Stress Verification of High-Speed Motor Rotors: From "Able To Calculate" To "Calculating Correctly"

I. Why Must High-Speed Motors Undergo Stress Verification?

For a high-speed motor with a rotational speed of 30,000 r/min, the centrifugal acceleration at the outer diameter of the rotor can reach tens of thousands of times the gravitational acceleration. Under such extreme operating conditions, the tensile stress faced by the permanent magnets will approach or even exceed their tensile limit—and the tensile strength of NdFeB permanent magnets is typically only about one-tenth of their compressive strength. Once the tensile stress exceeds the limit, the permanent magnets may fracture, and the high-speed rotating fragments are sufficient to destroy the entire motor within milliseconds.

At the same time, the rated speed of high-speed motors often approaches the critical speed of the rotor, and the risk of resonance rises sharply. Stress verification is not only about "whether it will break," but also about whether the stiffness design of the rotor can avoid dangerous resonance ranges.

For this reason, in the design process of high-speed motors, the first step after completing the electromagnetic design is to perform rotational stress analysis, checking the circumferential stress and von Mises stress distribution at each part of the rotor, and comparing them with the strength indices of the corresponding materials.

II. What Forces Does the Rotor Bear?

The stress sources of high-speed motor rotors mainly fall into three categories, which often act in superposition, forming a complex stress field.

Centrifugal force—the most important load. When the rotor rotates at high speed, every mass point is subjected to a centrifugal force directed radially outward, and its magnitude is proportional to the square of the rotational speed:

F = ρω⊃2;r⊃2;

where ρ is the material density, ω is the angular velocity, and r is the radial distance. The farther the material is from the rotating shaft, the greater the centrifugal load it bears. This means that the circumferential stress at the outer edge of the sleeve is usually the highest in the entire rotor.

Pre-stress generated by interference fit—a "double-edged sword." The permanent magnets of surface-mounted permanent magnet motors have low tensile strength and cannot withstand centrifugal force alone. Therefore, a high-strength sleeve must be fitted on the outside of the permanent magnets, applying pre-compression to the permanent magnets through an interference fit to offset the centrifugal tensile force during rotation. However, the interference amount is not as large as possible: excessive interference will introduce excessive compressive stress into the permanent magnets, and although permanent magnets have stronger compressive capacity than tensile capacity, it is not unlimited.

Thermal stress—an easily overlooked "hidden killer." During motor operation, temperature rise causes thermal expansion of various components, and different materials have different coefficients of linear expansion, so the resulting thermal stress may be considerable. At the mating surface between the sleeve and the permanent magnet, temperature changes will also alter the magnitude of the contact pressure: when the temperature rises, the contact pressure increases, and the sleeve stress also rises accordingly. Under low-temperature conditions, the difference in coefficients of linear expansion can even be utilized to realize the process scheme of "low-temperature assembly, room-temperature interference."

III. Verification Methods: Analytical Method and Finite Element Method

3.1 Analytical Method: A Sharp Tool for Rapid Estimation

The core idea of the analytical method is to simplify each rotor component into a thick-walled cylinder (or thick-walled ring) model, and establish a unified mathematical model of the stress field in two-dimensional polar coordinates. For the shaft, permanent magnet, and sleeve, the radial and tangential stress-strain relationships are listed separately, and the equations are solved simultaneously through the displacement continuity condition and force equilibrium condition at the interface.

Taking a 250 kW, 25 kr/min high-speed permanent magnet motor as an example: the permanent magnet outer radius is 44 mm, thickness is 8 mm, the carbon fiber sleeve thickness is 2 mm, and the interference between the sleeve and the permanent magnet is 0.3 mm. At 1.2 times the rated speed (30 kr/min), the rotor radial displacement, radial stress, and tangential stress distribution calculated by the analytical method are highly consistent with the finite element analysis results.

The advantages of the analytical method are its fast calculation speed and clear physical meaning, making it especially suitable for parameter scanning and optimization iteration in the early design stage. However, its limitations are also obvious: it is difficult to handle complex geometric shapes (such as the magnetic bridge structure of interior rotors) and the nonlinear behavior of anisotropic materials.

3.2 Finite Element Method: The Yardstick for Accurate Verification

The finite element method (FEM) can handle arbitrary geometric shapes, multiple material combinations, and complex boundary conditions, and is the "gold standard" for stress verification.

The key to modeling lies in the treatment of contact surfaces. The interference fit between the sleeve and the permanent magnet is usually simulated in finite element analysis using contact elements, and the contact stiffness, friction coefficient, and interference amount need to be set correctly. For anisotropic materials such as carbon fiber, the elastic modulus and Poisson's ratio in each direction also need to be input.

