Why Physical System Modeling Is Important to Industry: Bond Graph Models That Could Have Prevented Some Costly Mistakes
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Why Physical System Modeling Is Important to Industry: Bond Graph Models That
Could Have Prevented Some Costly Mistakes
Donald Margolis
Department of Mechanical and Aeronautical Engineering, University of California
Davis, CA 9561 6 USA(emai: dlmargolis@ucdavis.edu)
______________________________________________________________
Abstract: In order to develop new concepts into prototypes and ultimately into products, physical system
modeling is virtually a necessity. At the concept stage, low order models are needed to understand the
interactive dynamics of complex systems, and, as development proceeds into prototyping and manufacture,
more sophisticated models may be needed to size components, determine fatigue life, plus more.
Keywords: Dynamic systems, bond graphs, simulation, transfer functions, linear analysis
_______________________________________________________________________________________
The modeling discussed in this paper is specifically directed to the concept development of mechatronic systems. These
systems typically involve multiple energy domains where electro-mechanical, -pneumatic, and -hydraulic devices are involved.
Since the device or system is not well defined at this stage, the modeling must be handled by the inventors using physical
principles, and assembling the various pieces of the model into an overall dynamic model that can be simulated. Bond graphs
are particularly well suited for concept development of multi-energy domain systems. This is demonstrated here using several
examples where modeling at an early stage of development would have avoided some very costly mistakes.
1. INTRODUCTION
In this paper, several examples are presented where 2. TRUCK ISOLATION SYSTEM
production was reached for a particular product only to A company developed a vibration isolation system for use on
discover that the product did not work as expected. the back of heavy truck cabs. These cabs are pivoted with
Production was delayed, in some cases, for months, and the stiff mounts at the front, and the soft isolators are located on
costs were enormous. In some cases, once the physical the left and right side of the cab at the rear. The isolators
system was understood, a relatively simple fix could be consist of fluid filled displacers (piston/ cylinder devices)
employed. In other cases, the flaw was not addressable with attached by a fluid filled hose to an air charged accumulator.
a simple fix, and costly redesign was necessary. In all cases, The displacers are about 10 cm in diameter, the hose is about
physical system understanding at the concept stage would 2.5 cm in diameter, and the accumulators are 30 cm diameter
have avoided the problem. It is true that hindsight is perfect, spheres. A schematic of the system is shown in Fig. 1 along
and these examples are not to be misconstrued as criticisms with the system bond graph.
of any particular company or group. These examples point p (1)
out the necessity for modeling an overall system at the p f I f Ap vin m
m
concept level. Since real engineering systems involve many
complex components in several different energy domains,
The physical system modeling is completed when the
tools are needed that allow straightforward representation of
schematic of Figure 1 is drawn. Decisions have already been
complex systems in multiple energy domains and straight
made as to what dynamics are important and must be
forward assemblage of these components into overall,
included in the model and what dynamics can be neglected.
simulatable system models.
It has been decided that only vertical dynamics of the cab are
important, that the cab can be represented as a single point
Bond graphs are a concise pictorial representation of all types
mass, that the fluid inertia of the tube is important as well as
of engineering physical systems. With very few symbols, all
the fluid resistance of the valve, and that the volume
energetic systems can be represented. Once a bond graph has
compliance at the tube termination can be represented as a
been derived, state variables are automatically selected and
linear element.
formulation problems are exposed through assignment of
causality. Each system component can be derived separately
The company that proposed this isolation system went on to
and then the final overall bond graph can be assembled from
prototype, test, and commercialize this system. It was
the components. The end result is an analytical or
assessed by one customer as a superior isolation system and
computational model that contains all the interactive
purchased for inclusion in their product. However, when
dynamics of the overall, multi-energy domain system Bond
another customer tested the same product, their evaluation
Graph Compendium is a website that cites literally thousands
determined that the system was extremely poor and that ride
of technical papers that use bond graphs. Karnopp et al is a
was much better without the isolation system installed.
text that develops bond graph modeling from the elementary
Obviously, the developers of this product did not understand
concepts to the modeling of very complex mechatronic
the fundamental behavior of their system.
