Procedia Engineering 150 ( 2016 ) 384 – 392

1877-7058 © 2016 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).Peer-review under responsibility of the organizing committee of ICIE 2016doi: 10.1016/j.proeng.2016.06.730

ScienceDirectAvailable online at www.sciencedirect.com

International Conference on Industrial Engineering, ICIE 2016

Energy Analysis of Vehicle Suspension Oscillation Cycle

I.M. Ryabova, K.V. Chernyshova, A.V. Pozdeeva,* a Volgograd State Technical University, Lenin avenue, 28, Volgograd, 400005, Russia

Abstract

This article is devoted to conducting the energy analysis of the oscillation cycle in vehicle suspension based on the research of the dynamics equations of the linear single-support single-mass vibrating system with fixed elastic and damping characteristics at harmonic kinematic disturbance. The characteristics of the relative energy flows in the oscillation cycle of the suspension, taking into account the shock absorber inefficient work are received. The physical meaning of the overall characteristic point to the series of the sprung mass amplitude-frequency characteristics is identified. Analysis of vibration isolation properties of the suspension on the sprung mass amplitude-frequency characteristics and the frequency characteristics of the total width of the suspension ineffective work areas in the oscillation cycle is carried out in the damping regulation on the inefficient work areas and on different damping levels. © 2016 The Authors. Published by Elsevier Ltd. Peer-review under responsibility of the organizing committee of ICIE 2016.

Keywords: energy; oscillation cycle; suspension; shock absorber; spring; sprung mass; inefficient work area.

1. Introduction

One of the most perspective ways of increasing the vehicles smoothness is to control characteristics of the suspension elements in oscillation cycle. At present there are different principles for regulation of stiffness of elastic element and power of inelastic resistance of damping element [1-24]. However, many of them do not have a physical justification for using these methods of changing suspension characteristics in the oscillation cycle. In [25-29] it is shown that the damper effect in oscillation cycle is characterized by the presence of two areas in which its action is aimed at increasing the speed and amplitude of sprung mass displacement. In this connection it is necessary to carry out the energy analysis of oscillation cycle in the vehicle suspension.

* Corresponding author. Tel.: +7-987-643-8514.

E-mail address: pozdeev.vstu@gmail.com

© 2016 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).Peer-review under responsibility of the organizing committee of ICIE 2016

385 I.M. Ryabov et al. / Procedia Engineering 150 ( 2016 ) 384 – 392

2. Mathematical modelling

For execution the energy analysis of the oscillation cycle in the vehicles suspension we need to consider the linear single-mass singly-support vibration system with fixed elastic and damping characteristics at harmonic kinematic disturbance (Fig. 1).

Fig. 1. Design scheme of energy flows in linear suspension: m – sprung mass; c – stiffness of elastic element; k – inelastic resistance factor of damping element; z – displacement of sprung mass;

q – kinematic disturbance; Asz and As

q – spring work to the coordinates z and q; Aaz and Aa

q – shock absorber work on coordinates z and q.

The equation of the dynamics of this vibration system is as follows:

0qzcqzkzm (1)

where m – sprung mass; k – inelastic resistance factor of damping element (shock absorber); c – stiffness of elastic element (spring); z – absolute displacement of sprung mass; q – absolute displacement of disturbing ground (road surface irregularities).

After that it is necessary to consider the distribution of energy flows in oscillation cycle of linear suspension, taking into account the availability in oscillation cycle of inefficient work areas (IWA) of shock absorber [25-29]. Also we need to find out the physical meaning of the overall characteristic point A for the series of the sprung mass amplitude-frequency characteristics (AFC) of the absolute sprung mass displacement of the vehicle suspension (Fig. 2) [16].

Fig. 2. The series of amplitude-frequency characteristics of the sprung mass vertical oscillations: A – the overall characteristic point; i – relative vibration frequency; bz – sprung mass dynamic factor; – relative damping factor

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In the researched single-support single-mass suspension vibration system in the steady state oscillation cycle it can be identified the following five types of energy or work [16, 25, 28] without taking into account the availability of dry friction forces in the system:

qAs and zAs – work of the elastic element (spring) on coordinates q and z; qA and zA – work of the damping element (shock absorber) on coordinates q and z; xAs and xA – work of the spring and shock absorber on coordinate = z – q:

qzx AAA sss , qzx AAA ; (2)

zA – energy of sprung mass vibration on coordinate z:

zzz AAA s ; (3)

qA – energy of vibration exciter (road profile):

qqq AAA s . (4)

