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Active suspension control of flexible-bodied railway vehicles using electro-hydraulic and electro-magnetic actuators
Control Engineering Practice 8 (2000) 507}518
Active suspension control of #exible-bodied railway vehicles using electro-hydraulic and electro-magnetic actuators E. Foo*, R. M. Goodall Department of Electronic and Electrical Engineering, Loughborough University, Ashby Road, Loughborough, Leicestershire, LE11 3TU, UK Received 19 November 1998; received in revised form 6 October 1999
Abstract Lighter railway vehicles are going to become the norm in the future as the operator pushes towards a more economical operation. This paper looks into ways of applying classical control methods using `skyhook dampinga to minimise the #exibility e!ects that arise as a consequence of having lighter vehicles, and in particular includes the e!ects of real actuator dynamics. Two and three actuators schemes are considered: two of them are located at the front and rear secondary suspension pivot points, with a third actuator at the centre of the vehicle primarily targeted at reducing the e!ects of the main bending mode of the vehicle body. Hydraulic actuators are used at the front and rear while an electro-magnetic actuator is connected to the centre of the vehicle. 2000 Elsevier Science Ltd. All rights reserved. Keywords: Active vehicle suspension; Flexible; Railways; Actuators
1. Introduction Actively controlled suspensions have been widely studied theoretically and experimentally for automobiles and rail vehicles. These studies have demonstrated significant performance bene"ts. For railway vehicles, active suspensions that deal primarily with improvement in ride quality are now starting to be incorporated on a regular basis (Goodall, 1997). Furthermore railway engineers are also interested in using active control to achieve a functionality that is not possible with a purely passive (mechanical) suspension. The present railway operators are facing intense pressure due to competition from the air and road transport industries. In order for them to operate more e$ciently and e!ectively, they are investigating ways to reduce the travelling time, increase the number of passengers and reduce operating costs. The trend of faster and lighter vehicles are ways in which these objectives can be met. Therefore trains of the future will be designed with lighter bodies and bogies to enable them to operate at a much * Corresponding author. Tel.: #44-1509-228105; fax: #44-1509222854. E-mail addresses: [email protected] (E. Foo), r.m.goodall@ lboro.ac.uk (R. M. Goodall).
higher speed. At this moment, the highest speed for an in-service operation train is 300 km/h, which is achieved by TGV in France and Shinkansen in Japan, but as the speed of the train is increased above this level it will lead to more high-frequency vibrations which will certainly a!ect the ride quality and generate unacceptable levels of internal noise. Present vehicles are designed to have the body much heavier than the bogie, which prevents the bogie excitation frequencies from coinciding with the body excitation frequencies. However with lighter vehicles the #exible frequencies of the body will de"nitely coincide with the bogie excitation frequency, and this will cause an increase in the ride accelerations, and result in the reduction in the ride comfort (ride quality). An important application of active control is therefore for #exible railway vehicle bodies, not only to provide a better ride quality in general, but also to enable lighter bodies with lower #exibility frequencies to be taken advantage of. This paper reports the "rst and second phase of a research project. The "rst phase involves studying active suspension strategies speci"cally for more #exible vehicle bodies. Established `classical controla techniques based upon the idea of skyhook damping are extended to accommodate body #exibility, and the idea of an extra central actuator connected to an auxiliary mass is also included.
0967-0661/00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 7 - 0 6 6 1 ( 9 9 ) 0 0 1 8 8 - 4
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Presently the majority of the published work on active suspensions system assumes that ideal actuators are used, i.e. that the actuator is able to supply whatever amount of force is required by the controller, over an in"nite bandwidth and with no delay. Clearly this is not realisable in practice, as any real actuator will have a "nite bandwidth. The introduction of the actuator dynamics will certainly degrade the ride performance, when compared with ideal actuators. As a result the second phase of the research project involves modelling and applying realactuator dynamics to an existing controller designed in the "rst phase using ideal actuators only, so as to investigate the impact of the actuator dynamics on the ride quality.
Frequency analysis method is used to calculate the raw accelerations. The formula used is as follows: U "("H(u)"G J (u) du, (2) WW X where
is the r.m.s of the output power spectrum WW and H(u) is the frequency response of the vehicle dynamics. H(u) must take account of the time delay between the track inputs to the front and rear wheelsets, and Eq. (3) gives this relation where ¸ /v is the time delay between the two wheelsets. H (u) and H (u) are the transfer * 0 functions in response to inputs at the front and rear, respectively. H(u)"H (u)#H (u)e\HS* T. * 0
(3)
2. Method of ride assessment In order to determine how good the ride is, it is necessary to "nd ways of quantifying the ride quality. This section describes the random track inputs that create vibrations of the vehicle body, and explains how body accelerations calculated by frequency domain analysis can be used to assess ride quality. 2.1. Track proxle The random irregularities of a railway track can be represented by a spatial power spectrum, usually approximated as a fourth-order equation (Pollard, 1983). However for secondary-suspension assessment, the higher-order terms do not have any signi"cant e!ect above 10 Hz or so, and can therefore be neglected. It can be shown that a good approximation for the track vertical velocity input is Gaussian white noise with a #at spectrum (Pratt, 1996) given by G "(2p)A v(m/s)/Hz. (1) X The track velocity, v, is taken as 55 m/s and the roughness factor for a good quality mainline track, A , is taken as 2.5;10\ m (Pratt & Goodall, 1994). 2.2. Quantifying ride quality The standard method of quantifying ride quality is based upon calculating or measuring the root mean square (r.m.s) acceleration levels experienced on the vehicle body, normally measured above the front and rear bogies and at the centre of the vehicle. For absolute measurements it is preferable to include a frequency weighting before the r.m.s is calculated to allow for human sensitivity to vertical vibration. However, for comparative studies in which relative levels of ride quality are being assessed, the frequency weighting can be neglected. This is the approach adopted here, although the inclusion of frequency weighting is straightforward if required.
3. Modelling 3.1. Modelling of vehicle body with yexibility The vehicle is modelled with the rigid and #exible modes. The rigid body mode is implemented using normal Newtonian mechanics, while #exibility is modelled as a free}free beam using receptance principles (Richard, 1980). The sideview model of the train is given in Fig. 1. It is a three-mass model consisting of the body of the vehicle and the two bogies. The system has been simpli"ed to include only two wheelsets instead of four. This is because the time delay between the inputs to the two wheelsets are considered to be negligible. The bogie masses only have a vertical degree-of-freedom, whereas the body modelling includes both vertical (bounce) and pitch motions. The suspension between the bogie and body masses is representative of a conventional airspring with an auxiliary reservoir, in which damping is introduced by means of a suitable restriction (Tang, 1996). The parameter values given in Appendix A are typical for a modern inter-city vehicle. The #exible deformations (Hac, 1986) can be described by z(x, t), where t is the time and x is the distance along the beam. This can be represented by its mode shape and the principal co-ordinate modes, (x) and o (t), respecP P tively, in Eq. (4), where r is the mode number. z(x, t)" o (t) (x). (4) P P P Using the receptance functions de"ned in (Bishop, 1960), the following equation is obtained. M oK (t)#2mu o (t)#uo (t)"F (x )#F (x ), J P P P P P 0 0 0 * * * (5) where M is the mass of vehicle, m is the damping coe$c ient, u is the r mode frequency in rad/s. P
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509
Fig. 1. Simpli"ed side-view linearised model of a railway vehicle with centre mass and controllers.
