Nonlinear Forced Response of Electromechanical Integrated Toroidal Drive to Coupled Excitation

The electric excitation and the parameter excitation from mesh stiffness fluctuation are analyzed. The forced response equations of the drive system to the coupled excitations are presented. For the exciting frequencies far from and near natural frequencies, the forced responses of the drive system to the coupled excitations are investigated. Results show that the nonlinear forced responses of the drive system to the coupled excitations change periodically and unsteadily; the time period of the nonlinear forced responses depends on the frequencies of the electric excitation, the mesh parameter excitation, and the nonlinear natural frequencies of the drive system; in order to improve the dynamics performance of the drive system, the frequencies of the electric excitations should not be taken as integral multiple of the mesh parameter exciting frequency.


Introduction
Toroidal drive can transmit large torque in a very small size and is suitable for technical fields such as aviation and space flight [1][2][3]. As electrical and control techniques are utilized in mechanical engineering field widely, generalized composite drives become advancing edge of the mechanical science. So far, types of the generalized composite drives with integrated structure are still very limited.
The electromagnetic harmonic drive [4] and piezoelectric harmonic drive [5] are active drives in which the meshing forces between flexible gear and rigid one are controlled by electromagnetic force or piezoelectric one, and drive and power are integrated. Based on researching toroidal drive [6], the authors presented a kind of active generalized composite drive: electromechanical integrated toroidal drive. In the drive, the toroidal drive, power, and control are integrated [7].
The drive consists of four basic elements ( Figure 1) (a) the central worm, (b) radially positioned planets, (c) a toroidal shaped stator, and (d) a rotor, which forms the central output shaft upon which the planets are mounted. The central worm is fixed and coils are mounted in helical grooves of its surface. The planets have permanent magnets instead of teeth. The N and S polar permanent magnets are mounted alternately on a planet. And the stator has helical permanent magnets instead of helical teeth. In the same manner as planet, the N and S polar helical permanent magnets are mounted alternately on the stator.
If a specific parameter relation is realized, N pole of one element will correspond to S pole of the other one all along. The attractive forces between N and S pole of the different elements are driving forces and the meshes without contact are realized. When the alternating voltage source is connected to the coils of the worm, a toroidal circular field is formed. It drives several planets to rotate about their own axial. And by means of magnetic forces between teeth of the planet and stator, the rotor is driven to rotate about its own axial. Thus, a power of low speed and large torque is output.
Compared with toroidal drive, the new drive is easy to produce, without wear, and does not need lubrication. It can be substituted for a servo system to simplify the structure of the existing electromechanical systems. Beside the abovementioned fields that require compactness, the drive can be used in fields such as robots, which require accurate control.
The electromechanical integrated toroidal drive consists of a mechanical system, an electrical system and a coupled part. The mechanical vibration may occur in the mechanical system, and the electrical current oscillation may occur in the electrical system. With the coupled part, the mechanical vibration and the electrical current oscillation will influence on each other. Hence, the drive system is an electromechanical coupled dynamics system. The electromechanical coupled dynamics was first proposed for the motor [8]. Then, the electromechanical coupled dynamics of the electromechanical system consisting of the motor and mechanical system driven by the motor was developed. An electromechanical coupled dynamics model of the electromechanical system consisting of the several motors and mechanical system driven by these motors was proposed. Using the model, the natural frequency of the electromechanical system is analyzed [9]. The authors investigated nonlinear forced response of electromechanical integrated toroidal drive to voltage excitations [10]. However, not only the electric excitation occurs, but also the parameter excitations from mesh stiffness fluctuations occur. The nonlinear forced responses of the drive system to the coupled excitations consisting of the electric excitation and the parameter excitations have not been investigated yet. These nonlinear forced responses have important influence on the operating performance of the drive system. To design, evaluate, and control dynamics behavior of the drive system effectively, the nonlinear forced responses of the drive system to the coupled excitations should be developed.
In this paper, the electric excitation and the parameter excitation from mesh stiffness fluctuation are analyzed. The forced response equations of the drive system to the coupled excitations are presented. For the exciting frequencies far from and near natural frequencies, the forced responses of the drive system to the coupled excitations are investigated. The work can be used to predict the noise and dynamic load and are useful in maximizing the power density of the drive and reducing noise radiation.

