DYNAMIC MODELLING AND SIMULATION OF AHEAVY VEHICLE TRAILING ARM AIR SUSPENSIONBohao Li, Arnold G. McLean - Faculty of Engineering, University ofWollongong, Australia

ABSTRACTThis paper concentrates on the trailing arm air suspensions available on the rear tandem driveaxles of some heavy prime movers. On such vehicles, the application of air suspension isbelieved to have a series of advantages including road friendly characteristics, better loadsharing and self ride height adjustment. However, air suspensions are proved unstable underdynamic situations possessing inadequate support, harsh ride and chaotic response. In thispaper, some individual components of the air suspension are modeled as well as the suspensionvibration model and the pneumatic transmission line model. These models are then simulatedusing both SIMULINK and analytical techniques to find the causes for the adversecharacteristics. From the simulation it is identified, the harsh ride is caused by insufficient flowin the transmission line whereas the chaotic response, at least to some extent, is found to be dueto the haphazard design of the analogue feedback control system, especially the ride heightcontrol valve (HCV).

1. INTRODUCTION

1.1 BackgroundMechanical suspensions have dominated the area of heavy truck suspensions for many yearsand are still widely used on rigid trucks, especially construction trucks on which high stabilitiesare needed. This is because such suspensions have proven to be stable, reliable and relativelysimple and robust in design. However, the air suspension has been substituting the traditionalmechanical suspension (especially the leaf spring suspension) nowadays. In developed countriessuch as Europe countries, United States and Australia, air suspension has been the mainstay onlong-haul prime movers.

The idea of using air suspensions to replace mechanical suspensions was aroused by the naïveconcept of inherently non-linear, better load-sharing and self leveling characteristicsconceptually possible with air springs. The most significant feature of the air suspension is thatit can provide constant natural frequency under different loads, which is much lower comparedto that of the conventional suspension. Another minor advantage of air suspensions is that it istypically considerably lighter than mechanical suspension, which means the unsprung masses ofvehicles, using air suspension, are typically much smaller than those using mechanicalsuspensions.

1.2 The Structure of a Typical Trailing Arm Air SuspensionThe typical structure of a trailing arm air suspension system is relatively simple. The axle issupported on two rigid (Neway) (see Figure 1) or flexible (Freightliner, Hendrickson) (seeFigure 2) arms located under each side of the chassis rail. Each arm’s forward end is connectedto the chassis rail by shackle having a large diameter rubber bush, and its rear end is connectedto the bottom of the airbag. The axle is clamped to the link immediately forward of the airbag*.The shock absorber can be located either forward (Freightliner, Neway) or behind the airbag(Hendrickson), due to slightly different geometric details. The axle’s longitudinal motion is

limited by the trailing arms whereas its lateral motion is limited by a Panhard rod. This Panhardrod has one end connected to the differential housing and the other end connected to the innerside of one chassis rail. Another alternative design is to use the “V” type torsion bars toconstrain both longitudinal and lateral motions. In this paper, the trailing arm will be treated as arigid body for simplicity and the force transmitted through the Panhard rod to the chassis fromthe axle will be neglected.

1.3 Problem DefinitionSome unstable handling and rough ride characteristics have been experienced on those vehiclesequipped with factory fitted air suspension system. For example, it is reported that when primemovers negotiate corners after a period of sustained high speed cruising, the air suspension mayfail to respond.

Air suspension failure not only makes the prime mover tend to roll over, but also cause thesuspension components to prematurely fail. If one side of airbag fails, the loads on each sidewill be uneven. More seriously, more loads will compound on the failed side pedestal, shackleor torque rod, which will subsequently cause these components to fail. The outcome isdevastating namely the whole suspension will lose constraint presenting adverse kinematicalcharacteristics or loss of vehicle control.

Due to the poor design of the pneumatic plumbing system and feedback control system, factoryfitted air suspension systems often exhibit unstable or chaotic characteristics. The airbags arealso slow to respond causing uneven loads between the leading and trailing axles. This furthercauses poor ride quality and serious road damage.

Air suspension failure also causes the deviations in the universal joint drive angle. This is themain reason for transmission line vibrations [19].

Although air suspensions are often called “road friendly” suspensions, however, David Cebonpointed out that air suspensions are not actually “road friendly”, and they may cause even moreserious road surface fatigue than mechanical suspensions due to poor and haphazard designs [1].

