A truck with controlled semi-active suspensions traversing a bridge is examined for benefits to the bridge structure. The original concept of a road-friendly truck was extended to a bridge-friendly vehicle, using the same optimization tools. A half-car model with two independently driven axles is coupled with simply supported bridges (beam, slab model) with the span range from 5 m to 50 m. Surface profile of the bridge deck is either stochastic or in the shape of a bump or a pot in the mid-span. Numerical integration in the MATLAB/SIMULINK environment solves coupled dynamic equations of motion with optimized truck suspensions. The rear axle generates the prevailing load and to a great extent determines the bridge response. A significant decrease in contact road-tire forces is observed and the mid-span bridge deflections are on average smaller, when compared to commercial passive suspensions.

Máca, J.

Valášek, M.

Šmilauer, V.

Acta Polytechnica

English

A truck with controlled semi-active suspensions traversing a bridge is examined for benefits to the bridge structure. The original concept of a road-friendly truck was extended to a bridge-friendly vehicle, using the same optimization tools. A half-car model with two independently driven axles is coupled with simply supported bridges (beam, slab model) with the span range from 5 m to 50 m. Surface profile of the bridge deck is either stochastic or in the shape of a bump or a pot in the mid-span. Numerical integration in the MATLAB/SIMULINK environment solves coupled dynamic equations of motion with optimized truck suspensions. The rear axle generates the prevailing load and to a great extent determines the bridge response. A significant decrease in contact road-tire forces is observed and the mid-span bridge deflections are on average smaller, when compared to commercial passive suspensions

V. Šmilauer

J. Máca

M. Valášek

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Dynamic Bridge Response for a Bridge-friendly Truck

CTU Open Journal Systems (Czech Technical University, Prague / České vysoké učení technické v Praze)

