Bond Graph

A bond graph is a graphical representation of a dynamic system, used when describing one or another physical (mechanical, electrical, hydraulic, pneumatic, economic, etc.) system, that reflects the process of energy redistribution within that system. It resembles a graph better known as a block diagram, or a signal-flow graph, and relies on the law of conservation of energy. The main difference from block diagrams or signal-flow graphs is that in a bond graph the edges are associated with a flow of energy, which can be directed in either direction, whereas block diagrams and signal-flow graphs assume a unidirectional flow of information. The edges in bond graphs are equipped with symbols specifying either an energy flow or an information flow.

Compared with other visual representation tools such as block diagrams, bond graphs have many advantages:

• they distinguish between flows of energy and flows of information;
• since bond graphs rely on the law of conservation of energy, it turns out to be impossible to introduce energy that is not present in the system;
• they highlight the causal relationships between efforts (force, voltage, pressure) and flows (velocity, current, flow rate). Such causal relationships are set once, when the original diagram is created, which, among other things, makes it possible to detect modeled phenomena such as, for example, currents in a coil, the angular velocity of a flywheel, etc.;
• since each bond represents a flow in both directions, in systems with opposition, for example with an electromotive force, there is no need to add extra loops to describe the effect of an element on itself.

If the dynamics of the modeled system take place on different time scales, fast processes in real time can be treated as instantaneous phenomena using hybrid bond graphs.

In a bond graph a distinction is made between:

• nodes (vertices), which correspond to «physical phenomena» described by equations. This general concept can mean mechanical parts, electrical components, hydraulic devices, etc. A node can also correspond to a subset of parts, in other words, a node can itself be described as a nested bond graph. At the same time, however, the physical law applies to the system as a whole (for example, Kirchhoff's rules for electrical circuits) ;
• arcs (edges), which correspond to flows of energy. In other words, they define the action of one node on another. They are called " bonds ", from which the name of the graph is derived.

Exchanges between nodes are described by two parameters: flow and effort. Flow represents the change of a quantity per unit time: electric current I , volumetric flow rate of a liquid Qv , the velocity of an element v, etc. Effort represents the force by which the flow is driven: electrical voltage U , fluid pressure p , force F , etc. The product of flow and effort gives power, (measured in watts).

The edges of the graph — are half-arrows («harpoons»), whose barb elements are oriented downward or to the right: ⇁, ↽, ↾ ⇂. The direction of the arrow indicates the direction of power transfer, that is, power enters at the start of the arrow and leaves at its end. In the case of a measuring device (thermometer, tachometer, dynamometer, flow meter, manometer, voltmeter, ammeter, etc.) the energy flow is negligible, and a full arrow is used as the notation: →, ←, ↑ or ↓.

Bond graph for an electrical circuit with a resistance R and a voltage source (left) and a current source (right)

The laws governing the behavior at nodes often relate flow and effort. For example, for an electrical resistance, Ohm's law establishes the relationship between current and voltage :

U=R⋅I.

If the resistance is connected to a voltage source, then U is set at the source, and the resistance determines I . Conversely, if the resistance is connected to a current source, then I is set at it, and U is determined according to Ohm's law. Thus, there is causality. To indicate this on the diagram, a line is placed opposite the end of the arrow that defines the flow. This makes it possible to know the input value and the output value, obtained as a result of applying the law, that is, the value of the computed quantity: e=e(f) or f=f(e) .

Bond graph for an RLC circuit in the case of series (left) and parallel (right) connections

A node can also represent a physical law rather than a particular element. A law that delivers the same effort e to several other nodes is called a 0-junction. A law that delivers the same flow f to several other nodes is called a 1-junction.

For the series connection in an RLC circuit there is only one branch. According to Kirchhoff's rule, all elements in such a circuit are given the same value of intensity (flow, current). This is a 1-junction. For the parallel connection in an RLC circuit, Kirchhoff's rule imposes the same voltage value on all elements; this is a 0-junction.

