Fluid formulae for damping changeability conceptual design of railway semi-active hydraulic dampers

International Journal of Non-Linear Mechanics 44 (2009) 809 -- 819

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Fluid formulae for damping changeability conceptual design of railway semi-activehydraulic dampers

W.L. Wang∗, G.X. XuVehicle Engineering Laboratory, School of Mechanical and Electrical Engineering, Nanchang University, Nanchang 330031, China

A R T I C L E I N F O A B S T R A C T

Article history:Received 9 July 2008Received in revised form 19 April 2009Accepted 14 May 2009

Keywords:Semi-active hydraulic damperChangeable damping performancesFluid formulationConceptual design

Damping changeability design and evaluation is the most fundamental issue at the beginning of anynew railway semi-active hydraulic damper development. Therefore, physical fluid mechanics for thecalculation of basic structure and resistance parameters of the damper should be carefully studied in theconceptual phase. Fluid formulae for changeable damping performance evaluation of two commercialrailway semi-active hydraulic dampers are established. Simulation results show that the damper switchedby high-speed solenoid valves obtains a wide range of changeable damping coefficients, which guaranteesthe absorption of a wide spectrum of vibrations; however, a different low cost damper regulated with aninversely proportional relief valve, whose Force–velocity characteristics share the same rising curve, isrelatively limited in damping ability. In order to overcome the drawback of the latter one with no obviouscost increase, a new semi-active hydraulic damper which is regulated by a simple proportional throttlevalve is proposed. Continued fluid formulation and simulation suggests that the damper can changeits damping force rising curves or “effective” damping coefficients continuously, within a considerablywide range. Thus, fluid formulae explicitly established in this study are of significance in the dampingchangeability conceptual design, further refinement and control design for the three semi-active hydraulicdampers. The proposed new damper, which has both a simple configuration and an easy-to-controlability, might be feasible for industry applications.

© 2009 Elsevier Ltd. All rights reserved.

1. Introduction

There is a recent trend to incorporate active or semi-active damp-ing systems [1,2] in high-speed passenger train suspension to in-crease ride comfort [3] and stability [4]. For a compromise betweencost and performance, the semi-active damping systems [5–11] arepreferred and applied in industry, e.g., successful systems have beendeveloped and put into commercial use in Japanese SHINKANSENcars [7–9,11] since 2000. However, developing such a semi-activesystem can be multidisciplinary and a great challenge, especially interms of the semi-active hydraulic damper, which is the key com-ponent [12].

The development context of a new semi-active hydraulic damperusually involves three main issues: the first is the conceptual de-sign phase of the damper configuration and steady-state damp-ing changeability, which is determined by basic structure and fluidresistance parameters, such as the orifice diameters; the second

∗ Corresponding author. Tel.: +867913969635.E-mail address: [email protected] (W.L. Wang).

0020-7462/$ - see front matter © 2009 Elsevier Ltd. All rights reserved.doi:10.1016/j.ijnonlinmec.2009.05.003

concerns damper structure design and the development of high-speed switching valves [13], which enable the damper's inner fluidcircuit to change in real-time; the last is comprehensive prototyperefinement by means of transient close-loop simulation and testing[5–11] with the sensors, the electronic control unit (ECU) and the ve-hicle. Among these three issues, the first one is the most fundamen-tal, since the changeable damping performances for the automaticcontrol of any new damper should be properly evaluated and de-signed before the idea is implemented, i.e., the number of dampinglevels, concrete damping coefficient and relief point of every level,must be carefully calculated and evaluated.

Stribersky et al. [6], Sasaki [7], Tanifuji et al. [8], Codecà et al. [10]and Sugahara et al. [11] described the configuration and dampingcharacteristics of their changeable hydraulic dampers before vehiclesimulation or testing, but no special method to design the initial fluidparameters with the correct characteristics was introduced. Codecàet al. [14], Witters and Swevers [15] built black-box models for theirsemi-active hydraulic dampers by means of parameter identifica-tion based on test data, but black-box models [16,17] are implicitlyequivalent performances that are usually established after a proto-type or product was made. They are intrinsically non-parametric andgenerally suitable for vehicle dynamic simulations. Physical models

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810 W.L. Wang, G.X. Xu / International Journal of Non-Linear Mechanics 44 (2009) 809 -- 819

[18,19] are mostly employed in damper performance and structuredesign instead, but they are also limited in some aspects, such asin vehicle suspension optimization, because of too many physicalparameters.