The description which follows deals with two #exible modes only, but more can be incorporated into the model by adding more degrees-of-freedom to the #exibility equations. Eq. (5) can be changed into Eq. (6) with the symmetrical and asymmetrical modes separated. The former (excited by bounce inputs) has (x )" (x ), while the Q * Q 0 latter (excited by pitch inputs) has (x )"! (x ). ? * ? 0 !(F #F ) (x ) * 0 * , oK #2mu o #o u" P Aol !(F #F ) (x ) * 0 * , oK #2mu o #o u" Aol
3.2. Actuators (6)
where A, o and l are the cross section, density and length of the beam. Note that the subscript r in Eq. (5) has been replaced by subscripts s and a to represent the symmetrical and asymmetrical modes, respectively, in Eq. (6). Both modes are modelled with 5% damping, and frequencies of 8.43 and 23.23 Hz for the symmetrical and asymmetrical modes, respectively, are given, which are typical values for a modern railway vehicle. The matrices with rigid and #exible body modes can be combined as
XQ XQ
"
A
A } A A }
#
X X
B G [u]# [m] B 0
with A } and A } denoting the coupling of the #exible into the rigid body mode and the rigid into the #exible mode, respectively, u the control force and f is the track pro"le. The total order of the model (excluding the actuators dynamics) is therefore 14 and made up as follows: Eight body positions and velocity states (four rigid and four #exible). Four bogie position and velocity states. Two extra suspension states for the airspring model.
(7)
There are various rigorous ways of "nding the placement of actuators (Hac & Liu, 1992) but in this paper a third actuator is added at the centre of the vehicle to help suppress the "rst symmetrical #exible mode, which causes the biggest degradation on the ride performance. A mass is also included at the centre of the vehicle to provide an anchor point for the central actuator, which in practice will be an item of auxiliary equipment. A mass of 1 t supported by a spring to give a 1 Hz natural frequency has been assumed, although this will obviously be modi"ed by the control. A damper giving 5% damping is included to prevent oscillations when there is no actuator force input. Two types of actuators have been considered, the "rst one is servo-hydraulic, the other one is electromagnetic. The servo-hydraulic actuators are located across the secondary suspension at the front and rear of the vehicle, while the electromagnetic actuator is used in the centre location to control the #exible body modes. The servohydraulic actuator is used because there is a limited space
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Fig. 2. Model of a servo-hydraulic actuator.
Fig. 3. Servo-hydraulic actuator force control loop.
between the secondary suspension and the body. In principle an electromagnetic actuator could be used, but the size of the actuator would be large and would be di$cult to "t within the space allowable across the secondary suspension. For the centre actuator however, the electromagnetic actuator technology most suited as the force requirement for this actuator is around 1 kN r.m.s (Section 5.1.2), and with this force level the construction of the electro-magnetic actuator will be small enough to be "tted underneath the vehicle. Also the high-bandwidth capability of such actuators makes them very appropriate for controlling the vibration in #exible modes. Each of these actuators is force-controlled, using inner force-feedback loops carefully designed to achieve the required performance. Fig. 2 shows a typical model of a servo-hydraulic actuator (Thayer, 1958 and Goodall & Whit"eld, 1985). The inner-loop for the hydraulic actuators is shown in Fig. 3. A linear model is used here as the investigation is concentrated on the interaction between active suspension controllers and #exible modes of the body. The parameters of the PID controller are chosen to achieve a good performance. Fig. 4 shows the model used to represent the dynamics of the central electromagnetic actuator. Its inner loop is shown in Fig. 5, and again a linear model is used in this study (Goodall, Pearson & Pratt, 1993). The e!ect of air gap variations makes an electro-magnetic actuator on its own an unstable system, although "tting this in parallel with the auxiliary mass's own suspension will tend to
Fig. 4. Model of electromagnetic actuator.
Fig. 5. Electro-magnetic actuator force control loop.
overcome the stability. However a properly designed force feedback loop overcomes any residual instability, but it is important to model the characteristics properly to choose appropriate values for the PI controller and to give correct predictions of performance.
4. Suspension control A great variety of suspension control strategies are possible, but this paper concentrates upon the use of the so-called `skyhook dampinga control which is known to give excellent improvements in ride quality. The following section explain the principle and shows how it can be extended using modal control.
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4.1. Skyhook control Skyhook damping (Li & Goodall, 1997) provides a force dependent upon the absolute velocity of the vehicle body, instead of a force dependent upon the relative velocity given by a conventional damper. Of course, in practice absolute velocity is di$cult to measure, and so the velocity required by the skyhook damper is usually obtained by integrating the signal from an accelerometer. However a pure integrator is not practical, and as a result a second-order Butterworth highpass "lter (HPF) with a frequency of 0.1 Hz and a 70% damping is chosen because it will give a maximally #at frequency response at the low-frequency range. In fact this "lter's characteristics is also an important consideration with regard to the response to deterministic track inputs (Li & Goodall, 1999), but this paper deals with random track inputs only. The transfer function that is used to give a velocity estimate from the acceleration measurement is as follows: z( "
2.533s z( . 1#2.228s#2.533s
(8)
Fig. 6. Modal control structure.
Fig. 7. Example of a modal skyhook controller.
Table 1 Three actuators: location of sensors Strategy
In addition, controllers which use pure skyhook damping have been used as a starting point for the development of the controller. These assume that absolute velocity can be directly measured, and in the results section (Section 5.1.1) these are referred to as `Pskya controllers. 4.2. Modal control In principle, modal control (Williams, 1994) attempts to manage individual system modes. This is achieved by decomposing measurements into modal components. These can then be processed individually and recombined to drive the actuators. Fig. 5 shows the basic structure of a modal controller. In practice, linear accelerometers are normally located above the secondary suspensions at the front and rear of the vehicle, although the other option is to use one linear and one rotational accelerometer located at the vehicle's centre, but of course these will sense the #exibility e!ects di!erently. Using the relationship between the bounce/pitch accelerations and the front/rear accelerations, modal control can be applied to control the bounce and pitch separately. Fig. 6 illustrates a particular case where the input signals are the front and rear accelerations and the outputs are the front, centre and rear actuator forces demands for the actuator force-control loops. By changing the gain factors for the bounce, pitch and centre (G , G and G , respectively), di!erent amounts of skyhook damping can be applied to control the bounce and pitch accelerations (Fig. 7). Moreover di!erent variations of the modal control are investigated,
Actuator location
Accelerometers Front/rear
S1 S2 S3
Front/rear Centre Front/rear Centre Front/rear Centre
( ( (
Centre bounce
Centre pitch
( (
(
(
with two and three actuators and with di!erent combinations of sensors. For two actuators, the accelerometers (i.e. central bounce and pitch) are used in conjunction with the "lter in Eq. (8) to derive the control signals for the front and rear actuators (controller C1). For controller C2, notch "lters tuned to the relevant #exible frequencies have been included (the symmetric mode in the bounce loop, the asymmetric mode in pitch). Three control strategies (S1-3) for controlling three actuators are also investigated. Table 1 gives the location for the sensors. It is worthwhile to note that all the integrators (INT) include high pass "lters as described in Section 4.1, except for the integrators marked as INT* because these are associated with a bandpass "lter. For S1 (see Fig. 8) the central bounce and pitch accelerometers are used. Notch "lters are included in the bounce and pitch controllers (as C2), and the central actuator is controlled using skyhook damping through a bandpass "lter which has its centre frequency set to correspond to the main bending frequency. In this way,
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the central actuator is concentrated upon dealing with the resonance. For S2 (see Fig. 9) front and rear accelerometers are used. The bounce and pitch controllers have no notch "lters and the central actuator uses the bounce acceleration generated from the front and rear accelerations.
