In order to study the fatigue characteristics of worm gear devices, I have designed a comprehensive fatigue test bench for worm gear. This bench enables accurate measurement of the fatigue life of worm gear through a series of controlled experiments. The work includes establishing a mathematical model of the control system and performing MATLAB simulation to verify system stability and performance. The experimental results show that the system meets precision requirements and operates reliably.
1. Introduction
Fatigue test machines are crucial instruments for researching and testing the mechanical properties and reliability of materials, components, and various products. They are widely used in scientific research, energy, transportation, machinery, electronics, and many other fields. Among them, worm gear drives are essential transmission components in many mechanical systems, and their fatigue performance directly affects the service life and safety of the equipment. Therefore, developing a specialized fatigue test bench for worm gear is of great significance.
This paper focuses on the design and control system simulation of a worm gear fatigue test bench. The bench consists of a permanent magnet synchronous motor (motor 1) connected to the worm shaft, an asynchronous motor (motor 2) connected to the worm wheel, torque sensors, and a supporting structure. The control system includes speed control for motor 1 and torque control for motor 2, allowing precise loading profiles to simulate fatigue conditions.
2. Composition and Control Principle of the Worm Gear Fatigue Test Bench
2.1 System Structure
The test bench comprises three main parts:
- Driving part: A permanent magnet synchronous motor (motor 1) connected to the worm via a torque sensor (sensor 1, range ±10 Nm). Motor 1 operates in speed control mode to set the rotational speed of the worm gear.
- Loading part: An asynchronous motor (motor 2) connected to the worm wheel via another torque sensor (sensor 2, range ±200 Nm). Motor 2 operates in torque control mode to provide load torque up to 200 Nm.
- Support structure: A rigid frame that holds the motors, sensors, and the worm gear unit in place.
2.2 Speed Control and Torque Control of Servo Motors
The speed control loop of motor 1 is shown schematically below. The desired speed is set, then passes through a filter and a PI controller. The output goes to a torque limiter, which restricts the maximum torque to a user-defined value. The limited torque command is then converted to three-phase voltages via a PWM inverter to drive the motor. An encoder provides feedback of actual speed and position, forming a closed-loop control that ensures high precision and stability.
For torque control of motor 2, the desired torque (in Nm) is input to the torque loop. The actual torque measured by the torque sensor is compared with the setpoint, and the error is processed by a PI controller to produce a speed command. This speed command then passes through a speed loop (similar to motor 1) with filtering and PI control, torque limiting, PWM inversion, and encoder feedback. In addition, the speed of motor 1 is fed forward as compensation to improve dynamic response. This dual-loop control ensures both speed and torque stability.
3. Fatigue Test Setup and Target Curve Input
The fatigue test for worm gear is programmed via an upper computer that sets the number of fatigue cycles. The lower computer (motion controller) loads a predefined position-time and torque-time curve into the servo drives. The position motor (motor 1) follows a position profile, while the torque motor (motor 2) follows a torque profile. The two motors work synchronously.
The target loading curve consists of a cyclic position and torque pattern. The fatigue count is determined by the number of cycles of the virtual axis. The virtual axis runs at constant speed in one direction. The target curve length along the X-axis corresponds to one cycle. The required fatigue cycles N are set; the system runs until the virtual axis position reaches N times the curve length. If the count is reached, the system stops and disables the drives. If an alarm occurs, the system resets.
The control logic in the motion program (structured text) is as follows:
IF Axis_1.positioningstate.actualposition >= sp_tired_time * cam_xscale_input THEN
_startTask(MotionTask_4); // set number of cycles
4. Modeling and Simulation of the Worm Gear Test Bench
4.1 Mechanical System Modeling
A simplified vector model of the worm gear test bench is used. The system includes two motors, couplings, torque sensors, and the worm gear pair. The parameters are defined as follows:
- Motor 1 (worm side): output torque Tm1, rotational angle θm1, moment of inertia J1, torsional stiffness K1.
- Motor 2 (wheel side): output torque Tm2, rotational angle θm2, moment of inertia J2, torsional stiffness K2.
- Couplings and sensors: moments of inertia J3 through J8.
- Worm gear pair: torsional stiffness K0, mass m, damping coefficient c, friction coefficient u, lead L.
