A method for reducing electromagnetic vibration of a fractional slot permanent magnet motor

Abstract

This invention discloses a method for reducing electromagnetic vibration of a fractional-slot permanent magnet motor. By establishing an analytical model of electromagnetic force waves and radial force tooth modulation, the order of the radial force waves that have a significant impact on electromagnetic vibration is determined. Based on the number of pole slots, the optimization range of the stator tooth shoe size is determined. For various pole slot combinations of fractional-slot motors, different stator tooth shoe size constraints and optimization schemes are adopted to weaken the radial force wave modulation effect by adjusting the radial transmission path of the higher-order radial force waves. This minimizes the concentrated radial force generated by the higher-order radial force on each tooth, thereby effectively reducing the electromagnetic vibration caused by the higher-order radial force.

Description

Technical Field

[0001] This invention relates to the field of permanent magnet motor design and manufacturing, and in particular to a method for reducing electromagnetic vibration in a fractional slot permanent magnet motor. Background Technology

[0002] Fractional-slot permanent magnet motors possess advantages such as high torque density, high power density, and high efficiency, making them promising for applications in smart homes, servo actuation, and marine propulsion. However, the close proximity of pole and slot numbers in fractional-slot motors results in a low-order non-zero lowest-order radial force wave whose order equals the greatest common divisor of the pole and slot numbers, easily leading to significant electromagnetic vibration. Furthermore, when higher-order radial force waves propagate outward through the stator teeth, sampling aliasing occurs due to the Nyquist-Shannon theorem, causing the higher-order radial forces with larger amplitudes to modulate and act on the lower-order vibration response, greatly exacerbating the motor's electromagnetic vibration. Among the modulated higher-order radial force waves, the 2p-order force wave, generated by the interaction between the rotor's fundamental magnetic field and itself, has the largest amplitude and contributes the most to electromagnetic vibration. However, simultaneously, the rotor's fundamental magnetic field directly participates in the generation of electromagnetic torque; therefore, reducing the rotor's fundamental magnetic field would also significantly reduce the motor's electromagnetic output performance.

[0003] Chinese patent CN107579606B discloses a fractional-slot concentrated winding permanent magnet motor and its design method with low vibration and noise performance. By changing the stator structure, the content of tooth harmonics in the winding magnetomotive force in the air gap is reduced, thereby reducing the vibration and noise of the motor. However, this method increases the complexity of the motor and is only applicable to specific winding types, lacking versatility. Summary of the Invention

[0004] Purpose of the invention: To address the above problems, the purpose of this invention is to provide a method for reducing electromagnetic vibration of a fractional-slot permanent magnet motor, thereby weakening the tooth modulation effect of the high-order axial force wave in the motor's mid-diameter and reducing the amplitude of the second-order deformation generated by the high-order force wave through modulation, thus achieving the purpose of reducing electromagnetic vibration.

[0005] Technical solution: The present invention provides a method for reducing electromagnetic vibration of a fractional-slot permanent magnet motor, comprising the following steps:

[0006] Step 1: Establish an analytical model of the electromagnetic force wave and radial force tooth modulation of the fractional slot permanent magnet motor to obtain the spatiotemporal order characteristics of the radial force wave;

[0007] Step 2: Analyze the electromagnetic force wave radial transmission mechanism and stator tooth sampling aliasing conditions to obtain the high-order radial force wave that has the greatest impact on electromagnetic vibration by being applied to low-order vibration modes through tooth adjustment.

[0008] Step 3, based on the rotor pole number 2p and stator tooth number N of the fractional slot motor sThe relationship between N and other parameters is used to classify fractional slot permanent magnet motors into different types. s Motors with a speed of less than 2p adopt an optimized scheme with all teeth conforming to the same standard, for N s The >2p motor adopts an optimization scheme with different standards for adjacent teeth to reduce electromagnetic vibration.

