Introduction
Moment of inertia is a measure of the inertia (the property of a rotating object to maintain its uniform circular motion or remain at rest) of a rigid body rotating about an axis, denoted by the letters I or J. In rotational dynamics, the role of moment of inertia is analogous to mass in linear dynamics; it can be figuratively understood as the inertia of an object with respect to rotational motion. Inertia is a crucial parameter during acceleration and deceleration under load; therefore, a thorough understanding of the inertia calculation methods for commonly used transmission mechanisms is essential in motion control.
This article summarizes the inertia calculation methods for various common mechanisms and provides two application cases for calculating the selection of Leadshine servo motors.
1. Calculation methods for the load inertia of five common transmission mechanisms in servo drive systems.
1.1 Calculation of Inertia of Common Objects
Model 1
A thin rod of length L has its center of rotation passing through the center of the rod and perpendicular to it, as shown in the figure below.
Model 2
A thin rod of length L has its center of rotation passing through one end A of the rod and perpendicular to the rod, as shown in the figure below.
Similarly, it can be concluded that
Model 3
A thin, uniform circular ring of mass m has radius R. Its center of rotation passes through the center of the ring and is perpendicular to its surface.
Consider a length element dx, assuming the mass density of the rod is λ. Then the mass of the length element is dm = λdl. According to the formula for calculating the moment of inertia:
Model 4
A disk or solid cylinder of mass m, radius R, and thickness h rotates about its axis.
Consider a thin circular ring of arbitrary radius r and width dr. Let ρ be the density of the disk and dm be the mass of the thin circular ring. Then the moment of inertia of this ring is...
According to this formula, the moment of inertia of a cylinder with diameter D rotating about its central axis is:
Where L is the length of the cylinder, ρ is the density.
Model 5
Load inertia driven by the lead screw
Note: In the formula, Pb is the lead screw (pitch).
Summarize
Model 1 and Model 2 can be applied to the inertia calculation of uniform strip or rod-shaped load structures.
Model 3 can be applied to the inertia calculation of synchronous wheel load structures.
Model 4 can be applied to the calculation of the inertia of the lead screw itself or the inertia calculation of a cylindrical structure.
Model 5 can be applied to the calculation of load inertia driven by a lead screw.
Note: Mnemonic for calculating common rigid body inertia
1.2 Calculation Methods for Load Inertia of Five Common Transmission Mechanisms in Servo Drive Systems
Based on the above five models, we can provide methods for calculating the load inertia of five common transmission mechanisms in servo drive systems (screw mechanism, synchronous belt pulley mechanism, gear and rack structure, disc structure, and long arm structure).
Screw structure
Screw inertia
Synchronous belt pulley/rack structure
turntable structure
Long arm structure
2. Calculation and Selection Examples
Leadshine's AC servo motors are generally available in models with different inertia values, such as the 60 and 80 frame motors, which are offered in both medium and low inertia versions. The following section uses two common case studies to calculate load inertia and demonstrate appropriate motor selection, illustrating methods to reduce inertia mismatch.
2.1 Screw Structure
Given: load weight m = 200 kg, screw pitch P_b = 20 mm, screw diameter D_b = 50 mm, screw weight m_b = 40 kg, coefficient of friction µ = 0.002, mechanical efficiency η = 0.9, load moving speed V = 30 m/min, total moving time t = 1.4 s, acceleration/deceleration time t1 = t3 = 0.2 s, and stationary time t4 = 0.3 s. Please select the minimum power servo motor that meets the load requirements.
01 Calculate the load inertia referred to the motor shaft
02 Calculate motor speed
Required speed of motor
03 Calculate the torque required for the motor to drive the load.
Torque required to overcome friction
Selected motor option:
The total inertia of the motion system is 145.29 kg*cm^2, and the required maximum torque is 12.686 Nm. The Leadsai ACM13030M2E-51-B motor has a rated speed of 2500 RPM, a rated torque of 12 Nm, and a rotor inertia of 29 kg*cm^2. The load inertia ratio is approximately 145/29 ≈ 5 times, which meets the requirements.
2.2 Synchronous Pulley Structure
Given: In the rapid positioning motion model, the load weight M = 5kg, the diameter of the synchronous pulleys D = 60mm, D1 = 90mm, D2 = 30mm, the coefficient of friction between the load and the machine tool µ = 0.003, the maximum speed of the load is 2m/s, and the time for the load to accelerate from a standstill to the maximum speed is 100ms. Ignore the weight of each conveyor pulley and select a servo motor.
01 Calculate the load inertia referred to the motor shaft
02 Calculate the torque required for the motor to drive the load.
Torque required to overcome friction
Torque required during acceleration
03 Required Torque
04 Calculate the required speed of the motor
Selected motor option:
Based on the above calculation results, the Leadshine ACM6006L2H servo motor (rated torque 1.9NM, rated speed 3000RPM, motor inertia 0.6 kg*cm^2) can be selected, with an inertia ratio of 5 / 0.6 = 8.3 times.
During my observations at some customer sites, I found that some users were using the following motor model: ACM6004L2H (rated torque 1.27 Nm, peak torque 3.81 Nm, rated speed 3000 RPM, motor inertia 0.42 kg.cm^2). If this solution is chosen, the system inertia ratio is 5/0.42 = 11.9 times. The dynamic response performance and positioning completion time will be worse than if the ACM6006L2H servo solution is used. A reasonable inertia ratio significantly improves the dynamic performance of the entire motion system.
3. Conclusion
In servo drive systems, there are five common transmission mechanisms: lead screw mechanisms, synchronous belt pulley mechanisms, gear and rack structures, disc structures, and long-arm structures. Engineers should be proficient in calculating the load inertia of each mechanism. Only on this basis can the inertia ratio be calculated correctly. To improve the fast response characteristics of a servo system, it is first necessary to increase the resonant frequency of the mechanical transmission components, that is, to increase the rigidity of the mechanical transmission components and reduce their inertia. Secondly, increasing the damping to reduce the resonant peak value can also create conditions for improving fast response characteristics. Thirdly, if the load inertia is large, a reduction gear mechanism can be considered to achieve an inertia ratio between the load inertia and the motor inertia within a suitable range. In some application cases, a motor with a larger inertia can also be selected to meet the requirements of reducing the inertia ratio and improving acceleration performance and stability. Finally, the successful application of many new technologies in servo drive control algorithms has also made it possible for servo systems to operate with higher precision and higher stability. For a more detailed discussion on the reasonable value of the inertia ratio, please refer to the Leadshine Company article "Reasonable Value of Load Inertia Ratio for Servo Motors".