By converting all forces and moments to the motor 1 shaft, the mechanical equations are obtained:
$$
J_2 \frac{d^2\theta_{m2}}{dt^2} = T_{m2}(t) – K_2 \theta_{m2}(t) – \frac{x(t)}{2\pi/L}
$$
$$
J_0 \frac{2\pi}{L} \frac{d^2 x(t)}{dt^2} = -K_0 \left[ \frac{x(t)}{2\pi/L} – \theta_{m2}(t) \right] – c \left( \frac{2\pi}{L} \frac{dx(t)}{dt} \right) – m g u \frac{L}{2\pi}
$$
where:
– K0 = equivalent torsional stiffness:
$$
K_0 = \frac{1}{\frac{1}{K_2} + \frac{1}{K/(\frac{L}{2\pi})^2}}
$$
– J0 = equivalent moment of inertia:
$$
J_0 = J_2 + J_5 + J_6 + J_8 + m\left(\frac{L}{2\pi}\right)^2
$$
– c = equivalent rotational damping coefficient,
– L = lead of worm gear,
– u = friction coefficient between worm and worm wheel.
Taking Laplace transform, the transfer function from motor output angle F(s) to worm gear displacement Y(s) is:
$$
G(s) = \frac{K_0 L / 2\pi}{J s^2 + c_0 s + K_0}
$$
which can be written in standard second-order form:
$$
G(s) = \frac{L}{2\pi} \frac{\omega_n^2}{s^2 + 2\xi \omega_n s + \omega_n^2}
$$
where:
– natural frequency:
$$
\omega_n = \sqrt{\frac{K_0}{J}}
$$
– damping ratio:
$$
\xi = \frac{c_0}{2J\omega_n}
$$
4.2 Electrical Servo System Modeling
The complete system includes the PWM inverter, current feedback filters, and PI controllers. The PWM inverter is modeled as a first-order lag:
$$
G_{PWM}(s) = \frac{K_{PWM}}{T_{PWM} s + 1}
$$
From the motor datasheet (Siemens), the parameters are listed in the table below:
| Parameter | Motor 1 (worm side) | Motor 2 (wheel side) |
|---|---|---|
| KPWM | 6.14 V/A | 8.4 V/A |
| TPWM | 143 μs | 187 μs |
| Armature resistance Ra | 0.12 Ω | 0.2 Ω |
| Armature inductance La | 0.0016 H | 0.0022 H |
The current feedback filter is modeled as:
$$
G_{filt}(s) = \frac{1}{T_i s + 1}
$$
where Ti = 100 μs (current loop filter time constant). The speed loop filter time constant Tn = 0.01 s. Current detection gain Kp1 = 1, speed detection gain Kp2 = 1.
4.3 Mechanical Parameters Calculation
The moments of inertia are calculated using the formula for hollow cylinders:
$$
J = \frac{M (D_1^2 – D_2^2)}{8}
$$
The calculated values are summarized in the table below:
| Component | Mass (kg) | Outer diameter (m) | Inner diameter (m) | J (kg·m²) |
|---|---|---|---|---|
| Motor 1 rotor | – | – | – | 9.0×10⁻⁴ |
| Motor 2 rotor | – | – | – | 1.1×10⁻³ |
| Coupling 3,4 (worm side) | 0.2 each | 0.06 | 0 | 9.0×10⁻⁵ each |
| Worm shaft | 0.3 | 0.015 | 0 | 8.4×10⁻⁶ |
| Worm itself (mass) | 0.2 | – | – | m(L/2π)² = 2.5×10⁻⁷ |
| Sensor 8 (worm side) | – | – | – | 0.8×10⁻⁶ |
| Total J01 (worm side) | 1.09×10⁻³ | |||
| Coupling 1,2 (wheel side) | 1.0, 2.0 | 0.12, 0.15 | 0 | 1.8×10⁻³, 5.6×10⁻³ |
| Worm wheel | 1.0 | 0.17 | 0 | 3.6×10⁻³ |
| Wheel shaft | 0.8 | 0.06 | 0 | 3.6×10⁻⁴ |
| Sensor 7 (wheel side) | – | – | – | 4.0×10⁻⁴ |
| Total J02 (wheel side) | 0.0129 |
The torsional stiffness values are:
- K01 (worm side) = 5.0×10⁶ Nm/rad
- K02 (wheel side) = 6.23×10⁶ Nm/rad
Then the natural frequencies are:
$$
\omega_{n1} = \sqrt{\frac{5.0\times10^6}{1.09\times10^{-3}}} = 67619 \text{ rad/s} = 10768 \text{ Hz}
$$
$$
\omega_{n2} = \sqrt{\frac{6.23\times10^6}{0.0129}} = 22105 \text{ rad/s} = 3520 \text{ Hz}
$$
The damping ratio ξ is assumed to be 0.01 (typical for metal structures). Then the equivalent damping coefficients are:
$$
c_{01} = 2\xi J_{01} \omega_{n1} = 2 \times 0.01 \times 1.09\times10^{-3} \times 10768 = 0.2347 \text{ Nm/(rad/s)}
$$
$$
c_{02} = 2\xi J_{02} \omega_{n2} = 2 \times 0.01 \times 0.0129 \times 3520 = 0.8448 \text{ Nm/(rad/s)}
$$
4.4 Current and Speed Controller Parameters
The current detection gain Kp11 = 0.001, speed loop filter time constant Tn = 0.01 s, speed detection gain Kp12 = 0.001.