Electric Excitation of the Drive System
The magnetic energy storage of the electric system for the drive system is where λ is is magnetic linkage of the worm coils, λ is = n j=1 L i j I is , L i j is inductances of the ith phase worm coils, I is is current of the ith phase worm coils, and n is phase number of the worm coils.
From (1), the electromagnetic torque on the planet is given as where θ is the relative rotating angle between planet and worm.
The torque T p consists of the static torque T p0 and the dynamic torque δT p . The current I is consists of the static component I i and the dynamic one δi i . The angle θ consists of the static angle θ 0 and the dynamic angle δθ. The torque T p is written in series form as If δi j = δi i = 0, from (3), one can give Let ΔF wpi denote the dynamic magnetic meshing force between a planet tooth and worm, ΔF wpi = δT p /R. From (5), neglecting the high-order terms, one knows.
The Scientific World Journal 3 Here, ζ wpi = Rδθ; it is the dynamic relative displacement between planet and worm. Thus, (6) can be changed into the following form: Here, (δ 2 L i j /δθ 2 ) θ=θ 0 I is I js , it is considered as linear electromagnetic mesh stiffness between a tooth of the planet and worm; Δk Equation (3) shows that current fluctuation can produce electromagnetic torque fluctuation. The torque fluctuation from current fluctuation can be considered as equivalent exciting torque ΔT e . Let the current periodically change as below: where ΔI is the magnitude of the fluctuation current, ω e is the frequency of the fluctuation current, and t is time.
Substituting (8) into (3), neglecting high-order terms, yields Let ΔF e denote the equivalent exciting force between a planet tooth and worm; thus

Parametric Excitation from Stiffness Fluctuation
In operation of the drive system, the number of meshing tooth pairs between the worm and the planets is variable. It can cause fluctuation of the mesh stiffness between them. It is dependent on the number z 1 of planet teeth and the conditional face angles of the worm φ v . At φ v = 100 • and z 1 = 8, the changes of mesh stiffness along with the mesh tooth pair number are shown in Figure 2. In Figure 2, a typical stiffness variation through a mesh cycle of the drive system is given. The mesh stiffness at more contact regions is higher than that at less contact regions. If the drive rotates at appreciable speed, this time-varying stiffness will be a major excitation source of the drive system. The mesh stiffness between a planet and worm is considered to consist of its mean value k and time-varying one Δk(t). The average mesh stiffness between them through one periodic time can be given by π/36 where k 2 = 2(k wpi + Δk wpi ) and k 3 = 3(k wpi + Δk wpi ); they are two and three teeth mesh stiffness, respectively. The periodical time-varying portion of the mesh stiffness can be defined in the Fourier series form as where Δk n = (2/l) l 0 k(t)ω p cos nω p tdt, l is the period of the stiffness fluctuation. In Figure 2, l = π/4. For worm and planet, the periodical time-varying portion of the mesh stiffness is π/36 k 2 − k wpi ω p cos nω p tdt Substituting (13) into (12) yields Δk nw pi cos nωt.

