Numerical Challenges Posed by Modeling Hydraulic
Systems
C.W. Richards
Société Imagine
42300 ROANNE,
France.
Tel. +33 4 77 23 60 37
Fax. +33 4 77 23 60 31
Email imagine@https://www.doczj.com/doc/0713556710.html,
Abstract
This paper describes the characteristics of models of hydraulic systems which make them particularly challenging for numerical integrators. Problems associated with numerical stiffness, high index differential algebraic equations, extreme non-linearities and discontinuities are described. The consequences of hydraulic sub-systems in multi-domain simulation are discussed. Finally a plea for a new generation of integrator is made.
1 Introduction
The term hydraulic system must be taken broadly to include areas such as
aircraft and rocket fuel systems
vehicle fuel injection systems
cooling systems
lubrication systems
hydraulic braking systems
hydraulic power steering systems
employing a wide variety of fluids from water to liquid hydrogen as well as classical hydraulic systems with regular 'hydraulic' oil.
The author is a numerical mathematician who has been specializing in simulating hydraulic and pneumatic systems since 1978. In this time he has encountered many numerical problems. Most of these were due to old fashioned bugs in both coding and in
the underlying model. However, there have been other persistent problems which have to be attributed to the intrinsic nature of hydraulic systems. What are these special characteristic of models of hydraulic systems from a numerical point of view? Briefly the governing equations are numerically stiff, very non-linear, can have high index differential algebraic equations and are often modeled with discontinuities.
It is interesting that during meetings for the Toolsys project [1], the author in conversation with designers of multi-body and electrical software has heard almost identical characteristics attributed to these domains. Does this mean that there is no significance difference between the numerical characteristics of these domains? In the following sections the author attempts to show that there is.
Problems related to pneumatic systems are also mentioned briefly.
2 Hydraulic systems are multi-domain
A hydraulic system in isolation is singularly useless. It is necessary to do something with the hydraulic power. In practice this means moving some mechanical system. In addition it is necessary to control the hydraulic system. As a consequence even the simplest hydraulic systems will normally contain multi-body and control elements as well as hydraulic elements.
In consequence the solver must to some extent have a multi-domain capability. However, the 'foreign' domain elements are normally relatively simple. In the AMESim hydraulic software produced by Imagine, much effort has been extended in tuning the integrator to hydraulic systems.
3 Fluid Properties
Fundamental to the modeling of hydraulic systems are the fluid properties density, bulk modulus and viscosity. The hydraulic fluid will almost always contain air. This may be dissolved and/or free in the form of bubbles. At higher pressures the air will eventually become dissolved. At low pressure bubbles will form leading to a phenomenon known as air release. The process is relative slow (compared with cavitation) and with many systems the fraction of the air that is free can be taken to be constant.
Cavitation occurs when the pressure in the hydraulic fluid reaches the saturated vapor pressure. This is a much faster process than air release. Cavitation is very important because when it occurs, extensive damage can be done. Previously it was enough for
simulation to predict that cavitation occurred. Now users ask that it be quantitatively correct --- a very demanding requirement. If cavitation is present but not at too serious a level, a company may have a strong price advantage over a competitor that has chosen to eliminate cavitation completely and a strong reliability advantage over another that has severe cavitation.
Figures 1 to 3 show fluid properties of diesel fuel between 0 and 1000 bar gauge pressure. The fractional air content is 0.1 % by volume. In fig. 2, the plot of bulk modulus, the presence of the air greatly modifies the value.
Figure 1 Density
Figure 2 Bulk modulus
Figure 3 Absolute viscosity
Figures 4 to 7 show the plots for the pressure range -1 to 0 bar gauge. The dramatic effect of cavitation is evident in all these graphs. In particular the zoomed view in fig. 7 shows how as the liquid changes to vapor the bulk modulus becomes almost zero. As the integrator negotiates this region a lot of step size reductions and Jacobian re-evaluations will be necessary.
Figure 4 Density
Figure 5 Bulk modulus
Figure 6 Absolute viscosity
Figure 7 Bulk modulus
Note that, since all normal hydraulic liquids are not chemically pure, the boiling of the liquid does not take place at a single pressure but over a range of pressures. However, this range can be very narrow giving very non-linear fluid properties. Without careful rewriting, integrators cannot deal with these conditions.