systems. Some available commercial software is listed in the
References.m vm m I
p m Ap
I f , Rf Ca 1 TF
Ap
Q
vin
S f vin
Figure 1 Physical schematic and bond graph for the truck
isolation system
When queried about the effects of "fluid inertia", the A2 p
company stated that this was considered and deemed not to me m fl (10)
be a problem because the cab weighs about 1136 Kg and the At 2
total weight of the fluid is only 3 Kg. This constituted a
where At is the tube area.
fundamental misconception that would not have been made
had a hydro-mechanical model of their system been Using parameters taken from the actual device, the frequency
formulated. response of this isolator is shown in Fig. 2. As can be seen,
there is a high frequency asymptote that would be totally
From the bond graph of Fig. 1, using procedures from unacceptable in a vibration isolation system. The asymptote
Karnopp et al we write directly from the bond graph, is equal to G which is defined in Eq. (5). If the equivalent
The state equations become, mass is small, then the high frequency behavior will be small,
q p (2) but if the equivalent mass is large, then the response can
p A a A R v m A p become almost directly coupled to the input at high
m
m p p f in p f
Ca
frequencies. This would feel terrible in an actual vehicle test.
pm (3)
q a Ap vin
m 3
Since this is modeled as a linear system, a transfer function
and frequency response can be analytically evaluated. If the 2.5
derivative of (1) is substituted into (2) and the transfer
function derived, the result is, 2
vm ( s 2 00 s 0 )
2 2
G
v m /v i
1.5
vin ( s 2 2 nn s n 2 ) (4)
1
where,
0.5
me
G m (5) 0
me 0 0.5 1 1.5 2 2.5 3 3.5 4
1 f / fn
m
Fig. 2 Frequency response for the hydro-mechanical truck
2 cab isolation system
A p
mCa From (10) we see that the equivalent mass is far different
0 2
(6)
from just the mass of fluid. For the system that tested poorly,
me
m the equivalent mass was about 256 times the fluid mass,
which was effectively more mass than the mount supported.
n 2 G0 2 (7)
Rf A 2 So what was different between the vehicle that tested well
p
and those that tested poorly? The company advertised that
2 00 m (8) the accumulators could be placed anywhere desired by just
me changing the tube length. One customer placed the
m accumulators on the left and right side of the rear cab window
nn G 00 (9) using a tube length of about 1.5 meters. The other customer
decided to place the accumulators on the frame rails about 6
Finally, the equivalent mass me is related to the mass of fluid meters from the displacers. This placement changed the
in the tube, m fl , by, effective mass such that the high frequency behavior was
unacceptable. The fix was to simply increase the diameter of
the tube if long lengths were to be used. The costs to themanufacturer were substantial during the several months of exposed bonds, vt , t , vc , c attach to the corresponding
downtime needed to figure out this fundamental problem.
velocities and angular velocities in Fig. 4.
3. BACKHOE INSTABILITY
The previous example lent itself to linear modeling and
analysis. This is always a good thing when it is justified, as
physical insight is more easily gained when parameter groups
are exposed symbolically. This final example does not lend
itself to a linear analysis, and simulation is required to
identify the problem.
A backhoe is a tractor-like vehicle with a hydraulically driven
scoop on one end used typically to dig trenches and holes in
the ground. When this scoop, or hoe, is used, the tractor is
stationary with outriggers deployed for stability. The other
end of the tractor has a large hydraulically driven bucket that
can be raised and lowered. This implement is typically used
to load large amounts of dirt and debris onto dump trucks.
The vehicle is mobile while using the front bucket.
Fig. 3 Schematic of a backhoe showing cab, chassis, boom,
A company manufactures backhoes and designed a new, hydraulics, and dimensions
lighter vehicle for landscaping operations. The prototypes
worked fine and manufacturing was commenced. The first
production machines exhibited an interesting instability.
When the bucket was lifted, the vehicle became unstable and
literally jumped off the floor at a high frequency. The
shaking made it difficult for the operator to keep hands on the
controls, and, in some cases, the instability continued even
after the controls were released. The company engineers
pondered this problem as production continued, until, after
800 machines were manufactured, the plant was shut down.