1.4 Current Methods of ImprovementTo improve the performance of air suspension systems, different aftermarket modifications havebeen adopted. However, the effects of some modifications have not been verified.

The most common modification is using double action shock absorbers to replace the singleaction shock absorber. This is a simple way of increasing damping to dispel vibration quickly,although it will transmit extra force to both the chassis and the road surface.

Another common modification is to use fast response, no delay, and minimum deadband height control valve to replace the original valve. For prime movers that use two individualHCVs to control each side of airbags, it is not unusual to observe that operators install just onefast response valve to replace the original two. Operators and drivers’ report indicate this is arelatively effective modification. However, such near standard systems remain far from optimal.______________________________________________________________________

*In this paper the term airbag is used interchangeable with air spring or air bag. For this reason, theterm airbag should not be confused with passenger vehicle airbags used for passenger protection duringmotor vehicle accidents.

One further modifications is to enlarge the diameter of the transmission line to increase airflow.On some prime movers, the original capillary sized transmission line has been replaced by muchlarger orifice sized transmission line of 50 mm diameter, which can increase the airflowsignificantly.

To improve the feedback signal, Bill Haire of Haire Truck & Bus Repairs designed a uniquemean ride height feedback linkage system to replace the ubiquitous rear axle feedback system[20]. This design ensures that the feedback signal represents the mean ride height of the leadingand trailing axles rather than crudely the rear axle ride height. The Haire system also uses largeor fast response 50 mm diameter transmission line with novel biased airflow orifices to generateinherent system damping.

1.5 Aims of This PaperThe aim of this paper is to develop a series of dynamic models of a typical trailing arm airsuspension including the vibration model and the pneumatic plumbing model involving bothslow and fast response components. These models will then be used to investigate the systembehaviour subjected to typical operation situations. SIMULINK is used to simulate the timeresponse of these models under dynamic situations. This initial paper is aimed to find theadverse operation circumstances under which the trailing arm air suspension possessesdangerous behaviour and the possible causes for such responses. Therefore, some useful designrecommendation result and some reasonable modifications can be applied to existing designs.

2. MODELLING OF RIDE HEIGHT FEEDBACK LINKAGE

2.1 Trailing Axle Ride Height Feedback OnlyThe following part of analysis is based on the assumption that the Hendrickson height controlvalve is used, which is a rotation block type valve. (see Figure 3)

On most new-built prime movers, the feedback link is directly connected to the trailing axleonly, so the feedback signal is actually from the trailing axle exclusively. (see Figure 4) Hencewhen the leading axle hits a bump, no active port flow through the HCV occurs. Consequentlyonly flow in the transmission line between the leading axle and the trailing axle occurs toachieve load sharing. After a small time delay determined by the vehicle road speed and bogieaxle spacing, the trailing axle will hit the same bump, at which time the HCV will be actuated.This system response implies that the flow rate from the orifice port of the HCV can beexpressed as:

−−

=−= dbv

aacv

dbvv rrryy

KKq θπ

θθ)(180

)(

1

(2-1)

Where:qv – Flow rate from the orifice port, m3/sK – Gain of the valve characteristicsθv – Valve spindle angular displacement, degreeθdb – Valve dead band angular displacement, degreeycv – Vertical displacement of the chassis at the valve, myal – Vertical displacement of the feedback point on the trailing arm, mya – Axle differential vertical displacement, mr1 – Radius from trailing arm connection point to centerline axle, m

ra – Radius from trailing arm connection point to feedback point, mrv – Radius of valve control arm, m

2.2 Mean Ride Height FeedbackSome prime movers have been modified by replacing the original trailing axle feedback systemwith the innovative mean ride height feedback system. The arrangement of this mean ride heightfeedback system is depicted in the following schematic. (see Figure 5)

The advantages of this mean ride height feedback system are obvious. Firstly, not only thetrailing axle but also the leading axle will actuate the HCV instantly. Secondly, because thefeedback link is connected to the near central point of the leading and trailing axles, thefeedback signal always presents the mean height of the bogie rather than single axle. This meanheight feedback minimizes the transient error signal generated by high frequency chassisvibrations, axles moving over bumps and chassis whip.