This study establishes physical fluid formulae for dampingchangeability evaluation or the conceptual design of several trainsemi-active hydraulic dampers. Two commercial semi-active hy-draulic dampers, which were introduced by Sasaki [7], are analysedfirst; fluid formulae for their steady-state damping performancesevaluation under different control modes are established. The fol-lowing simulation results show that the semi-active hydraulicdamper, which is switched using high-speed solenoid valves, hasexcellent damping changeability to absorb a wide spectrum of vibra-tions, while another low cost one, which is regulated by an inverselyproportional relief valve, is relatively limited in damping changing.Thus, explicit fluid formulae established in this study are signifi-cant to their damping changeability evaluation, further refinementand control design. In an attempt to overcome the changeabilitylimitation of the latter one, a new semi-active hydraulic damperconfiguration, which is simply regulated with a proportional throttlevalve, is proposed; a pertinent study shows that this configurationhas obtained obvious better changeability, as well as a low cost.Concluding remarks are given in the last section.

2. Changeable damping performances formulation underdifferent control modes

2.1. The semi-active hydraulic damper switched by high-speedsolenoid valves

Fig. 1 shows the principle of a high-speed passenger train sec-ondary lateral suspension employing two semi-active hydraulicdampers, which are switched using high-speed solenoid valves. Twochangeable dampers are installed between the bogie and car bodyto damp lateral vibrations, and a “Sky-hook” [7,8] control strategy

Car body

Low-set

High-set

Solenoid valve 2

Orifice 1

Unloading valve 1

(stretch)

Unloading valve 2 (compression)

Bogie

Orifice 2

Orifice 3

Solenoid valve 1

Solenoid valve 3

Check valve 2

Tank

Check valve 1

relief valve

relief valve

V

Fig. 1. Train secondary lateral suspension employing two semi-active hydraulic dampers that are switched by high-speed solenoid valves.

is implemented by controlling three high-speed solenoid valves andtwo unloading valves of each damper. When the three solenoidvalves are controlled with specific logic, a combination of orificesand relief valves is formed and specific level of damping is achieved.When one of the two unloading valves is excited under certain vi-bration conditions, the damper is unloaded to avoid producing neg-ative damping. Table 1 lists all the combinations and damping levelsunder different control modes.

In the following formulation, denote the damping force in thecontrol mode i as Fmodei and pressure limits of the low-set and high-set relief valves as PLset and PHset, respectively. Denote critical strokevelocities when the low-set and high-set relief valves are about toopen in control mode i as VLmodei and VHmodei, respectively. Denotethe diameter of orifice i as di.

2.1.1. Control mode 1Referring to Table 1 and Fig. 1, one observes that orifices 2, 3

and the low-set relief valve are “shorted” under control mode 1, i.e.,orifice 1 and the high-set relief valve are selected and combined towork under this mode. Therefore, the equivalent resistance networkunder control mode 1 can be simply sketched as in Fig. 2a, whereQ denotes instantaneous stroke flow and P denotes instantaneouspressure built in the damper.

Hydraulic dampers with symmetric damping characteristics inboth the stretch and compression directions have a parameter pre-requisite of D=√

2d [20], where D and d are the diameters of the pis-ton and piston rod, respectively. Thus, by omitting various leakageflows, oil compressibility, set pressure error of the relief valves andback pressure of the tank, one easily obtains the following equation,which is essential to the coming formulation process:

Q = �4d2V , F = �

4d2P, (1)

where V is the stroke velocity and F the produced instantaneousdamping force.

W.L. Wang, G.X. Xu / International Journal of Non-Linear Mechanics 44 (2009) 809 -- 819 811

Table 1A summary of valves, orifices combinations and resulting damping levels under different control modes.

Control modes 1 2 3 4 5 6 7 8

Unloading valve 1 − − − − − − + −Unloading valve 2 − − − − − − − +Solenoid valve 1 + + + + − − − −Solenoid valve 2 − − + + − − − −Solenoid valve 3 + − + − − + − −Orifice(s) working 1 1, 3 1, 2 1–3 3 × × ×Relief valve(s) working H H, L H H, L L × × ×Damping level Hard Harder 1 Harder 2 Harder 3 Normal Soft Unloaded Unloaded

Where “+” and “−” denote “on” and “off”, respectively; “×” denotes none member is working; “H” and “L” denote the “high-set relief valve” and “low-set relief valve”,respectively.

P1

Q , P

Q , PQ , P

PLset

PLset

PHset

PHset PHset

PLset

PHset

Tank

d1

d2

d3

P1

P2

Q , P

Q , P

d1

Tank

P2

d3

d1

Tank

d2

d1

Tank

Tank

d3

Fig. 2. Equivalent resistance networks under different control modes: (a) mode 1, (b) mode 2, (c) mode 3, (d) mode 4 and (e) mode 5.