The directions of the arrows depend on the conventions chosen for the circuit.

Bond graphs characterize the transfer of power between the elements of a system, so they are ideally suited for modeling systems that connect several different domains of physics, such as, for example, electricity and mechanics. Before beginning to model, it is necessary to recall how the concept of power is introduced for each of these domains.

Power

Power is defined as the product of flow and effort:

Momentum

A causal quantity defined by effort and related to it through integration:

Displacement

A causal quantity defined by flow and related to it through integration:

Many systems can be described using the terms used in bond graphs. These terms are presented in the table below.

Conventions for the table below:

• Pis the active power ;
• X^is a matrix object;
• x→is a vector object;
• is the Hermitian conjugate of x ; it is the complex conjugate of the transposed matrix x . If x — is a scalar, then the Hermitian conjugate coincides with the complex conjugate;
• — is the Euler notation for differentiation , where:Dtnf(t)={∫−∞tf(s)ds,n=−1f(t),n=0∂nf(t)∂tn,n>0
• 
• Vergent-factor:φL={Prismatic: length/cross−sectional area Cylinder: ln⁡(radius_out/radius_in)2π⋅length Sphere: 14π(radius_in∥−radius_out)

	Generalized flow	Generalized displacement	Generalized effort	Generalized momentum	Generalized power (in watts for power systems)	Generalized energy (in joules for power systems)
Name				p→(t)	P=f→(t)†e→(t)	E=q→(t)†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 hyperrigitance	ComplianceCrigitance	Resistance	InertanceI(orL)	AbrahanceA	MagnanceM
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 one-dimensional systems (linear): For one-dimensional systems: Impedance:	Potential energy for N-dimensional systems: Potential energy:V=∫0qe(q)dq Potential coenergy :V¯=∫0eq(e)de For one-dimensional systems: Impedance:Z(s)=1sC=1sk	For one-dimensional systems (linear) Power for one-dimensional non-linear resistances (this is the effort applied by the element):P=e(f)f Rayleigh power:R=12R⋅f(t)2 Rayleigh power for non-linear resistances:R=∫0fe(f)df Rayleigh effort:eR=dRdf=e(f) For N -dimensional systems: ) For one-dimensional systems: Impedance:	Kinetic energy for N -dimensional systems: Kinetic energy:T=∫0ρf(ρ)dρ Kinetic coenergy:T¯=∫0fρ(f)df For one-dimensional systems:e(t)=L⋅dfdt+f⋅dLdt Impedance:Z(s)=sL	For one-dimensional systems (linear): For one-dimensional systems:e(t)=A⋅γ[Dt3q(t)] ImpedanceZ(s)=s2A	For one-dimensional systems (linear)P=A⋅(Dt3q(t))2 For one-dimensional systemse(t)=M⋅γ[Dt5q(t)] ImpedanceZ(s)=s4M
Generalized behaviour	Energy obtained from sources of active efforts: Lagrangian : Hamiltonian Hamiltonian effort Lagrangian effort: Passive effort: Power equation: Effort equation: Lagrangian equation: Hamiltonian equation: If is the coenergy is the energy, is the state variable and is the costate variable, For linear elements