On a mean ride height feedback system the HCV port flow rate can be expressed as:

v

dba

cv

dbvv r

yyK

Kq])

2[(

)(θ

θθ−−

=−= (2-2)

An examination of equation (2-2) suggests the response of this feedback is equivalent to addinga gain with value of 0.5, on the unit feedback loop for the trailing axle feedback system (seeFigure 6). This relatively low gain may be allayed by using a Hendrickson HCV whichpossesses favourable porting characteristics and internal details (Figure 3).

It should be noted the mean height feedback linkage also occupies slightly more room, isslightly more complex, requires slightly greater maintenance and may be difficult to apply onvehicles with low ride height. Furthermore, the more complex linkage structure means it isslightly heavier than trailing axle ride height feedback system, and this, in turn, will have somevery slight adverse effects on ride quality because the feedback linkage is additive to thevehicle’s unsprung mass.

3. SIMULATION OF THE SUSPENSION MODEL

3.1 Transient Response of the SuspensionThe diagram (see Figure 7) shows the model of the two-degree of freedom suspension model. Itpresents the suspension of one ‘corner’ of the whole vehicle. This model consists of springs,dampers and masses, and it will be used to develop the dynamic differential equations.

The model notation is as follows:Mu – The total unsprung mass, comprising of the axle, wheels, hubs, tyres and brakeassembly mass, 800 kg assumedMs – The proportion of the sprung mass supported by each air spring, 3200 kg assumedkt – Tyre spring stiffness, 0.81 MN/m under tyre pressure of 759 kPa (110 psi)kp – Airbags (air springs) spring stiffness, 125.6 kN/m assumedct – Tyre damping coefficient, 0.003 Ns/m assumedcp – Shock absorber damping coefficient, 6013.62 N-s/m assumed, equivalent to 15%

damping ratiox1 – Road signal input, mx2 – Axle displacement, m

x3 – Chassis displacement, mFw – Road roughness input signal, m

3.1.1 Single Axle Suspension Vibration Model

According to the 2 DoF vibration model, the dynamic differential equation is:

=−

+−

=−

−−−−

+−

23

232

32

22

232

3221

21

)()(

)()()()(

dtxd

Mdt

xxdCxxk

dtxdM

dtxxd

Cxxkdt

xxdCxxk

spp

upptt (3-1)

Hence the block diagram of the suspension system incorporating wheel unbalance can bedeveloped using SIMULINK. (see Figure 8)

The system response subject to excitation from passing over a 1 unit step, which isdimensionless, is shown in Figure 9. The forces transmitted to the chassis rail when Cp equalsto 15% and 20% damping ratio are shown in Figure 10 and Figure 11, respectively. It can beseen that adoption of the shock absorber with larger damping coefficient do dissipate vibrationenergy rapidly, but it also transmits larger force to the chassis .

3.1.2 Grouped Axle Vibration ModelThe combination of two of the previous single axle models connected by a time delay block canbe used to simulate tandem axles used on prime movers. The resulting block diagram is shownin Figure 12.

The signal delay time is the time interval for the trailing drive wheel to be excited by a bumppreviously contacted by the leading drive wheel. If the wheelbase is 1295.4 mm, which is astandard value of Freightliner prime movers, the signal delay is 0.047s at a road speed of100km/h. The response of the model subjected to a step input is shown in Figure 13.

The dark trace represents the displacement of the chassis excited by leading axle while the lightone represents the displacement of the chassis excited by trailing axle. The third trace is theinput signal. It can be clearly seen that there is some time delay between two response curvesalthough this time interval is very short at the speed of 100 km/h

3.2 Suspension Frequency ResponseThe vibration model can also be expressed as shown in Figure 14.

Substituting all parameters into this block diagram, the transfer function of the wholesuspension can be achieved. Thus, the “bode” command in MATLAB can be used to get thesystem frequency response. (see Figure 15)

It can be seen that there is a resonance occurring at 6 rad/s, which equals a frequency of 0.95Hz. Two conclusions can be drawn: (1) If the system excitation source is wheel unbalance,adverse conditions will occur at a speed of 11.28 km/h; (2) If the excitation source is roadundulations, road wavelengths of approximately 30 m will excite the vehicle when operating at100km/h. Furthermore, in the high frequency range, the vibration amplitude decreases rapidly.This verifies that standard air suspensions have low frequency characteristics.

