# Calculation of interference fit of motorized spind

2022-08-17
• Detail

The calculation of the interference fit of the motorized spindle of high-speed machine tools

1 preface

high-speed machine tools are the needs of the development of modern high-speed machining technology. As the core component of high-speed machine tools, high-speed spindle unit should have the following two basic characteristics:

first, it should have a high speed. The index to measure the spindle speed is dm N value, DM of high-speed spindle The value of n is generally required to be above 1.0106, and has a wide constant power speed regulation range, good speed up and down characteristics and fast quasi stop function. At present, the maximum spindle speed of medium-sized high-speed machining centers is 12000 ~ 60000 R/min

secondly, the high-speed spindle should have large power. Although the cutting force will decrease and the cutting process will become easier under the condition of high-speed cutting, due to the high cutting speed, the energy consumed per unit time will also increase significantly, so a larger power is required. At present, the main motor power of the high-speed machining center is 10 ~ 45 kW

the traditional main transmission mode of main motor + belt transmission + gear speed change, due to the long transmission chain, low transmission efficiency, and the large moment of inertia of the transmission system at high speed, it is difficult for the system to obtain a higher angular acceleration (deceleration); The dynamic and thermal characteristics are poor, so it is difficult to meet the requirements of high-speed NC machining

at present, the spindle type that can better meet the needs of high-speed machining is the electric spindle, that is, it is directly driven by the motor without housing. In the gd-ii electric spindle developed by our school, the rotor of the motor is directly assembled on the spindle, and the stator of the motor is matched with the shell of the spindle unit, eliminating a complete set of intermediate transmission links such as pulley and gearbox. The stepless speed regulation of the spindle is carried out by the frequency converter, and the zero transmission of the main shaft system is realized 

due to the absence of intermediate transmission chain, the spindle system has compact structure, small inertia, good dynamic response characteristics, can achieve rapid start, speed change, quasi stop, and has good c-axis control function

in order to ensure high-speed cutting, the spindle should have good motion accuracy and transmission capacity, and the spindle parts should have good machining accuracy and surface quality, as well as good assembly accuracy. In order to easily achieve accurate dynamic balance, keyless interference connection is adopted between the motor rotor and the machine tool spindle, so as to form torque transmission capacity. The size of the interference directly affects the performance of the motorized spindle. The interference will make the assembly of the spindle difficult, affect the assembly accuracy, and even destroy the mating surface; Too small interference will affect the ability of the spindle to transmit torque. Therefore, the interference between the motor rotor and the machine tool spindle must be carefully studied to meet the needs of the design of high-speed motorized spindle

2 principle of interference calculation of spindle rotor

when there is interference between motor rotor and machine tool spindle, positive pressure will be generated on the mating surface, which will expand the internal and external diameters of motor rotor and compress the internal and external diameters of spindle. Due to the large axial size of the motor rotor and main shaft, the axial elongation can be ignored. Therefore, the fit of the motorized spindle can be simplified as the interference fit of two thick walled cylindrical sleeves. The magnitude of the interference between the motor rotor and the machine tool spindle is not only closely related to the transmission capacity of the mating surface, but also affected by the motion state of the electric spindle. At low speed, the influence of centrifugal force can be ignored; For high-speed spindle, centrifugal force is the key factor that affects the coordination between motor rotor and machine tool spindle. Therefore, the interference calculation of motorized spindle should not only meet the requirements of static transmission capacity, but also consider the influence of spindle dynamic characteristics

2.1 calculation principle of static interference of electric spindle 

as shown in Figure 1, set the inner hole radius of motor rotor as a, the outer circle radius as B, the radius of spindle mating surface as a, and the inner hole radius as C. If the axial end effect is ignored, it can be considered that the mating surface is under uniformly distributed pressure. Since the motor rotor and the main shaft are axisymmetric, in the polar coordinate system, the stress and displacement are axisymmetric, and the tangential displacement and shear stress r of the fitting are zero. According to the principle of elasticity, the stress function is only a function of radial dimension R, independent of polar angle

Figure 1 Stress Analysis of the mating surface between the main shaft and the rotor

(a) electronic rotor (b) machine tool spindle

the inner hole of the motor rotor is under the action of positive pressure P, and the outer circular surface is a free surface, as shown in Figure 1a. Therefore, the boundary conditions are:

(1)

where p is the positive pressure, N/m2

re is the radial normal stress of the rotor, N/m2

under static conditions, the stress and micro displacement of the motor rotor can be determined by the following formula :

(2)

where e (R) is the tangential normal stress of the rotor, N/m2

CE is the internal/external diameter ratio of the motor rotor, CE = A/b

UE (R) is the radial displacement of the motor rotor, M

EE elastic modulus of rotor material, N/m2

e Poisson's ratio of rotor material

the outer cylindrical surface of the main shaft is subjected to positive pressure P, and the inner hole of the main shaft is a free surface, as shown in Figure 1b. Therefore, the boundary conditions are:

(3)

in the formula RI (R) radial normal stress of the spindle, N/m2

under static conditions, the stress and micro displacement at the fitting position of the machine tool spindle are determined by the following formula :

(4)

in the formula I (R) tangential normal stress of the spindle, N/m2

ratio of internal and external diameters of the fitting surface of the spindle, CI = C/a

UI (R) radial displacement of the spindle, m

EI.Elastic modulus of spindle material, N/m2

I the Poisson's ratio of the spindle material

both the spindle and the motor rotor are made of steel, and their elastic modulus and Poisson's ratio are basically the same, which are e and e respectively

according to formula (2) and formula (4), under the condition of static fit, the interference amount s between the machine tool spindle and the motor rotor can be determined by the following formula:

(5)