Returning to control mode 1, according to the flow continuityprinciple, one easily writes

�4d2VHmode1 = Cd

(�4d21)( 2

�PHset

)1/2

, (2)

where Cd is the discharge coefficient and � the oil density. The rightterm of Eq. (2) formulates the flow passing through resistance orifice1 while the high-set relief valve is about to open, but is not yet open.

We solve Eq. (2) to obtain the critical stroke velocity under controlmode 1

VHmode1 = Cd

(d1d

)2( 2�PHset

)1/2

. (3)

Thus, when 0�V <VHmode1, flow continuity and damping forceof the damper can be formulated by⎧⎪⎪⎨⎪⎪⎩

�4d2V = Cd

(�4d21)( 2

�P)1/2

Fmode1 = �4d2P.

(4)

Solving this united equation by removing parameter P gives

Fmode1 = ��8C2

d

(1d41

)d6V2. (5)

When V �VHmode1, the high-set relief valve is open and the pres-sure built in the damper is limited by PHset, then

Fmode1 = �4d2PHset. (6)

Therefore, damping force under control mode 1 can be formulatedpiecewise as

Fmode1 =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

��8C2

d

(1d41

)d6V2 when 0�V <VHmode1

�4d2PHset when V �VHmode1,

(7)

where VHmode1 is described by Eq. (3).

2.1.2. Control mode 2Under control mode 2, orifices 1, 3 and both the high-set and low-

set relief valves are selected and combined. The equivalent resistancenetwork is sketched by Fig. 2b, where P2 denotes the instantaneouspressure before orifice 3.

Assuming that V = VLmode2, the low-set relief valve is then aboutto open, while the high-set one is still closed, so one has

�4d2VLmode2 = Cd

(�4d23)( 2

�PLset

)1/2

. (8)


The right term of Eq. (8) formulates the flow passing through resis-tance orifice 3 at this instant. Solving Eq. (8), one obtains

VLmode2 = Cd

(d3d

)2( 2�PLset

)1/2

. (9)

Likewise, assuming V = VHmode2, then the high-set relief valve isabout to open, while the low-set one has already been open for awhile. Referring to Fig. 2b, according to the flow continuity principle,

�4d2VHmode2 = Cd

(�4d21) [ 2

�(PHset − PLset)

]1/2. (10)

The right term of Eq. (10) describes the flow passing through resis-tance orifice 1 at this stroke velocity. Solving Eq. (10), one obtains

VHmode2 = Cd

(d1d

)2[ 2�(PHset − PLset)

]1/2. (11)

Thus, when 0�V <VLmode2, flow continuity and damping forceof the damper can be formulated by⎧⎪⎪⎨⎪⎪⎩

�4d2V = Cd

(�4d21) [ 2

�(P − P2)

]1/2= Cd

(�4d23)( 2

�P2

)1/2

Fmode2 = �4d2P.

(12)

Solving this united equation by removing parameters P and P2 gives

Fmode2 = ��8C2

d

(1d41

+ 1d43

)d6V2. (13)

When VLmode2 �V <VHmode2, the low-set relief valve has beenopen, while the high-set one is still closed, so one has

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

�4d2V = Cd

(�4d21) [ 2

�(P − P2)

]1/2P2 = PLset

Fmode2 = �4d2P.

(14)

Solving the united equation Eq. (14) by removing P and P2 gives

Fmode2 = ��8C2

d

(1d41

)d6V2 + �

4d2PLset. (15)

When V �VHmode2, both the two relief valves are open. BecauseP is limited by PHset, one directly writes


All the above, damping force under control mode 2 can be for-mulated as

Fmode2=

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩

��8C2

d

(1d41

+ 1d43

)d6V2 when 0�V <VLmode2

��8C2

d

(1d41

)d6V2 + �

4d2PLset when VLmode2 �V <VHmode2


(17)

where VLmode2 and VHmode2 are described by Eqs. (9) and (11), re-spectively.

2.1.3. Control mode 3Orifices 1, 2 and the high-set relief valve are selected and com-

bined to work under control mode 3. The equivalent resistance

network is illustrated by Fig. 2c, where P1 denotes the instantaneouspressure before orifice 2.

Assuming that V=VHmode3, the high-set relief valve is then aboutto open, one has the following flow continuity equation:

�4d2VHmode3 = Cd

(�4d21) [ 2

�(PHset − P1)

]1/2

= Cd(�4d22)( 2

�P1

)1/2

. (18)

Remove P1 to get

VHmode3 = Cd

(d1d2d

)2[

2�(d41 + d42)

PHset

]1/2. (19)

Thus, when 0�V <VHmode3, flow continuity and damping forceof the damper can be similarly formulated by

⎧⎪⎪⎨⎪⎪⎩

�4d2V = Cd

(�4d21) [ 2

�(P − P1)

]1/2= Cd

(�4d22)( 2

�P1

)1/2

Fmode3 = �4d2P.