The instability was associated with an interaction between the
hydraulics and the mechanical system, and a "mechatronic"
model was needed to understand this problem. Fig. 3 shows
a schematic of the system. The cab sits above the chassis
attached by cab mounts with some stiffness and damping.
The boom is also connected to the chassis. Hydraulic
cylinders, one on each side of the vehicle, raise and lower the
boom. The controls for the boom are inside the cab and are
not shown in Fig. 3. Fig. 4 Bond graph of the cab, chassis, and boom. The
modulated transformers, -MTF-, indicate the nonlinear
The model for this complex interactive system is kinematics between the chassis and boom, and the exposed
accomplished using bond graphs. These allow each part to be bond, S , is the relative velocity across the hydraulic cylinder
modeled and tested separately, and then assembled into an
overall computational model. Fig. 4 shows the bond graph The overall system model was assembled, state equations
for the cab, chassis, and boom. The nonlinear kinematics derived from the bond graph, and simulated using a
between the chassis motion and boom motion are indicated commercial equation solver. The instability was nicely
by the modulated transformers, -MTF-. The relative velocity exposed by this model, and one simulation result is shown in
across the hydraulic cylinder was constructed and exposed as Fig. 7. This shows the angular motion of the chassis using
indicated in Fig. 4. This is where the hydraulic model will the cab mount parameters for the manufactured system and
attach when the overall model is assembled. Figure 5 shows using cab mounts with double the stiffness. The system is
the hydraulic system schematic and bond graph along with unstable using the manufactured mounts, and stable using
stiffer mounts. Obviously there are many stabilizing
the exposed bond with flow variable S . This bond graph
scenarios to try. The availability of a model that exposes the
fragment attaches to Fig. 4 at the corresponding S -junction. instability allows concepts to be tested before they are built.
The controls for the hydraulic valve are shown in Fig.6 along The instability in the backhoe is fundamentally due to having
with the bond graph for this interaction. As can be seen, the the control for the valve stem mounted on the cab and the
valve motion is affected by the operator inputs, but also by valve body mounted on the chassis. This allows relative
the relative motion between the cab and chassis. The motion between cab and chassis to modulate the valve. The
company wanted a lighter machine. Reduction in the weightof the cab meant softening the cab mounts in order to have of these machines of course knew about this possibility. So
low accelerations of the cab while the machine is idle with why did it become a problem with this machine? By
the engine running. softening the mounts, increased relative motion between cab
and chassis allowed the valve to move beyond its "dead"
zone, and trigger the instability. Each of the 800 machines
had to be brought back into the factory for a "fix”.
Fig 7 Angular motion of the backhoe chassis using
conventional and double stiffness cab mounts.
4. AIRCRAFT FUEL PUMP FAILURE
A vane pump is used to pump fuel from the fuel tanks to the
injectors of most jet engines on jet aircraft. The pump is
typically driven from a direct drive using a gearbox or
electrically driven using an electric motor. The pump is a
positive displacement device that has multiple trapped
volume segments that expand and draw fuel into the volume
followed by a compression where the volume segment gets
smaller and ultimately discharges the fuel into a volume
collector that operates at a regulated pressure. The vane
Fig. 5 Model of hydraulic cylinder and valve. Note the bond pump that is the subject of this paper regulates at about 30
labled s . This bond graph attaches to the mechincal bond atm pressure.
graph at this location.
The motivation for this paper is a fleet of business jets that
were experiencing premature failures of the vane pumps.
Since vane pumps are so prevalent in jet aircraft, it was
puzzling as to what was different about the particular
application that was failing. There are some configurational
differences between this vane pump and others that are
typical for aircraft. This vane pump has a hydraulic pump
“piggy-backed” to it and both pumps are driven by a common
shaft. Also, this pump is a “double pumper” meaning that 2
intakes and 2 discharges take place for each rotation of the
pump.