4 Implicit and explicit systems
We can normally express the governing equations for a hydraulic system either by a system of ordinary differential equations (ODEs)
),(y t f dt
dy =
or by a system of differential algebraic equations (DAEs) of the form
0,,(=dt
dy y t F
where t is time and y is the vector of state variables. For DAEs there is an important measure known as the index of nilpotency. In simple terms we can always convert DAEs to ODEs by differentiation and manipulation. The smallest number of differentiations required is the index of nilpotency. ODEs have index 0. Normally index 1 problems are easy to solve, index 2 problems more difficult and index 3 problems very difficult. This is because the Jacobian matrices employed to solve the systems of equations become progressively more badly conditioned as the index increases. With high index problems (certainly 3 and perhaps 2) it is not a good idea to employ very small integration steps.
As will be shown in the next section, hydraulic system governing equations can be very stiff. This means that during the initial transient behavior, very small integration steps must be used. If there is also a high index, there is a natural conflict and failure is possible.
At Imagine, where the AMESim fluid power software is produced, modelers take a pride in reducing the number of DAEs used to a minimum. Other AMESim users show no such restraint and the DAE solver DASSL is used almost all the time.
For a discussion of nilpotency and the DASSL integrator see reference [2].
5 Hydraulic systems as electrical circuits
It is very easy to see the analogy between hydraulic systems and electrical systems. Engineering courses often stress this point. It is not surprising, therefore, to see attempts to deal with modeling and simulation of the two domains in the same way. Pressure is the same as voltage and flow rate the same as current. However, problems arise very quickly with this approach.
When we join two hydraulic components we use a pipe or hose. The walls of this will expand with pressure (especially hoses) and the fluid has some compressibility (especially if it contains significant quantities of air). The most commonly used equation in hydraulic modeling is probably
V Q B dt dP ∑= 1
where the time derivative of pressure is expressed in terms of the effective bulk modulus B, the volume V and the sum of the inflow Q . This equation is often used to model the compressibility effects in the pipe. Since B for a hydraulic fluid is of the order 91017x Pa, a very small time constant is introduced and the equations are numerically very stiff.
In contrast we join electrical components by a wire. There is very little compressibility which is like saying B in equation 1 is infinity. Hence we use
∑=0i
2 This is a constraint equation whereas 1 is an ODE.
Figure 8 Electrical system in AMESim
Figure 8 shows an electrical system modeled using AMESim. (Some AMESim users do simulate electrical systems using AMESim.) By performing a linearization it is possible to compute the index of nilpotency which turns out to be 1.
The hydraulic equivalent of equation 2 is
∑=0
Q 3
This can be thought of as a pipe in which the effective bulk modulus is infinity or the volume is zero. This model is not without merit as using equation 1 with a very large value of B/V can give very slow simulation whereas switching to equation 3 speeds things up. Figure 9 shows an injection system in which there is a small volume, called the sac and indicated by V in the figure, next to the actual injection holes. It can be useful to model this using equation 3 resulting in faster simulation. The equations are index 2.
Figure 9
This model has been used internally at Imagine but has never been released as a standard AMESim model because it gets into big trouble when cavitation occurs. The problem is not just that it is index 2. Other index 2 problems are solved with contemptuous ease. The problem is that other factors are involved.
If we determine a voltage V such that such that equation 2 is satisfied, there are no special restrictions on the voltage. Before convergence is obtained on an integration step, V may gets some quite extreme values. As long as convergence is obtained in the end, this is not a problem.
In contrast if a pressure P is obtained using equation 3, there are fundamental restrictions on the values that P may have. We cannot have a pressure less that 0 absolute. Thus we have a combination of
an index 2 problem for which a very small step size is not a good idea;
severe non-linearities in the fluid properties due to cavitation tending to reduce the integration step;
restriction on the admissible value of pressure.
These in combination reduce the reliability of the model.
Pressure ultimately is not like voltage.
6 Causal and acausal modeling
There has been some interest in recent years in acausal modeling and acausal libraries of hydraulic libraries have been proposed. Unfortunately many hydraulic components are not acausal by nature. The relief valve and check valve are good examples. If all we know is that the flow rate through them is zero, we cannot compute the pressure drop across them.
A acausal model tries to get over this by assuming that these valves always leak [3]. However, real units do not suffer from this incontinent behavior and hence such modeling techniques are best avoided in hydraulic systems.
7 Get the physics right
One of the most common errors in modeling hydraulic system occurs in an orifice. All good text books on hydraulic systems give the flow rate Q through the orifice in terms of the pressure drop P ?, density ρ, cross-sectional area A and a flow coefficient flow C as
ρ?P C A Q flow 2.= 4
The numerical problem with this formula is that the graph of Q against P ? has an infinite slope at the origin. This shows up as problems when P ? crosses zero. If a linearization is made at the equilibrium point, very strange results are obtained.