1 IntroductionOne of the most important factors determining road de-sign is the intensity of heavy trucks which is often higher thanoriginally assumed by the designers. According to the trafficprognosis for Europe and the USA, there is an annual in-crease in such vehicles, and new trucks with multiple axles ortrailers appear. There are additional costs for road repair andmaintenance.Unevenness on the road surface generates, aside fromstatic, a dynamic contact force which can be controlled andreduced possibly. While direct reduction of bridge deflectionswas found to be complicated, e.g. tuned mass dampers orintelligent stiffeners [1], new trends in suspension develop-ment follow the concept of semi-active suspensions, directlyon the vehicle axles. The driving force that controls thedamping properties of a suspension is negligible when com-pared to the active damper force, while semi-active suspen-sions after a good compromise between performance andprice. A mechatronic solution combined with a controlleddamper provides the a tool for its optimization and forreducing the dynamic contact force. The concept of road--friendliness has been extended to bridge-friendliness byoptimizing the damper parameters on bridges [2, 3, 4]. Theaim of this work is to explore the benefits of bridge-friendlytrucks on ordinary, simply supported bridges.The results fromprevious studies with a quarter-carmodelwere so promising that a more accurate model with a half-carand a slab bridge was assembled [3]. A similar model wasproduced for a simply supported bridge with a specific roadprofile, traversed by a quarter-car model with a passive,sky-hook and ground-hook control configuration. The effectof a semi-controlled damper on the bridge response is signifi-cant for close natural frequencies of the vehicle and thebridge [5].2 The half-car modelThe proposed half-car model is based on the parame-ters of the commercially available LIAZ truck, simplifiedto four DOF [6]. Fig. 1 displays the configuration of thetruck together with a bridge. The front axle comprisestwo axles of a real car, and the rear axle comprises thefour axles. The model parameters were set to: m1 15 t,m2  0.75 t, m3  1.5 t, a  4 m, b  1.3 m, k12  430 kN/m,k13  650 kN/m, k20  1700 kN/m, k30  4900 kN/m,I 50 tm2, and the damping factors of the tires were set tozero. Damper forces Fd12 and Fd13 result from the movement152 ©  Czech Technical University Publishing House http://ctn.cvut.cz/ap/Acta Polytechnica Vol. 44  No.5–6/2004Dynamic Bridge Response for aBridge-friendly TruckV. Šmilauer, J. Máca, M. ValášekA truck with controlled semi-active suspensions traversing a bridge is examined for benefits to the bridge structure. The original concept of aroad-friendly truck was extended to a bridge-friendly vehicle, using the same optimization tools. A half-car model with two independentlydriven axles is coupled with simply supported bridges (beam, slab model) with the span range from 5m to 50 m. Surface profile of the bridgedeck is either stochastic or in the shape of a bump or a pot in the mid-span. Numerical integration in the MATLAB/SIMULINKenvironment solves coupled dynamic equations of motion with optimized truck suspensions. The rear axle generates the prevailing load andto a great extent determines the bridge response. A significant decrease in contact road-tire forces is observed and the mid-span bridgedeflections are on average smaller, when compared to commercial passive suspensions.Keywords: half-car model, semi-active suspension, bridge-friendly truck, bridge response, dynamics.Fig. 1: Nonlinear half-car model traversing the bridgeof the damper attachment, and they depend on an operativealgorithm, as will be explained further. All springs in the carmodel are considered to be linear without hysteresis.The road profile superimposed on the bridge deck deflec-tion determines the position of the tire contact area. Theequations of motion describing car the dynamic behavior andthe damper force are as follows:m z k z z a Fk z z b Fdd1 1 12 1 2 1213 1 3 1324 ( )( ) ,        (1)I ak z z a aFbk z z b bFdd01 1 12 1 2 1213 1 324 ( )( )        13 ,(2) m z k z z a F k z z tb zd2 2 12 1 2 12 20 2 0220 22 ( ) ( )          ( ) ,z t02(3) m z k z z b F k z z tb zd3 3 13 1 3 13 30 3 0330 34 ( ) ( )          ( ) ,z t03(4)F b z z b z a b z a zd f f f12 1 2 02 2 1 12 1 2       (   ) (   ) (    )     k z z k z a zf f10 2 02 12 1 2( ) ( ) ,(5)F b z z b z b b z b zd r r r13 1 3 03 2 1 12 1 3       (   ) (   ) (    )     k z z k z b zr r10 3 03 12 1 3( ) ( ) ,(6)z t z zr02 4( ) ,  (7)z t z zr03 5( ) ,  (8)where z4, z5 are the bridge displacements and zr is a road ir-regularity. This extended ground-hook model consists ofthree damping rates, where bf(r)1 and bf(r)2 corresponds to thedamping factor of the ground hook and sky-hook respec-tively, and the passive damper corresponds to the dampingfactor bf(r)1. The semi-active damper forces Fd12 and Fd13 arechanged with the setting of the damping rate bf(r)1, bf(r)2 andbf(r)12 in such a manner that the damping force approachesthe desired value. The typical 17 ms delay response of thedamper is further considered. Four values can be assignedto each damping