For S3 (see Fig. 10) three accelerometers at the front, centre and rear are used. The end actuators are controlled using modal signals just like S2, but the central actuator is controlled using the signal from the central accelerometer because this is much more e!ective for measuring the main structural resonance.
5. Results and discussion The suspension design problem is concerned with minimising the acceleration experienced by passengers without causing large suspension de#ections. For an active suspension, the dynamics of the actuator and its force requirement are also important, and so the assessments have included all three issues. Before any comparisons can be made the ride performances of the passive system are obtained (see Table 2). The table shows that including the #exibility generally worsens the ride quality, particularly at the centre of the vehicle. The results tabulated in Tables 3 and 4 show the r.m.s acceleration levels using ideal actuators in %g and the r.m.s force and suspension displacement values at the front, centre and the rear of the vehicle. They indicate the ride performance results, which are the optimum values found by varying the di!erent parameters (i.e. bandwidth of the bandpass "lter, gains of the pitch, bounce and the centre controllers).
Fig. 8. S1: controller structure.
Fig. 9. S2: controller structure.
5.1. Ideal actuators 5.1.1. Two actuator control Table 3 summarises the active results. Overall pure skyhook (Psky) gives the best results, but this is a theoretical value which cannot be achieved in practice.
Fig. 10. S3: controller structure.
Table 2 RMS acceleration and suspension de#ection for a passive system
Passive rigid Passive #exible
Front acc (%g)
Cent acc (%g)
Rear acc (%g)
Front suspension De#ection (mm)
Rear suspension De#ection (mm)
3.4 3.04
1.65 3.04
2.68 3.75
7.8 7.8
11 11.35
Table 3 Results with two actuators Strategy
Front acc (%g)
Cent acc (%g)
Rear acc (%g)
Front suspension De#ection (mm)
Rear suspension De#ection (mm)
Front act Force (kN)
Rear act Force (kN)
Psky C1 C2
2.24 2.46 2.11
2.35 3.03 2.89
2.33 2.55 2.34
7.79 9.87 9.98
7.96 6.17 5.53
1.93 1.85 2.96
2.11 1.8 2.33
5.1.2. Three actuator control With a central actuator, the modal control strategy can be fully optimised, and Table 4 shows that further reductions in acceleration level can be achieved compared with two actuators. The results show that S3 gives the lowest ride accelerations, which is because this strategy uses the three accelerations, namely front, centre and the rear, to derive the control signals. This is the best possible solution, as the
3.2 0.67 1.28 9.66
Fig. 11. Two actuators: front acceleration psds.
1.9 1.33 1.84 S3
513
Controller C1 (without notch "lter) gives reasonable results for the front and rear, but the acceleration level at the centre is not so good. For controller C2, notch "lters are added to avoid exciting the #exible modes, and overall the results come quite close to Psky, but with higher actuator forces. Figs. 11}13 give power spectral densities (psds) for the front, centre and rear accelerations, respectively. They show a signi"cant reduction of the accelerations in the bounce frequency region of 1 Hz, but the #exibility e!ect just below 10 Hz is still very obvious.
10.29
2.95
2.33
2.33
0.8
0.16 2.97
2.95 5.59
5.82
1.65
4.46
10.05
10.21
1.78
1.96
2.04
1.7
2.53
1.83
S2
S1
Centre suspension De#ection (mm) Cent acc (%g) Front acc (%g) Strategy
Table 4 Result with three actuators
Rear acc (%g)
Front suspension De#ection (mm)
Rear suspension De#ection (mm)
Front actuator Force (kN)
Centre actuator Force (kN)
Rear actor Force (kN)
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Fig. 12. Two actuators: centre acceleration psds.
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Fig. 14. Three actuators: front accelerations psds. Fig. 13. Two actuators: rear acceleration psds.
three sensors that measure the three accelerations will de"nitely pick up the #exibility e!ect associated with the three controlling points (front, centre and rear). Strategy S2, which uses measurements from front and rear accelerometers, is not so good for the centre of the vehicle because they are less e!ective at detecting the main resonance, and for this reason the strategy requires a lower central actuator force. Note that the force and displacement required for the centre actuator are quite small, which makes practicable implementation of the third actuator feasible. The psds for the front, centre and rear accelerations are plotted in Figs. 14}16, respectively. In these "gures, the accelerations for the passive (#exible), S1, S2 and S3 are plotted. They show that the #exibility frequency at 8.43 Hz is greatly attenuated in all cases.
Fig. 15. Three actuators: centre accelerations psds.
5.2. Ewect of actuator dynamics Table 5 contains results with actuator dynamics for three actuator strategies only, with the ideal actuator results from Table 4 given in brackets, and shows the degradation in ride quality caused by the actuator dynamics. In addition there is an increase in the amount of force produced compared with the ideal, which generally correlates with the worse rail quality. Fig. 17 shows the e!ect upon the frequency responses of the three actuators' output for the front, centre and rear actuators. The y-axis is the ratio of the real actuator force output with the force generated for an ideal actuator, in both cases derived from the responses with respect to the track input. The "gure shows that the response is substantially equal to unity up to 1 Hz, meaning that the actuator is able to supply the correct amount of force at low frequencies. Thereafter, the response deviates from the ideal response as now the actuator is unable to produce the correct amount of force due to the e!ects of the actuator dynamics. The high deviations, at high
Fig. 16. Three actuators: rear acceleration psds.
frequency, of the front and rear hydraulic actuators are caused by the coupling of the bogie excitation to the vehicle body as the actuator now is unable to respond fast enough due to the limited bandwidth of the hydraulic actuator, and in general produces more force than is required. It is worth noting that the e!ects are a consequence of the track input as a disturbance into the loop * the closed-loop bandwidth of the force controllers
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Table 5 Results of three actuators with actuators dynamics Strategy
Front acc (%g)
Cent acc (%g)
Rear acc (%g)
Front act Force (kN)
Centre act Force (kN)
Rear actuator Force (kN)
S1 S2 S3
2.15(1.83) 2.33(1.96) 2.14(1.84)
1.96(1.7) 3.05(2.53) 1.54(1.33)
2.09(1.78) 2.45(2.04) 2.22(1.9)
2.91(2.95) 2.93(2.97) 2.97(2.95)
0.97(0.8) 0.2(0.16) 0.82(0.67)
2.5(2.3) 2.29(2.33) 3.22(3.2)
Fig. 17. Ratio of front, centre and rear real force/ideal force.