(20)

Solving this united equation by removing parameters P and P1, oneobtains

Fmode3 = ��8C2

d

(1d41

+ 1d42

)d6V2. (21)

When V �VHmode3, P is limited by PHset, one directly has


In this way, the damping force under control mode 3 can beformulated by

Fmode3 =

⎧⎪⎪⎨⎪⎪⎩

��8C2

d

(1d41

+ 1d42

)d6V2 when 0�V <VHmode3


(23)

where VHmode3 is described by Eq. (19).

2.1.4. Control mode 4All the orifices and relief valves are combined to work under

control mode 4. The corresponding equivalent resistance network isillustrated by Fig. 2d.

Provided that V=VLmode4, referring Eq. (9) of mode 2, one directlywrites

VLmode4 = VLmode2 = Cd

(d3d

)2( 2�PLset

)1/2

. (24)

Provided that V = VHmode4, one has the flow continuity equation

�4d2VHmode4 = Cd

(�4d21) [ 2

�(PHset − P1)

]1/2

= Cd(�4d22) [ 2

�(P1 − PLset)

]1/2, (25)

from which we obtain

VHmode4 = Cd

(d1d2d

)2[2(PHset − PLset)

�(d41 + d42)

]1/2. (26)


Thus, when 0�V <VHmode4, flow continuity and damping forceof the damper can be formulated by

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

�4d2V = Cd

(�4d21) [ 2

�(P − P1)

]1/2

=Cd(�4d22) [ 2

�(P1 − P2)

]1/2

=Cd(�4d23)( 2

�P2

)1/2

Fmode4 = �4d2P.

(27)

We solve this united equation by removing parameters P, P1 to obtain

Fmode4 = ��8C2

d

(1d41

+ 1d42

+ 1d43

)d6V2. (28)

Continuing, when VLmode4 �V <VHmode4, the low-set relief valvehas already been open, so

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

�4d2V = Cd

(�4d21) [ 2

�(P − P1)

]1/2

=Cd(�4d22) [ 2

�(P1 − P2)

]1/2P2 = PLset

Fmode4 = �4d2P.

(29)

We solve Eq. (29) by removing parameters P and P1 to get

Fmode4 = ��8C2

d

(1d41

+ 1d42

)d6V2 + �

4d2PLset. (30)

When V �VHmode4,


Thus, the damping force under controlmode 4 can be summarizedpiecewise as

Fmode4 =

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

��8C2

d

(1d41

+ 1d42

+ 1d43


��8C2

d

(1d41

+ 1d42

)d6V2 + �

4d2PLset when VLmode4 �V <VHmode4


(32)

where VLmode4 and VHmode4 are described by Eqs. (24) and (26), re-spectively.

2.1.5. Control mode 5Control mode 5 is a “fail-safe” mode that is designed to work

under circumstances when the entire control system is out of controlor powered off. Under this mode, all the solenoid valves are switchedoff which causes orifice 3 and the low-set relief valve to work incombination; thus, the semi-active hydraulic damper performs asa passive one. The corresponding equivalent resistance network issketched in Fig. 2e.

Since mode 5 has the same resistance network configuration asthat of mode 1, the damping force under this mode can be formulated

as Eq. (33) by analogy to Eq. (7).

Fmode5 =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩

��8C2

d

(1d43


where VLmode5 = Cd

(d3d

)2( 2�PLset

)1/2

�4d2PLset when V �VLmode5.

(33)

2.1.6. Control mode 6Under control mode 6, only the solenoid valve 3 is switched on, so

all the orifices and relief valves are “shorted”. In this way, the dampercan supply a very low level of damping force, which is producedby the resistance of the hydraulic circuit itself. Assuming that aneffective orifice diameter of the hydraulic circuit is d0 that is usuallydetermined by the bench test of a prototype, then the damping forceof mode 6 can be described as

Fmode6 = ��8C2

d

(1d40

)d6V2. (34)

2.1.7. Control modes 7 and 8According to “Sky-hook” control rationale [21], an imaginary

“Sky-hook” damper always exists and produces a damping forceagainst vibration, but a real damper cannot do this all the time. Forinstance, by referring to Fig. 1, we assume that the car body andbogie frame both vibrate laterally to the right, but the bogie framehas a larger vibration speed than that of the train body, i.e., thebogie frame moves faster than the train body. In this way, both ofthe two hydraulic dampers produce damping forces to the right,which cannot damp the car body motion, but may even intensify it.This phenomenon is referred to as “negative damping”.