It is often true with hydraulic systems that, if you get the physics wrong, you end up with numerical problems. The solution is simple. An orifice has linear behavior when P
? is small. By modifying the model good results can be obtained.
8 Discontinuities
It is very convenient to include discontinuities in models of hydraulic and associated components. Strictly speaking a discontinuity involves a jump change in the value of a state variable and we could call this a hard discontinuity. These discontinuities will normally kill an integrator unless very careful coding is employed.
An example of a hard discontinuity is in the model of a hydraulic jack in which, when the piston reaches the end of its stroke, the velocity comes instantaneously to rest. This implies a perfectly inelastic collision.
Models where there is a jump change in the first or second derivative of a state variable are also used in models of hydraulic components. These 'soft' discontinuities can also give problems for the integrator.
Most simulation software except multi-body software now have refined methods for handling discontinuities. For a description of running a mixed hydraulic multi-body system in a multi-body software see reference [4].
9 Pneumatic systems
In some respect modeling of pneumatic systems is easier than hydraulic systems since the governing equation are less stiff. The fluid properties are also less non-linear but the restrictions on pressures described in section 5 apply equally to pneumatic systems. In addition there are similar restrictions on temperature values.
It is interesting that there is an ISO 6358 standard on the modeling of pneumatic orifices. Unfortunately the model described introduces precisely the problems described in section 7.
10 Domain Specific Integrators
As mentioned in section 2 the integrator of AMESim has been extensively tuned for solving systems models of the domain for which it was designed. This is probably true of all electrical and multi-body software.
In work for the Toolsys project the author has imported and run multi-body systems within AMESim. These were developed in COMPAMM [5]. Very simple examples ran with no problems but bigger systems required much manual adjustment to run parameters. The problem was that the multi-body sub-system model contained an iterative procedure employing a Newton type method to solve iteratively non-linear equations. The incomplete convergence of these iterations was seen by the AMESim integrator as noise which lead to much reduction in the step size, re-evaluation of Jacobians and very slow simulation. Some re-tuning of the integrator lead to very fast runs. Unfortunately this new state of tune proved very unsuitable for regular hydraulic systems.
Experience running hydraulic systems created in AMESim within the Adams software [4] lead to a conflict in the needs of the two domains. The multi-body sub-system needed to avoid small integration steps whereas the hydraulic sub-system needed small step sizes. In addition all hard discontinuities had to be eliminated from the hydraulic sub-system model and in particular
valves which were normally 'stepped' open had to be 'ramped' open;
cavitation had to be either avoided completely or a much simpler model had to
be employed;
if actuator models employed hard discontinuities due to end-stop modeling
either great care had to be taken to ensure that there was no contact with end-
stops or a difficult form of end-stop modeling had to be employed;
sometimes hydraulic volumes had to be increased somewhat to 'de-stiffen' the
equations.
11 Conclusions
There are numerical difficulties peculiar to hydraulic systems and these impose a severe test on the solver employed in simulation software. These solver tend to be highly tuned to a specific domain and problems occur when a non-trial foreign system is imported. To
make multi-domain a routine activity, a new generation of numerical integrators need to be developed capable of coping efficiently with multi-body, hydraulic, pneumatic, electrical and electronic sub-systems.
Development of these new range of integrators is a real challenge. If this task is left to the software suppliers, a lot of state of the art numerical analysis may be lost. If it is done by specialist numerical analysts working in universities or research institutes, contact with real industrial problems may be lost. The author believes that the best solution is for specialist numerical analysts to work very close with the simulation software suppliers.
12 References
[1] Toolsys, Open Toolset For Mixed Simulation of Multi-domain Systems, BRITE-EURAM Project CEC Ref No: BRPR CT96 0303
[2] Brenan K.E., S.L. Campbell and L.R. Petzold, Numerical Solution of Initial-value Problems in Differential-Algebraic Equations, North Holland, 1989
[3] Beater P., Object-orientated Modeling and Simulation of Hydraulic Drives, Simulation News Europe, No. 22, March 1998
[4] Jansson A.J., M. Yahiaoui and C.W. Richards, Running Combined Multi-Body Hydraulic System Simulations within Adams, Proceedings of International Adams User Conference, Ann Arbor, July 1998
[5] COMP uter A nalysis of M achines and M echanisms, Centro de Estudios e Investigaciones Técnicas, San Sebastian, Spain.