For determining these works we need to compose and use integrals of corresponding forces from the spring and damper of suspension:

dtqxcAqs , dtzxcAz

s , dtxxcAxs , dtqxkA q , dtzxkA z , dtxxkA x ,

where x – suspension deformation, and also known ratio from [30]:

2222

2

0041 ii

iqx , 2222

22

0041

41

ii

iqz , 22

2

0041 i

izx ,

622222

222

1441

41cosiii

ii , 622222

3

1441

2siniii

i , 2222

2

241

1cosii

i ,

22222

41

2sinii

i,

2241

1cosi

, 2241

2sini

i,

where – total width of inefficient areas; 1, 2 – width of inefficient area on the compression and extension strokes. In view of the existence in the cycle of inefficient and efficient areas [25-29] the following relationship is

obtained for the relative energy or relative work:

)( 20xkAa , (5)

where A – absolute energy or work; k – inelastic resistance factor of damping element; – vibration frequency. The expressions for the relative energies and works of spring, shock absorber, sprung mass and vibration exciter

in the oscillation cycle of vehicle suspension are shown below. The relative energy of the sprung mass vibration is

in the inefficient area 2ineff 2 iia z (6)

387 I.M. Ryabov et al. / Procedia Engineering 150 ( 2016 ) 384 – 392

in the efficient area 2eff 2 iia z (7)

total 0za (8)

The relative energy of the vibration exciter is

in the inefficient area i

a q 2ineff (9)

in the efficient area i

a q 2eff (10)

total qa (11)

The relative work of spring on the coordinate z is

in the inefficient area 2s.ineff ia z (12)

in the efficient area 2s.eff )( ia z (13)

total 2s iaz (14)

The relative work of spring on the coordinate q is

in the inefficient area 222s.ineff 412

ii

iaq (15)

in the efficient area 222s.eff 412

ii

iaq (16)

total 2s iaq (17)

The relative work of spring on the coordinate x is

in the inefficient area 22s.ineff 412

iia x (18)

in the efficient area 22s.eff 412

iia x (19)

total 0sxa (20)

The relative work of shock absorber on the coordinate z is

in the inefficient area 2a.ineff )2( iia z (21)

in the efficient area 2a.eff )2( iia z (22)

total 2ia z (23)

The relative work of shock absorber on the coordinate q is

in the inefficient area 22

222

2

2

a.ineff 414121

iii

iiiaq (24)

388 I.M. Ryabov et al. / Procedia Engineering 150 ( 2016 ) 384 – 392

in the efficient area 22

222

2

2

a.eff 414121)(

iii

iiia q (25)

total 2

21i

iaq (26)

The relative work of shock absorber on the coordinate x is

in the inefficient area 22

222

2

2

a.ineff 414122

iii

iiiiax (27)

in the efficient area 22

222

2

2

a.eff 41412)(2

iii

iiiiax (28)

total xa (29)

3. Simulation results

Fig. 3 shows graphs of relative total work for the oscillation cycle, depending on the relative frequency of oscillation – )(fa (solid lines). The graphs show that the relative spring work zq aa ss (curve 1), i.e. the energy of the vibration exciter fully transmitted to the sprung mass through the spring. Therefore, the work of the spring per oscillation cycle is 0s

xa (axis i at a = 0). The relative energy of road profile is qa , and the relative energy of shock absorber is xa (line 2) , i.e. energy of kinematic exciter delivered to the vibration system is completely absorbed by the shock absorber.

Fig. 3. Distribution of relative energy in the oscillation cycle of vehicle suspension: 1 – total relative work of spring on the coordinates q and z

( qas and zas ); 2 – total relative energy of the vibration exciter ( qa ) and total relative work of damper on the coordinate x ( xa ); 3, 5 and 7 –

total, efficient and inefficient relative works of damper on the coordinate q ( qa , qa .eff and qa .ineff ); 4, 6 and 8 – total, efficient and inefficient

relative works of damper on the coordinate z ( za , za .eff and za .ineff ); axis i for a = 0 – total relative energy of spring on the coordinate x

( xas ) and sprung mass on the coordinate z ( za )

Unlike spring shock absorber performs various kinds of work qa and za on the coordinates x and z (curves 3 and

4). Thus it shows that zz aa s (curves 4 and 1), i.e. the work absorbed by the damper on the coordinate z is equal

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to the energy supplied to the sprung mass from the spring. Therefore, the relative energy of the sprung mass oscillations is 0za (axis i).

Before resonance ( 1i ) the energy per oscillation cycle in the vibration system circulates in one direction (solid arrows lines in Fig. 1), i.e. the energy from the vibration exciter through the spring and shock absorber partially returned to exciter.