Despite the difficulty in obtaining “positive damping” duringthe foregoing-like vibration situations, “negative damping” can beavoided by unloading the semi-active hydraulic dampers at thatmoment. Control modes 7 and 8 are the correct engineering ap-proaches to realize this strategy. For instance, unloading valve 1while the damper is being stretched causes fluid to be conductedto the other chamber of the damper with very low level resistance,which is below half of Fmode6; on the contrary, valve 2 is excitedwhile the damper is being compressed and fluid is conducted tothe tank that also has very low resistance. In control modes 7 and8, one usually says that the damper produces no damping forces.

However, if the two unloading valves were excited simultane-ously, the two chambers of the damper would be “communicated”simultaneously and this could be a very dangerous situation. There-fore, an electronic interlock control circuit [7] should be integratedin the valve driver to prevent such wrong operations.

2.2. The semi-active hydraulic damper regulated by an inverselyproportional relief valve

As shown by the preceding established fluid formulae, automaticcontrol of the preceding semi-active hydraulic damper is velocity-based, so the stroke velocity must be sampled, i.e., the relative vibra-tion velocity between the car body and bogie frame must be sam-pled instantaneously during operation, and this is usually done byembedding a sensor in the piston rod [7,8]. In this way, the damperstructure becomes more complex, which leads not only to highercost but also to maintenance trouble.

A relatively simple semi-active hydraulic damper shown byFig. 3 was developed in an attempt to cut down cost and mainte-nance inconvenience [7]. The damper is comprised of a constantorifice, an inversely proportional relief valve and two unloading


d4

Check valve2

Check valve1

Unloading valve 2(compression)

Unloading valve 1(stretch)

V

Orifice

Inversely proportional

relief valve

Pset(I)Tank

Fig. 3. The semi-active hydraulic damper, which is regulated by an inversely proportional relief valve.

valves. Changeable damping forces are obtained by real-time con-trol of the inversely proportional relief valve according to car bodyvibration acceleration, instead of damper stroke velocity, so a sen-sor does not have to be installed in the piston rod. In the out ofcontrol or powered off states, the inversely proportional relief valvebehaves in its maximum pressure set status, and the entire damperbecomes an ordinary passive damper, whose damping force is pro-duced by the constant orifice and limited by the consequent reliefvalve. The two unloading valves perform the same function as thatof the preceding damper.

The semi-active hydraulic damper regulated by an inversely pro-portional relief valve has an analogous resistance network to that ofthe former damper under control mode 1 or 5, which is shown inFig. 2a or e, except its relief pressure is controllable. Therefore, re-ferring to Eq. (7) or Eq. (33), one can similarly formulate the con-trollable damping force as

F4 =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩

��8C2

d

(1d44

)d6V2 when 0�V <Vcritical4

where Vcritical4 = Cd

(d4d

)2( 2�Pset(I)

)1/2

�4d2Pset(I) when V �Vcritical4,

(35)

where F4 is the controlled changeable damping force, d4 the diameterof the constant orifice, Vcritical4 the critical stroke velocity while theinversely proportional relief valve is about to open and Pset(I) thecontrolled set-pressure function to current I.

3. Simulation results and discussion

The damping changeability simulation of the two preceding for-mulated semi-active hydraulic dampers is conducted. Pertinent pa-rameters and their values for simulation are given by Table 2.

Fig. 4 graphically shows the changeable damping performancesof the semi-active hydraulic damper, which is switched by high-speed solenoid valves when subjected to a sinusoidal excitation ofdisplacement amplitude 10mm and frequency 3.2Hz. Force–velocity(F–v) characteristics shown in Fig. 4a illustrate that the semi-active damper has obtained a wide range of changeable damping

Table 2Parameters and values for simulation.

Parameter Value

Cd 0.82� (kg/m3) 872d (m) 34.7e−003d1 (m) 0.923e−003d2 (m) 0.952e−003d3 (m) 0.982e−003d0 (m) 2.2e−003PHset (N/m2) 15e+006PLset (N/m2) 4e+006d4 (m) 1.2e−003

coefficients (generally defined by slope of each curve before relief),and this changeability would adequately meet the requirements ofabsorbing wide spectrum lateral vibrations of the car body.