[0009] Furthermore, the analytical model expression in step 1 is as follows:

[0010]

[0011] In the formula, f r (θ s ,t) represents the radial force function, θ s Indicates the angular position along the circumference, t represents time, and B r (θ s ,t) represents the radial magnetic flux density of the air gap, μ0 represents the relative permeability, f pm (θ s ,t) represents the permanent magnet magnetomotive force, λ(θ) s ) represents the magnetic permeability function.

[0012] Furthermore, step 2 includes:

[0013] According to Jordan's formula, the amplitude of vibration displacement caused by radial forces of order 2 or higher is expressed as:

[0014]

[0015] In the formula, Y n R is the radial displacement of the motor stator core under the action of a radial electromagnetic force wave of spatial order n. is R is the inner radius of the stator core. yoke h is the average radius of the yoke of the stator core. yoke Let f be the height of the yoke of the stator core, E be the Young's modulus of the stator core, and f be the height of the yoke of the stator core. r This represents the radial force amplitude.

[0016] The amplitude of non-zero order vibration is approximately inversely proportional to the fourth power of the spatial order of the electromagnetic force wave. Lower order force waves are more likely to cause larger vibrations in the motor, so the lowest non-zero radial force, i.e., GCD(N), needs to be given special attention. s ,2p) order radial force; although the vibrational deformation produced by higher order force waves is very small, if the order m of the radial force satisfies m>N s / 2 will cause radial force modulation when propagating outwards radially, causing higher-order radial forces to be modulated into lower-order radial forces, thus generating large electromagnetic vibrations. The order of the modulated radial force is:

[0017] m'=|m-kN s |

[0018] In the formula, m represents the spatial order before radial force wave modulation, and k represents the order of magnetic permeability harmonics;

[0019] Based on analytical calculations of the amplitude and order of the radial force, the radial force generated by the rotor's fundamental magnetomotive force itself has the largest amplitude, corresponding to an order of 2p. Therefore, the modulated order is:

[0020] m'=|2p-N s |

[0021] Furthermore, based on the design of fractional slot motors and the principle of pole-slot matching, the modulated order satisfies:

[0022] m'=|2p-N s |=GCD(Ns,2p)

[0023] That is, the 2p-order radial force, after being modulated, acts on the non-zero lowest order, i.e., GCD(N s The 2p order also has a significant impact on electromagnetic vibration.

[0024] Furthermore, step 3 includes:

[0025] When N s When the value is less than 2p, all stator teeth of the fractional slot permanent magnet motor are defined as type I teeth and type I optimization is performed. When performing type I optimization, the circumferential angle corresponding to the tooth shoe satisfies the following formula:

[0026] θ I =π / p

[0027] Where p is the number of rotor pole pairs.

[0028] Furthermore, step 3 also includes:

[0029] When N s When the number of poles is greater than 2p, calculate the greatest common divisor (GCD) of the number of poles and slots, GCD(N). s ,2p), when GCD(N s When ,2p) is an even number, adjacent stator teeth are arranged alternately in the circumferential direction using type I teeth and type II teeth respectively;

[0030] When GCD(N) s When ,2p) is odd, then GCD(N) is used. s 2p) toothed shoes form a unit, and GCD(N) is used within each unit. s ,2p)+1) / 2 type I teeth and (GCD(N s ,2p)-1) / 2 Type II teeth are arranged alternately along the circumferential direction;

[0031] Type I teeth are optimized using Type I optimization, and Type II teeth are optimized using Type II optimization.

[0032] Furthermore, the toothed shoe width during Type II optimization satisfies the following formula:

[0033] θ Ⅱ <(2π / p-2π / N s ).

[0034] Beneficial effects: Compared with the prior art, the significant advantages of this invention are:

[0035] 1. This invention optimizes the stator gear shoe size by developing different optimization schemes for motors with different pole slot combinations, which greatly weakens the modulation effect of high-order radial forces and reduces the electromagnetic vibration caused by the large amplitude high-order radial forces dominated by 2p-order radial forces.