factor, depending on the velocity and direc-tion of the damper attachment [6]. The numerical experi-ments proved small dependence on the fictitious change instiffness kf(r)10 and kf(r)12, therefore only damping factorsare employed in the optimization process. During optimiza-tion, 4*3  12 free damping parameters are involved foreach axle. Since the half-car model holds two independ-ently controlled dampers, 24 free parameters in total areoptimized, using genetic algorithms [4]. The multi objec-tive parameter optimization method (MOPO) within theMATLAB/SIMULINK environment was found appropriatefor such a large task [4]. The first part of the objective func-tion for optimization on both axles took the form of thesquare root of the time integral of the dynamic contact force:RF F tSUM RMS dynt, ,  1220d . (9)The second performance criterion considered drivercomfort in the form of truck sprung mass acceleration:ACC z a tSUMt  (  )1 20d . (10)The truck moved at a velocity of 50 km/h during all simu-lations, and passive or bridge-friendly damper control wasadopted for the purposes of comparison. The unevennessamplitudes of the road profile were set to 20 mm for a bumpor a pot in the mid-span, or for a stochastic road, Fig. 2.3 The bridge modelThe bridge is modeled as a simply supported Euler--Bernoulli beam or as a slab bridge for shorter spans, in orderto include the bridge torsional effect. The bridge span variesfrom 5 to 50 m, covering the majority of real bridges madefrom concrete or steel with such a statical system [7]: reinforced concrete bridges – span of 5 to 12 m prestressed concrete bridges – span of 12 to 30 m composite steel-concrete bridges – span of 15 to 50 m©  Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 153Acta Polytechnica Vol. 44  No.5–6/2004Fig. 2: Bump and pot used for damper response on the bridgesFig. 3: Bending stiffness and the first eigenfrequency of bridges(5–50) mEach bridge consists of two road lanes and two sidewalks,and the preliminary design calculations provide bridge pa-rameters for the simulation [7, 8]. The bending stiffness andthe first eigenfrequency for all considered bridges are given inFig. 3. The first truck eigenvalue is 9.3 Hz.It is assumed that the vehicle passes over the beam bridgeon the symmetry axis and on the slab bridge as close as possi-ble to the sidewalks, Fig. 5. FEM with equally spaced nodes incombination with the bridge parameters provides the equa-tion of motion in the form:M B K r r  r F, (11)where M, B, K, r, F are the mass, damping, stiffness matrices,the displacement vector and the vector of contact axle forces.Rayleigh damping with a logarithmic decrement of 0.05 wasused for all bridges for the damping matrix assemblage. Twocontact forces from the truck axles are linearly distributedbetween adjacent nodes to vector F:F k z zdyn1 20 2 02 ( ) , (12)F k z zdyn2 30 3 03 ( ) . (13)The connection links between the bridge and the carmodel are the bridge deflections below the axle and the con-tact force between the tire and the road, Fig. 4.The equations of motion (1)–(8) and (11) are simulta-neously solved in the MATLAB/SIMULINK environmentwith an implicit trapezoidal integration scheme, using avariable time step. Fig. 6 displays the SIMULINK scheme ofthe bridge model as an example. The lowest critical speedof the vehicle is over 220 km/h and the first natural fre-quency is higher than all first bridge frequencies considered.No frequency-matching phenomenon is observed during thesimulations.4 Dynamic contact forceDynamic contact force is a variable part of the total contactforce between the tire and the road, with the positive directionupward. The slab bridge model, Fig. 5, as a short bridge withthe same parameters as the beam model was proposed andverified. In all such cases, the bump is placed in the mid-spanof a beam or slab bridge. Fig. 7 shows similar behavior of bothbridges with a different damper control strategy. Even whencompared to a road on a solid base, only the damper controlmode plays a significant role in reducing the dynamic contactforce.154 ©  Czech Technical University Publishing House http://ctn.cvut.cz/ap/Acta Polytechnica Vol. 44  No.5–6/2004Fig. 4: Interconnection of truck and bridge model Fig. 5: Slab bridge model with 10 m span and the traversing axlepathFig. 6: The SIMULINK scheme of the bridge model©  Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 155Acta Polytechnica Vol. 44  No.5–6/2004Fig. 7: Comparison of semi-active and passive damper performance on the bridge type across a bump: the bridge span is 10 mFig. 8: Contact dynamic force of the rear axle across a bump on bridges (5–50) mFig. 9: Contact dynamic force of the rear axle across a pot on bridges (5–50) mFor the half-car model, the results for the rear axle arein Figs. 8 and 9 for the bump and the pot. The change ofdynamic contact force is evident shortly after unevennesspassing, reducing force value in the next peek. Again, thebridge parameters have minor effect on the truck response.It is evident that damper functionality for the pot is notas much effective as for the bump. The reason lies in lowdamping values when the damper is under contraction. Nev-ertheless, the