Modes 5 and 1, which are the two most frequently usedcontrol modes, are referred to as “Normal” and “Hard” modes,respectively. Under “Normal” mode, the damper supplies normaland relatively soft damping forces described by the polynomialcurve “mode 5” shown in Fig. 4a when the stroke velocity is below0.063m/s, and experiences a constant relief damping force of about3.783kN when stroke velocity exceeds that value. The “Normal”mode, which also performs as a redundant “fail-safe” mode, caneffectively absorb vibrations that are mainly caused by track irreg-ularities and transmitted from the bogie to the car body. However,under certain circumstances, such as high train speed, the “hard”mode, i.e., mode 1 is switched on. Under mode 1, the damper sup-plies relatively high damping forces, which are prescribed by thepolynomial curve “mode 1” shown in Fig. 4a when stroke velocityis below 0.108m/s, and experiences a constant relief damping forceof about 14.185kN when the stroke velocity exceeds that value toabsorb induced high frequency vibrations of the car body.

Modes 2–4 are “Harder” control modes with graduallyincreasing levels of damping force to resist high-frequency lateralvibrations of the car body which are mainly induced by variousaerodynamic excitations. Fig. 4a illustrates that damping forces of“mode 2” and “mode 3” are analogous in the low stroke velocityinterval, but the former's slope increases more slowly when thestroke velocity exceeds 0.063m/s and is relieved when the stroke


-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2-15

-10

-5

0

5

10

15

Dam

ping

forc

e (k

N)

mode 1

mode 2

mode 3

mode 4

mode 5

mode 6

-0.015 -0.01 -0.005 0 0.005 0.01 0.015-15

-10

-5

0

5

10

15

Stroke (m)

Dam

ping

forc

e (k

N)

mode 5

mode 6

mode 1

mode 2

mode 3

mode 4

Stroke velocity (m/s)

Fig. 4. Changeable damping performances of the semi-active hydraulic damper switched by high-speed solenoid valves when subjected to a sinusoidal excitation of displacementamplitude 10mm and frequency 3.2Hz: (a) damping force versus stroke velocity, i.e., F–v characteristics and (b) damping force versus stroke, i.e., F–s characteristics.

velocity exceed about 0.092m/s; this causes the F–v characteristicof mode 2 to be approximately linear before relief. However, “mode3” does not change its polynomial characteristic until the strokevelocity exceeds 0.078m/s or so. The slope of the damping forceof “mode 4” also increases more slowly when the stroke velocityexceeds 0.063m/s and is relieved when the stroke velocity exceedsabout 0.067m/s. All three “Harder” modes share the same reliefdamping force level at about 14.185kN, as shown in Fig. 4a.

Mode 6 is a “Soft” mode designed to absorb long stroke and lowvelocity vibrations; for example, when the train is at low speed andturning, it makes the secondary lateral suspensionmore flexible witha slight damping hold.

Fig. 4b shows the F–s characteristics under various controlmodes. It is obvious that the theoretical F–s characteristics shown by

Fig. 4b are less round and smoother than ordinary rig tested ones,this might be due to the omission of various leakage flows, oilcompressibility, set error of the relief valve or back pressure ofthe tank during formulation. Because this study concentrates onmacro damping changeability design in the conceptual phase, thepreceding omission is tolerable. The area enclosed by an F–s curverepresents the work done or energy absorbed by the damper inone vibration cycle; one can see “Hard” modes, such as modes 1–4perform more efficiently in vibration absorption than those of the“Normal” and “Soft” ones.

Fig. 5 graphically shows the simulated changeable dampingperformances of the semi-active hydraulic damper regulated byan inversely proportional relief valve when subjected to the sameexcitation as the input to the former damper. Fig. 5a defines the


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

2

4

6

8

10

12

14

Current (A)

Set

pre

ssur

e (M

Pa)

P1

P3

P2

P(I)

-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2

-10

-5

0

5

10

Dam

ping

forc

e (k

N)

C2

C3

C1

Curve 2

Curve 3

Curve 1


-0.01 -0.005 0 0.005 0.01

-10

-5

0

5

10

Stroke (m)

Dam

ping

forc

e (k

N)

Curve 1

Curve 3

Curve 2

Fig. 5. Changeable damping performances of the semi-active hydraulic damper regulated by an inversely proportional relief valve when subjected to a sinusoidal vibrationinput with displacement amplitude 10mm and frequency 3.2Hz: (a) inversely proportional property of the controlled set pressure P (MPa) to current I (A) of the reliefvalve, (b) F–v characteristics and (c) F–s characteristics.

controlled function of the set pressure P (MPa) to current I (A) ofthe inversely proportional relief valve, while Fig. 5b and c are F–vcharacteristics and F–s characteristics of the damper, respectively.

