As the core component of CNC machine tools, electric spindle has the advantages of compact structure, stable operation and high speed, and is widely used in the field of CNC machining [1]. In order to ensure the machining accuracy and transmission capacity of the electric spindle, an interference connection is adopted between the spindle and the rotor, and the amount of interference will affect the overall cutting performance and processing efficiency of the machine tool [2]. If the interference is selected too large, it will increase the difficulty of assembly between the spindle and the rotor, and even cause the collapse of the mating surface, which will affect the machining accuracy of the machine tool [3] [4]. If the interference is selected too small, the extrusion force between the spindle and the rotor will be small, and sufficient torque will not be transmitted, or the mating surface will slide relatively, which will aggravate the wear of the spindle and affect the service life of the machine tool [5] [6]. When the electric spindle is running at high speed, due to the centrifugal effect, the spindle and the rotor will expand to different degrees, resulting in a great change in the interference fit characteristics between the spindle and the rotor [7] [8]. The letter provides a theoretical formula showing the relationship of the speed, stress on the rotation speed, and also establishes a finite element model of the interference fit [9] [10]. Knowing the interference between the spindle and rotor during the high speed, one can analyze the optimum transmission capacity to improve the processing quality of the parts and hence life of the machine tools.

When the spindle and the rotor are in an interference fit, a positive pressure is generated at the spindle surface, the spindle is compressed, and the rotor is expanded. Since the axial size of the spindle and the rotor is relatively large, and the size along the axial direction is almost unchanged, the axial elongation can be ignored. The matching size of the spindle and the rotor is shown in Figure 1.

The outer surface of the spindle is squeezed, and the inner surface has no load. The boundary conditions are:

{ σ ρ i ( ρ 1 = b ) = − P σ ρ i ( ρ 1 = a ) = 0 ( a ≤ ρ 1 ≤ b ) (1)

where P is positive pressure, N/m²; σ ρ i is the radial normal stress of the spindle, N/m².

The inner surface of the rotor is subject to expansion, and the outer surface has no load. The boundary conditions are:

{ σ ρ e ( ρ 2 = b ) = − P σ ρ e ( ρ 2 = c ) = 0 ( b ≤ ρ 2 ≤ c ) (2)

where σ ρ e is the radial normal stress of the rotor, N/m².

In polar coordinate form, take out the differential body at the angle φ and radius ρ, and the thickness of the differential body is equal to 1. Since the spindle and the rotor are both axisymmetric cylindrical sleeves, the tangential displacement and shear stress at the mating surface are zero in the polar coordinate system. The stress distribution of this differential body is shown in Figure 2. The forces on the differential body are projected onto the radial axis of the differential body to obtain a balanced differential equation.

d σ ρ d ρ + σ ρ − σ φ ρ = 0 (3)

According to the literature [11], the relationship between stress and displacement under plane strain is as follows:

{ σ ρ = E ( 1 − υ ) ( 1 + υ ) ( 1 − 2 υ ) ( d u ρ d ρ + υ 1 − υ u ρ ρ ) σ φ = E ( 1 − υ ) ( 1 + υ ) ( 1 − 2 υ ) ( u ρ ρ + υ 1 − υ d u ρ d ρ ) (4)

Substituting (4) into (3), we get:

d 2 u ρ d ρ 2 + 1 ρ d u ρ d ρ − u ρ ρ 2 = 0 (5)

The general solution of this second-order differential equation is:

u ρ = C 1 ρ + C 2 ρ (6)

According to Formulas (1) (4) (6), the relationship between the stress and displacement of the spindle can be obtained:

{ σ ρ i = b 2 P ( a 2 − b 2 ) ( 1 − a 2 ρ 1 2 ) σ φ i = b 2 P ( a 2 − b 2 ) ( 1 + a 2 ρ 1 2 ) u ρ i = ( 1 + υ i ) b 2 P ρ 1 E i ( a 2 − b 2 ) ( 1 − 2 υ i + a 2 ρ 1 2 ) (7)

where σ φ i is the tangential normal stress of the spindle, N/m²; u ρ i is the radial displacement of the spindle, m; E i is the elastic modulus of the spindle material, N/m²; υ i is the poisson’s ratio of the spindle material.

Similarly, the rotor stress and displacement can be determined by the following formula:

{ σ ρ e = b 2 P ( c 2 − b 2 ) ( 1 − c 2 ρ 2 2 ) σ φ e = b 2 P ( c 2 − b 2 ) ( 1 + c 2 ρ 2 2 ) u ρ e = ( 1 + υ e ) b 2 P ρ 2 E e ( c 2 − b 2 ) ( 1 − 2 υ e + c 2 ρ 2 2 ) (8)

where σ φ e is the tangential normal stress of the rotor, N/m²; u ρ e is the radial displacement of the rotor, m; E e is the elastic modulus of the rotor material, N/m²; υ e is the poisson’s ratio of the rotor material.