2.2 calculation principle of dynamic interference amount of electric spindle

when the electric spindle is running at high speed, the influence of centrifugal force on the interference amount must be considered. Suppose that the thick walled cylinder is subjected to uniformly distributed pressure, the rotation speed of the cylinder is n, and the balance differential equation of the rotating body is:

(6)

in the formula, the angular velocity, rad/S

material density, kg/m3

the relationship between normal stress and radial displacement is :

there is still a large market space for the development of the extruder industry

(7)

due to the uniformly distributed compressive stress in the inner hole of the motor rotor, the outer circular surface is a free surface, according to formula (1) Equations (6) and (7) are obtained by solving the differential equation:

(8)

in the equation, the coefficient K is:

according to equations (3), 6) and (7), the stress and displacement formula of the spindle obtained by solving the differential equation is:

(9)

under the condition of high-speed operation, the interference amount of the motorized spindle can be determined by the following formula:

(10)

because the motor power is transmitted through the keyless interference fit between the motor rotor and the spindle of the machine tool, Therefore, the power transmission capacity between mating surfaces is determined by the friction torque between mating surfaces:

(11)

in the formula, the transmission torque of MT spindle, NM

the friction force DF of the unit mating surface between machine tool spindle and motor rotor is:

(12)

in the formula, the effective contact length of B mating surface, the unit center angle of M

the friction coefficient between mating surfaces

KC safety factor, Generally, take 2 ~ 4

according to formula (11) and formula (12), the relationship between the torque MT of the motor and the positive pressure P between the mating surface can be determined by the following formula:

(13)

according to formula (10) and formula (13), under high-speed operation conditions, the relationship between the interference of the mating surface of the electric spindle and the torque of the motor and the speed of the spindle can be determined by the following formula:

(14)

we can compare formula (14) and formula (5), The interference of high-speed motorized spindle is composed of static component s and dynamic component D. The static component s is determined by the following formula:

(15)

the dynamic component D is determined by the following formula:

(16)

it can be seen from formula (15) and formula (16) that the static component is related to the transmission capacity of the spindle, which is directly proportional to the torque of the spindle. The dynamic component is mainly determined by the centrifugal force, which is proportional to the square of the spindle speed. When the spindle speed is low, the d dynamic component can be ignored, and the interference of the spindle is determined by the static part; However, at high speed, the spindle composite polyurethane adhesive industry is a sub industry of fine chemical industry, and the interference of the adhesive industry is greatly affected by centrifugal force, so the interference of the high-speed spindle is mainly determined by the dynamic part

3 specific calculation and strength check

the basic dimensions of the rotor of gd- Ⅱ electric spindle motor are: the outer diameter of the rotor 2B = 134.2mm, the diameter of the inner hole of the rotor 2A = 66mm, the axial length of the rotor is 260mm, and the effective contact length of the rotor mating surface B = 104mm. The basic dimensions of the spindle mating surface are: outer diameter 2A = 66mm, inner hole diameter 2C = 25mm. Therefore, CE = 0.4918, CI = 0.3788. The maximum speed of the motor is 18000r/min, so its maximum angular speed Max is 1884 rad/s. The rated power is 13.5 kW and the rated torque is 85 nm. The structure of the motorized spindle is shown in Figure 2

Figure 2 the friction coefficient of gd- Ⅱ high-speed electric plastic film is a main target spindle structure

the motor rotor and spindle are made of steel, the elastic modulus of the material e = 2.1011 n/m2, the Poisson's ratio = 0.3, the friction coefficient between the spindle mating surfaces = 0.09, the allowable stress of the motor rotor bushing material [] is 287 n/mm2, and the allowable stress of the spindle material [] is 567 n/mm2

according to formula (16), to meet the high-speed performance of the electric spindle, the minimum value Dmin of the dynamic interference component between the motor rotor and the spindle mating surface can be obtained from the following formula:

according to formula (15), to meet the torque transmission capacity of the electric spindle, the minimum value smin of the static interference component between the motor rotor and the spindle mating surface can be obtained from the following formula:

according to the calculation, The required dynamic interference Dmin of high-speed motorized spindle is more than 6 times of its required static interference. It can be seen that the interference of high-speed spindle is mainly determined by the dynamic interference. According to formula (14), the minimum interference min of high-speed motorized spindle is:

min = Dmin + smin = 0.03404 (mm)

accordingly, in the design of gd-ii motorized spindle, the interference fit of 66h6/S6 is adopted for the fit of spindle and motor rotor, and the actual minimum interference of this fit is 0.040 mm (> 0.03404 mm), which can meet the high-speed transmission requirements of motorized spindle in July this year. The actual maximum interference is 0.078mm. According to formula (5), the maximum positive pressure actually generated by the mating surface is:

according to formula (2), the mating surface of the inner hole of the motor rotor has the maximum tangential tensile stress E and the maximum radial compressive stress REMAX, whose value is: Max

according to formula (4), RI (R) and I (R) of the main shaft are compressive stresses, in which the mating surface of the main shaft has the maximum radial compressive stress rimax, The maximum tangential compressive stress I at the inner hole wall of the spindle is: Max

the assembly stress distribution of the motorized spindle is shown in Figure 3

Figure 3 stress distribution of interference fit between main shaft and rotor

it can be seen that the danger point of electric spindle is on the inner side of motor rotor. According to the third strength theory :

the allowable stress of motor rotor bushing material [] is 287 n/mm, R3 <[], which is safe to use

4 conclusion

motorized spindle is a new spindle component developed for high-speed CNC machine tools. It is very important to correctly calculate the interference fit between spindle and motor rotor. It is a key technology that affects the rotation performance of spindle. Through the research on the interference fit of gd-ii motorized spindle, this paper draws the following conclusions:

(1) according to the mechanics of materials and elasticity, the static and

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