Page 152 - The Mechatronics Handbook

= p
F 14 14 F 15 15
S 1 I:m
e V 15 15
13
S p rung V 15 C:k -1
F 12
mass m 15
Ac t ive 12 V 12 = z 12
su sp e n sio n 0 1
syst e m 10 F 11
11
V V 11 Ac t ive
Uns prung 8 su sp e n sio n
9
mass m 8 F 7 7 F 8 = p 8
Tire s t iffn ess S e 1 V 8 I:m
an d da m p in g 6 8
mo d e l C:k -1
F 5 Tire s t iffn e s s
V 5
1 V = z a nd da m pin g
F 3 5 5
0 1 mo d e l
3 V F 4
The causality assignment shows that the 3 4
mechanical system (including tire) has 4 dynamic states. V
4 R:b
2
Ve r t ical ve lo cit y a t 1 S
g ro u n d -t ire in t e rfa c e 1 V 1 f
FIGURE 9.25 Example of model for vertical vibration in a quarter-car suspension model with an active suspension
element. This example builds on the simple mass-spring-damper model, and shows how to integrate an actuator
into a bond graph model structure.
displacement variable is the state z 5 , which here represents the change in length of the spring. The state
T
vector is x = [p 8 , z 5 ].
A mathematical model can be derived by referring to this bond graph, focusing on the independent
energy storage elements. The rate law (see Tables 9.4 and 9.5) for each energy storage element in
integral causality constitutes one first-order ordinary differential state equation for this system. In
order to formulate these equations, the right-hand side of each rate law must be a function only of
states or inputs to the system. The process is summarized in the table of Fig. 9.24. Note that the example
assumes linear constitutive relations for the elements, but it is clear in this process that this is not
necessary. Of course, in some cases nonlinearity complicates the analysis as well as the modeling process
in other ways.
Quarter-car Active Suspension: Bond Graph Approach
The simple mass-spring-damper system forms a basis for building more complex models. A model for
the vertical vibration of a quarter-car suspension is shown in Fig. 9.25. The bond graph model illustrates
the use of the mass-spring-damper model, although there are some changes required. In this case, the
base is now moving with a velocity equal to the vertical velocity of the ground-tire interface (this requires
knowledge of the terrain height over distance traveled as well as the longitudinal velocity of the vehicle).
The power direction has changed on many of the bonds, with many now showing positive power flowing
from the ground up into the suspension system.
The active suspension system is isolated to further illustrate how bond graph modeling promotes a
modular approach to the study of complex systems. Most relevant is that the model identifies the required
causal relation at the interface with the active suspension, specifying that the relative velocity is a causal
input, and force is a causal output of the active suspension system. The active force is exerted in an equal
and opposite fashion onto the sprung and unsprung mass elements.
The causality assignment identifies four states (two momentum states and two spring displacement
states). Four first-order state equations can be derived using the rate laws of each of the independent
energy-storing elements (C 5 , I 8 , C 12 , I 15 ). At this point, depending on the goals of the analysis, either the
nonlinear equations could be derived (which might include an active suspension force that depends on
the velocity input), or a linearized model could be developed and impedance methods applied to derive
a transfer function directly.

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