The materials used for the spindle and the rotor are steel materials, and their elastic modulus and poisson’s ratio can be considered the same. Therefore, the amount of interference at the mating surface can be determined by the following formula:

δ s = 2 ( 1 − υ 2 ) P b E ( c 2 + b 2 c 2 − b 2 + b 2 + a 2 b 2 − a 2 ) (9)

When the spindle is running at high speed it is subjected to centrifugal force. The balanced differential equation at this time is:

d σ ρ d ρ + σ ρ − σ φ ρ + ρ 0 ω 2 ρ = 0 (10)

where ρ 0 is the material density, kg/m³; ω is the angular velocity, rad/s.

According to Formulas (1) (4) (10), the spindle stress and displacement equations are obtained:

{ σ ρ i = ( 3 − 2 υ ) b 2 ρ 0 ω 2 8 ( 1 − υ ) ( 1 + a 2 b 2 − a 2 ρ 1 2 − ρ 1 2 b 2 ) + b 2 P ( a 2 − b 2 ) ( 1 − a 2 ρ 1 2 ) σ φ i = ( 3 − 2 υ ) b 2 ρ 0 ω 2 8 ( 1 − υ ) [ 1 + a 2 b 2 + a 2 ρ 1 2 − ( 1 + 2 υ ) ρ 1 2 ( 3 − 2 υ ) b 2 ] + b 2 P ( a 2 − b 2 ) ( 1 + a 2 ρ 1 2 ) u ρ i = ( 3 − 2 υ ) ( 1 + υ ) ( 1 − 2 υ ) b 2 ρ 0 ω 2 ρ 1 8 E ( 1 − υ ) [ 1 + a 2 b 2 + a 2 ( 1 − 2 υ ) ρ 1 2 − ρ 1 2 ( 3 − 2 υ ) b 2 ]                 + b 2 P ( 1 + υ ) ρ 1 ( a 2 − b 2 ) E ( 1 − 2 υ + a 2 ρ 1 2 ) (11)

According to Formulas (2) (4) (10), the rotor stress and displacement equations are obtained:

{ σ ρ e = ( 3 − 2 υ ) b 2 ρ 0 ω 2 8 ( 1 − υ ) ( 1 + c 2 b 2 − c 2 ρ 2 2 − ρ 2 2 b 2 ) + b 2 P ( c 2 − b 2 ) ( 1 − c 2 ρ 2 2 ) σ φ e = ( 3 − 2 υ ) b 2 ρ 0 ω 2 8 ( 1 − υ ) [ 1 + c 2 b 2 + c 2 ρ 2 2 − ( 1 + 2 υ ) ρ 2 2 ( 3 − 2 υ ) b 2 ] + b 2 P ( c 2 − b 2 ) ( 1 + c 2 ρ 2 2 ) u ρ e = ( 3 − 2 υ ) ( 1 + υ ) ( 1 − 2 υ ) b 2 ρ 0 ω 2 ρ 2 8 E ( 1 − υ ) [ 1 + c 2 b 2 + c 2 ( 1 − 2 υ ) ρ 2 2 − ρ 2 2 ( 3 − 2 υ ) b 2 ]               + b 2 P ( 1 + υ ) ρ 2 ( c 2 − b 2 ) E ( 1 − 2 υ + c 2 ρ 2 2 ) (12)

When running at high speed, the interference of the electric spindle can be determined by the following formula:

δ = ( 3 − 2 υ ) ( 1 + υ ) ( c 2 − a 2 ) b ρ 0 ω 2 2 E + 2 ( 1 − υ 2 ) P b E ( c 2 + b 2 c 2 − b 2 + b 2 + a 2 b 2 − a 2 ) (13)

In the finite element analysis software ANSYS Workbench, the static structural module is used to simulate the interference fit model of the spindle and the rotor. The contact between the spindle and the rotor is frictional contact, where the outer surface of the spindle is the contact and the inner surface of the rotor is the target.

When the electric spindle is running at a low speed, the coordination characteristics between the spindle and the rotor are affected by centrifugal force very little and can be ignored. When the electric spindle is running at high speed, the coordination characteristics between the spindle and the rotor are greatly affected by the centrifugal force. As the rotation speed increases, the interference and contact pressure between the spindle and the rotor decrease, and it is proportional to the square of the rotation speed. Therefore, the influence of centrifugal force on the interference fit of the electric spindle must be considered.

The authors declare no conflicts of interest regarding the publication of this paper.

Ding, C., Qi, X.Y., Gao, Z. and Chang, L. (2020) The Influence of Centrifugal Force on the Interference Fit of High-Speed Electric Spindle. World Journal of Engineering and Technology, 8, 792- 799. https://doi.org/10.4236/wjet.2020.84058