Forensic investigation of the failed pumps did not provide
conclusive evidence as to the cause. The problem was
investigated for quite some time when it was decided that a
dynamic model of the complete system, from mechanical
drive, through internal fluid mechanics, regulation, and fuel
injection, was needed to determine the cause of failure. Such
multi-energy domain systems are the perfect application of
bond graph modeling. A cross-section of the vane pump is
shown schematically in Figure 8. The vanes are pressed
Fig. 6 Schematic and partial bond graph of the controls for against the cam profile from slots (not shown) below the
front bucket vanes that are pressurized from the discharge pressure. This
pressure below the vanes is referred to as the undervane
Thus boom motion causes chassis motion which causes cab pressure. For this pump there are 12 vanes and thus 12
motion which causes valve motion which causes boom trapped volumes between each vane pair. A vane pair rotates
motion. The overall feedback is unstable. The manufacturer past the inlet port as the volume is expanding and draws fuelinto the trapped volume. As rotation continues the volume
decreases as the vanes are pushed inward by the cam profile. The spring in the relief valve is preloaded such that the desire
The vanes rotate past the boost port and some high pressure regulated pressure will lift the valve from its seat and allow
fuel is pumped back to the inlet port to aid the inlet process. fluid to flow past the valve and return to the low pressure side
The compression continues until the trapped volume rotates of the pump. When working properly the flow downstream of
past the discharge port and high pressure fuel is discharged the valve is regulated at a fairly steady pressure with small
into an accumulating volume outside the discharge port. As oscillations superimposed due to the vane passing frequency
indicated in the figure, this process repeats itself 2 times per at the discharge port. The dashpot is composed of a piston in
revolution of the pump. cylinder and the fuel is intended to be trapped in the cylinder.
As the valve moves vertically the trapped fuel is pumped in
vanes
the clearance between piston and cylinder thus providing
Boost port
Cam profile damping.
Downstream of the vane pump the fluid dynamics are
Inlet port Discharge
port One trapped volume
included in a very approximate way. There are wave
p
between a vane pair dynamics to consider, but the first resonant frequency of the
fuel delivery system can be reasonably approximated by the
Discharge
the single fluid inertia and single fluid compliance shown in
port Inlet port Figure 11.
Load compliance
Boost port CL
single fluid inertia and single fluid compliance shown in
Figure 8 Schematic cross-section of a vane pump Figure 14. C dis IL
Figure 9 shows the mechanical dynamics of the fuel pump Load fluid inertia
and hydraulic pump.
Figure 11 Fluid dynamics downstream of the vane pump
Rotational unit of the
hydraulic pump The bond graph for the entire system was constructed from
Rotational unit Jh bond graph fragments for each of the sub-models. For space
of the fuel pump Jp reasons the bond graph fragments for each part are not shown
k 2
and the resulting overall system bond graph is shown in
in k1 Figure 12.
h
p As mentioned above, there are 12 trapped volumes in this
vane pump. Shown in the bond graph are the first and twelfth
trapped volumes. Each trapped volume is represented using a
Torsional stiffness of the
2-port compliance element. On the mechanical side of the 2-
shafts port is the rate at which the volume is changing due to the
cam profile and on the other side of the 2-port is the volume
Figure 9 Mechanical dynamics of the drive shaft for the vane compliance of the almost incompressible fluid. The
pump and hydraulic pump pressures P1 through P12 are the respective pressures in the
The relief valve located outside the discharge port of the trapped volumes as these volumes change and move through
pump is shown schematically in Figure 10. the pump. On the left side of Figure 12 is the mechanical
dynamics as indicated in Figure 9. The output from this
segment is the angular velocity of the vanes. This angular
kv
Spring preload sets th velocity is converted to the rate of change of volume through
relief valve pressure
the MTF elements.