At resonance ( 1i ) the energy from the vibration exciter is completely extinguished as 0qa . Herewith the

relative energy of the shock absorber is aa z , and the relative energy of the spring is zas . After resonance ( 1i ) the energy from the vibration exciter changes direction and is applied to the shock

absorber (dashed arrow line in Fig. 1). At the common characteristic point A ( 2i ) it can be observed the following – 2ss

zqq aaa , and 2za . As zq aa , then it follows that the energy supplied to the shock absorber from the vibration exciter

and the sprung mass per oscillation cycle is the same in absolute value and different in sign. This energy is completely absorbed by the shock absorber. Thus, the physical sense of the existence of a characteristic common point for series of AFC is that the shock absorber does not transmit energy from the vibration exciter to the sprung mass regardless of the damping level. Therefore, when 2i vibration protection factor of linear suspension always equals 1.

Fig. 3 also shows the graphs of the relative inefficient and efficient work of the shock absorber with 5,0 (dotted and dash-dotted lines). The graphs show that the relative efficient work of the shock absorber on the coordinates q and z ( qa .eff and za .eff , curves 5 and 6) is greater than its total relative work ( qa and za , curves 3 and

4) on the value of the relative inefficient work ( qa .ineff za .ineff, curves 7 and 8). Moreover, calculations show that this difference increases with increasing of the relative frequency i and the relative damping factor . Besides at resonance qq aa .ineff.eff , i.e. when 1i the same energy per oscillation cycle from the road profile to the shock absorber is brought or taken regardless of the factor . It ultimately explains zero value of the total relative work of the shock absorber on the coordinate q.

Thus, if to turn the shock absorber off in the inefficient areas, i.e. to control its inelastic resistance on vibration phase, then the energy transfer between the damper and the vibration exciter per oscillation cycle will be reduced. Herewith the energy of the shock absorber on the coordinate x will be reduced and the total work of the shock absorber on the coordinate z will be increased.

Fig. 4 shows the damping characteristic of the linear vibration system (Fig. 1) with the control of inelastic resistance on inefficient work areas.

Fig. 4. Damping characteristic of the shock absorber in controlled linear vibration system: I, II, III, IV – areas of damping characteristic with constant damping; Pd – damping force; qz – deformation velocity of vibration system

As it can be seen from Fig. 4, areas I and III of controlled damping characteristic correspond to the time intervals in the oscillation cycle of suspension with increased damping. Areas II and IV correspond to the periods of time in the oscillation cycle of suspension with reduced damping. The vertical sections of the damping characteristic

390 I.M. Ryabov et al. / Procedia Engineering 150 ( 2016 ) 384 – 392

correspond to the moments of inelastic resistance switching in damping element of suspension from larger to smaller, which coincide with the beginning of inefficient work area of suspension in the oscillation cycle.

The control principle shown in Fig. 4 can be presented in detailed form:

,00:,00:,00:,00:

min11

max11

min11

max11

zqzuuIVzqzuuIIIzqzuuIIzqzuuI

(30)

where conditions I, II, III, IV correspond to damping characteristic areas of the shock absorber in which the damping level does not change; u1 – damping control parameter, which is subjected to conditions max11 uu or min11 uu .

Fig. 5 shows the influence of the dissipative forces on the width of the inefficient work areas in linear suspension with different resistance of the shock absorber.

Fig. 5. Frequency characteristics of the total width of inefficient work areas in low-frequency vibration cycle of the suspension with various options of damping control: 1 and 1/ – max = 0,25; 2 and 2/ – max = 0,5; 3 and 3/ – max = 0,75; 1, 2 and 3 – damping, controlled on vibration phase ( max = 0,25; 0,5; 0,75 and min = 0); 1/, 2/ and 3/ – uncontrolled damping ( max = min = 0,25; 0,5; 0,75); 4 – optimum damping, controlled on

vibration phase and frequency ( max = f(i))

As it can be seen from Fig. 5, the width of inefficient work areas of the suspension go up with increasing of kinematic disturbance frequency (curves 1/, 2/ and 3/). Increasing of damper resistance leads to increasing of the phase shift between the sprung mass vibration and the relative vibration of suspension.

Comparison of the curves shows that the damping control on vibration phase (curves 1, 2 and 3) leads to increasing of the width of the inefficient work areas in the suspension. Herewith the greater resistance of the shock absorber is, the wider the inefficient area becomes with control on vibration phase. Optimum control of damping on vibration phase and frequency (curve 4) leads to a quite different character of changing the considered phase shift in the low frequency range. The width of the areas has maximum at resonance zone in comparison with options of damping control on vibration phase, but at resonance and after-resonance zones the value of the phase shift is constant and equal to radian, that corresponds to a half of the vibration period.