Longitudinal mechanical power system ^{[ 2 ]}

Flow-related variables	Dt6x: pounce / pop[m/s6]	Dt5x: flounce / crackle[m/s5]	Dt4x: jounce / snap[m/s4]	Dt3x: jerk [m/s3]
Dt2x: acceleration (x2t) [m/s2]	Dt1x: velocity (xt) [m/s] (flow)	Dt0x: displacement(x) [m] (displacement)	Dt−1x: absement [m⋅s]
Dt−2x: absity[m⋅s2]	Dt−3x: abseleration[m⋅s3]	Dt−4x: abserk[m⋅s4]
Effort-related variables	Dt1F: yank [N/s]	Dt0F: force (Pxt) [N] (effort)	Dt−1F: linear momentum (Px2t) [kg⋅m/s] (momentum)
Passive elements
Compliance (C)	Resistance (R)	Inertance (I)	Abrahance (A)	Magnance (M)
Spring〈Pxt〉=k〈x〉wherekis the spring stiffness	Damper〈Pxt〉=b〈xt〉wherebis the damper parameter	Mass〈Pxt〉=m〈x2t〉wheremis the mass	Abraham-Lorentz force 〈Pxt〉=μ0q26πc〈x3t〉where μ0is the permeability​ qis the electric charge c— is the speed of light.	Magnetic radiation reaction force 〈Pqt〉=μ0q2R24πc3〈x5t〉where μ0is the permeability​ qis the electric charge c— is the speed of light. R— is the radius of the magnetic moment
Cantilever beam〈F〉=3EI/L3〈x〉 L: length of the cantilever EYoung's modulus I: second moment of area	Resistance to cyclotron radiation 〈F〉=σTB2cμ0〈v〉where σT: Thomson cross section cspeed of light μ0: permeability B: magnetic field density
Prismatic float in a large pond 〈F〉=ρLgA〈x〉where ρL: liquid density A: area g: gravitational acceleration	Viscous friction〈F〉=b〈v〉whereb— is the viscous friction parameter
Elastic rod〈F〉=EA/L〈x〉 where EYoung's modulus A: area L: rod length	Inverse of kinetic mobility〈F〉=1μ〈v〉whereμis the kinetic mobility
Newton's law of universal gravitation 〈F〉=GMm〈x〉−2where G: gravitational constant M: mass of body 1 m: mass of body 2	Interaction of geometry with air (e.g., air resistance) 〈F〉=12cρA〈v〉2where A: contact area caerodynamic shape coefficient ρ: fluid density
Coulomb's law 〈F〉=q1q24πε0〈x〉−2where ε0permittivity q1: charge of body 1 q2: charge of body 2	Absquare [ clarification needed ] damper〈F〉=B〈v〉2whereB— is the parameter of the Absquare damper.
Casimir force 〈F〉=Aℏcπ2240〈x〉−4where A: plate area ℏ: reduced Planck constant cspeed of light	dry friction 〈F〉=μFn〈v〉0where μcoefficient of kinetic friction Fn: normal force
Biot–Savart law 〈F〉=μ0I1I2l2π〈x〉−1where μ0: permeability I1: current in wire 1 I2: current in wire 2 llength of the wires
A piston compresses fluid inside an adiabatic chamber. F=AP0(1+xx0)−γwhere A: piston area P0initial internal pressure x0: initial piston position γadiabatic index

Angular mechanical power system

Flow-related variables	d3θ/dt3angular jerk [rad/s3]	d2θ/dt2angular acceleration (θ2t) [rad/s2]
dθ/dtangular velocity (θt) [rad/s] (flow)	θangular displacement (θ) [rad] (displacement)
Effort-related variables	dτ/dt: rotatum [V/rad]	τ: torque (θt) [J/rad] (effort)
∫τdtangular momentum (θ2t) [J⋅s/rad] (momentum)
Passive elements
Compliance (C)	Resistance (R)	Inertia (I)
Inverse of the angular spring constant〈τ〉=k〈θ〉wherek— is the angular spring constant.	Angular damping〈τ〉=R〈θt〉whereR— is the damping constant	Mass moment of inertia Type〈τ〉=J〈θ2t〉 whereJmass moment of inertia
Torsion of a rod 〈τ〉=GJL〈θ〉where Gshear modulus J: polar moment of area L: rod length	Governor (e.g., used in music boxes)〈τ〉=R〈θt〉2whereRis the governor constant
Bending moment (cantilever) 〈τ〉=EJL〈θ〉where EYoung's modulus J: second moment of area Lbeam length
Field of parallel forces 〈τ〉=〈F〉sin⁡(α−θ)where α: angle (positive counterclockwise) between the polar axis and the force field θ: current tilt angle of the object (e.g., a pendulum)