Fig. 19. Ratio of front, centre and rear real force/ideal force.
Fig. 18. Front acceleration psds for strategy S3.
Fig. 20. Front acceleration psds for strategy S3.
without track inputs would be signi"cantly higher than these "gures indicate. Notice how much better the electromagnetic actuator is at high frequency. The acceleration psds for ideal and real actuators are only given for strategy S3 in Figs. 18}20, because the rest of the strategies show the same trend. These "gures con"rm that the ride degradation due to the inclusion of the dynamics of the actuators occurs from 0 to 25 Hz (where the main #exible frequencies lie), and there is very little to be seen of the "rst asymmetric mode around 23 Hz.
5.3. Robustness of the controller It is inevitable that the physical parameters will change. As a result a robustness check was carried out using models that include the dynamics of the actuators. The change in the #exibility frequency is considered an important issue and therefore has to be investigated, although the mode shape has been kept constant. The range of the #exibility frequency considered is$5% variation from the "rst symmetrical and asymmetrical modes of 8.43 and 23.23 Hz.
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Numerous runs were carried out and the results obtained show that for strategy S1 and S3 (Figs. 21 and 22) the variation in the ride results obtained is around 3%, compared to 10% for strategy S2. The results for S2, which is the most sensitive to changes in the #exibility frequency, are given in Figs. 23 and 24. The "rst plots the change in acceleration as the frequency changes by $5%, while the second shows the changes in the centre actuator force. The di!erences in sensitivity arises because, for both S1 and S3, the controller has some information about the centre acceleration, whereas in the case of strategy S2, only two sensors are considered, one at the front and rear of the vehicle. One interesting result that emerges from this investigation is that it seems that a slight mis-tuning of the controller gives noticeably better results. With the #exibility frequency of the vehicle
model set at 0.5 Hz lower than the nominal frequency of 8.43 Hz for symmetrical and 23.23 Hz for the asymmetrical, Fig. 23 shows reduced acceleration levels, and this is achieved with a lower actuator force (Fig. 24). On the whole strategy S3 perform the best in terms of the ride results and the robustness check.
Fig. 21. Centre acceleration psds for strategy S3.
Fig. 22. Rear acceleration psds for strategy S3.
6. Conclusion In this paper classical active suspension controllers for railway vehicles with #exible bodies have been investigated and an overall summary of the results is presented in Table 6. The addition of a central actuator enables the #exibility e!ect to be e!ectively reduced. With the centre actuator, the percentage improvement for the ride
Fig. 23. Variation of ride acceleration with frequency variation for strategy S2.
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Fig. 24. Variation of centre actuator force with frequency variation for strategy S2.
Table 6 Percentage improvement in ride quality compared with the passive System Strategy
C1 C2 S1 S2 S3
Front
Centre
Rear
Ideal (%)
Real act (%)
Ideal (%)
Real act (%)
Ideal (%)
Real act (%)
19.08 30.59 39.80 35.53 39.47
9.87 16.78 29.28 23.03 29.61
0.33 4.93 44.08 16.78 56.25
!3.62 !14.14 35.53 0.00 49.34
19.2 22.93 54.67 32.53 64.53
16 7.47 47.73 18.93 58.93
compared with a passive suspension averages 40%, and for the best strategy an improvement of around 60% is attained in some parts of the vehicle. Force levels for the central actuator have proved to be quite modest. However, when the dynamics of the actuators are considered, the improvements are reduced. For example, in the case of strategy S2 the centre acceleration has no improvement at all. However strategy S3 still manages to o!er reasonable improvements even with real actuators. The results emphasise the importance of considering the dynamics of real actuators even during the theoretical investigation stage, otherwise the controllers might not perform as expected. In addition the robustness check carried out found that S3 performed the best in comparison with the other two strategies. The research is continuing to assess the use of optimal control strategies together with the actuator dynamics,
and a "nal stage of experimental evaluation using a laboratory rig is planned.
Appendix A. Train parameters Mass of the vehicle Mass of the vehicle, M , Mass of the bogie, M , Body pitch Inertia, I ,
38000 kg 2500 kg 2.31;10 kg m
Primary components Primary spring sti!ness per axle, K , 4.935 MN/m Primary damping per axle, B , 0.05074 MNs/m
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Secondary components
PID controller (servo-hydraulic)
Secondary spring sti!ness per bogie, K , Secondary reservoir sti!ness per bogie, K , Secondary damping per bogie, B ,
1.016 MN/m 1.175;10\ 0.508 MN/m 0.06411 MNs/m References
Centre actuator parameters Mass for the centre actuator, to react, M , Centre spring, K , Centre damper, B ,
1000 kg 39000 N/m 628.32 Ns/m
Appendix B. Actuator parameters Electromagnetic actuator parameters Pole area, A , No. of turns, N, Nominal airgap, G , Nominal #ux density, B , Nominal current, I , Nominal force, F , Coil resistance, R,
62.8 cm 1432 15;10\ m 0.6 T 10 A 1000 N 2.66 )
PI controller parameters (electromagnetic) 2.05
(1#1.25;10\s) (1#0.011s) A/N (1#1.86;10\s) (1#1.1s)
(1#0.002s)
Hydraulic actuator's parameters Frequency of the servo-valve, u , Damping of the servo-valve, f, Area of the actuator spool, area, Flow/pressure constant, K , Servo-valve #ow gain, K , Oil column sti!ness, K , Oil column damping, B ,
628.32 rad/s 0.4 7.26;10\ m 20.61;10\ Ams\ 4.74;10\ P ms\ 90.95;10 N/m 471.24 N/m
Bishop, R. E. D. (1960). The mechanics of vibration. Cambridge, UK: University Press. Goodall, R. M. (1997). Active railway suspensions: Implementation status and technological trends. Vehicle System Dynamics, 28(2-3), 87}117. Goodall, R. M., Pearson J. T., & Pratt I., (1993). Actuator technologies for secondary active suspensions on railway vehicles. Proceedings of International Conference of Speedup Technology for Railway and Maglev Vehicles, vol 2, (pp. 377}382). Goodall, R. M., Whit"eld, A. H., (1985). Automation of wood-tuning machine, hydraulic or electric, Proceedings of Institute of Mechanical Engineers Conference, C269/85, 59}62. Hac, A. (1986). Stochastic optimal control of vehicles with elastic body and active suspension. Journal of Dynamics Systems, Measurement and Control, 108, 106}110. Hac, A., & Liu, L. (1992). Sensor and actuator locations in motion control of #exible structures. Journal of Sound and Vibration, 167(2), 239}261. Li, H., & Goodall, R. M. (1997). Practical implementations of skyhook damping for active railway suspensions. 12th International Conference on System Engineering ICSE'97 (pp. 425}430). Li, H., & Goodall, R. M. (1999). Linear and non-linear skyhook damping control laws for active railway suspensions. Control Engineering Practice, 7, 843}850. Pollard, M. G. (1983). Active suspensions enhance ride quality, Railway Gazette International, 138, 850}853. Pratt, I., (1996). Active suspensions applied to railway trains. Loughborough University Ph.D. thesis. Pratt, I., & Goodall, R. M., (1994). Control strategies to improve ride quality in railway trains, Proceedings of Control'94 Conference, vol. 389 (pp. 350}355). Richard, J. (1980). Natural frequencies of Bernoulli}Euler beam resting on two elastic supports: Application to railway vehicles. Vehicle System Dynamics, 9, 309}326. Tang, J. S. (1996). Passive and semi-active airspring suspensions for rail passenger vehicles * theory and practice. Journal of Rail and Rapid Transit Part F, 210, 103-117. Thayer, W. J. (1958). Transfer functions for moog servovalves. In Moog Technical Bulletin 103. East Auro NY: Moog Inc. Williams, (1994). Control of low frequency active suspension. Proceedings of Control 94 Conference (pp. 338-342).