Fig. 6 shows the amplitude-frequency characteristics (AFC) of the absolute displacement of the sprung mass M = 1500 kg with different control parameters of shock absorber damping on inefficient work areas ( max = 0,25; 0,5 and 0,75) at double amplitude of kinematic disturbance 2q0=15 mm.

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Analysis of the graphs data showed that when the damping is controlled on vibration phase (curves 1, 2 and 3), in the area of low-frequency resonance efficiency of vibration damping is increased with increasing of the relative damping factor. When the damping factor is max = 0,25 vibration protection factor is 1,7, when max = 0,75 – vibration protection factor approaches to 1. In the case of optimal damping control on vibration phase and frequency (curve 4) we can get AFC absolutely without resonance and with vibration protection factor less than 1. In the resonance zone vibration system with damping control shows practically the same results. Herewith the vibration gain with frequency increasing is not observed unlike Fig. 2.

Consequently, the calculated and theoretical research of the optimal damping control algorithm has shown the high efficiency of the considered method of vibration damping in vehicles suspension. This method provides potentially high vibration protection properties of the suspension.

Fig. 6. AFC of the sprung mass absolute displacement with various options of shock absorber damping control under harmonic kinematic disturbance: 1, 2 and 3 – damping, controlled on vibration phase ( max = 0,25; 0,5; 0,75 and min = 0); 4 – optimum damping, controlled on vibration

phase and frequency ( max = f(i))

4. Conclusion

Thus, conducted energy analysis of suspension oscillation cycle showed that as a result of the switching shock absorber off in inefficient areas it should be expected a decrease of the sprung mass displacement amplitude and raise of vehicle smoothness. Improving of the suspension vibration protection properties should also be expected from using of shock absorbers with energy recovery in the oscillation cycle, which provide changing resistance force action in the inefficient work areas to the opposite direction.

References

[1] A.N. Gustomyasov, Vibration system analysis of car suspension with a discrete stiffness changing, Proceedings of the universities. 5 (1978) 187–188.

[2] B.N. Belousov, Synthesis of dynamic control system of vehicle active suspension, Automotive industry. 4 (2004) 15–17. [3] B.N. Belousov, Merkulov, I.V. Fedotov, Controlled car suspensions, Automotive industry. 1 (2004) 23–24. [4] E.M. Gneusheva, O.V. Fominova, V.I. Chernyshev, Dynamic properties of vibroprotective systems with additional elastic-damping element

of intermittent action, Handbook, Engineering magazine. 6 (2006) 59–64. [5] B.A. Kalashnikov, Objects damping systems with discrete switching of elastic elements, OmSTU Publ., Omsk, 2008. [6] E.M. Gneusheva, O.V. Fominova, V.I. Chernyshev, Systematics of vibroprotective systems with additional elastic-damping element of

intermittent action, Handbook, Engineering magazine. 9 (2004) 31–35.

392 I.M. Ryabov et al. / Procedia Engineering 150 ( 2016 ) 384 – 392

[7] K.V. Chernyshov, V.V. Novikov, I.M. Ryabov, Optimal control of suspension damping on the basis of maximum principle, Tractors and agricultural machinery. 2 (2006) 13–15.

[8] A.V. Pozdeev, V.V. Novikov, A.S. Diakov, A.V. Pokhlebin, I.M. Ryabov, K.V. Chernyshov, Adjustable pneumatic and hydro-pneumatic springs of motor vehicles suspensions, VSTU Publ., Volgograd, 2013.

[9] V.V. Novikov, A.V. Pozdeev, Determination of the optimal control algorithms for actively-controlled air suspensions, Truck&Mechanical engineering. 5 (2010) 6–10.

[10] I.M. Ryabov, K.V. Chernyshov, A.M. Kovalev, A.V. Pozdeev, Development and research of the new damping control algorithm in the oscillation cycle of vehicle suspension, Design of wheeled vehicles, in: Proceedings of the scientific and technical conference devoted to the 100th anniversary of beginning of engineers training in the automotive specialty in the Bauman MSTU. (2010) 2–5.

[11] K.V. Chernyshov, A.V. Pozdeev, V.V. Novikov, I.M. Ryabov, Determination of the optimal conditions of stiffness control in vehicle air suspension, Truck. 11 (2010) 2–5.

[12] K.V. Chernyshov, A.V. Pozdeev, V.V. Novikov, Optimal stiffness control of air springs in vehicle suspension, Russia of reform period: the formation of market-oriented organization model as an element of crisis management Gazprom, in: Proceeding of Mater. XIV Intern. scientific-practical. conf., Volgograd College of gas and oil Gazprom, Volgograd. (2010) 163–167.

