Abstract
Nonlinear differential equations for an electromechanical integrated magnetic gear (EIMG) system are developed by considering the nonlinearity in the magnetic force of the system components. Expressions for the main resonances and superharmonic resonances are obtained for output wave frequencies close to the natural frequency and half the natural frequency of the derived EIMG system. The response laws are discussed in detail. The magnetic coupling stiffness among the components is found to exhibit distinct nonlinearity, leading to strong main resonances and superharmonic resonances. Smaller values of the magnetic coupling stiffness and damping result in larger response amplitudes and transient responses that slowly decay to zero. When the main resonances and superharmonic resonances occur, the dominant frequency of the response is the natural frequency of the derived EIMG system, and the amplitudes of different components of the resonance display large differences.
1. Introduction
Magnetic drives, and especially magnetic gear boxes, are extensively used in the medical, food, and chemical industries. These systems have the advantages of being contactless, wear-free, and lubrication-free, with low levels of vibration and noise, and built-in overload protection [1]. The field-modulated magnetic gear (FMMG) has a coaxial topology, which significantly improves the permanent magnet (PM) utilization rate and can generate much higher torque and torque density than traditional magnetic gears that use a parallel shaft topology [2]. Therefore, FMMGs are widely used in the vehicle, wind, marine, and aerospace fields [3–5]. Various high-performance FMMG-based systems have been proposed, such as axial-flux magnetic gears, linear magnetic gears, and intersecting axes magnetic gears [6–8]. Various direct current PM rotors and alternating current PM motors have been developed for use in electric vehicles and wind power systems [9, 10]. Together, these innovations have significantly promoted the development of magnetic gears.
Electrical and control techniques are widely used in mechanical engineering, resulting in the development of many cutting-edge composite drives such as the electromagnetic harmonic drive and electromechanical integrated toroidal drive. One of the authors has previously presented a drive system known as the electromechanical integrated magnetic gear (EIMG) [11]. This integrates power and control systems into FMMG. Thus, EIMG possesses the advantages of FMMG alongside a compact structure with torque and speed control. Compared with FMMG, EIMG can generate higher torque at lower speeds. EIMG can transmit torque by magnetic field coupling, unlike mechanical gears, which require direct contact to do so. The magnetic coupling stiffness of EIMG is much lower than the contact stiffness of mechanical gears [12, 13], and EIMG has different dynamic characteristics compared with mechanical gear systems. Because EIMG system has a lower magnetic coupling stiffness, its natural frequencies are much lower than those of mechanical gear systems. As a result, low-frequency resonance can easily occur. When such resonance occurs, the amplitudes of the EIMG system’s components slowly attenuate, degrading the dynamics of the system. To overcome this problem, Frank et al. added damper windings to synchronous generators to suppress the oscillations caused by transients [14]. Montague et al. used servo control in magnetic gears to overcome the disadvantages of low stiffness [15].
As EIMG is a relatively new type of magnetic gear, its dynamics should be thoroughly studied to control the behavior and improve the service performance of such a system. This study develops a dynamic model and differential equations for EIMG system by considering the nonlinearity of the magnetic coupling stiffness. The main resonances and superharmonic resonances produced by a torque wave are discussed, and these provide the foundation for the parameter optimization and performance improvement of the EIMG system.
2. Nonlinear Dynamic Model of EIMG
Figure 1 shows the inner rotor, inner and outer stator, and inner and outer ferromagnetic pole-pieces (FP) that comprise an EIMG system. All components are concentric, and the EIMG has four air gaps. Windings made of insulated wire are mounted in the dew-drop slot of the inner stator. PMs are positioned uniformly on the inner surface of the outer stator and the inner and outer surfaces of the inner rotor. The inner and outer FP comprise permeable and nonpermeable magnetic materials layered at regular intervals. FP modulate the magnetic fields in the two air gaps next to them, thus ensuring that the number of pole pairs of PMs on the inner rotor and outer stator agrees with the number of pole pairs of the space harmonic flux density of the air gaps.