Active suspension control of #exible-bodied railway vehicles using electro-hydraulic and electro-magnetic actuators E. Foo*, R. M. Goodall Department of Electronic and Electrical Engineering, Loughborough University, Ashby Road, Loughborough, Leicestershire, LE11 3TU, UK Received 19 November 1998; received in revised form 6 October 1999
Abstract Lighter railway vehicles are going to become the norm in the future as the operator pushes towards a more economical operation. This paper looks into ways of applying classical control methods using `skyhook dampinga to minimise the #exibility e!ects that arise as a consequence of having lighter vehicles, and in particular includes the e!ects of real actuator dynamics. Two and three actuators schemes are considered: two of them are located at the front and rear secondary suspension pivot points, with a third actuator at the centre of the vehicle primarily targeted at reducing the e!ects of the main bending mode of the vehicle body. Hydraulic actuators are used at the front and rear while an electro-magnetic actuator is connected to the centre of the vehicle. 2000 Elsevier Science Ltd. All rights reserved. Keywords: Active vehicle suspension; Flexible; Railways; Actuators
1. Introduction Actively controlled suspensions have been widely studied theoretically and experimentally for automobiles and rail vehicles. These studies have demonstrated significant performance bene"ts. For railway vehicles, active suspensions that deal primarily with improvement in ride quality are now starting to be incorporated on a regular basis (Goodall, 1997). Furthermore railway engineers are also interested in using active control to achieve a functionality that is not possible with a purely passive (mechanical) suspension. The present railway operators are facing intense pressure due to competition from the air and road transport industries. In order for them to operate more e$ciently and e!ectively, they are investigating ways to reduce the travelling time, increase the number of passengers and reduce operating costs. The trend of faster and lighter vehicles are ways in which these objectives can be met. Therefore trains of the future will be designed with lighter bodies and bogies to enable them to operate at a much * Corresponding author. Tel.: #44-1509-228105; fax: #44-1509222854. E-mail addresses: [email protected] (E. Foo), r.m.goodall@ lboro.ac.uk (R. M. Goodall).
higher speed. At this moment, the highest speed for an in-service operation train is 300 km/h, which is achieved by TGV in France and Shinkansen in Japan, but as the speed of the train is increased above this level it will lead to more high-frequency vibrations which will certainly a!ect the ride quality and generate unacceptable levels of internal noise. Present vehicles are designed to have the body much heavier than the bogie, which prevents the bogie excitation frequencies from coinciding with the body excitation frequencies. However with lighter vehicles the #exible frequencies of the body will de"nitely coincide with the bogie excitation frequency, and this will cause an increase in the ride accelerations, and result in the reduction in the ride comfort (ride quality). An important application of active control is therefore for #exible railway vehicle bodies, not only to provide a better ride quality in general, but also to enable lighter bodies with lower #exibility frequencies to be taken advantage of. This paper reports the "rst and second phase of a research project. The "rst phase involves studying active suspension strategies speci"cally for more #exible vehicle bodies. Established `classical controla techniques based upon the idea of skyhook damping are extended to accommodate body #exibility, and the idea of an extra central actuator connected to an auxiliary mass is also included.
0967-0661/00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 7 - 0 6 6 1 ( 9 9 ) 0 0 1 8 8 - 4
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Presently the majority of the published work on active suspensions system assumes that ideal actuators are used, i.e. that the actuator is able to supply whatever amount of force is required by the controller, over an in"nite bandwidth and with no delay. Clearly this is not realisable in practice, as any real actuator will have a "nite bandwidth. The introduction of the actuator dynamics will certainly degrade the ride performance, when compared with ideal actuators. As a result the second phase of the research project involves modelling and applying realactuator dynamics to an existing controller designed in the "rst phase using ideal actuators only, so as to investigate the impact of the actuator dynamics on the ride quality.
Frequency analysis method is used to calculate the raw accelerations. The formula used is as follows: U "("H(u)"G J (u) du, (2) WW X where
is the r.m.s of the output power spectrum WW and H(u) is the frequency response of the vehicle dynamics. H(u) must take account of the time delay between the track inputs to the front and rear wheelsets, and Eq. (3) gives this relation where ¸ /v is the time delay between the two wheelsets. H (u) and H (u) are the transfer * 0 functions in response to inputs at the front and rear, respectively. H(u)"H (u)#H (u)e\HS* T. * 0
(3)
2. Method of ride assessment In order to determine how good the ride is, it is necessary to "nd ways of quantifying the ride quality. This section describes the random track inputs that create vibrations of the vehicle body, and explains how body accelerations calculated by frequency domain analysis can be used to assess ride quality. 2.1. Track proxle The random irregularities of a railway track can be represented by a spatial power spectrum, usually approximated as a fourth-order equation (Pollard, 1983). However for secondary-suspension assessment, the higher-order terms do not have any signi"cant e!ect above 10 Hz or so, and can therefore be neglected. It can be shown that a good approximation for the track vertical velocity input is Gaussian white noise with a #at spectrum (Pratt, 1996) given by G "(2p)A v(m/s)/Hz. (1) X The track velocity, v, is taken as 55 m/s and the roughness factor for a good quality mainline track, A , is taken as 2.5;10\ m (Pratt & Goodall, 1994). 2.2. Quantifying ride quality The standard method of quantifying ride quality is based upon calculating or measuring the root mean square (r.m.s) acceleration levels experienced on the vehicle body, normally measured above the front and rear bogies and at the centre of the vehicle. For absolute measurements it is preferable to include a frequency weighting before the r.m.s is calculated to allow for human sensitivity to vertical vibration. However, for comparative studies in which relative levels of ride quality are being assessed, the frequency weighting can be neglected. This is the approach adopted here, although the inclusion of frequency weighting is straightforward if required.