[0036] 2. This invention divides the tooth modulation channels of radial force waves into two categories by optimizing the tooth width of adjacent teeth. The number of modulation channels in each category is equal, which is half of the total number of teeth. By changing the number of effective modulation channels, the low-order component of the force wave generated by the 2p radial force wave after modulation is converted into a high-order component, thereby reducing its contribution to electromagnetic vibration. Attached Figure Description

[0037] Figure 1 A flowchart illustrating a method for reducing electromagnetic vibration in a fractional-slot permanent magnet motor.

[0038] Figure 2 In the example, N s <2p Stator tooth type classification structure diagram;

[0039] Figure 3 In the example, N s >2p Stator tooth type classification structure diagram;

[0040] Figure 4 This is a schematic diagram showing the relationship between the width of the Type II tooth shoe and the radial component of the concentrated force on a single tooth in the embodiment;

[0041] Figure 5 In the example, N s >Diagrams showing the concentrated force distribution of each tooth in a 2P type motor before and after optimization;

[0042] Figure 6 In the example, N s <2p type motor concentrated force distribution diagram before and after optimization.> Detailed Implementation

[0043] To make the objectives, technical solutions, and advantages of this application clearer, the following detailed description is provided in conjunction with the accompanying drawings and embodiments.

[0044] like Figure 1The diagram shown is a flowchart of a method for reducing electromagnetic vibration of a fractional-slot permanent magnet motor according to this embodiment, including the following steps:

[0045] Step 1: Establish an analytical model of electromagnetic force wave and radial force tooth modulation of fractional slot permanent magnet motor to obtain the spatiotemporal order characteristics of radial force wave.

[0046] The analytical model expression in step 1 above is as follows:

[0047]

[0048] In the formula, f r (θ s ,t) represents the radial force function, θ s Indicates the angular position along the circumference, t represents time, and B represents the position along the circumference. r (θ s ,t) represents the radial magnetic flux density of the air gap, μ0 represents the relative permeability, f pm (θ s ,t) represents the permanent magnet magnetomotive force, λ(θ) s ) represents the magnetic permeability function;

[0049] After sampling aliasing occurs, the originally higher-order radial force, after being modulated by the stator teeth, acts as a lower-order radial force on the stator yoke and housing. Based on analytical calculations of the amplitude and order of the radial force, the radial force generated by the rotor's fundamental magnetomotive force itself has the largest amplitude, corresponding to an order of 2π. The close pole-slot pairing, modulated by the stator teeth, acts on the N... s -2p mode shape.

[0050] As the primary source of electromagnetic vibration in a fractional-slot permanent magnet motor, the radial force wave generated in the air gap acts on the stator teeth and is transmitted radially through the stator teeth, subsequently acting on the stator yoke and the motor housing, thus generating electromagnetic vibration. The magnitude of the vibration is related to the amplitude and order of the radial force; therefore, an analytical model of the radial force wave must first be established. The radial force wave can be represented by a simplified model. We obtain, where the magnetic permeability function is After substituting the permeability function, the analytical model is as follows:

[0051]

[0052] In the formula, f r-noload (θ s ,t) represents the radial force wave under no-load conditions, μ is the spatial order of the rotor magnetomotive force, the subscript numbers distinguish the two magnetic fields that form this radial force wave, ω1 is the electric angular velocity, and θ s Let t be the spatial angular position along the circumference, μ0 be the relative permeability, v be the spatial order of the stator magnetomotive force, p be the number of pole pairs, and λ be the spatial position along the circumference. kN represents the permeability amplitude of the low-k-th order tooth harmonic. s This refers to the number of stator teeth.

[0053]

[0054] In the formula, f r-load (θ s ,t) represents the radial force wave that intersects with the unloaded state under load, v represents the spatial order of the stator magnetomotive force, and Ψ represents the initial position of the stator magnetomotive force.

