Generalized Energy VariablesIdeal Energy-Storage ElementsGeneralized CapacitorGeneralized InertiaModulated Energy Storage is ProhibitedIntegrated Modeling of Physical System Dynamics © Neville Hogan 1994 page 1 Generalized Energy Variables Energetic interactions are mediated by the flow of power. Power flow through an interaction port may be expressed as the product of two real-valued variables, an effort and a flow, and all instantaneous interactions between systems or elements may be described in terms of these conjugate power variables. However, to define the energy stored in a system (i.e. its instantaneous energetic state) it is necessary to define energy variables. Just as we may define two power variables, we may define two dual or conjugate energy variables, obtained by integrating the power variables with respect to time. The first of these is generalized momentum1, p. p ∆__ ⌡⌠tot e(t)dt + p(to) (4.1) It will be associated with kinetic energy storage. This relation may be differentiated. dp = e dt (4.2) The conjugate variable is generalized displacement, q. q ∆__ ⌡⌠tot f(t)dt + q(to) (4.3) It will be associated with potential energy storage. This relation, too, may be differentiated. dq = f dt (4.4) The following tables provide a partial list of energy variables and notation we will use for several energetic media. 1 Sometimes known as impulse.Integrated Modeling of Physical System Dynamics © Neville Hogan 1994 page 2 Table 4.1 Generalized Momenta and Notation ENERGY MEDIUM EFFORT SYMBOL MOMENTUM SYMBOL General e p Mechanical translation force F momentum or impulse p Fixed-axis mechanical rotation torque or momentτ (or µ) angular momentum η Electrical voltage or potential difference e (or v) flux linkage2λ Magnetic magnetomotive force F not defined Incompressible fluid flow pressure difference P pressure momentum Γ Compressible fluid flow enthalpy h not defined Thermal temperature θ (or T) not defined Note that generalized momentum is not defined in some media. The reason is because there is no known kinetic energy storage phenomenon in those media. We will return to this point in a subsequently. 2 Strictly speaking, this variable should be associated with displacement in the magnetic medium.Integrated Modeling of Physical System Dynamics © Neville Hogan 1994 page 3 Table 4.2 Generalized Displacements and Notation ENERGY MEDIUM FLOW SYMBOL DISPLACEMENT SYMBOL General f q Mechanical translation speed or velocity v position or deflection x Fixed-axis mechanical rotation angular speed or velocity ω (or Ω) angle θ Electrical current i charge q Magnetic flux rate ϕ˙ flux ϕ Incompressible fluid flow volumetric flow rate Q volume V Compressible fluid flow mass flow rate m˙ mass m Thermal entropy flow rate s˙ entropy s, Ideal Energy-Storage Elements We are now in a position to define ideal energy-storage elements. (Ideal in the sense of not being contaminated by dissipation or any other non-storage phenomenon). The energy in a system may be determined from the power flux across its boundaries3. E = ⌡⌠tot Pdt + E(to) (4.5) Using equations 4.2 and 4.4 and the definition of effort and flow, this may be rewritten in the following ways. 3 Once again we assume the convention that power is positive inwards.Integrated Modeling of Physical System Dynamics © Neville Hogan 1994 page 4 E - E(to) = ⌡⌠tot e f dt = ⌡⌠tot e dq = ⌡⌠tot f dp (4.6) Now, if we encounter a phenomenon characterized by a relation which permits the integral in either of the latter two forms to be evaluated so that it is not an explicit function of time, that phenomenon may be regarded as energy storage. As you might expect, there are two possibilities. Generalized Capacitor A ideal generalized capacitor is defined as any phenomenon characterized by an algebraic relation (possibly nonlinear) for which effort is an integrable (single-valued) function of displacement. e = Φ(q) (4.7) The algebraic function Φ(·) is the constitutive equation for this element. Note that although we will use energy storage elements to describe dynamic behavior, this constitutive equation is a static or memory-less function. The constitutive equation permits us to evaluate the generalized potential energy, Ep Ep ∆__ ⌡⌠ e dq = ⌡⌠ Φ(q) dq = Ep(q) (4.8) For this element, potential energy is a function of displacement alone. It is a generalized potential energy storage element. The displacement, q, plays the same role as the specific entropy and specific volume do for a pure thermodynamic substance: it is sufficient to define the energy in the system. By convention we will define Ep = 0 at q = 0 as shown in figure 4.1. It will turn out to be important to distinguish potential energy from a related quantity, (generalized) potential co-energy, E*p , which is a function of effort. E*p ∆__ ⌡⌠ q de = E*p(e) (4.9) Energy and co-energy are related by a Legendre transformation: E*p(e) = e q - Ep(q) (4.10)Integrated Modeling of Physical System Dynamics © Neville Hogan 1994 page 5 displacement, q*EpEp Figure 4.1: Sketch of a possible potential energy storage constitutive equation. The relation between the two quantities is illustrated in figure 4.1. We will postpone further discussion of co-energy until later. Taken together, the definitions of generalized displacement and the constitutive equation for a generalized capacitor specify a set of relations between flow, displacement and effort. These are represented by the symbol shown in figure 4.2. Note that as this is a passive element, power flow has been depicted as positive into it. Figure 4.2: Bond graph symbol for an ideal capacitor. An ideal capacitor may have a nonlinear constitutive relation. We may also define an ideal linear capacitor, one with a linear constitutive relation. e = q/C (4.11) The parameter C is termed the capacitance of this ideal linear element. The potential energy stored in an ideal linear capacitor is a quadratic function of displacement. Ep = q2/2C (4.12) Aside: The reason for writing equation 4.11 with the proportionality constant, C, dividing the argument is largely historical. The generalized capacitor is based on an electrical capacitor, usually described by a linear relation between charge, q, (displacement) and voltage, e, (effort). q = C e (4.13) However, in the nonlinear case it may not always be possible to