An inversely proportional relief valve usually consists of a mainregulating unit and a supplementary one [13]. Under normal oper-ations, the main regulating unit regulates set pressure continuouslyfrom P2 to P1 with a negative slope, as shown in Fig. 5a, while thesupplementary one does not function. However, if the entire damperor relief valve itself was out of control or powered off, the main reg-ulation unit would not work, while the supplementary one, whichis actually an ordinary relief valve, sets the pressure to a constant P3that was manually preset between P2 and P1 before service. Thus, theentire damper is still safe when in failure. As Fig. 5a shows, the sim-ulation example relief valve has a controlled set pressure function,P = −22I + 15.96, in current interval [0.18A, 0.68A] and a constantset pressure P3 = 8MPa.

Curves 1–3 shown in Fig. 5b are F–v characteristics of the change-able damper when relief pressures are controlled at P1, P2 and P3, re-spectively. Though the damper can change its relief damping forcescontinuously between curve 1 (the softest) and curve 2 (the hard-est) according to car body vibration acceleration, all its changeabledamping forces share the same rising polynomial curve, that is to say,

the damper can only change its relief damping forces, or change its“effective” damping coefficients, which are graphically exemplifiedby C1, C2 and C3 shown in Fig. 5b. Therefore, this simulated drawbacksuggests that the damping changeability of the low cost semi-activehydraulic damper is limited relative to that of the former, especiallywhen the damper works in the normal velocity range. Fig. 5c showsthe corresponding F–s characteristics of the damper.

The above simulation and discussion verifies that the fluid for-mulae explicitly established in this study contain detailed dampingchangeability information of the two commercial semi-active hy-draulic dampers, which is necessary for their controllable dampingperformances evaluation, further refinement and control design.

4. Conceptual design of a new semi-active hydraulic damper

The motivation to overcome damping changeability limitation ofthe semi-active hydraulic damper, which is regulated by an inverselyproportional relief valve with no obvious cost increase, suggests anew semi-active hydraulic damper, which is simply regulated by aproportional throttle valve with “Sky-hook” control; Fig. 6 shows theconcept. The proportional throttle valve is continuously controlled toperform as a “changeable orifice”, which combines a constant orifice


Check valve1

Check valve2

Unloading valve 1(stretch)

Unloading valve 2(compression)

V

P set

Reliefvalve

Orifice

d5

Tank

Proportional throttle valve

Fig. 6. A new semi-active hydraulic damper which is regulated by a proportional throttle valve.

in parallel to yield changeable damping forces, while an ordinaryrelief valve limits the maximum damping force when the strokevelocity becomes high enough. The two unloading valves unload thedamper under circumstances that may produce negative damping.

Provided that the proposed damper has the same basic construc-tion parameters as those of the former ones and denoting Av(I) ascontrolled function of flow cross section area of the proportionalthrottle valve to current I, denoting d5 as the diameter of the con-stant orifice, Pset as the preset pressure of the relief valve, Vcritical5as the critical stroke velocity while the relief valve is about to open,referring to Fig. 6 and according to flow continuity principle, oneeasily formulates

�4d2Vcritical5 = Cd

[�4d25 + Av(I)

]( 2�Pset

)1/2

(36)

from which we get

Vcritical5 = Cd

[�d25 + 4Av(I)

�d2

](2�Pset

)1/2

. (37)

Thus, if denoting F5 as the produced instantaneous damping force,when 0�V <Vcritical5, flow continuity and damping force of the newdamper can be formulated by⎧⎪⎪⎨⎪⎪⎩

�4d2V = Cd

[�4d25 + Av(I)

]( 2�P)1/2

F5 = �4d2P.

(38)

Removing medium parameter P gives the result

F5 = �3�8C2

d

1

[�d25 + 4Av(I)]2 d

6V2. (39)

When V �Vcritical5, P is limited by Pset, so

F5 = �4d2Pset. (40)

Thus, changeable damping forces of the proposed semi-activehydraulic damper can be summarized as

F5 =

⎧⎪⎪⎨⎪⎪⎩

�3�8C2

d

1

[�d25 + 4Av(I)]2 d

6V2 when 0�V <Vcritical5

�4d2Pset when V �Vcritical5.

(41)

Fig. 7 graphically shows the simulated changeable damping per-formances of the proposed new damper. The damper is based on theabove conceptual design formula when subjected to a sinusoidal ex-citation of displacement amplitude 3.2mm and frequency 12.7Hz. Inthe simulation, d5 = 1.26e-0.03m, Pset = 13MPa, the controlled flowcross section area function Av to current I of the proportional throt-tle valve is Av = 1.985I − 0.317 in interval [0.16A, 0.76A], which isshown by Fig. 7a.

F–v characteristics of the proposed semi-active hydraulic dampershown in Fig. 7b suggest that the damper can change its dampingforce rising curves or “effective” damping coefficients dynamicallyand continuously between curve 4 (the softest) and curve 1 (thehardest) according to the vibration conditions. C1–C4 are “effective”damping coefficients of curves 1–4, respectively. Therefore, it hasobtained better damping coefficient changeability than that of thepreceding damper which is regulated by an inverse proportionalrelief valve. One shortcoming of the new damper might be that itcontains only one relief damping force level within the entire strokevelocity interval.