[0055] Step 2: Analytical analysis is performed using the electromagnetic force wave radial transmission mechanism and stator tooth sampling aliasing conditions to obtain the high-order radial force wave that has the greatest impact on electromagnetic vibration by acting on low-order vibration modes through tooth tuning.

[0056] Furthermore, step 2 includes:

[0057] According to Jordan's formula, the amplitude of vibration displacement caused by radial forces of order 2 or higher is expressed as:

[0058]

[0059] In the formula, Y n R is the radial displacement of the motor stator core under the action of a radial electromagnetic force wave of spatial order n. is R is the inner radius of the stator core. yoke h is the average radius of the yoke of the stator core. yoke Let f be the height of the yoke of the stator core, E be the Young's modulus of the stator core, and f be the height of the yoke of the stator core. r Indicates the radial force amplitude;

[0060] The amplitude of non-zero order vibration is approximately inversely proportional to the fourth power of the spatial order of the electromagnetic force wave. Lower order force waves are more likely to cause larger vibrations in the motor, so the lowest non-zero radial force, i.e., GCD(N), needs to be given special attention. s ,2p) order radial force; although the vibrational deformation produced by higher order force waves is very small, if the order m of the radial force satisfies m>N s / 2 will cause radial force modulation when propagating outwards radially, causing higher-order radial forces to be modulated into lower-order radial forces, thus generating large electromagnetic vibrations. The order of the modulated radial force is:

[0061] m'=|m-kN s |

[0062] In the formula, m represents the spatial order before radial force wave modulation, and k represents the order of magnetic permeability harmonics;

[0063] Based on analytical calculations of the amplitude and order of the radial force, the radial force generated by the rotor's fundamental magnetomotive force itself has the largest amplitude, corresponding to an order of 2p. Therefore, the modulated order is:

[0064] m'=|2p-N s |

[0065] Furthermore, based on the design of fractional slot motors and the principle of pole-slot matching, the modulated order satisfies:

[0066] m'=|2p-N s |=GCD(Ns,2p)

[0067] That is, the 2p-order radial force, after being modulated, acts on the non-zero lowest order, i.e., GCD(N s The 2p order also has a significant impact on electromagnetic vibration.

[0068] Although radial force waves act on the surface of the stator tooth shoe in the form of distributed forces, their propagation in the radial direction is still transmitted through sampling channels on a per-tooth basis. The distributed force form can only observe the distribution of radial force waves on the tooth shoe and cannot meet the requirements for calculating the amplitude of radial force waves transmitted outwards. Therefore, the distributed force is usually converted into a concentrated force form to calculate the force wave transmitted through a single tooth. After conversion to a concentrated force, there are three components: radial component, tangential component, and bending moment. Among them, the radial component plays a major role in electromagnetic vibration. The concentrated force on a single tooth can be calculated as the integral of the radial force wave within the corresponding tooth shoe region.

[0069]

[0070] In the formula, θ s Indicates the angular position along the circumferential direction, where L is the axial length of the motor, and R is the angular position along the circumference. in_s Indicates the stator inner diameter, σ r This represents the amplitude of the radial force density, where m is the spatial order of the radial force wave, and ω1 represents the electric angular velocity. This indicates the initial position of the radial force.

[0071] As can be seen from the above equation, the electromagnetic vibration of a fractional-slot permanent magnet motor is actually the result of the combined effect of the radial components of the concentrated forces on each stator tooth. Although the vibration response is approximately inversely proportional to the fourth power of the force wave order, the extremely close number of poles and slots in the fractional-slot motor results in a limited number of sampling channels equal to the number on the teeth. According to the Nyquist-Shannon sampling theorem, high-frequency radial force waves will experience sampling aliasing during radial propagation. After sampling aliasing, the order of the modulated radial force wave is:

[0072] m'=|m-kN s |

[0073] After aliasing occurs, the originally higher-order radial force, after being modulated by the stator teeth, acts as a lower-order radial force on the stator yoke and housing, resulting in a significant vibration response. Based on analytical calculations of the amplitude and order of the radial force, the radial force generated by the rotor's fundamental magnetomotive force itself has the largest amplitude and corresponds to an order of 2p. Due to the close pole-slot fit, the vibration is modulated by the stator teeth and applied to the N... s The -2p mode shape is one of the main vibration sources in fractional slot motors. Under low load conditions, its contribution to electromagnetic vibration can even be greater than that generated by the non-zero lowest order radial force.

[0074] To address the issue of significant electromagnetic vibration in fractional-slot motors, this embodiment employs a low-electromagnetic-vibration permanent magnet motor design concept to mitigate the modulation effect of the 2p-order radial force wave. This minimizes the concentrated force obtained by integration on each tooth, reducing the amplitude of the low-order force wave after modulation, thereby weakening the electromagnetic vibration. The contribution of a specific-order radial force wave to electromagnetic vibration depends on the amplitude of the radial component of the concentrated force on each stator tooth. This amplitude is influenced by both the amplitude of the radial force wave itself and the stator modulation effect. The width of the stator tooth shoe determines the integration range of the force wave on that tooth and directly affects the modulation effect. Under different tooth shoe width ranges, the peak value of the radial component of the concentrated force obtained by integrating the radial force wave satisfies the following... Figure 4 The relationship shown is that when the width of the toothed shoe is π / p, the integral of the 2p-order radial force on the tooth is 0 for a complete cycle. As the width of the toothed shoe decreases, the radial component of the concentrated force gradually increases first, reaches its maximum when the width of the toothed shoe is π / 2p, and then continuously decreases.

[0075] Step 3, based on the rotor pole number 2p and stator tooth number N of the fractional slot motor s The relationship between N and other parameters is used to classify fractional slot permanent magnet motors into different types. s Motors with a speed of less than 2p adopt an optimized scheme with all teeth conforming to the same standard, for N s The >2p motor adopts an optimization scheme with different standards for adjacent teeth to reduce electromagnetic vibration.

[0076] Specifically, step 3 above includes:

[0077] When N s When the value is less than 2p, all stator teeth of the fractional slot permanent magnet motor are defined as type I teeth and type I optimization is performed. When performing type I optimization, the circumferential angle corresponding to the tooth shoe satisfies the following formula:

[0078] θ I =π / p

[0079] Where p is the number of rotor pole pairs.

[0080] In one example, such as Figure 2 As shown, to maximize the reduction of the modulation effect of the 2p-order radial force and thus reduce electromagnetic vibration, this example proposes a type I optimization, which limits the angle of the tooth shoe 3, defined as type I tooth 2, to θ. Ⅰ =π / p, at this time Δθ = θ Ⅰ / 2=π / 2p, the type I toothed shoe corresponds to a complete cycle of the force wave at any time, and the integral under a complete cycle is always 0, as shown in the following formula. Therefore, the radial component of the concentrated force is also approximately 0, so that the 2p-order radial force wave cannot be modulated by the type I toothed shoe (specifically acting on the stator yoke 1 and the housing) to generate a low-order force wave, thus reducing its influence on electromagnetic vibration.

[0081]

[0082] Specifically, step 3 above also includes:

[0083] When N s When the number of poles is greater than 2p, calculate the greatest common divisor (GCD) of the number of poles and slots. s ,2p), when GCD(N s When ,2p) is an even number, adjacent stator teeth are arranged alternately in the circumferential direction using type I teeth and type II teeth respectively;

[0084] When GCD(N) s When ,2p) is odd, then GCD(N) is used. s 2p) toothed shoes form a unit, and GCD(N) is used within each unit. s ,2p)+1) / 2 type I teeth and (GCD(N s ,2p)-1) / 2 Type II teeth are arranged alternately along the circumferential direction;

[0085] Type I teeth are optimized using Type I optimization, and Type II teeth are optimized using Type II optimization.

