Bond graph

A bond graph is a graphical representation of a physical dynamic system. It allows the conversion of the system into a state-space representation. It is similar to a block diagram or signal-flow graph, with the major difference that the arcs in bond graphs represent bi-directional exchange of physical energy, while those in block diagrams and signal-flow graphs represent uni-directional flow of information. Bond graphs are multi-energy domain (e.g. mechanical, electrical, hydraulic, etc.) and domain neutral. This means a bond graph can incorporate multiple domains seamlessly.

The bond graph is composed of the "bonds" which link together "single-port", "double-port" and "multi-port" elements (see below for details). Each bond represents the instantaneous flow of energy ( $dE / dt$ ) or power. The flow in each bond is denoted by a pair of variables called power variables, akin to conjugate variables, whose product is the instantaneous power of the bond. The power variables are broken into two parts: flow and effort. For example, for the bond of an electrical system, the flow is the current, while the effort is the voltage. By multiplying current and voltage in this example you can get the instantaneous power of the bond.

A bond has two other features described briefly here, and discussed in more detail below. One is the "half-arrow" sign convention. This defines the assumed direction of positive energy flow. As with electrical circuit diagrams and free-body diagrams, the choice of positive direction is arbitrary, with the caveat that the analyst must be consistent throughout with the chosen definition. The other feature is the "causality". This is a vertical bar placed on only one end of the bond. It is not arbitrary. As described below, there are rules for assigning the proper causality to a given port, and rules for the precedence among ports. Causality explains the mathematical relationship between effort and flow. The positions of the causalities show which of the power variables are dependent and which are independent.

If the dynamics of the physical system to be modeled operate on widely varying time scales, fast continuous-time behaviors can be modeled as instantaneous phenomena by using a hybrid bond graph. Bond graphs were invented by Henry Paynter.^[1]

Systems for bond graph

Many systems can be expressed in terms used in bond graph. These terms are expressed in the table below.

Conventions for the table below:

$P$ is the active power;
${\hat {X}}$ is a matrix object;
${\vec {x}}$ is a vector object;
$x^{\dagger }$ is the Hermitian conjugate of $x$ ; it is the complex conjugate of the transpose of $x$ . If $x$ is a scalar, then the Hermitian conjugate is the same as the complex conjugate;
$D_{t}^{n}$ is the Euler notation for differentiation, where: $D_{t}^{n}f(t)={\begin{cases}\displaystyle \int _{-\infty }^{t}f(s)\,ds,&n=-1\\[2pt]f(t),&n=0\\[2pt]{\dfrac {\partial ^{n}f(t)}{\partial t^{n}}},&n>0\end{cases}}$
${\begin{cases}\langle x\rangle ^{\alpha }:=|x|^{\alpha }\operatorname {sgn}(x)\\\langle {a}\rangle =k\langle b\rangle ^{\beta }\implies \langle b\rangle =\left({\frac {1}{k}}\langle a\rangle \right)^{1/\beta }\end{cases}}$
Vergent-factor: $\phi _{L}={\begin{cases}{\textrm {Prismatic}}:\ {\dfrac {\textrm {length}}{{\textrm {cross-sectional}}\ {\textrm {area}}}}\\{\textrm {Cylinder}}:\ {\dfrac {\ln \left({\frac {\mathrm {radius_{out}} }{\mathrm {radius_{in}} }}\right)}{2\pi \cdot {\textrm {length}}}}\\{\textrm {Sphere}}:\ {\dfrac {1}{4\pi \left(\mathrm {radius_{in}} \parallel \mathrm {-radius_{out}} \right)}}\end{cases}}$