If the damper was in failure or powered off, the proportionalthrottle valve and two unloading valves would be shut off whichcauses only the constant orifice and relief valve to work in the hy-draulic circuit. In this way, the entire damper would behave as anordinary passive hydraulic damper whose damping performance ischaracterized by “curve 2” in Fig. 7b.

The F–s characteristics of the new damper shown by Fig. 7c alsoillustrate that there exists an obvious wide range of energy absorp-tion ability from curves 4 to 1. The “Hard” modes, such as mode 1,perform more efficiently in vibrations absorption than the “Normal”(mode 2) and “Soft” ones (modes 3 and 4).


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

0.2

0.4

0.6

0.8

1

1.2

Current (A)

Con

trolle

d flo

w c

ross

sec

tion

area

(mm

2 )

-0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25-15

-10

-5

0

5

10

15


Dam

ping

forc

e (k

N)

Curve 1

Curve 4

Curve 3Curve 2

C1C3

C2

C4

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4

-10

-5

0

5

10

Stroke (cm)

Dam

ping

forc

e (k

N)

Curve 1

Curve 2Curve 3

Curve 4

Av(I)

Fig. 7. Changeable damping performances of the semi-active hydraulic damper regulated by a proportional throttle valve when subjected to a sinusoidal excitation ofdisplacement amplitude 3.2mm and frequency 12.7Hz: (a) controlled function of flow cross section area Av (mm2) of the proportional throttle valve to current I (A), (b)F–v characteristics and (c) F–s characteristics.

In the conceptual design, one should also estimate the control re-sponse lag of the proposed new damper. The semi-active hydraulicdamper switched by high-speed solenoid valves works well in com-mercial use without an obvious time delay in a controlled cycle of10ms [7], while another one introduced by [10,14] has a laboratorytested functional switching time between 30 and 100ms, which suf-fices for the need of normal 0.5–3Hz train lateral vibration absorp-tion. Because the proportional throttle valve spool of the proposednew damper can “tremble” at 15–20Hz near its neutral position,just like that of a servo-valve in normal operations, not only can alarge spool displacement or response lag can definitely be avoided,a 50–70ms functional switching time can be obtained that is ca-pable of absorbing 10–15Hz very high frequency lateral vibrations.On the other hand, a small diameter and smart proportional throttlevalve that is less sensitive to fluid contamination than the high-speedsolenoid valve is commonly available in industry; the proposed newdamper would be of high reliability as well as low cost.

5. Conclusions

The damping changeability conceptual design, which involves de-termination of the number of damping levels, as well as the con-crete damping coefficient and relief point of every level, is the most

basic issue in the development of any new railway semi-active hy-draulic damper. Therefore, the basic structure and fluid resistanceparameters, such as the orifice diameters of the damper, should becalculated with detail in the conceptual phase.

Fluid formulae for the changeable damping performances eval-uation of two commercial semi-active hydraulic dampers for sec-ondary lateral suspension of a high-speed passenger train areestablished. Simulation results show that the damper switched byhigh-speed solenoid valves obtains a wide range of changeabledamping coefficients, which guarantees the absorption of a widespectrum of lateral vibrations, while another low cost one regulatedby an inversely proportional relief valve, whose F–v characteristicsshare the same rising curve, can only change its relief dampingforces, so it is relatively limited in damping changing.

In an effort to overcome the changeability limitation of thedamper regulated by an inversely proportional relief valve with noobvious cost increase, a new semi-active hydraulic damper, whichis regulated by a simple proportional throttle valve, is proposed;continued fluid formulation and simulation demonstrate that thedamper can change its damping force rising curves or “effective”damping coefficients continuously in a considerably wide range.

Thus, fluid formulae explicitly established in this study are ofsignificance in the damping changeability conceptual design, further


refinement and control design for the three semi-active hydraulicdampers. The proposed new damper, which is short in changingdelay and insensitivity to fluid contamination, might be feasible inindustry applications.

Various leakage flows, oil compressibility, and back pressure ofthe tank are not considered in the formulation, which might af-fect the simulation accuracy to some extent. However, this studyaddresses damping changeability design at the conceptual phase,which is relatively qualitative, so these unremarkable effects can beneglected. However, the above factors should not be omitted in thenext development steps, such as in damper dynamic performancesmodelling and full vehicle close-loop simulation.

References

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Fluid formulae for damping changeability conceptual design of railway semi-active hydraulic dampers

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