The selection of verification conditions is equally important. In engineering practice, the following combinations usually need to be verified:

• Sleeve strength: von Mises stress at high temperature + 1.2 times rated speed;

• Permanent magnet tensile strength: maximum principal stress at low temperature + 1.2 times rated speed;

• Contact pressure: minimum contact pressure at low temperature + 1.2 times rated speed, to ensure that the sleeve does not separate from the permanent magnet.

These three types of verification cover the most severe combination of operating conditions, and none of them can be omitted.

IV. Sleeve Design: The Rotor's "Seatbelt"

The sleeve is the most critical protective component of the high-speed permanent magnet motor rotor. Currently, mainstream sleeve materials fall into two major categories:

Metal sleeves (titanium alloy, Inconel alloy, etc.) have good electrical conductivity and will generate eddy current losses in the alternating magnetic field, reducing motor efficiency; however, their high thermal conductivity is beneficial for heat dissipation. Metal sleeves are more suitable for scenarios with demanding air-gap requirements and limited heat dissipation conditions.

Carbon fiber composite sleeves are currently the mainstream choice for high-speed motors. Their tensile strength can reach more than 3500 MPa (as required by group standards), their density is only about one-quarter that of titanium alloy, and they are non-conductive and do not generate eddy current losses. However, the anisotropy of carbon fiber sleeves brings additional complexity to stress analysis, requiring the use of three-dimensional elasticity theory for analysis.

The core contradiction in sleeve design lies in thickness: the thicker the sleeve, the stronger the constraint on the permanent magnet and the lower the rotor stress; but an increase in sleeve thickness will increase the equivalent air gap and weaken electromagnetic performance. Therefore, sleeve design is essentially a multi-objective optimization problem—it is necessary to find the optimal balance between structural strength and electromagnetic performance.

V. How Should the Safety Factor Be Selected?

The value of the safety factor directly determines whether the verification conclusion is reliable. If the value is too small, it is risky; if it is too large, it leads to excessive design redundancy and degraded performance.

In engineering practice, the safety factor for high-speed motor rotor verification is usually determined according to the following principles:

• Sleeve material: based on yield strength or tensile strength, a safety factor of 1.5–2.0 is taken. For example, when the maximum stress of the sleeve of a certain ultra-high-speed permanent magnet motor is 967 MPa, a safety factor greater than 2 is considered acceptable.

• Permanent magnet: because NdFeB is a brittle material and the dispersion of its tensile strength is relatively large, a safety factor of 2.0–3.0 is usually taken. In some design cases, the safety factor of the permanent magnet can reach more than 6, leaving ample margin.

• Shaft: shaft materials are usually alloy steel, with good toughness and high strength, and the stress level is far below the allowable value, so a safety factor of about 1.5 is sufficient.

• Contact pressure: to ensure that the sleeve and the permanent magnet do not separate at the highest speed, the minimum contact pressure is usually required to be no less than several megapascals. In a certain design case, the minimum contact pressure is required to be no less than 5.4 MPa at the lower limit of the rated speed and no less than 2.7 MPa at the upper limit.

VI. From Verification to Validation

Stress verification is ultimately "paper calculation." In engineering practice, the verification results also need to be validated through experiments.

Common validation methods include:

Rotational testing is the most direct validation method. The rotor is accelerated to an overspeed condition (usually 1.2 times the rated speed) to observe whether structural failure occurs. Some studies have specifically designed prototypes in which the carbon fiber sleeve fails at 30,000 r/min, in order to observe the behavioral characteristics of the rotor during actual failure.

Strain measurement can capture strain data at key positions in real time during rotation. Using strain gauges and telemetry systems, strain at key rotor positions can be measured at 18,000 r/min or even higher speeds, and correlation validation can be performed with finite element calculation results.

Static tensile testing is an indirect but safe validation method. By performing static tensile tests on the rotor core, the danger of high-speed rotation can be avoided, and the accuracy of the analytical model and finite element results can be indirectly verified.

Conclusion

High-speed motor rotor stress verification is a systematic engineering task that integrates elasticity, materials science, and numerical computation. From understanding the superimposed effects of centrifugal force, interference force, and thermal stress, to selecting appropriate verification methods and safety factors, to validating the reliability of the design through experiments, every step is related to whether the motor can operate stably under extreme conditions of tens of thousands of revolutions per minute.

As the speed of drive motors for new energy vehicles exceeds 20,000 r/min and the speed of flywheel energy storage motors approaches 70,000 r/min, the accuracy and efficiency requirements for rotor stress verification continue to increase. New technologies such as carbon fiber composite sleeves, multilayer sleeve structures, and multiphysics coupling optimization are constantly expanding the performance boundaries of high-speed motors, and stress verification is always the safety bottom line for all these innovations.