Forced Response Equation to Coupled Excitation
The dynamic model for the electromechanical integrated toroidal drive (see Figure 3) allows rotor and each planet to rotate about their own rotating axes and allows each planet to translate in x i and z i directions. The rotations are replaced by the corresponding translational mesh displacements as u j = r j θ j , j = 1, . . . , m, r (here, m is planet number, θ j the rotation of planet or rotor, r j is the rolling circle radius for planet and the radius of the circle passing through planet centers for the rotor). A displacement vector q j and a mass matrix m j are defined for each planet j as Here, J j and m j are polar mass moment of inertia and mass for planet j, respectively. M r (M r = J r /r 2 r ) is equivalent mass of rotor corresponding to its displacement u r . Thus, the motion equations of the drive system are where p wpi , p spi and p cpzi are relative displacements between planet-i and worm, stator, or rotor, respectively, p wpi = u i sin γ wpi − z i cos γ wpi , p spi = −u i cos γ spi − z i sin γ spi and p cpzi = z i − u r . γ wpi and γ spi are lead angles at contact points between planet-i and worm or stator, respectively, tan γ wpi = 1/[i wp (a/R − 1)], and tan γ spi = 1/[i sp (a/R + 1)]. Here, a is center distance between worm and planet R is reference circle radius of planet i wp and i sp are speed ratios between planet and worm or stator, respectively. k wpi and k spi are mesh stiffness between planet-i and worm or stator, respectively, k czi and k cxi are planet support stiffness in z i and x i directions, respectively, and T r is torque transmitted by rotor. Equation (15) can be written in matrix form as where X and F are displacement and static load vectors; respectively, M and K are mass and stiffness matrix, respectively, ΔF is nonlinear component of the force vector, ΔF e is equivalent force vector caused by current fluctuation: The Scientific World Journal 5 it is a disturbing parameter, ΔF e = D cos(ω e t) sin γ wpi 0 cosγ wpi sin γ wpi 0 cosγ wpi sin γ wpi 0 cosγ wpi · · · 0 The total displacement of the each component consists of the static displacement and the dynamic one. Hence, the displacement vector is where X and ΔX are static and dynamic displacement vectors; respectively, Considering the fluctuation of the mesh stiffness caused by the changes of the mesh tooth pair number, the stiffness matrix can be expressed as where K is the mean stiffness matrix, and ΔK(t) is timevarying stiffness matrix. Substituting (20)-(22) into (16) yields where ΔF p = −EΔK(t)X; it is considered as the equivalent exciting force vector caused by mesh stiffness excitation; E = R/r. Equation (24) is just the forced response equation of the drive system to coupled excitation.

Solution of the Nonlinear Forced
Response Equation 5.1. Far from Natural Frequencies. First, we consider to resolve the nonlinear forced response equation of the drive system to electric excitation (let ΔF p = 0). For simplicity purposes, (24) should be transmitted into equations independent on each other. Then, (24) is changed into the following form: where K N is the diagonal mean stiffness matrix, respectively. ΔX N is transmitted dynamic displacement vector. ΔF eN and ΔF N are transmitted exciting force vectors of the forces ΔF e and ΔF, respectively. Matrix K N and vector ΔX N are given by K N = A T N KA N and ΔX N = A T N ΔX. The transmitted nonlinear force vector ΔF N and transmitted equivalent exciting force vector ΔF eN can be given as below: where A N is the mode matrix of (24): At m = 3, ΔP N = [P N1 P N2 · · · P Ni · · · ] T , P Ni = −3A i N1 sin γ wpi + 3A i N3 cos γ wpi (i = 1, 4, . . . , 3m − 2), P Ni = 0 (i = 2, 3, . . . , 3m + 1, i / = 1, 4, . . . , 3m − 2); ΔP eN = [P eN1 0 0 P eN2 0 0 P eN3 0 0 0] T , P Ne = 3A i N1 sin γ wpi + 3A i N3 cos γ wpi (i=1, 4, . . . , 3m−2), P Nei = 0 (i = 2, 3, . . . , 3m+ 1, i / = 1, 4, . . . , 3m − 2). Let where ω i is natural frequency of ith order mode for the nonlinear drive system, and ω 0i is natural frequency of ith order rotational mode for the linear drive system. Substituting (28) and (29) into (25), let sum of the coefficients with the same-order power of the parameter ε equal zero, following equations can be given  Here, initial conditions are The solution of zero-order equation under the above initial conditions is Substituting (31) into the second equation of (30) yields N0 +P N1 Bu 2 0i +P N1 Cu 3 0i +D cos(ω e t)P Ne , N0 + P N2 Bu 2 0i +P N2 Cu 3 0i +D cos(ω e t)P Ne , The rotational displacement u i is Substituting (33) into (30) yields In order to remove secular item, let Substituting (35) into (34), the solutions of the firstorder equations can be obtained. As the equations of the solutions are relatively complicated, it is not given here.
In a same manner, the solution of nth order equation can be obtained as well. Substituting these solutions into (28) and (29), the solutions of the regular nonlinear forced response equations and natural frequencies of the drive system can be given. Then, the real solutions of the nonlinear forced responses can be calculated as below:

Near Natural Frequencies.
When the exciting frequency is near natural frequency, the nonlinear forced response equation can be resolved as below. Considering damping of the drive system, (25) can be changed into The Scientific World Journal 7 where C N is regular damping matrix, When exciting frequency is near natural frequency, the exciting frequency can be written as Substituting (38) and (28) into (37), let sum of the coefficients with the same-order power of the parameter ε equal zero, the following equations can be given: where C Ni = 2ζω i , The solution of zero-order equation under the above initial conditions is Substituting (41) into the second equation of (43) yields 2 N0 + P N2 B 2 u 2 01 + P N2 C 2 u 3 01 + D 2 P Ne2 cos(ω e2 t + θ), Substituting rotational displacement 3 + D 1 P Ne1 (cos ω e1 t cos θ + sin ω e1 t sin θ), In order to remove secular item, let where ]. From (44), it is known that Thus where s i = ω ei /ω i and D i = εD i P Nei . From (46), the changes of the nonlinear vibrating magnitudes along with exciting frequencies can be given.

Results and Discussions
When exciting frequency is far from natural frequency, from the above equations, the nonlinear forced vibrations for the drive system are analyzed. The parameters of the numerical example are shown in Table 1. Figure 4 shows changes of the forced vibrations of the transmitted variables along with nonlinear parameter ε. From Figure 4, the following are known.

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The Scientific World Journal  (1) Under the coupled excitations, the nonlinear forced responses of the drive system change periodically and unsteadily. The time period of the nonlinear forced responses depends on the frequencies of the electric excitation the mesh parameter excitation, and the nonlinear natural frequencies of the drive system.
(2) The vibrating amplitudes of the nonlinear forced responses of the drive system to the coupled excitations are larger than those of the nonlinear forced responses to the single excitation. The vibrating amplitudes of the nonlinear forced responses of the planet are larger than that of the rotor. It is because the exciting frequencies are near to the vibrating frequency of the planet modes.
(3) The vibrating amplitudes of the tangent vibration for the planet are larger than that of the axial vibration for the planet, and the frequency of the tangent vibration for the planet is smaller than that of the axial vibration for the planet. electric excitations should not be taken as integral multiple of the mesh parameter exciting frequency.
When exciting frequency is near natural frequency, changes of the nonlinear vibrating magnitudes along with exciting frequencies and the drive parameters are given in Figure 5. From Figure 5, the following are known.
(1) As exciting frequency increase, the vibrating magnitudes of all the modes increases, and at point s i = ω ei /ω i ≈ 1, the vibrating magnitudes get to the maximum, and then they decrease with increasing exciting frequency. For different modes, the previously mentioned curves bend toward the direction of the exciting frequency increase. The results are typical nonlinear character of the drive system.
(2) For the first and third modes, as nonlinear parameter ε increases, their vibrating magnitudes increase. For the second mode, as nonlinear parameter ε increases, its vibrating magnitude decreases. The vibrating magnitudes are large for the first and third modes and small for the second mode. The vibrating magnitude for the first mode is larger than that for the third mode.

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The Scientific World Journal (3) For a given exciting frequency, the vibrating amplitudes of the nonlinear forced responses of the drive system to the coupled excitations for modes 1 and 3 are larger than those of the nonlinear forced responses to the single excitation; for mode 2, the vibrating amplitude of the nonlinear forced responses of the drive system to the coupled excitations is smaller than that of the nonlinear forced responses to the single excitation.
(4) Under the coupled excitations, the nonlinearity has obvious effects on the relationship between the frequency and the amplitudes; it should be considered.

Conclusions
In this paper, the electric excitation and the parameter excitation from mesh stiffness fluctuation are analyzed. The forced response equations of the drive system to the coupled excitations are presented. For the exciting frequencies far from and near to natural frequencies, the forced responses of the drive system to the coupled excitations are investigated. Results show the following.
(1) Under the coupled excitations, the nonlinear forced responses of the drive system change periodically and unsteadily.
(2) The time period of the nonlinear forced responses depends on the frequencies of the electric excitation the mesh parameter excitation, and the nonlinear natural frequencies of the drive system.
(3) In order to improve the dynamics performance of the drive system, the frequencies of the electric excitations should not be taken as integral multiple of the mesh parameter exciting frequency.