4. SIMULATION OF THE PNEUMATIC TRANSMISSION LINECHARACTERISTICS

4.1 Transient Response of a Transmission Line Subjected to a PressureSignal Input

Anderson [3] provides the relationship between the supply pressure change and the terminalpressure change in a pneumatic transmission line with one end connected to an airbag. Thissystem is shown in the schematic. (see Figure 16)

The response of this system is given by:

})](12)/(21

)/(411[){21(1 2

22

*

*

2

*1

2

ββsE

ALVALVsE

ALV

pp

++++++

= (4-1)

Where:p1 – Supply pressure change, Pap2 – Terminal (airbag) pressure change, Paw12 – Weight flow rate from supply end to terminal end, kg/sV* - Effective volume = (1 + kp/k)V2, m3

kp – Airbag spring stiffness, 125.6 kN/mk – Connected mechanical spring stiffness. Here it is roughly the tyre spring stiffness,

which is 0.81 MN/mV2 – Volume of airbag under set height, 0.01227 m3 if both the diameter and the set

height of the airbag are assumed equalling to 250 mmA – Section area of transmission line, m2

L – Length of transmission line, 1.65 m

E2 – Equals 222

1

)10/()/(3200 PDLngRTµµ – Air viscosity, 1.81e-5 Pa.secP2 – Pressure of airbag under 12.8t sprung mass, 639.5 kPa

β – Equals LngRT /)( 21

s – Laplace operator

If the transmission line is a capillary with inner diameter of 4.15mm, equation (4-1) can berewritten as:

122 1200099.00566.01 p

ssp

++= (4-2)

Similarly, if the transmission line is an orifice tube with inner diameter of 50.8mm, equation (4-1) can be rewritten as:

122 1001375.0000389.01 p

ssp

++= (4-3)

The transient responses of these two different transmission line systems, to a step input pressurechange, are presented in Figure 17 and Figure 18.

From an examination of the above figures, it can be concluded that the orifice transmission linedoes have a shorter rise time, but the pressure response exhibits high frequency oscillationbecause the damping ratio is extremely small. This finding is consistent with observed heavyvehicle behaviour in so far as vehicles fitted with simple orifice transmission lines exhibit highfrequency oscillations and require the fitment of heavy-duty shock absorbers.

4.2 Frequency Response of Different Diameter Transmission LinesFrom equation (4-2) and (4-3), the Bode diagrams of the capillary transmission line system andthe orifice transmission line system can be plotted using MATLAB. (see Figure 19 and Figure20)

It can be seen that the capillary resonance frequency occurs at about 4 rad/s, while the orificeresonance frequency occurs at about 50 rad/s.

4.3 Backflow in Capillary Transmission Line SystemsOne problem of the capillary transmission line is that when the airbag is compressed rapidly,there is insufficient flow through the transmission line causing extremely high pressure in theairbag, which may exceed the supply pressure of the reservoir. Thus, when the HCV valveopens, the air will flow from the airbag due to the pressure differential rather than flow into theairbag. This is an adverse situation, which will cause the airbag to deflate further. Thus, not onlywill the ride height deviate from the set value, but also the suspension components will beoverloaded.

Because the transmission line is a capillary, when compressed rapidly, there is insufficientairflow from it. This implies that when the airbag is subjected to rapid compressions, there isalmost no gas mass transmitted out of the airbag through the capillary. On the other hand,should the compression process be sufficiently rapid to exceed 60 Hz the process can beassumed to be polytropic.

The gas law for a polytropic process is given by:

nbpb

nsps

ns hAPhAPVP )()(2 == (4-4)

Where Ps – Static pressure of the airbag, 639.5 kPa assumed Pb – Pressure of airbag when back flow occurs, 828 kPa assumed which equals to the

reservoir supply pressure hs – Airbag set height, 250 mm assumed hb – The height of the airbag when back flow occurs, mm Ap – Airbag effective area, 250 mm assumed n – Polytropic exponent, 1.4 in this case

Thus, hb is given by

4.1

b

ssb P

Phh = (4-5)

Substitution of all known variables into equation (4-5) yields hb equal to 207.04 mm, whichcorresponds to critical ∆h, where backflow differential between Ps and Pb occurs, is 42.96 mm

(i.e. 250 - 207.4 mm). This implies that when the compression displacement exceeds about 43mm, the phenomena of back flow may occur in the transmission line.