	Generalized flow	Generalized displacement	Generalized effort	Generalized momentum	Generalized power (in watts for power systems)	Generalized energy (in joules for power systems)
Name	${\vec {f}}(t)$	${\vec {q}}(t)$	${\vec {e}}(t)$	${\vec {p}}(t)$	$P={\vec {f}}(t)^{\dagger }{\vec {e}}(t)$	$E={\vec {q}}(t)^{\dagger }{\vec {e}}(t)$
Description	Time derivative of displacement	A quality related to static behaviour.	The energy per unit of displacement	Time integral of effort	Transformation of energy from one to another form	Conserved quantity in closed systems
Elements
Name	Hyperance $H$ , hyperrigitance $P=H^{-1}$	Compliance $C$ , rigitance $K=C^{-1}$	Resistance $R$	Inertance $I$ (or $L$ )	Abrahance $A$	Magnance $M$
Properties	Power dissipative element	Charge storage element (State variable: displacement) (Costate variable: effort)	Power dissipative element	Momentum storage element (State variable: momentum) (Costate variable: flow)	Power dissipative element	Power dissipative element
Quantitative behaviour	For 1-dimension systems (linear): $P=H\cdot \left(D_{t}^{0}q(t)\right)^{2}$ For 1-dimension systems: $e(t)=P\gamma \left[D_{t}^{-1}q(t)\right]$ Impedance: $Z(s)={\frac {1}{s^{2}H}}={\frac {1}{s^{2}}}P$	Potential energy for N-dimension systems: ${\begin{array}{lcl}V&=&{\frac {1}{2}}{\vec {q}}(t)^{\dagger }{\vec {e}}(t)\\{\vec {q}}(t)&=&{\hat {C}}{\vec {e}}(t)\end{array}}$ Potential energy: $V=\int _{0}^{q}e(q)\,dq$ Potential coenergy: ${\overline {V}}=\int _{0}^{e}q(e)de$ For 1-dimension systems: $f(t)=C\cdot {\frac {de}{dt}}+e{\frac {dC}{dt}}$ Impedance: $Z(s)={\frac {1}{sC}}={\frac {1}{s}}k$	For 1-dimension systems (linear): $P=R\cdot \left(D_{t}q(t)\right)^{2}$ Power for 1-dimension non-linear resistances ( $e(f)$ is the effort developed by the element): $P=e(f)f$ Rayleigh power: ${\mathfrak {R}}={\frac {1}{2}}R\cdot f(t)^{2}$ Rayleigh oower for non-linear resistances: ${\mathfrak {R}}=\int _{0}^{f}e(f)\,df$ Rayleigh effort: $e_{\mathfrak {R}}={\frac {d{\mathfrak {R}}}{df}}=e(f)$ For N-dimension systems: ${\begin{array}{lcr}P&=&{\vec {f}}(t)^{\dagger }{\vec {e}}(t)\\{\vec {e}}(t)&=&{\hat {R}}{\vec {f}}(t)\end{array}}$ For 1-dimension systems: $e(t)=R\cdot \gamma \left[D_{t}^{1}q(t)\right]$ Impedance: $Z(s)=R$	Kinetic energy for N-dimension systems: ${\begin{array}{lcl}T={\frac {1}{2}}{\vec {\rho }}(t)^{\dagger }{\vec {f}}(t)\\{\vec {\rho }}(t)={\hat {L}}{\vec {f}}(t)\end{array}}$ Kinetic energy: $T=\int _{0}^{\rho }f(\rho )\,d\rho$ Kinetic coenergy: ${\overline {T}}=\int _{0}^{f}\rho (f)\,df$ For 1-dimension systems: $e(t)=L\cdot {\frac {df}{dt}}+f\cdot {\frac {dL}{dt}}$ Impedance: $Z(s)=sL$	For 1-dimension systems (linear): $P=A\cdot \left(D_{t}^{2}q(t)\right)^{2}$ For 1-dimension systems: $e(t)=A\cdot \gamma \left[D_{t}^{3}q(t)\right]$ Impedance $Z(s)=s^{2}A$	For 1-dimension systems (linear) $P=A\cdot \left(D_{t}^{3}q(t)\right)^{2}$ For 1-dimension systems $e(t)=M\cdot \gamma \left[D_{t}^{5}q(t)\right]$ Impedance $Z(s)=s^{4}M$
Generalized behaviour	Energy from active effort sources: $W=\int _{0}^{q}{e_{\text{source}}}\,dq$ Lagrangian: ${\mathfrak {L}}=T-(V-W)$ Hamiltonian: ${\mathfrak {H}}=T+(V-W)$ Hamiltonian effort: $e_{\mathfrak {H}}={\frac {d{\mathfrak {H}}}{dq}}$ Lagrangian effort: $e_{\mathfrak {L}}={\frac {d{\mathfrak {L}}}{dq}}$ Passive effort: $e_{\mathfrak {LH}}={\frac {d}{dt}}{\frac {d{\mathfrak {L}}}{df}}$ Power equation: ${\frac {d{\mathfrak {L}}}{dt}}+{\frac {d{\mathfrak {H}}}{dt}}=0$ Effort equation: $e_{\mathfrak {L}}+e_{\mathfrak {H}}=0$ Lagrangian equation: $e_{\mathfrak {R}}+e_{\mathfrak {LH}}=e_{\mathfrak {L}}$ Hamiltonian equation: $e_{\mathfrak {R}}+e_{\mathfrak {LH}}=-e_{\mathfrak {H}}$ If ${\overline {W}}$ is the coenergy, $W$ is the energy, $S$ is the state variable and ${\overline {S}}$ is the costate variable, ${\begin{aligned}{\overline {W}}+W=S\cdot {\overline {S}}\\{\overline {W}}=\int _{0}^{\overline {S}}S({\overline {S}})\,d{\overline {S}}\\W=\int _{0}^{S}{\overline {S}}(S)dS\end{aligned}}$ For linear elements: ${\overline {W}}=W={\frac {1}{2}}S\cdot {\overline {S}}$