3. Modelling 3.1. Modelling of vehicle body with yexibility The vehicle is modelled with the rigid and #exible modes. The rigid body mode is implemented using normal Newtonian mechanics, while #exibility is modelled as a free}free beam using receptance principles (Richard, 1980). The sideview model of the train is given in Fig. 1. It is a three-mass model consisting of the body of the vehicle and the two bogies. The system has been simpli"ed to include only two wheelsets instead of four. This is because the time delay between the inputs to the two wheelsets are considered to be negligible. The bogie masses only have a vertical degree-of-freedom, whereas the body modelling includes both vertical (bounce) and pitch motions. The suspension between the bogie and body masses is representative of a conventional airspring with an auxiliary reservoir, in which damping is introduced by means of a suitable restriction (Tang, 1996). The parameter values given in Appendix A are typical for a modern inter-city vehicle. The #exible deformations (Hac, 1986) can be described by z(x, t), where t is the time and x is the distance along the beam. This can be represented by its mode shape and the principal co-ordinate modes, (x) and o (t), respecP P tively, in Eq. (4), where r is the mode number. z(x, t)" o (t) (x). (4) P P P Using the receptance functions de"ned in (Bishop, 1960), the following equation is obtained. M oK (t)#2mu o (t)#uo (t)"F (x )#F (x ), J P P P P P 0 0 0 * * * (5) where M is the mass of vehicle, m is the damping coe$c ient, u is the r mode frequency in rad/s. P
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509
Fig. 1. Simpli"ed side-view linearised model of a railway vehicle with centre mass and controllers.
The description which follows deals with two #exible modes only, but more can be incorporated into the model by adding more degrees-of-freedom to the #exibility equations. Eq. (5) can be changed into Eq. (6) with the symmetrical and asymmetrical modes separated. The former (excited by bounce inputs) has (x )" (x ), while the Q * Q 0 latter (excited by pitch inputs) has (x )"! (x ). ? * ? 0 !(F #F ) (x ) * 0 * , oK #2mu o #o u" P Aol !(F #F ) (x ) * 0 * , oK #2mu o #o u" Aol
3.2. Actuators (6)
where A, o and l are the cross section, density and length of the beam. Note that the subscript r in Eq. (5) has been replaced by subscripts s and a to represent the symmetrical and asymmetrical modes, respectively, in Eq. (6). Both modes are modelled with 5% damping, and frequencies of 8.43 and 23.23 Hz for the symmetrical and asymmetrical modes, respectively, are given, which are typical values for a modern railway vehicle. The matrices with rigid and #exible body modes can be combined as
XQ XQ
"
A
A } A A }
#
X X
B G [u]# [m] B 0
with A } and A } denoting the coupling of the #exible into the rigid body mode and the rigid into the #exible mode, respectively, u the control force and f is the track pro"le. The total order of the model (excluding the actuators dynamics) is therefore 14 and made up as follows: Eight body positions and velocity states (four rigid and four #exible). Four bogie position and velocity states. Two extra suspension states for the airspring model.
(7)
There are various rigorous ways of "nding the placement of actuators (Hac & Liu, 1992) but in this paper a third actuator is added at the centre of the vehicle to help suppress the "rst symmetrical #exible mode, which causes the biggest degradation on the ride performance. A mass is also included at the centre of the vehicle to provide an anchor point for the central actuator, which in practice will be an item of auxiliary equipment. A mass of 1 t supported by a spring to give a 1 Hz natural frequency has been assumed, although this will obviously be modi"ed by the control. A damper giving 5% damping is included to prevent oscillations when there is no actuator force input. Two types of actuators have been considered, the "rst one is servo-hydraulic, the other one is electromagnetic. The servo-hydraulic actuators are located across the secondary suspension at the front and rear of the vehicle, while the electromagnetic actuator is used in the centre location to control the #exible body modes. The servohydraulic actuator is used because there is a limited space
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Fig. 2. Model of a servo-hydraulic actuator.
Fig. 3. Servo-hydraulic actuator force control loop.
between the secondary suspension and the body. In principle an electromagnetic actuator could be used, but the size of the actuator would be large and would be di$cult to "t within the space allowable across the secondary suspension. For the centre actuator however, the electromagnetic actuator technology most suited as the force requirement for this actuator is around 1 kN r.m.s (Section 5.1.2), and with this force level the construction of the electro-magnetic actuator will be small enough to be "tted underneath the vehicle. Also the high-bandwidth capability of such actuators makes them very appropriate for controlling the vibration in #exible modes. Each of these actuators is force-controlled, using inner force-feedback loops carefully designed to achieve the required performance. Fig. 2 shows a typical model of a servo-hydraulic actuator (Thayer, 1958 and Goodall & Whit"eld, 1985). The inner-loop for the hydraulic actuators is shown in Fig. 3. A linear model is used here as the investigation is concentrated on the interaction between active suspension controllers and #exible modes of the body. The parameters of the PID controller are chosen to achieve a good performance. Fig. 4 shows the model used to represent the dynamics of the central electromagnetic actuator. Its inner loop is shown in Fig. 5, and again a linear model is used in this study (Goodall, Pearson & Pratt, 1993). The e!ect of air gap variations makes an electro-magnetic actuator on its own an unstable system, although "tting this in parallel with the auxiliary mass's own suspension will tend to
Fig. 4. Model of electromagnetic actuator.
Fig. 5. Electro-magnetic actuator force control loop.
overcome the stability. However a properly designed force feedback loop overcomes any residual instability, but it is important to model the characteristics properly to choose appropriate values for the PI controller and to give correct predictions of performance.
4. Suspension control A great variety of suspension control strategies are possible, but this paper concentrates upon the use of the so-called `skyhook dampinga control which is known to give excellent improvements in ride quality. The following section explain the principle and shows how it can be extended using modal control.
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511
4.1. Skyhook control Skyhook damping (Li & Goodall, 1997) provides a force dependent upon the absolute velocity of the vehicle body, instead of a force dependent upon the relative velocity given by a conventional damper. Of course, in practice absolute velocity is di$cult to measure, and so the velocity required by the skyhook damper is usually obtained by integrating the signal from an accelerometer. However a pure integrator is not practical, and as a result a second-order Butterworth highpass "lter (HPF) with a frequency of 0.1 Hz and a 70% damping is chosen because it will give a maximally #at frequency response at the low-frequency range. In fact this "lter's characteristics is also an important consideration with regard to the response to deterministic track inputs (Li & Goodall, 1999), but this paper deals with random track inputs only. The transfer function that is used to give a velocity estimate from the acceleration measurement is as follows: z( "
2.533s z( . 1#2.228s#2.533s
(8)
Fig. 6. Modal control structure.
Fig. 7. Example of a modal skyhook controller.