5. CONCLUSIONSome typical symptoms found on air suspensions include:Harsh ride dues to insufficient flow rate in the pneumatic transmission line.Chaotic response dues to poor design of the analogue ride height feedback system.By carefully modelling and simulation, some conclusions are listed below:To overcome the shortcomings of the capillary transmission line, it is simple to increase thediameter of the transmission line. This can not only increase the flow rate but also can reducethe restriction in the transmission line and reduce the delay time. However, the simple largediameter transmission line has another problem which is rapid fluctuation in the transientresponse. In practice, a compromise must be considered or preferably a large diameter biasedorifice controlled transmission line should be utilized.

The mean height feedback linkage does not possess the shortcomings of trailing axle feedbackcontrol. These shortcomings are generated by decreasing the amplitude of the feedback signalby half. For example, if the leading axle has a vertical displacement of 10cm while the trailingaxle has no vertical motion, the vertical displacement of the feedback link is 5cm. This effect isequivalent to a gain of 0.5 relative to the unit feedback loop of the trailing axle feedback system.In some contrived situations this may cause steady state error. The mean height feedbacklinkage also occupies slightly more room, and it is slightly more complex and may be somewhatdifficult to apply on vehicles with low ride height.

Double action shock absorbers are effective to dissipate vibration energy rapidly, but willtransmit larger forces to the chassis and road surface. The large dynamic forces may causechassis component fatigue, tyre and road damage. Therefore, caution should be applied whensubstituting the original single action units with suitable double action shock absorbers.

This work forms the basis of ongoing research into the dynamics of heavy vehicles at theUniversity of Wollongong. Preparation is well advanced in regard publication of results ofdynamic roll over simulations. Further work is in progress to accurately incorporate the effectsof chaotic lever actions, biased flow orifice controlled transmission lines, vehicle speeddependent delay time, reservoir, air compressor and tyre compound hysteresis into theSIMULINK based dynamic simulation. This work should prove most significant for the safetyand well being of the vital components in the overall heavy vehicle transport system

REFERENCES

1. HANDBOOK OF VEHICLE – ROAD INTERACTION – David Cebon (Swets & ZeitlingerB.V. – 1999 – ISBN 90-265-1554-5)

2. DIESEL EQUIPMENT II – Erich j. Schulz (McGraw Hill Book Co. – 1988 –ISBN 0-07-055708-X (v.2))

3. THE ANALYSIS AND DESIGN OF PNEUMATIC SYSTEMS – Blaine W. Andersen (JohnWiley and Sons inc. – 1967)

4. MODERN CONTROL ENGINEERING 4th edition – Katsuhiko Ogata (Prentice Hall, Inc. –2002 – ISBN 0-13-043245-8)

5. INTRODUCTION TO CONTROL SYSTEM ANALYSIS AND DESIGN 2nd edition – FrancisJ. Hale (Prentice Hall, Inc. – 1988 – ISBN 0-13-479767-1)

6. ENGINEERING VIBRATION – Daniel J. Inman (Prentice Hall, Inc. – 1994 – ISBN 0-13-951773-1)

7. MODELING, ANALYSIS, AND CONTROL OF DYNAMIC SYSTEMS 2nd Edition – WilliamJ. Palm III (John Wiley & Sons, Inc. ISBN 0-471-07370-9)

8. MODELING AND SIMULATION OF DYNAMIC SYSTEM – Robert L. Woods, Kent L.Lawrence (Prentice Hall, Inc. – 1997 – ISBN 0-13-337379-7)

9. MASTERING SIMULINK® 2 – James B. Dabney, Thomas L. Harman (Prentice Hall, Inc. –1998 – ISBN 0 –13-243767-8)

10. MATLAB (VERSION 5) USER’S GUIDE – STUDENT EDITION – Hanselman Littlefield,Duane C. Bruce (MathWorks, Inc and Prentice Hall – 195 – ISBN 0-1327-2550-9)

11. THE SHOCK ABSORBER HANDBOOK – John C. Dixon (Society of Automotive Engineers,Inc. – 1999 – ISBN 0-7680-0050-5)

12. ROAD VEHICLE SUSPENSION – Wolfgang Matschinsky (Professional EngineeringPublishing Limited – 1998 – ISBN 1-86058-202-8)

13. FUNDAMENTALS OF VEHICLE DYNAMICS – Thomas D. Gillespie (Society ofAutomotive Engineers, Inc. – 1992 – ISBN 1-56091-199-9)