Electrical power system

Flow-related variables	d2q/dt2: rate of current change[A/s]	dq/dtelectric current (qt) [A] (flow)	qelectric charge (q) [C] (displacement)
Effort-related variables	dV/dt: voltage rate[V/s]	V: Voltage (qt) [V] (effort)	∫Vdt: flux linkage (q2t) [V⋅s] or [Wb⋅turn] (momentum)
Elements
Hyperactance (H)	Compliance (C)	Resistance (R)	Inertia (I)	Abrahance (A)
Frequency-dependent negative resistor (FDNR) 〈V〉=1H〈i〉	Linear capacitor 〈V〉=1εφL〈q〉where ε: permittivity φL: geometry factor	Linear resistor 〈V〉=ρφL(1+α(T−T0))〈i〉where ρ: resistivity φL: geometry factor αtemperature coefficient of resistance T: current temperature T0reference temperature	Linear inductor (solenoid) 〈V〉=μ0N2AL〈q2t〉where μ0: permeability N: number of turns A: area L: length	Frequency-dependent negative conductance (FDNC) Typeγ1⇔V=Rd2i/dt2[H⋅s]
		Diode V=nVTln⁡(1+iis)where nideality factor VT: thermal voltage isleakage current	Toroid 〈V〉=μN2A2πr〈q2t〉where μ: permeability N: number of turns A: cross-sectional area rtoroid radius relative to the centerline
Intra-gyrator	Inter-gyrator
Compliant gyrator	Resistive gyrator	Inertial gyrator
	Hall effect device ex=RHBz/tz⋅iy ey=RHBz/tz⋅ixwhere RHHall coefficient Bz: vertical magnetic flux density tz: vertical thickness	Induction motor er1=Mcos⁡(θ)dφs1/dt+φs1d[Mcos⁡(θ)]/dt es1=Mcos⁡(θ)dφr1/dt+φr1d[Mcos⁡(θ)]/dtwhere θelectrical angle er1: rotor bar voltage es1: stator bar voltage φs1flux through the stator bar φr1flux through the rotor bar	DC motor τe=kaφa(if)ia ωe=1kaφa(if)eawhere τeelectromagnetic torque ωe: shaft angular velocity if: field current iaarmature current	Faraday vibration generator F=Bli V=Blvwhere F: force v: rod velocity V: rod voltage Bmagnetic flux density l: rod length
			Faraday disk V=12Br2ω τ=12Br2iwhere Bmagnetic flux density rdisk radius
Intra-transformer	Inter-transformer
Electrical transformer (AC signals only) V2=N2N1V1 φ2=N1N2φ1

Hydraulic/pneumatic power system

Flow-related variables	dV/dtvolumetric flow rate (Vt) [m3/s] (flow)	V: volume (V) [m3] (displacement)
Effort-related variables	P: pressure (PVt) [Pa] (effort)	∫Pdt: fluid momentum(PV2t) [Pa⋅s] (momentum)
Elements
Compliance (C)	Resistance (R)	Inertia (I)
Pipe elasticity 〈P〉=(tVE2r0V0)〈V〉where r0nominal pipe radius V0: pipe volume (unstressed) tV: wall thickness EYoung's modulus	Darcy sponge 〈P〉=(μkφL)〈Vt〉where μdynamic viscosity k: permeability φL: geometry factor	Fluid inertia in pipes 〈P〉=(ρφL)〈V2t〉where ρ: fluid density φL: geometry factor
Compressible fluid (approximation) 〈P〉=(BV0)〈V〉=(ρ0c2V0)⏟AcousticApproximation〈V〉where V0: pipe volume (unstressed) Bbulk modulus ρ0

продолжение следует...

Продолжение:

Часть 1 Bond Graph
Часть 2 Tetrahedron of State - Bond Graph
Часть 3 State equations - Bond Graph