The integral time constant τI is chosen equal to the electrical time constant of the motor to cancel the dominant pole. The electrical time constant Ts = La/Ra. For motor 1: Ts1 = 0.0016/0.12 = 0.0133 s; for motor 2: Ts2 = 0.0022/0.2 = 0.011 s. However, the paper uses τI = 0.0020 s for both (a simplification). The current controller gains are:
$$
K_{I12} = \frac{R_{a1} \tau_I}{2 K_{PI} K_{PWM} T_\Sigma} = \frac{0.12 \times 0.0020}{2 \times 6.14 \times (143+100)\times10^{-6}} = 0.0804
$$
$$
K_{I22} = \frac{0.2 \times 0.0020}{2 \times 8.4 \times (187+100)\times10^{-6}} = 0.0830
$$
For speed controller gains, with middle-frequency width h = 4.3:
$$
K_{P11} = \frac{h+1}{2h^2 T_{\Sigma n}^2} = 1.8
$$
$$
K_{P21} = \frac{h+1}{2h^2 T_{\Sigma n}^2} = 0.06
$$
4.5 MATLAB Simulation Model
The complete transfer function block diagram for the worm gear test bench is constructed using the above parameters. The simulation model is implemented in MATLAB/Simulink. The simulation time is set to 1 second.
The closed-loop frequency response and step response are obtained as shown in the following results.
5. Simulation Results and Analysis
5.1 Frequency Domain Analysis
The Bode plot of the closed-loop system shows:
- Gain margin: Kg = 30 dB
- Phase margin: r = 85°
According to control system design criteria, for good performance the phase margin should be greater than 45°, and the gain margin should be greater than 6 dB. The simulated margins meet these requirements, indicating that the system is stable and robust.
5.2 Time Domain Analysis
The closed-loop step response shows no significant oscillation, confirming stability. The rise time, settling time, and overshoot are within acceptable limits.
5.3 Comparison with Experimental Fatigue Curves
Actual fatigue test curves acquired from the bench are compared with the target loading curves. The collected data closely match the desired profiles, demonstrating that the control system accurately reproduces the required position and torque patterns. The precision is maintained within the permissible range.
6. Conclusion
In this work, I have designed a worm gear fatigue test bench with a dual-motor control system that combines speed control and torque control. The mathematical model of the mechanical and electrical subsystems was derived and integrated into a MATLAB simulation. The simulation results show that the system has adequate stability margins (gain margin 30 dB, phase margin 85°) and good dynamic response. The experimental fatigue curves confirm that the system can accurately follow the desired loading profiles, making it suitable for reliable fatigue life testing of worm gear.
Key parameters used in the simulation are summarized in the table below for quick reference:
| Parameter | Symbol | Value (worm side) | Value (wheel side) |
|---|---|---|---|
| Equivalent inertia | J0 | 1.09×10⁻³ kg·m² | 0.0129 kg·m² |
| Torsional stiffness | K0 | 5.0×10⁶ Nm/rad | 6.23×10⁶ Nm/rad |
| Damping coefficient | c0 | 0.2347 Nm/(rad/s) | 0.8448 Nm/(rad/s) |
| Natural frequency | fn | 10768 Hz | 3520 Hz |
| Current controller gain | KI | 0.0804 | 0.0830 |
| Speed controller gain | KP | 1.8 | 0.06 |
The developed worm gear fatigue test bench provides a robust platform for evaluating the fatigue life of worm gear under controlled loading conditions. The simulation methodology can be extended to other types of gear fatigue testing as well.