A three-phase alternating current is supplied to the inner stator coils to generate a rotary magnetic field with a main harmonic of . After being modulated by the inner FP, the main harmonic of the magnetic field in the second air gap is consistent with the number of pole pairs of PMs on the inner surface of the inner rotor. Therefore, the inner rotor can be actuated, thereby realizing the first drive. When the inner rotor runs, a magnetic field with a main harmonic of in the third air gap is generated and modulated by the outer FP. As a result, the main harmonic of the magnetic field in the fourth air gap becomes equal to the number of pole pairs of PMs on the outer stator. Then, the second drive is achieved.
To realize equal pole coupling among the EIMG components, , , , and have the following constraint relations:where is the number of pole pairs of the three-phase alternating current and , , and are the number of pole pairs of PMs on the stator and the inner and outer surfaces of the inner rotor, respectively.
The magnetic coupling torques among the components are calculated as follows:where , , and are the torques on the inner stator, inner rotor, and outer stator, respectively; , , and are the maximum torques on the inner stator, inner rotor, and outer stator, respectively; and are the relative rotation angles between the inner stator and inner rotor and between the inner rotor and outer stator, respectively; and , , , and are the relative torsional displacements between the inner stator and inner FP, inner FP and inner rotor, inner rotor and outer FP, and outer FP and outer stator, respectively.
Figure 2 shows the dynamic model of the EIMG system. This model contains four subsystems: inner stator/inner FP subsystem, inner FP/inner rotor subsystem, inner rotor/outer FP subsystem, and outer FP/outer stator subsystem. Because the magnetic coupling stiffness is much lower than the supporting stiffness [12], only torsional displacements are considered, and transverse vibration displacements are assumed to be negligible.
(a) Inner stator/inner FP subsystem
(b) Inner FP/inner rotor subsystem
(c) Inner rotor/outer FP subsystem
(d) Outer FP/outer stator subsystem
The tangential magnetic coupling forces among the components are calculated as follows:where , , , and are the tangential magnetic coupling forces between the inner stator and inner FP, inner FP and inner rotor, inner rotor and outer FP, and outer FP and outer stator, respectively; , , and are the equivalent radius of gyration of the inner stator, inner rotor, and outer stator, respectively; and () is the torsional angular displacement of the inner stator, inner FP, inner rotor, outer FP, and outer stator, respectively.
For convenience, the torsional angular displacements are all replaced by their corresponding torsional displacements aswhere () is the torsional displacement of the inner stator, inner FP, inner rotor, outer FP, and outer stator, respectively, and .
The magnetic coupling forces among the components are functions of the relative torsional displacements. When , namely, , the magnetic coupling forces are expanded using the Taylor series aswhere , , , , , and ; , , , , , and ; , , , , , and ; and , , , , , and .
The magnetic coupling forces among the components are nonlinear functions of the relative torsional displacements. The constant components of the Taylor series expansions of the magnetic coupling forces will lead to relative static displacements rather than dynamic displacements. Therefore, is neglected. Based on Newton’s second law of motion, the nonlinear dynamic differential equations of an EIMG system can be expressed as follows:where and are the masses of the inner stator and inner FP, respectively; () and () are the equivalent masses of the inner stator and inner FP along their torsional vibration direction, respectively; () and () are the mass moments of inertia of the inner stator and inner FP, respectively; and () are the mass and equivalent mass of the inner rotor, respectively; () is the mass moment of inertia of the inner rotor; and () are the mass and equivalent mass of the outer FP, respectively; () is the mass moment of inertia of the outer FP; and () are the mass and equivalent mass of the outer stator, respectively; is the mass moment of inertia of the outer stator; , , and are the torsional support stiffness values of the inner stator, inner FP, and outer stator, respectively; , , , , and are the damping coefficients of the inner stator, inner FP, inner rotor, outer FP, and outer stator, respectively (, , , , and , where , , , , and are the damping ratios of the inner stator, inner FP, inner rotor, outer FP, and outer stator, resp.).