Longitudinal mechanical power system^[2]
Passive elements
Flow-related variables	$D_{t}^{6}x$ : pounce / pop $\mathrm {[m/s^{6}]}$	$D_{t}^{5}x$ : flounce / crackle $\mathrm {[m/s^{5}]}$	$D_{t}^{4}x$ : jounce / snap $\mathrm {[m/s^{4}]}$	$D_{t}^{3}x$ : jerk $\mathrm {[m/s^{3}]}$
	$D_{t}^{2}x$ : acceleration $(x_{2t})\ \mathrm {[m/s^{2}]}$	$D_{t}^{1}x$ : velocity $(x_{t})\ \mathrm {[m/s]}$ (flow)	$D_{t}^{0}x$ : displacement $(x)\ \mathrm {[m]}$ (displacement)	$D_{t}^{-1}x$ : absement $\mathrm {[m\cdot s]}$
	$D_{t}^{-2}x$ : absity $\mathrm {[m\cdot s^{2}]}$	$D_{t}^{-3}x$ : abseleration $\mathrm {[m\cdot s^{3}]}$	$D_{t}^{-4}x$ : abserk $\mathrm {[m\cdot s^{4}]}$
Effort-related variables	$D_{t}^{1}F$ : yank $\mathrm {[N/s]}$	$D_{t}^{0}F$ : force $(P_{x}^{t})\ \mathrm {[N]}$ (effort)	$D_{t}^{-1}F$ : linear momentum $(P_{x}^{2t})\ \mathrm {[kg\cdot m/s]}$ (momentum)
Compliance (C)	Resistance (R)	Inertance (I)	Abrahance (A)	Magnance (M)
Spring $\langle P_{x}^{t}\rangle =k\langle x\rangle$ where $k$ is the spring stiffness	Damper $\langle P_{x}^{t}\rangle =b\langle x_{t}\rangle$ where $b$ is the damper parameter	Mass $\langle P_{x}^{t}\rangle =m\langle x_{2t}\rangle$ where $m$ is the mass	Abraham–Lorentz force $\langle P_{x}^{t}\rangle ={\frac {\mu _{0}q^{2}}{6\pi c}}\langle x_{3t}\rangle$ where $\mu _{0}$ is the permeability $q$ is the electric charge $c$ is the speed of light	Magnetic radiation reaction force $\langle P_{q}^{t}\rangle ={\frac {\mu _{0}q^{2}R}{24\pi c^{3}}}\langle x_{5t}\rangle$ where $\mu _{0}$ is the permeability $q$ is the electric charge $c$ is the speed of light $R$ is the radius of the magnetic moment
Cantilever $\langle P_{x}^{t}\rangle ={\frac {3EI}{L^{3}}}\langle x\rangle$ $L$ : cantilever length $E$ : Young's modulus $I$ : second moment of area	Cyclotron radiation resistance $\langle P_{x}^{t}\rangle ={\frac {\sigma _{t}B^{2}}{c\mu _{0}}}\langle x_{t}\rangle$ where $\sigma _{t}$ : Thompson cross-section $c$ : speed of light $\mu _{0}$ : permeability $B$ : magnetic field density
Prismatic floater in a wide waterbody $\langle P_{x}^{t}\rangle =\rho _{L}gA\langle x\rangle$ where $\rho _{L}$ : liquid density $A$ : area $g$ : gravitational acceleration	Viscous friction $\langle P_{x}^{t}\rangle =b\langle x_{t}\rangle$ where $b$ is the viscous friction parameter
Elastic rod $\langle P_{x}^{t}\rangle ={\frac {EA}{L}}\langle x\rangle$ where $E$ : Young's modulus $A$ : area $L$ : rod length	Inverse of kinetic mobility $\langle P_{x}^{t}\rangle ={\frac {1}{\mu }}\langle x_{t}\rangle$ where $\mu$ is the kinetic mobility
Newton's law of gravitation $\langle P_{x}^{t}\rangle =GMm\langle x\rangle ^{-2}$ where $G$ : gravitational constant $M$ : mass of body 1 $m$ : mass of body 2	Geometry interacting with air (e.g. drag) $\langle P_{x}^{t}\rangle ={\frac {1}{2}}c\rho A\langle x_{t}\rangle ^{2}$ where $A$ : contact area $c$ : aerodynamic geometry coefficient $\rho$ : fluid density
Coulomb's law $\langle P_{x}^{t}\rangle ={\frac {Qq}{4\pi \varepsilon _{0}}}\langle x\rangle ^{-2}$ where $\varepsilon _{0}$

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