Page 134 - The Mechatronics Handbook

The example in Fig. 9.6(c) illustrates how causality “propagates” through a bond graph of intercon-
nected bonds and systems. Note that a 1-junction with multiple ports can only have one bond specifying
flow at that junction, so the other bonds specify effort into the 1-junction. A 0-junction requires one
bond to specify effort, while all others specify flow. Also note that a direction for positive power flow has
not been assigned on these bonds. This is intentional to emphasize the fact that power sense and causality
assignment on a bond are independent of each other.
Causality assignment in system models will be applied in examples that follow. An extensive discussion
of the successive cauality assignment procedure (sometimes referred to as SCAP) can be found in Rosenberg
and Karnopp [32] or Karnopp, Margolis, and Rosenberg [17]. By using the defined bond graph elements,
causality assignment is made systematically. The procedure has been programmed into several commer-
cially available software packages that use bond graphs as formal descriptions of physical system models.
Because it reveals the input–output relationship of variables on all the bonds in a system model,
causality can infer computational solvability of a bond graph model. The results are used to indicate the
number of dynamic states required in a system, and the causal graph is helpful in actually deriving the
mathematical model. Even if equations are not to be derived, causality can be used to derive physical
insight into how a system works.

9.3 Descriptions of Basic Mechanical Model Components

Mechanical components in mechatronic systems make their presence known through motional response
and by force and torque (or moment) reactions notably on support structures, actuators, and sensors.
Understanding and predicting these response attributes, which arise due to combinations of frictional,
elastic, and inertial effects, can be gained by identifying their inherent dissipative and energy storing
nature. This emphasis on dissipation and energy storage leads to a systematic definition of constitutive
relations for basic mechanical system modeling elements. These model elements form the basis for
building complex nonlinear system models and for defining impedance relations useful in transfer
function formulation. In the following, it is assumed that the system components can be well represented
by lumped-parameter formulations.
It is presumed that a modeling decision is made so that dissipative and energy storing (kinetic and
potential) elements can be identified to faithfully represent a system of interest. The reticulation is an
essential part of the modeling process, but sometimes the definition and interconnection of the elements
is not easy or intuitive. This section first reviews mechanical system input and output model elements,
and then reviews passive dissipative elements and energy-storing elements. The section also discusses
coupling elements used for modeling gears, levers, and other types of power-transforming elements. The
chapter concludes by introducing impedance relationships for all of these elements.

Defining Mechanical Input and Output Model Elements

In dynamic system modeling, initial focus requires defining a system boundary, a concept borrowed
from basic thermodynamics. In isolating mechanical systems, a system boundary identifies ports through
which power and signal can pass. Each port is described either by a force–velocity or torque–angular
velocity power conjugate pair. It is helpful, when focusing on the mechanical system modeling, to make
a judgement on the causality at each port. For example, if a motor is to be attached to one port, it may
be possible to define torque as the input variable and angular velocity as the output (back to the motor).
It is important to identify that these are model assumptions. We define specific elements as sources
of effort or flow that can be attached at the boundary of a system of interest. These inputs might be
known and or idealized, or they could simply be “placeholders” where we will later attach a model for
an actuator or sensor. In this case, the causality specified at the port is fixed so that the (internal) system
model will not change. If the causality changes, it will be necessary to reformulate a new model.
In bond graph terminology, the term effort source is used to define an element that specifies an effort,
such as this force or torque. The symbol S e or E can be used to represent the effort source on a bond graph.

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