[0086] Furthermore, the toothed shoe width during Type II optimization satisfies the following formula:

[0087] θ Ⅱ <(2π / p-2π / Z s ).

[0088] In one example, such as Figure 3 As shown, for N s For motors with a diameter of less than 2p, due to sufficient circumferential space, all stator teeth can be defined as type I teeth and optimized using type I, thereby minimizing radial force wave tooth modulation; for N... sMotors with more than 2 poles are limited by circumferential space, making it impossible to optimize all stator teeth using Type I. In this case, the stator teeth are divided into Type I and Type II teeth, which are periodically distributed along the circumference. The distribution characteristics depend on the greatest common divisor (GCD) of the number of poles and slots (N). s ,2p), and perform Type I and Type II optimizations respectively. At this time, the high-order radial force wave can still transmit concentrated force through the Type II tooth. Although the concentrated force on the stator tooth is time-varying, the degree of electromagnetic vibration response caused is determined only by the maximum value of the concentrated force at all times. If the concentrated force on all teeth is regarded as a time-varying force wave, then the amplitude of the force wave is the maximum value of the concentrated force. Therefore, for Type II tooth 4, the calculation of its maximum concentrated force is as follows:

[0089]

[0090] The width of type II tooth 4 is usually 0 < θ Ⅱ Within the range <π / 2p, when integrating the radial force wave within the type II tooth range, Δθ=θ Ⅱ / 2, this type of size optimization method is called Type II optimization. In this region, the smaller the corresponding toothed shoe width 5, the smaller the radial component of the concentrated force; therefore, θ must be satisfied. n <(2π / p-2π / Z s To minimize the modulation effect of radial force waves. The arrangement of type I and type II teeth depends on GCD(N) s Parity of ,2p), GCD(N s Motors with an even number of teeth (2p) use alternating Type I and Type II teeth, GCD(N) s Motors with odd numbers of ,2p) are governed by GCD(N s A cycle consists of 2p teeth, and follows GCD(N) s ,2p)+1) / 2 type I teeth and (GCD(N s The number of Type II teeth (2p)-1) / 2 is staggered. Type I teeth are optimized using Type I optimization, and Type II teeth are optimized using Type II optimization.

[0091] Taking a 12-slot, 10-pole motor as an example, the concentrated forces on each tooth caused by the 2p-order radial force wave before and after applying the method described in this invention are as follows: Figure 5-6 As shown, where Figure 5 Corresponding to N s 2P type motor, Figure 6 Corresponding to N sThe 2p-type motor, after adopting the method described in this invention, exhibits the following improvements: Firstly, the peak value of the concentrated force on all teeth is reduced; secondly, compared to the motor before optimization where the concentrated force on all teeth exhibited a clear second-order force wave pattern, the concentrated force on each stator tooth after this optimization method significantly weakens the second-order radial force wave and increases the fourth-order force wave. However, due to the larger order of the fourth-order force wave, its contribution to electromagnetic vibration is far less than that of the second-order radial force wave. Therefore, it can be seen that the method proposed in this invention has a good suppression effect on low-order electromagnetic vibration caused by the modulation effect of the 2p-order radial force wave.

