Examples

Reference diagrams support visual comparison of schematic structure and notation. This does not establish model equivalence or numerical agreement. The automated checks described below apply to the implemented models with the stated parameters and assumptions.

Frozen bench schedule

At a switch, states remain continuous and algebraic outputs use the new command. A source step follows the same rule. The solver adds switching times to the output grid when needed.

Electrical domain

Effort is voltage in volts (V); flow is current in amperes (A). The palette offers common voltage (0), common current (1), resistor R (Ω), capacitor C (F), inductor I (H), voltage source Se and current source Sf. I is the bond-graph symbol; L is its inductance. Electrical ports use E and violet bonds. Advanced currently has no electrical components.

Capacitor state is charge q in coulombs (C): v = q/C and q₀ = C·v₀. Inductor state is flux linkage λ in webers (Wb): i = λ/L and λ₀ = L·i₀. Enter SI values, not mF or mH. The common return conductor is the implicit zero-voltage reference; no hydraulic reference pressure applies.

C1 dv1/dt = (vi − v1)/R1 − iL
L diL/dt = v1 − v2
C2 dv2/dt = iL − v2/R2

For the RLC example, v1 = b4.e, v2 = b9.e, i1 = b1.f, i2 = b4.f, i3 = b6.f, i4 = b9.f, i5 = b8.f. At DC, both voltages tend to 6.667 V and inductor current to 0.333 A. Parameters are illustrative because the supplied schematic has no numerical values.

Electrical circuit with an ideal transformer

electrical_transformer. Electrical circuit with an ideal transformer TF, resistors R1, R2, R3, capacitors C1, C2, and inductor L. Numerical values are chosen for demonstration and implementation checks. Select Electrical: ideal transformer in the example list.

Vi = 10 V; R1 = 10 Ω; R3 = 5 Ω; R2 = 20 Ω; C1 = 0.01 F; C2 = 0.02 F; L = 0.5 H; n = 2. Zero initial states; duration 5 s.

Se → 1 → 0 → 1 → TF → 1
     R1  C1  R3,L     R2,C2
n = N1/N2
C1 du1/dt = (vi − u1)/R1 − i1
L di1/dt = u1 − (R3 + n² R2) i1 − n u2
C2 du2/dt = n i1

u1 = b4.e; u2 = b11.e; i1 = b8.f; i2 = b9.f. Steady state: u1 = 10 V, u2 = 5 V, i1 = i2 = 0 A.

Verification checks: independent Kirchhoff equations, energy balance, the circuit referred to the primary winding, causality, and reversal of bond references.

E = C1 u1²/2 + L i1²/2 + C2 u2²/2
dE/dt = vi i_in − R1 i_in² − R3 i1² − R2 i2²
R2_referred = n² R2; C2_referred = C2/n²

Geared elastic shaft

Open Rotational mechanics: geared elastic shaft. Torque source Se drives inertia Jm through a common angular-velocity junction, with damping Bm to the support. Ideal gear TF connects to a common-torque junction that splits angular velocity into load speed and relative shaft speed. Spring K and damper B act in parallel on that relative motion. The load contains JL and damping BL.

Illustrative SI values, not measured data: Jm = 1 kg·m², JL = 2 kg·m², Bm = 1, B = 2, BL = 3 N·m·s/rad, K = 20 N·m/rad, n = 0.5, Tm = 1 N·m. All initial states are zero. Duration: 10 s; 1,001 points; rtol = 1e-7. The spring stores C = 0.05 rad/(N·m) in JSON.

Effort is torque, flow is angular velocity, and the applied torque uses Se. The dual representation with exchanged variables is not used. Shaft axes are chosen so ωT = n·ωm, although external gears physically rotate in opposite directions. Gear losses, backlash, dry friction and tooth inertia are omitted. Absolute shaft angles are not integrated; spring twist δ is available.

Verification compares independent integration of the shaft equations at n = 0.5, 1 and 2, with nonzero initial states and a torque step. Tests cover gear power conservation, energy balance with all three dampers, integral causality, excitation from either TF port and bond-reference reversal. A mixed-domain model is checked separately.

Rotational component reference

Ideal hydraulic pressure transformer

Choose Hydraulics: ideal pressure transformer (hydraulic_transformer). Two pistons on a common rod illustrate ideal conversion: Aₛ/Aₚ = n = 0.5. The piston backs are at reference pressure. This is a power-conversion sketch; piston dynamics and stroke limits are excluded.

Pressure:Se ──b1──→ P [TF: n=0.5] S ──b2──→ Resistance:R

Source pressure is zero before 1 s and 100000 Pa afterward. R = 10⁹ Pa·s/m³. No stored states; duration 3 s, 301 points. All quantities change immediately at the step.

Default charts show pressures b1.e and b2.e. Select flows b1.f, b2.f and powers b1.power, b2.power to compare. All are zero before 1 s. See the TF reference.

Four-strand polyspast lift

Choose Translational mechanics: four-strand polyspast lift (mechanical_transformer). Four parallel rope strands support the moving block; the final fixed pulley redirects the pulled end. P represents the pulled rope and S the lifted load. n = 1/4 gives four times the pulling force and one-quarter of the rope speed at the load.

Pull:Se ──b1──→ P [TF: n=0.25] S ──b2──→ 1:Load
                                                   ├──b3──→ I:Mass
                                                   └──b4──→ Se:Gravity

m = 10 kg; g = 9.80665 m/s²; weight = 98.0665 N; initial momentum is zero. Pull = 24.516625 N before 1 s and 30 N afterward. Gravity is a constant effort source on an outgoing junction bond, opposing the lift. Duration 3 s, 301 points. Force excitation preserves integral causality for the mass.

Default charts show b1.f (rope speed) and b2.f (load speed). At 3 s these are 17.5468 and 4.3867 m/s. b2.e is the lifting force; b3.e is the net force on the mass after subtracting weight. Mass.state is momentum, not height. Powers b1 and b2 agree.

Assumes a taut, inextensible rope and frictionless, massless pulleys. The positive input axis follows the rope pull; the positive output axis points upward. Height, travel limits and slack-rope behavior are not modeled. The same TF primitive also models an ideal lever with a constant motion ratio.