Page 140 - The Mechatronics Handbook

Kinetic Energy Storage

All components that constitute mechanical systems have mass, but in a system analysis, where the concern
is dynamic performance, it is often sufficient to focus only on those components that may store relevant
amounts of kinetic energy through their motion. This presumes that an energetic basis is used for
modeling, and that the tracking of kinetic energy will provide insight into the system dynamics. This is
the focus of this discussion, which is concerned for the moment with one-dimensional translation and
fixed-axis rotation. Later it will be shown how the formulation presented here is helpful for understanding
more complex systems.
The concept of mass and its use as a model element is faciliated by Newton’s relationship between the
rate of change of momentum of the mass to the net forces exerted on it, F = , where p is the momentum.
p ˙
The energy stored in a system due to translational motion with velocity V is the kinetic energy. Using
the relation from Newton’s law, dp = Fdt, this energy is E(p) = T(p) = T p = ∫Pdt = ∫FV dt = ∫V dp.
If the velocity is expressed solely as a function of the momentum, p, this system is a pure translational
mass, V = Φ(p). If the velocity is linearly proportional to the momentum, then V = p/m, where m is the
mass. Similar basic definitions are made for a body in rotation about a fixed axis, and these elements are
summarized in Table 9.5.
For many applications of practical interest to engineering, the velocity–momentum relation, V = V(p)
(the constitutive relation), is linear. Only in relativistic cases might there be a nonlinear relationship in
the constitutive law for a mass. Nevertheless, this points out that for the general case of kinetic energy
storage a constitutive relation is formed between the flow variable and the momentum variable, f = f(p).
This should help build appreciation for analogies with other energy domains, particularly in electrical
systems where inductors (the mass analog) can have nonlinear relationships between current (a flow)
and flux linkage (momentum).
The rotational motion of a rigid body considered here is constrained thus far to the simple case of
planar and fixed-axis rotation. The mass moment of intertia of a body about an axis is defined as the
sum of the products of the mass-elements and the squares of their distance from the axis. For the discrete
2
2
case, I = ∑r ∆m, which for continuous cases becomes, I = ∫r dm (units of kg m ). Some common shapes
2
TABLE 9.5 Mechanical Kinetic Energy Storage Elements (Integral Form)
Physical System Fundamental Relations Bond Graph
Generalized Kinetic State: = momentum
p
Energy Storage Rate: = e 1 e = 1 p n e = n p
p
Element 1 f n f
f
p
e I Constitutive: =Φ ( ) 2 e = 2 p I ...
f Energy: T = f d ⋅ p 2 f
∫
p
3 e = 3 p 3 f
Inertive element Co-energy: T = p d ⋅ f
∫
Inertance, I f Generalized multiport I-element
Mechanical Translation State: p = momentum p = F
I: M
mass, M Rate: p = F V
1 F 2 F p
Constitutive: V = V ( ) Linear: V = M
p
1 V 2 V Energy: T = f dp Energy: T = p 2
p ∫
⋅
1 F − 2 F = F p 2M
Mass = Co-energy: T = p dV Co-energy: V T = 1 MV 2
⋅
V ∫
2 V
mass, m 1 VV = 2
=
Mechanical Rotation State: h = angular momentum hT
ω 2 ω I:J
inertia, J Rate: h= T h
1 ω 2 T Constitutive: = ( ) Linear: ω = J
h
ω
ω
h ∫
1 TT− 2 = T Energy: T = ω ⋅ dh Energy: T = h 2
1 T 1 ω = ω 2 = ω h 2J
Rotational inertia Co-energy: T = h dω Co-energy: T = 1 J ω 2
ω ∫
⋅
mass moment of inertia, J ω 2
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