14. INVESTIGATION INTO THE SPECIFICATION OF HEAVY TRUCKS ANDCONSEQUENT EFFECTS ON TRUCK DYNAMICS AND DRIVERS: FINAL REPORT –Peter F Sweatman and Scott McFarlane, report prepared for FORS by RoaduserInternational Pty Ltd

15. A STUDY OF DYNAMIC WHEEL FORCES IN AXLE GROUP SUSPENSION OF HEAVYVEHICLES – P. F. Sweatman, Published by Australian Road Research Board

16. CHARACTERISTICS OF FAST RESPONSE MEAN RIDE HEIGHT ANALOGUECONTROLLED HEAVY VEHICLE AIR SPRING SUSPENSION SYSTEM – Dr ArnoldMcLean, J Lambert and W Haire Proceedings ATRF 2001 Hobart

17. PRIME MOVER AIR SUSPENSION RIDE HEIGHT CONTROL MALFUNCTION – DrArnold McLean Proceedings SAE Paper 99085 Melbourne

18. ACTIVE COMPUTER CONTROLLED AIR-SUSPENSION SYSTEMS FOR HEAVY PRIMEMOVERS – A CONCEPTUAL EVALUATION – Dr Arnold McLean Proceedings ITSA 99Adelaide

19. HENDRICKSON REFERENCE WHITE PAPER – REDUCING THE EFFECTS OFSUSPENSION-RELATED DRIVELINE VIBRATION (Published by HendricksonInternational website, http://www.hendrickson.com/reference/white/04079801.htm)

20. BE THESIS – DYNAMIC CHARACTERISTICS OF HEAVY PRIME MOVERS –KENWORTH AIRGLIDE 100 BOGIE SUSPENSION – Rod Visman, UoW, 1999

21. BE THESIS – EVALUATION OF HEAVY PRIME MOVER BOGIE SUSPENSIONMECHANIS – Gary Sawyer, UoW, 1998

22. STERLING ACTERRA – Cab and Chassis Vocational Reference Guide, Section 4 –Suspension (Published by Sterling Trucks, October 2000)

23. AirLiner Rear Suspension Specifications, available on Freightliner Trucks’ website,http://www.freightlinertrucks.com/components/

PRESENTING AUTHOR BIOGRAPHYBohao Li – Mr Bohao Li was awarded the Bachelor of Engineering in Automobile Engineeringfrom Shanghai University of Engineering Science, China in 2001. He was involved in theproject of modelling and simulation of the heavy vehicle air suspension as a postgraduatestudent in the Faculty of Engineering, University of Wollongong, Australia under thesupervision of the co author, graduating with a Master of Engineering Practice in Mechatronicswith Distinction in 2002. The presenting author will continue this paramount work as a MEHonours candidate in 2003 with conversion to a PhD supervised by Dr Arnold McLean from2004 onwards.

FIGURES

Figure 1: Neway Air Ride AD-200 air suspension(From http://www.rvcalifornia.com)

Figure 2: Freightliner AirLiner 46 K air suspension(From http://www.freightlinertrucks.com/components)

Figure 3: Hendrickson height control valve rotation block details

Figure 4: Trailing axle ride height feedback schematic

Figure 5: Mean ride height feedback schematic

Figure 6: The block diagrams for the trailing axle ride height feedback system (above) andmean ride height feedback system (below)

Figure 7: 2 DoF Suspension vibration model

Figure 8: Block diagram for a single axle suspension

Figure 9: System transient responses at 15% damping ratio

Dark continuous line: vibration amplitudeof the chassisLight continuous line: vibration amplitudeof the axleRed line: the step input

Figure 10: Force transmitted to the chassis at 15% damping ratio

Figure 11: Force transmitted to the chassis at 20% damping ratio

Figure 12: Block diagram for grouped axles

Figure 13: Transient response of grouped axles

Figure 14: Block diagram for the single axle suspension – Alternative expression

Figure 15: The Bode diagram of the vibration model

Figure 16: The pneumatic transmission line system schematic

Dark continuous line: vibration amplitudeof the chassisLight continuous line: vibration amplitudeof the axleRed line: the step input

AIR SUPPLY TRANSMISSION LINE

AIRBAG

TYRESTIFFNESS

Figure 17: Pressure transient response of the capillary transmission line system

Figure 18: Pressure transient response of the orifice transmission line system

Figure 19: The Bode diagram of the capillary transmission line system

Figure 20: The Bode diagram of the orifice transmission line system