Claims

1. A method for reducing electromagnetic vibration of a fractional-slot permanent magnet motor, characterized in that, Includes the following steps: Step 1: Establish an analytical model of the electromagnetic force wave and radial force tooth modulation of the fractional slot permanent magnet motor to obtain the spatiotemporal order characteristics of the radial force wave; Step 2: Analyze the electromagnetic force wave radial transmission mechanism and stator tooth sampling aliasing conditions to obtain the high-order radial force wave that has the greatest impact on electromagnetic vibration by being applied to low-order vibration modes through tooth adjustment. Step 3, based on the rotor pole number 2p and stator tooth number N of the fractional slot motor s The relationship between N and other parameters is used to classify fractional slot permanent magnet motors into different types. s Motors with a speed of less than 2p adopt an optimized scheme with all teeth conforming to the same standard, for N s The >2p motor adopts an optimization scheme with different standards for adjacent teeth to reduce electromagnetic vibration; Step 3 includes: When N s When the value is less than 2p, all stator teeth of the fractional slot permanent magnet motor are defined as type I teeth and type I optimization is performed. When performing type I optimization, the circumferential angle corresponding to the tooth shoe satisfies the following formula: i I =π / p Where p is the number of rotor pole pairs; When N s When the number of poles is greater than 2p, calculate the greatest common divisor (GCD) of the number of poles and slots. s ,2p), when GCD(N s When ,2p) is an even number, adjacent stator teeth are arranged alternately in the circumferential direction using type I teeth and type II teeth respectively; When GCD(N) s When ,2p) is odd, then GCD(N) is used. s 2p) toothed shoes form a unit, and each unit uses (GCD(N) s ,2p)+1) / 2 type I teeth and (GCD(N s ,2p)-1) / 2 Type II teeth are arranged alternately along the circumferential direction; Type I teeth are optimized using Type I optimization, and Type II teeth are optimized using Type II optimization. When performing Type II optimization, the toothed shoe width satisfies the following formula: θⅡ<(2π / p-2π / N s )。 2. The method for reducing electromagnetic vibration of a fractional-slot permanent magnet motor according to claim 1, characterized in that, The analytical model expression in step 1 is as follows: In the formula, f r (θ s ,t) represents the radial force function, θ s Indicates the angular position along the circumference, t represents time, and B r (θ s fpm(θ) represents the radial magnetic flux density of the air gap, μ0 represents the relative permeability, and fpm(θ) represents the relative magnetic flux density of the air gap. s ,t) represents the permanent magnet magnetomotive force, λ(θ) s ) represents the magnetic permeability function.

3. The method for reducing electromagnetic vibration of a fractional-slot permanent magnet motor according to claim 2, characterized in that, Step 2 includes: According to Jordan's formula, the amplitude of vibration displacement caused by radial forces of order 2 or higher is expressed as: In the formula, Y n R is the radial displacement of the motor stator core under the action of a radial electromagnetic force wave of spatial order n. is R is the inner radius of the stator core. yoke h is the average radius of the yoke of the stator core. yoke Let f be the height of the yoke of the stator core, E be the Young's modulus of the stator core, and f be the height of the yoke of the stator core. r This represents the radial force amplitude. The amplitude of non-zero order vibration is approximately inversely proportional to the fourth power of the spatial order of the electromagnetic force wave. Lower order force waves are more likely to cause larger vibrations in the motor, so the lowest non-zero radial force, i.e., GCD(N), needs to be given special attention. s ,2p) order radial force; although the vibrational deformation produced by higher order force waves is very small, if the order m of the radial force satisfies m>N s / 2 will cause radial force modulation when propagating outwards radially, causing higher-order radial forces to be modulated into lower-order radial forces, thus generating large electromagnetic vibrations. The order of the modulated radial force is: m'=|m-kN s | In the formula, m represents the spatial order before radial force wave modulation, and k represents the order of magnetic permeability harmonics; Based on analytical calculations of the amplitude and order of the radial force, the radial force generated by the rotor's fundamental magnetomotive force itself has the largest amplitude, corresponding to an order of 2p. Therefore, the modulated order is: m'=|2p-N s | Furthermore, based on the design of fractional slot motors and the principle of pole-slot matching, the modulated order satisfies: m'=|2p-N s |=GCD(Ns,2p) That is, the 2p-order radial force, after being modulated, acts on the non-zero lowest order, i.e., GCD(N s The 2p order also has a significant impact on electromagnetic vibration.

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