phase of dynamic contact force is shifted and itsvalue slightly reduced as well in the case of MOPO controlstrategy [8]. The front axle carries about a third of the totalstatic truck load and an effect of damper is lower than it is forthe rear axle.5 Bridge deflectionsBridge deflections express the effect of the truck on thebridge structure itself. The difference for beam and slabbridge response reveals the qualitative behavior of these twomodels. The torsional effect is taken into account for the slabbridge, hence higher deflection values are expected. Fig. 10shows similar behavior of both bridge systems, and approx.15 % difference in bridge deflection is observed. The beambridge model is found to be sufficient even for a bridge ofnearly square shape desk.Short and long bridges are compared as an example ofbridge excitation, using the same truck. The majority of thecar weight is located in the rear axle, and the reading on thegraph is therefore from this axle. Short bridges are mainly in-fluenced by the shape of the unevenness, Fig. 11. There aretwo reasons for their excitation: the truck load prevails onshort bridges because of the available space and the bridgestiffness, and also because its mass is low. No significant forceimpulse would appear for a stochastic road, and the deflec-tions are then close to the static values.An overall response of 5 m–50 m spans is illustrated inFig. 12, depending on the damper control strategy. An aver-age semi-active dampers reduce maximum deflections on astochastic road by 2.5 %, and on bump unevenness by 3.6 %.156 ©  Czech Technical University Publishing House http://ctn.cvut.cz/ap/Acta Polytechnica Vol. 44  No.5–6/2004Fig. 10: Deflections of the beam and slab bridge under one axle of 10 m spanFig. 11: Mid-span bridge deflections with spans 5m - 50 m and a comparison with static displacements6 ConclusionsThe paper describes the effect of a bridge-friendly truckon a road and bridge structures, and proves that the conceptof road-friendliness can be extended to bridges. A semi-activeoptimized damper, when compared to a passive damper, re-duces the local contact force in all cases and shifts the contactforce peaks after passing the unevenness. Bridge-friendlytruck dampers are beneficial for decreasing road damage,mainly the rear axle, which bears the prevailing truck load.The dynamic contact force is influenced mainly by the shapeof unevenness and the control strategy of the damper. Thebridge span plays a minor role in this case.A beam bridge for a span of 10 m captures the qualitativebehavior of the truck well when compared to themore sophis-ticated slab bridge. The average reduction of deflections onbridge spans of 5 m–50 m using semi-active suspensions isfound to be an average of 2.5 % lower for a stochastic road,and 3.6 % for a bump, respectively, in comparison with pas-sive dampers.7 AcknowledgmentsThe authors gratefully acknowledge support for this workfrom the Ministry of Education, Youth and Sports as a part ofgrant MSM 210000003.References[1] Kwon H. C., Kim M. C., Lee I. W.: “Vibration Control ofBridges under Moving Loads.” Computers & Structures,Vol. 66 (1998), p. 473–480.[2] Valášek M., Kejval J.: “New Direct Synthesis of Nonlin-ear Optimal Control of Semi-Active Suspensions.” Proc.of AVEC 2000, Ann Arbor, 2000, p. 691–697.[3] Máca J., Valášek M.: “Dynamic interaction of trucks andbridges.” Advanced Engineering Design, 3rd Interna-tional Conference, Prague, 2003.[4] Valášek M., Kejval J., Máca J.: “Control of truck-suspen-sion as bridge-friendly,” In: Structural DynamicsEurodyn 2002. (Editors: H. Grundmann and G. I.Schueller), Munich, Balkema, 2002, p. 1015–1020.[5] Chen Y. et al.: “Smart Suspension Systems for Bridge--Friendly Vehicles.” Paper 4696-06. Proc. of 9th SPIEAnnual International Symposium on Smart Structuresand Materials, San Diego, 2002.[6] Valášek M., Kejval J.: “Bridge-friendly Truck Suspen-sion,” In: Proc. of Mechatronics, Robotics andBiomechanics, VUT Brno, Trest, 2001, 277-284.[7] Šmilauer V., Máca J.: “Dynamic Interaction of Bridgeand Truck with Semi-active Suspension.” EngineeringMechanics 2003, Svratka, Czech Republic, 2003.[8] Šmilauer V., Máca J., Valášek M.: “Dynamic BridgeResponse for a Truck with Controlled Suspensions.” En-gineering Mechanics 2004, Svratka, Czech Republic,2004.Ing. Vít Šmilauere-mail: vit.smilauer@fsv.cvut.czDoc. Ing. Jiří Máca, CSc.phone: +420 224 354 500e-mail: maca@fsv.cvut.czDepartment of Structural MechanicsCzech Technical University in PragueFaculty of Civil EngineeringThákurova 7166 29 Prague 6, Czech RepublicProf. Ing. Michael Valášek, DrSc.phone: +420 224 357 361fax: +420 224 916 709e-mail: valasek@fsik.cvut.czDepartment of MechanicsCzech Technical University in PragueFaculty of Mechanical EngineeringKarlovo nám. 13121 35 Prague 2, Czech Republic©  Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 157Acta Polytechnica Vol. 44  No.5–6/2004Fig. 12: Maximum bridge deflection on a stochastic road on (5 m–50) m spans for various bridges and two control strategies

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Dynamic Bridge Response for a Bridge-friendly Truck

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