904 - Mechanical Systems¶

Motor-load dynamics and electromechanical simulation.

Overview¶

Power electronics often drive mechanical loads through electric machines. GeckoCIRCUITS supports: - Rotating machine models (PMSM, BLDC, IM) - Mechanical load models - Multi-mass systems - Gear boxes and couplings

Mechanical-Electrical Analogy¶

Force/Torque Analogy¶

Mechanical (Rotational)	Electrical	Unit
Torque τ	Voltage V	Nm / V
Angular velocity ω	Current I	rad/s / A
Inertia J	Capacitance C	kg·m² / F
Friction b	Conductance G	Nm·s / S
Stiffness k	1/L	Nm/rad / 1/H

Basic Mechanical Models¶

Single Inertia¶

\[J \frac{d\omega}{dt} = \tau_{motor} - \tau_{load} - b\omega\]

In GeckoCIRCUITS: Capacitor with value J

Two-Mass System¶

Models shaft compliance:

    Motor        Shaft         Load
   ┌─────┐    ┌────────┐    ┌─────┐
   │  J1 │────│ k, b   │────│  J2 │
   └─────┘    └────────┘    └─────┘

Equations: $$J_1 \frac{d\omega_1}{dt} = \tau_m - \tau_s$$ $$J_2 \frac{d\omega_2}{dt} = \tau_s - \tau_L$$ $$\tau_s = k(\theta_1 - \theta_2) + b(\omega_1 - \omega_2)$$

Load Types¶

Constant Torque¶

\[\tau_L = \tau_0\]

Applications: Conveyors, hoists, cranes

Quadratic (Fan/Pump)¶

\[\tau_L = k \cdot \omega^2\]

Applications: Fans, pumps, blowers

Linear (Friction)¶

\[\tau_L = b \cdot \omega\]

Viscous friction model

Constant Power¶

\[\tau_L = \frac{P_0}{\omega}\]

Applications: Winders, machine tools

Motor Models¶

PMSM (Permanent Magnet Synchronous Motor)¶

Electrical equations (dq frame): $$v_d = R_s i_d + L_d \frac{di_d}{dt} - \omega_e L_q i_q$$ $$v_q = R_s i_q + L_q \frac{di_q}{dt} + \omega_e L_d i_d + \omega_e \psi_m$$

Torque: $$\tau_e = \frac{3}{2} p [\psi_m i_q + (L_d - L_q)i_d i_q]$$

BLDC (Brushless DC)¶

Trapezoidal back-EMF model: $$e_{ph} = k_e \cdot \omega \cdot f(\theta_e)$$

Where f(θe) is trapezoidal function of electrical angle.

Induction Motor¶

Rotor flux model: $$\frac{d\psi_r}{dt} = \frac{L_m}{\tau_r}i_s - \frac{1}{\tau_r}\psi_r + j\omega_{slip}\psi_r$$

Torque: $$\tau_e = \frac{3}{2}p\frac{L_m}{L_r}Im\{\psi_r^* i_s\}$$

Gearbox Modeling¶

Ideal Gear Ratio¶

\[\omega_2 = \frac{\omega_1}{n}$$ $$\tau_2 = n \cdot \tau_1\]

Reflected Inertia¶

\[J_{eq} = J_1 + \frac{J_2}{n^2}\]

Inertia seen from motor side

Gear Efficiency¶

$$\tau_2 = \eta \cdot n \cdot \tau_1$$ (motoring) $$\tau_2 = \frac{n \cdot \tau_1}{\eta}$$ (regenerating)

GeckoCIRCUITS Setup¶

Adding Mechanical Domain¶

Insert motor model from component library
Set parameters:
Electrical: Rs, Ld, Lq, ψm
Mechanical: J (motor inertia), b (friction)
Pole pairs p
Add load model:
Connect mechanical port to load
Set load type and parameters

Multi-Mass Systems¶

Use mechanical nodes (angular velocity)
Connect:
Inertia elements (capacitors)
Shaft compliance (spring-damper)
Gearbox (transformer)

Simulation Exercises¶

Startup transient of PMSM with fan load
Compare single vs two-mass system response
Analyze gear ratio selection
Simulate regenerative braking

Design Considerations¶

Inertia Ratio¶

\[\sigma = \frac{J_{load}}{J_{motor}}\]

Ratio	Application	Response
σ < 3	Servo	Fast
3 < σ < 10	Industrial	Moderate
σ > 10	High inertia	Slow, oscillation risk

Resonance Frequency¶

For two-mass system: $$f_{res} = \frac{1}{2\pi}\sqrt{\frac{k(J_1 + J_2)}{J_1 J_2}}$$

Control bandwidth should be below resonance.