and criterion for evaluation of optimum power distribution to veicle weels, Int. J. Veicle Dign, Vol. 5, No. 3, pp ,

17 APPENDIX ө s Seat x F F s1 x s a 1i Suspension ө s F s ө u S x u i F t1 F t Tir & axle F t3 F t4 A i Fig. 13: Five-degree-of-freedom roll and bounce motion configuration of te system to a sudden impact T i COMMON EQUATION FOR REPRESENTING NONLINEAR FORCE Nonlinear tire forc, suspension forc, and driver seat forc can be obtained by substituting appropriate coefficients to te following nonlinear equation tat covers wide range of operating conditions for reprenting dynamical beavior of te system. 3 F = k( d + d ) + C(1 + d& ) d& Were F Force k Spring constant C Damping coefficient d Deflection d& Rateof cangeof deflection For te suspension: F = k ( d + d ) + C (1 + d& ) d&, i = 1, 3 si si si si si si si For te tir: 3 & & ti ti ti ti ti ti ti F = k ( d + d ) + C (1 + d ) d, i = 1,,3,4 For te seat: 3 = ( + ) + (1 + & ) & F k d d C d d 17

18 DEFLECTION OF THE SUSPENSION SPRINGS AND DAMPERS Based on te matematical model developed, deflection of te suspension system on te axle is found for bot sid as follows: Deflection of side1, ds1 = ( xs xu) + Si(sinθs sin θu) Deflection of side, d = ( x x ) S (sinθ sin θ ) s s u i s u DEFLECTION OF THE SEAT SPRINGS AND DAMPERS By considering te free body diagram in Fig. 13, deflection of te seat is obtained as follows: d = ( x x ) a sinθ s 1i s TIRE DEFLECTIONS Te tir are modeled by using springs and dampers. Deflections of te tir to a road disturbance are given by te following equations. Deflection of tire1, dt1 = xu + ( Ti + Ai)sinθu Deflection of tire, dt = xu + Ti sinθu Deflection of tire3, dt3 = xu Tisinθu Deflection of tire 4, d = x ( T + A)sinθ t4 u i i u EQUATIONS OF MOTION Based on te matematical model developed above, te equations of motion for eac of te sprung mass, unsprung mass, and te seat are written by utilizing te free-body diagram of te system in Fig. 13 as follows: t Vertical and roll motion for te i axle (unsprung mass) m && uxu = Fs + Fs Ft + Ft + Ft + Ft J && θ = S F F θ + T F F θ + T + A F F θ ( 1 ) ( 1 3 4) ( 6) cos cos cos ( 7) u u i s1 s u i t3 t u i i t4 t1 u Vertical and roll motion for te sprung mass mx && s s = ( Fs 1 + Fs ) + F ( 8) J && θ = S F F θ + a F θ cos cos ( 9) s s i s s1 s 1i s Vertical motion for te seat mx && = F ( 30) Equations (6)-(30) ave to be solved simultaneously, since tere exist many parameters and nonlineariti. Nonlinear effects can better be understood by varying te parameters and 18

19 examining relevant dynamical beavior, since cang in parameters cange te dynamics of te system. Furtermore, Equations (6)-(30) can be reprented in te pase plane, wile varying te parameters of te truck, since eac and every trajectory in te pase portrait caracteriz te state of te truck. Equations above can be converted to te state space form and te solutions can be obtained using MATLAB. Pase portraits are used to observe te nonlinear effects wit te cange of te parameters. Cange of initial conditions clearly cang te pase portraits and te important effects on te dynamical beavior of te truck can be understood. 19