[13] A.S. Dyakov, A.V. Pozdeev, A.V. Pokhlebin, Optimal stiffness and damping control of vehicle suspension based on the maximum principle of Pontryagin L.S., Bulletin of the Military Sciences Academy. 2 (2011) 132–139.

[14] A.V. Pozdeev, V.V. Novikov, K.V. Chernyshov, I.M. Ryabov, Synthesis of optimal control algorithms for damping and stiffness of vehicle suspension, Truck. 6 (2011) 2–6.

[15] A.S. Dyakov, M.O. Kotlyarenko, A.V. Pozdeev, A.V. Pokhlebin, Yu.O. Shakhmardanov, Hydropneumatic spring of high-speed tracked vehicle, Bulletin of the Military Sciences Academy. 2 (2011) 125–132.

[16] V.V. Novikov, I.M. Ryabov, K.V. Chernyshov, The vibration isolation properties of the motor vehicles suspensions, VSTU Publ., Volgograd, 2009.

[17] V.V. Novikov, Yu.G. Lapynin, I.M. Ryabov, K.V. Chernyshov, A.V. Podzorov, Shock absorber, RU Patent 2426921. (2011). [18] V.V. Novikov, A.V. Podzorov, K.V. Chernyshov, Self-controlled gas-filled hydraulic shock absorber, Truck. 9 (2011) 2–6. [19] I.M. Ryabov, V.V. Novikov, K.V. Chernyshov, A.V. Pozdeev, Sh.M. Mukhuchev, Hydraulic shock absorber with increased efficiency,

Modern Railways: achievements, problems, education, in: Proceeding of mater. of 8th Int. scientific-practical. conf., dedicated 70th anniversary of the Victory in the Great Patriotic War, MIIT, the Volgograd branch, Volgograd. (2015) 125–129.

[20] V.V. Novikov, I.M. Ryabov, K.V. Chernyshov, V.V. Vorobiev, A.V. Gallov, Shock absorbers with energy recovery in an oscillation loop, Handbook,Engineering magazine. 7 (2001) 31–34.

[21] V.V. Novikov, O.V. Smolyanov, Vibrational protection provided by automobile suspensions with both hydraulic and inertial-frictional shock absorbers, Russian Engineering Research. 1 (2009) 68–70.

[22] I.M. Ryabov, K.V. Chernyshov, A.V. Pozdeev, Types of designs of inertial-frictional shock absorbers, modeling and testing, Tractors and agricultural machinery. 4 (2013) 23–26.

[23] V.V. Novikov, A.V. Pozdeev, K.V. Chernyshov, A.S. Dyakov, D.A. Chumakov, I.A. Golyatkin, RU Patent 157974. (2015). [24] V.V. Novikov, A.V. Pozdeev, A.V. Pokhlebin, I.M. Ryabov, K.V. Chernyshov, A.S. Dyakov, D.A. Chumakov, I.A. Golyatkin, A.V.

Bondarenko, RU Patent 158085. (2015). [25] V.V. Novikov, Improving the efficiency of vehicle suspension, Tractors and agricultural machinery. 9 (2005) 29–32. [26] V.V. Novikov, New ways to improve the efficiency of shock absorbers, Progress of vehicles and transport systems – 2005, in: Proceedings

of the international scientific-practical conference, VSTU Publ., Volgograd. (2005) 267–269. [27] I.M. Ryabov, V.V. Novikov, K.V. Chernyshov, Efficiency factor of vehicle damper at resonance, Efficiency of transport exploitation:

interuniversity scientific collection, SGTU, Saratov, 1995, pp. 81–86. [28] I.M. Ryabov, V.V. Novikov, K.V. Chernyshov, A.V. Vasiliev, O.V. Osintsev, The distribution of energy in the oscillation cycle of vehicle

suspension, Handbook, Engineering magazine. 4 (1998) 31–33. [29] V.V. Novikov, I.M. Ryabov, A.S. Dyakov, A.V. Pokhlebin, A.V. Pozdeev, Research of inefficient work areas of air suspension in co-

operation with hydraulic shock absorbers and air dampers, Russia of reform period: the formation of market-oriented organization model as an element of crisis management Gazprom, in: Mater. XIII Intern. sectoral scientific-practical. conf., Volgograd College of gas and oil Gazprom, Volgograd. (2009) 214–219.

[30] A.D. Derbaremdiker, Hydraulic shock absorbers of vehicles, Engineering Publ., Moscow, 1979.