Table 1 Three actuators: location of sensors Strategy
In addition, controllers which use pure skyhook damping have been used as a starting point for the development of the controller. These assume that absolute velocity can be directly measured, and in the results section (Section 5.1.1) these are referred to as `Pskya controllers. 4.2. Modal control In principle, modal control (Williams, 1994) attempts to manage individual system modes. This is achieved by decomposing measurements into modal components. These can then be processed individually and recombined to drive the actuators. Fig. 5 shows the basic structure of a modal controller. In practice, linear accelerometers are normally located above the secondary suspensions at the front and rear of the vehicle, although the other option is to use one linear and one rotational accelerometer located at the vehicle's centre, but of course these will sense the #exibility e!ects di!erently. Using the relationship between the bounce/pitch accelerations and the front/rear accelerations, modal control can be applied to control the bounce and pitch separately. Fig. 6 illustrates a particular case where the input signals are the front and rear accelerations and the outputs are the front, centre and rear actuator forces demands for the actuator force-control loops. By changing the gain factors for the bounce, pitch and centre (G , G and G , respectively), di!erent amounts of skyhook damping can be applied to control the bounce and pitch accelerations (Fig. 7). Moreover di!erent variations of the modal control are investigated,
Actuator location
Accelerometers Front/rear
S1 S2 S3
Front/rear Centre Front/rear Centre Front/rear Centre
( ( (
Centre bounce
Centre pitch
( (
(
(
with two and three actuators and with di!erent combinations of sensors. For two actuators, the accelerometers (i.e. central bounce and pitch) are used in conjunction with the "lter in Eq. (8) to derive the control signals for the front and rear actuators (controller C1). For controller C2, notch "lters tuned to the relevant #exible frequencies have been included (the symmetric mode in the bounce loop, the asymmetric mode in pitch). Three control strategies (S1-3) for controlling three actuators are also investigated. Table 1 gives the location for the sensors. It is worthwhile to note that all the integrators (INT) include high pass "lters as described in Section 4.1, except for the integrators marked as INT* because these are associated with a bandpass "lter. For S1 (see Fig. 8) the central bounce and pitch accelerometers are used. Notch "lters are included in the bounce and pitch controllers (as C2), and the central actuator is controlled using skyhook damping through a bandpass "lter which has its centre frequency set to correspond to the main bending frequency. In this way,
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the central actuator is concentrated upon dealing with the resonance. For S2 (see Fig. 9) front and rear accelerometers are used. The bounce and pitch controllers have no notch "lters and the central actuator uses the bounce acceleration generated from the front and rear accelerations.
For S3 (see Fig. 10) three accelerometers at the front, centre and rear are used. The end actuators are controlled using modal signals just like S2, but the central actuator is controlled using the signal from the central accelerometer because this is much more e!ective for measuring the main structural resonance.
5. Results and discussion The suspension design problem is concerned with minimising the acceleration experienced by passengers without causing large suspension de#ections. For an active suspension, the dynamics of the actuator and its force requirement are also important, and so the assessments have included all three issues. Before any comparisons can be made the ride performances of the passive system are obtained (see Table 2). The table shows that including the #exibility generally worsens the ride quality, particularly at the centre of the vehicle. The results tabulated in Tables 3 and 4 show the r.m.s acceleration levels using ideal actuators in %g and the r.m.s force and suspension displacement values at the front, centre and the rear of the vehicle. They indicate the ride performance results, which are the optimum values found by varying the di!erent parameters (i.e. bandwidth of the bandpass "lter, gains of the pitch, bounce and the centre controllers).
Fig. 8. S1: controller structure.
Fig. 9. S2: controller structure.
5.1. Ideal actuators 5.1.1. Two actuator control Table 3 summarises the active results. Overall pure skyhook (Psky) gives the best results, but this is a theoretical value which cannot be achieved in practice.
Fig. 10. S3: controller structure.
Table 2 RMS acceleration and suspension de#ection for a passive system
Passive rigid Passive #exible
Front acc (%g)
Cent acc (%g)
Rear acc (%g)
Front suspension De#ection (mm)
Rear suspension De#ection (mm)
3.4 3.04
1.65 3.04
2.68 3.75
7.8 7.8
11 11.35
Table 3 Results with two actuators Strategy
Front acc (%g)
Cent acc (%g)
Rear acc (%g)
Front suspension De#ection (mm)
Rear suspension De#ection (mm)
Front act Force (kN)
Rear act Force (kN)
Psky C1 C2
2.24 2.46 2.11
2.35 3.03 2.89
2.33 2.55 2.34
7.79 9.87 9.98
7.96 6.17 5.53
1.93 1.85 2.96
2.11 1.8 2.33
5.1.2. Three actuator control With a central actuator, the modal control strategy can be fully optimised, and Table 4 shows that further reductions in acceleration level can be achieved compared with two actuators. The results show that S3 gives the lowest ride accelerations, which is because this strategy uses the three accelerations, namely front, centre and the rear, to derive the control signals. This is the best possible solution, as the
3.2 0.67 1.28 9.66
Fig. 11. Two actuators: front acceleration psds.
1.9 1.33 1.84 S3
513
Controller C1 (without notch "lter) gives reasonable results for the front and rear, but the acceleration level at the centre is not so good. For controller C2, notch "lters are added to avoid exciting the #exible modes, and overall the results come quite close to Psky, but with higher actuator forces. Figs. 11}13 give power spectral densities (psds) for the front, centre and rear accelerations, respectively. They show a signi"cant reduction of the accelerations in the bounce frequency region of 1 Hz, but the #exibility e!ect just below 10 Hz is still very obvious.
10.29
2.95
2.33
2.33
0.8
0.16 2.97
2.95 5.59
5.82
1.65
4.46
10.05
10.21
1.78
1.96
2.04
1.7
2.53
1.83
S2
S1
Centre suspension De#ection (mm) Cent acc (%g) Front acc (%g) Strategy
Table 4 Result with three actuators
Rear acc (%g)
Front suspension De#ection (mm)
Rear suspension De#ection (mm)
Front actuator Force (kN)
Centre actuator Force (kN)
Rear actor Force (kN)
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Fig. 12. Two actuators: centre acceleration psds.
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Fig. 14. Three actuators: front accelerations psds. Fig. 13. Two actuators: rear acceleration psds.
three sensors that measure the three accelerations will de"nitely pick up the #exibility e!ect associated with the three controlling points (front, centre and rear). Strategy S2, which uses measurements from front and rear accelerometers, is not so good for the centre of the vehicle because they are less e!ective at detecting the main resonance, and for this reason the strategy requires a lower central actuator force. Note that the force and displacement required for the centre actuator are quite small, which makes practicable implementation of the third actuator feasible. The psds for the front, centre and rear accelerations are plotted in Figs. 14}16, respectively. In these "gures, the accelerations for the passive (#exible), S1, S2 and S3 are plotted. They show that the #exibility frequency at 8.43 Hz is greatly attenuated in all cases.
Fig. 15. Three actuators: centre accelerations psds.