Return to inlet
1
line Trapped volume 1
Discharge port 1 Relief valve
mv x dV1
R
d1 Se I mrv
P1 P1 1
MTF 0 C 0 1
Qd1 xrv s
V1
R brv
Dv 1 R 1
J hI k h C ApTF A
1
Inlet port 1 Boost port 1 1 C krv
From discharge b h R
R 1 S e Ps I 1 R
p
TF
ports Dd 1 0 R 1
Pin 0 PB 0 R Rd
1 0 1 0 C 0 Pdis
Dashpot in S f
0 1 1
R R C Cd
C QL
k p C JpI R Inlet ejector
b p 0 1 0 R
12
Figure 10 Pressure relief valve schematic 1
1 R
Shaft dynamics C I C
dV12 R 12
12R
d12 P12 P12
MTF 0 C 0 1Q Dynamics of the downstream load
V12 d12(a) clearance of 0.001 (b) clearance of 0.0001 in
Figure 12 Bond graph of the entire system constructed from 2
Figure 13 Area of the relief valve, m for varying dashpot
bond graph segments of sub-models clearance
The port area modulations are modeled as nonlinear Figure 13 shows the flow area in the relief valve for 2
resistance elements that follow an “abs-square” constitutive different clearances in dashpot. The relief valve exhibited a
law. They are modulated by the rotation angle of the resonant unstable behavior for the smaller clearance in the
machine which dictates how the vanes rotate past a given port dashpot. For a clearance of 0.001 in., the discharge pressure
and thus modulate its flow area. is well regulated and exhibits a small cycle to cycle variation
as the vanes rotate past the discharge port.
The flow passing through the discharge port enters into a For the resonant condition associated with the reduced
collector volume that is represented by the first C element clearance the pressure fluctuates in excess of 200 psi. It was
in the bond graph fragment for the downstream dynamics. determined that the clearance specification for the dashpot
The pressure in the discharge volume is regulated by the had been changed on a drawing and was specified at 0.0001
pressure relief valve modeled as shown in Figure 10. The inch. This specification was changed and the pump failures
spring krv is preloaded to set the desired pressure level. When ceased.
the discharge pressure creates a force bigger than the spring 5. CONCLUSIONS
preload force, the valve lifts from its seat and the valve Several examples were presented where lack of physical
understanding of an overall system has led to costly errors
position xrv modulates the flow resistance that allows fuel to
and downtime. These systems are typical of most products
return to the fuel tanks. The valve itself is modeled as an where many components interact to produce an overall effect.
inertial element of mass mrv . Real engineering systems typically have interacting electrical,
thermal, hydraulic, and mechanical components. The
The dashpot for the relief valve has both fluid
engineering development that takes a concept to proof-of-
compliance Cd to account for the compressibility of the concept to prototype to production should always include
trapped fuel and fluid resistance Rd to account for the energy physical system modeling as part of the design process.
dissipation due to moving fluid through the clearance
Bond graphs are perfectly suited for developing models of
between piston and cylinder. The dashpot is filled by gravity multi-energy domain systems. Bond graph fragments can be
and there are times when it is suspected that some air is created for sub-structures of a system and then put together
trapped in the volume below the piston. We can test for the into an overall system model. These component models can
effect of this eventuality by varying the bulk modulus of the be developed by the experts in a particular area with the bond
fluid in the dashpot. Additionally there is some mechanical graph language being the common thread among all the
damping due to shearing the fluid in the annulus between the experts in all the component areas. Causality is dictated as
piston and cylinder. Even if there was little fuel trapped in part of the modeling process so there are no surprises as to
the dashpot there would still be some fuel in the clearance whether a computational model results.
and shear damping would be present. This mechanical
damping is represented by the damping element brv in Figure It is true that hindsight is a great predictor of what has
12. already happened, and the examples here are not intended to
The plumbing that leads from the vane pump to the engine reflect poorly on the engineers that developed these systems.
inlet is shown as the bond graph fragment of the downstream The conclusion here is that physical system modeling is an
load in Figure 12. This fragment comes straightforwardly essential part of system development and can help prevent
from the schematic of Figure 11. costly mistakes in the product development.
The overall system bond graph yields causal information, REFERENCES
identifies the state variables and directly yields nonlinear Karnopp, D.C., Margolis, D. L., and Rosenberg, R. C.,
state equations using procedures from Karnopp et al. The System Dynamics: Modeling and Simulation of
system was simulated using a commercial equation solver Mechatronic Systems, Wiley and Sons, NY, 2000
and parameter studies were undertaken. AMESim, Imagine Company, Roanne, France, located at
http://amesim.com
Bond Graph Compendium, located at
http://www.ece.arizona.edu/~cellier/bg.html
The CAMP-G Users Manual, Cadsim Engineering, located at
http://www.bondgraph.com
Twente Sim, Controllab Products B. V., located at
http://www.20sim.comYou can also read