Equation (6) can be rewritten in matrix form as
The mass matrix , displacement vector , damping matrix , stiffness matrix , load vector , and equivalent load vector have the following forms:where is the amplitude of the torque fluctuation and is the excitation frequency of the torque wave.
Equation (7) can be normalized aswhere , , and are the normal displacement vector, normal stiffness matrix, and normal equivalent load vector, respectively; these are expressed as follows:where is the natural frequency of the linear system derived for the EIMG.
Because the damping matrix is diagonal, the elements on the primary diagonal of the normal damping matrix are much larger than the other elements. Therefore, the normal damping matrix in (9) is simplified into the following diagonal matrix:
3. Main Resonance
A multiscale method is used to solve (9). To balance the effects of the damping forces and nonlinear elements and express them in the same perturbation equation, the following assumptions are made:
When the wave frequency of the output torque is close to the natural frequency of the outer FP rotational mode, the following assumption is introduced:
By substituting (12) and (13) into (9) and setting the power coefficients of the small parameters on both sides of the equations to be equal, the following differential equations are obtained: Zero power First powerwhere
Under the normal coordinate system, the general solution of (14) is expressed as
Under the regular coordinate system, the solutions of (14) arewhere , , , , and ; , , , , and ; , , , , and ; , , , , and ; and , , , , and .
Here, denotes the element in the th row and th column of the normal modal matrix :
By substituting (18), (15) can be rewritten aswhere is the complex conjugate of the right-hand-side expression in (20) and is a constant associated with , , , and that is expressed as follows:
There are multiple frequency components in the right-hand-side expressions of (20), such as , , and , where and . When there is no internal resonance, the secular terms can be eliminated to obtainwhere , and 5.
Solutions of (22) are obtained as follows:where and are constants related to the output torque, is a constant related to and , , and .
Under the normal coordinate system, the zero-order approximate analytical solution of the EIMG system is expressed as
After eliminating the secular terms, the first-order approximate solutions are obtained by substituting (23) and (24) into (20):where
will approach zero with increasing time because of the damping among components. The stable responses of the first-order approximate analytic solution of the EIMG system can be obtained by substituting (24) and (25) into (12):
Based on (27), approximate analytic solutions of the EIMG system under the regular coordinate system are given by
When the torque wave frequency approaches a certain frequency of the derived EIMG system, resonance will occur. In this situation, stable responses of the EIMG system under the normal coordinate system are calculated aswhere .
4. Superharmonic Resonance
When the wave frequency of the output torque is close to half of the natural frequency of the outer FP rotational mode, the following assumption is introduced:
By substituting (30) into (9) and setting the power coefficients of the small parameters on both sides of the equation to be equal, the following differential equations are obtained: Zero power First powerwhere
Under the normal coordinate system, the general solution of (31) is expressed aswhere .
Under the regular coordinate system, solutions of (31) are calculated aswhere , , , , and .
After substituting (35) into (32), there are multiple frequency components such as , , , and , where and . When there is no internal resonance, the secular terms can be eliminated from (32) to obtain
The solutions of (36) are written aswhere is a constant related to and , , and .
Because of the damping in the EIMG system, the initial vibration displacements will decrease to zero. Under the normal coordinate system, the zero-order stable analytical solutions are given by
After eliminating the secular terms, the first-order approximate solutions are obtained by substituting (38) into (32):where , , , , , , and ; , , , and ; and , , , and .
Under the normal coordinate system, the approximate solutions are then calculated by substituting (38) and (39) into (29):
When the wave frequency of the torque wave is close to half of the natural frequency of the EIMG system, namely, , the approximate solutions of the EIMG system under the normal coordinate system are