5.2. Ewect of actuator dynamics Table 5 contains results with actuator dynamics for three actuator strategies only, with the ideal actuator results from Table 4 given in brackets, and shows the degradation in ride quality caused by the actuator dynamics. In addition there is an increase in the amount of force produced compared with the ideal, which generally correlates with the worse rail quality. Fig. 17 shows the e!ect upon the frequency responses of the three actuators' output for the front, centre and rear actuators. The y-axis is the ratio of the real actuator force output with the force generated for an ideal actuator, in both cases derived from the responses with respect to the track input. The "gure shows that the response is substantially equal to unity up to 1 Hz, meaning that the actuator is able to supply the correct amount of force at low frequencies. Thereafter, the response deviates from the ideal response as now the actuator is unable to produce the correct amount of force due to the e!ects of the actuator dynamics. The high deviations, at high
Fig. 16. Three actuators: rear acceleration psds.
frequency, of the front and rear hydraulic actuators are caused by the coupling of the bogie excitation to the vehicle body as the actuator now is unable to respond fast enough due to the limited bandwidth of the hydraulic actuator, and in general produces more force than is required. It is worth noting that the e!ects are a consequence of the track input as a disturbance into the loop * the closed-loop bandwidth of the force controllers
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515
Table 5 Results of three actuators with actuators dynamics Strategy
Front acc (%g)
Cent acc (%g)
Rear acc (%g)
Front act Force (kN)
Centre act Force (kN)
Rear actuator Force (kN)
S1 S2 S3
2.15(1.83) 2.33(1.96) 2.14(1.84)
1.96(1.7) 3.05(2.53) 1.54(1.33)
2.09(1.78) 2.45(2.04) 2.22(1.9)
2.91(2.95) 2.93(2.97) 2.97(2.95)
0.97(0.8) 0.2(0.16) 0.82(0.67)
2.5(2.3) 2.29(2.33) 3.22(3.2)
Fig. 17. Ratio of front, centre and rear real force/ideal force.
Fig. 19. Ratio of front, centre and rear real force/ideal force.
Fig. 18. Front acceleration psds for strategy S3.
Fig. 20. Front acceleration psds for strategy S3.
without track inputs would be signi"cantly higher than these "gures indicate. Notice how much better the electromagnetic actuator is at high frequency. The acceleration psds for ideal and real actuators are only given for strategy S3 in Figs. 18}20, because the rest of the strategies show the same trend. These "gures con"rm that the ride degradation due to the inclusion of the dynamics of the actuators occurs from 0 to 25 Hz (where the main #exible frequencies lie), and there is very little to be seen of the "rst asymmetric mode around 23 Hz.
5.3. Robustness of the controller It is inevitable that the physical parameters will change. As a result a robustness check was carried out using models that include the dynamics of the actuators. The change in the #exibility frequency is considered an important issue and therefore has to be investigated, although the mode shape has been kept constant. The range of the #exibility frequency considered is$5% variation from the "rst symmetrical and asymmetrical modes of 8.43 and 23.23 Hz.
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Numerous runs were carried out and the results obtained show that for strategy S1 and S3 (Figs. 21 and 22) the variation in the ride results obtained is around 3%, compared to 10% for strategy S2. The results for S2, which is the most sensitive to changes in the #exibility frequency, are given in Figs. 23 and 24. The "rst plots the change in acceleration as the frequency changes by $5%, while the second shows the changes in the centre actuator force. The di!erences in sensitivity arises because, for both S1 and S3, the controller has some information about the centre acceleration, whereas in the case of strategy S2, only two sensors are considered, one at the front and rear of the vehicle. One interesting result that emerges from this investigation is that it seems that a slight mis-tuning of the controller gives noticeably better results. With the #exibility frequency of the vehicle
model set at 0.5 Hz lower than the nominal frequency of 8.43 Hz for symmetrical and 23.23 Hz for the asymmetrical, Fig. 23 shows reduced acceleration levels, and this is achieved with a lower actuator force (Fig. 24). On the whole strategy S3 perform the best in terms of the ride results and the robustness check.
Fig. 21. Centre acceleration psds for strategy S3.
Fig. 22. Rear acceleration psds for strategy S3.
6. Conclusion In this paper classical active suspension controllers for railway vehicles with #exible bodies have been investigated and an overall summary of the results is presented in Table 6. The addition of a central actuator enables the #exibility e!ect to be e!ectively reduced. With the centre actuator, the percentage improvement for the ride
Fig. 23. Variation of ride acceleration with frequency variation for strategy S2.
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Fig. 24. Variation of centre actuator force with frequency variation for strategy S2.
Table 6 Percentage improvement in ride quality compared with the passive System Strategy
C1 C2 S1 S2 S3
Front
Centre
Rear
Ideal (%)
Real act (%)
Ideal (%)
Real act (%)
Ideal (%)
Real act (%)
19.08 30.59 39.80 35.53 39.47
9.87 16.78 29.28 23.03 29.61
0.33 4.93 44.08 16.78 56.25
!3.62 !14.14 35.53 0.00 49.34
19.2 22.93 54.67 32.53 64.53
16 7.47 47.73 18.93 58.93
compared with a passive suspension averages 40%, and for the best strategy an improvement of around 60% is attained in some parts of the vehicle. Force levels for the central actuator have proved to be quite modest. However, when the dynamics of the actuators are considered, the improvements are reduced. For example, in the case of strategy S2 the centre acceleration has no improvement at all. However strategy S3 still manages to o!er reasonable improvements even with real actuators. The results emphasise the importance of considering the dynamics of real actuators even during the theoretical investigation stage, otherwise the controllers might not perform as expected. In addition the robustness check carried out found that S3 performed the best in comparison with the other two strategies. The research is continuing to assess the use of optimal control strategies together with the actuator dynamics,
and a "nal stage of experimental evaluation using a laboratory rig is planned.
Appendix A. Train parameters Mass of the vehicle Mass of the vehicle, M , Mass of the bogie, M , Body pitch Inertia, I ,
38000 kg 2500 kg 2.31;10 kg m
Primary components Primary spring sti!ness per axle, K , 4.935 MN/m Primary damping per axle, B , 0.05074 MNs/m
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Secondary components
PID controller (servo-hydraulic)
Secondary spring sti!ness per bogie, K , Secondary reservoir sti!ness per bogie, K , Secondary damping per bogie, B ,
1.016 MN/m 1.175;10\ 0.508 MN/m 0.06411 MNs/m References
Centre actuator parameters Mass for the centre actuator, to react, M , Centre spring, K , Centre damper, B ,
1000 kg 39000 N/m 628.32 Ns/m
Appendix B. Actuator parameters Electromagnetic actuator parameters Pole area, A , No. of turns, N, Nominal airgap, G , Nominal #ux density, B , Nominal current, I , Nominal force, F , Coil resistance, R,
62.8 cm 1432 15;10\ m 0.6 T 10 A 1000 N 2.66 )
PI controller parameters (electromagnetic) 2.05
(1#1.25;10\s) (1#0.011s) A/N (1#1.86;10\s) (1#1.1s)
(1#0.002s)
Hydraulic actuator's parameters Frequency of the servo-valve, u , Damping of the servo-valve, f, Area of the actuator spool, area, Flow/pressure constant, K , Servo-valve #ow gain, K , Oil column sti!ness, K , Oil column damping, B ,
628.32 rad/s 0.4 7.26;10\ m 20.61;10\ Ams\ 4.74;10\ P ms\ 90.95;10 N/m 471.24 N/m
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