Page 139 - The Mechatronics Handbook

TABLE 9.4 Mechanical Potential Energy Storage Elements (Integral Form)
Physical System Fundamental Relations Bond Graph
Generalized Potential State: q = displacement 1 e
Energy Storage 1 f = 1 q n e
Rate: q = f
Element n f = n q
q
Constitutive: e =Φ ( ) 2 e
e C 2 f = 2 q C ...
∫
f Energy: U = e d ⋅ q
q
∫
Capacitive element Co-energy: U = q d ⋅ e 3 e 3 f = 3 q
e
Capacitance, C Generalized multiport C-element
Mechanical Translation State: x = displacement F
C:1/C=k
=
stiffness, k = 1/C Rate: xV xV
=
1 F 2 F
k x
Constitutive: F = F ( ) Linear: F =⋅
x
2 V Energy: U = F dx Energy: U = 1 kx 2
⋅
1 V 1 F = 2 F = F x ∫ x 2
spring 1 VV V Co-energy: U = x dF Co-energy: U = F 2 2k
⋅
F ∫
−
2 =
F
stiffness, k, compliance, C
Mechanical Rotation State: θ = angle T C:1/C=K
=
1 T stiffness, K= 1/C 2 T Rate: = ω θ ω
θ
⋅
Constitutive: = ( ) Linear: T = K θ
TT θ
1 ω ω 2 Energy: U = 1 kθ 2
⋅
1 T T= 2 = T Energy: U θ = ∫ T dθ θ 2
1 ω − ω 2 = ω T 2
Torsional spring Co-energy: U = θ ⋅ dT Co-energy: U = 2K
T ∫
T
stiffness, K, compliance, C
F F T
C
=
x = V θ ω
x
θ
T F k 11 k 12 x
=
T k k θ
21 22
(a) (b)
FIGURE 9.12 Example of two-port potential energy storing element: (a) cantilevered beam with translational and
rotational end connections, (b) C-element, 2-port model.
x ˙
variable of interest is either translational, x, or angular, θ, and the associated velocities are V = and
ω = θ, respectively. A generalized potential energy storage element is summarized in Table 9.4, where
examples are given for the translational and rotational one-port.
The linear translational spring is one in which F = F(x) = kx = (1/C)x, where k is the stiffness and
C ≡ 1/k is the compliance of the spring (compliance is a measure of “softness”). As shown in Table 9.4, the
1
2
potential energy stored in a linear spring is Ux = ∫ F dx = ∫ kx dx = kx , and the co-energy is U F = ∫
--
2
2
F dx = ∫ (F/k) dF = F /2k. Since the spring is linear, you can show that U x = U F . If the spring is nonlinear
due to, say, plastic deformation or work hardening, then this would not be true.
Elastic potential energy can be stored in a device through multiple ports and through different energy
domains. A good example of this is the simple cantilevered beam having both tip force and moment
(torque) inputs. The beam can store energy either by translational or rotational displacement of the tip.
A constitutive relation for this 2-port C-element relates the force and torque to the linear and rotational
displacments, as shown in Fig. 9.12. A stiffness (or compliance) matrix for small deflections is derived
by linear superposition.
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