# A Comparative Study Based on Different Control Algorithms for Suppressing Multistage Gear Transmission System Vibrations

Authors: Wenhao Sun; Feng Zhang; Weidong Zhu; Han Wang; Shunan Luo; Haiyan Li
Journal: Shock and Vibration
 (2018)

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## Abstract

A modal analysis (MA) was preconsidered to determine a novel active vibration control (AVC) structure of multistage gear transmission system (MGTS) and an appropriate actuating position for the piezoelectric actuator (PZT); the results of the calculating method and the finite element method (FEM) were compared to validate the reliability of MA. The controllers based on different control algorithms were designed to drive the PZTs to output the control force for suppressing the host structure vibrations. To analyze the feasibility of the applied control schemes and discuss the control effects dominated by the different control algorithms, a series of active vibration control numerical simulations were studied. The cosimulation results validate the practicability of the proposed control schemes and provide a forcible guidance for the further experimental works.


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## Body

## 1. Introduction

The presence of the vibrations is a usual problem in mechanical structures, especially in meshing gears. The vibrations of multistage gear transmission system (MGTS) are closely related to the time-varying meshing stiffness, transmission errors, and meshing impacts [1, 2]. The gear meshing vibrations will be transmitted to the whole housing structure via the transmission shafts and supporting parts, which will damage the properties of the entire transmission system seriously. Over the years, many researchers devoted themselves to solve the critical MGTS active vibration control (AVC) problems. Montague et al. [3] employed an analog phase shifter with an amplifier to drive the piezoelectric actuators (PZT) mounted onto the shaft in the gearbox. The vibration reduced up to 70% at the fundamental gear mesh frequency (FGMF) in that work. Rebbechi et al. [4] applied two pairs of magnetostrictive actuators fixed on a double row bearing to isolate the vibration transmitting from the supporting structure passing to the housing. An attenuation of 20–28 dB at FGMF was achieved, but only the first three harmonics were controlled. Chen and Brennan [5] developed a new control scheme that used three magnetostrictive actuators mounted on the gear body. In his research, the actuator according to the control scheme produced circumferential forces to suppress the torsional vibration, and about 7 dB attenuation was obtained at FGMF of 250 Hz. Guan et al. [6] proposed an active shaft transverse vibration control concept that resides within the gearbox. The system they developed contains a PZT for applying active control force to the shaft via an additive bearing, and an enhanced controller based on filtered-x least mean square (FxLMS) algorithm was designed to generate the actuation control signal. The results show the vibrations reduction up to about 20 dB at the first two mesh harmonics. Li et al. [7–9] adopted the FxLMS control algorithm to actuate the PZT to produce the active control force for reducing the gearbox vibrations, and the simulation work successfully demonstrated the feasibility of applying the proposed control scheme to gear vibration problems. Based on a combination of feedback and repetitive control strategy, Pinte et al. [10] applied a modular bearing using piezostacks to generate secondary forces for suppressing the noise and vibration of the rotation machinery. In addition, Belanger et al. [10–12] developed a finite element model of a Bell 407 transmission and simplified roof structure for investigating the active vibration control problems. An improved LMS control strategy was implemented to reduce the gearbox harmonic tones successfully. Gao et al. [13] studied an active shaft transverse vibration control structure using a modified FxLMS algorithm with online identification to achieve the gearbox vibration suppression, and the simulation results validate the efficiency of the proposed approach. In order to suppress the structure-borne noise and vibration of the rotating machinery, Zhao et al. [14] considered an active vibration control approach using two piezo-based inertial actuators directly mounted on the rotational shaft, and a multi-input-multi-output FxLMS algorithm is employed to generate the appropriate actuation signal. The results show that when the rotational speed is below 180 rpm, the plate vibrations reduced more than 7 dB. Recently, Dogruer and Pirsoltan [2] designed a nonlinear controller which can adjust the torque acting on the driving gear to minimize the disadvantageous impact of the time-varying mesh stiffness.Comparing to the single stage gear transmission system, the vibrations of MGTS aroused by the gear meshing excitations will be more complex. Thus, in the research present here, it is necessary to design a AVC structure of MGTS applying the built-in PZTs. As is known, the positions where the actuator acts on will also have an important influence on the control effect of the whole system. Therefore, confirming a AVC structure of MGTS and an accurate position for the PZT is the primary prerequisite for the following further study. A modal analysis (MA) using calculating method and finite element method (FEM), respectively, was carried out to analyze the natural characteristics of the MGTS. And the AVC structure of MGTS and an appropriate actuating position for the PZT can be determined according to the analysis results of MA. In addition, the Proportional Integral Derivate (PID) algorithm and FxLMS algorithm were applied individually to drive the PZTs to generate control forces for suppressing the complex multi-harmonic vibrations of MGTS. And a more effective control scheme was ascertained by analyzing the control effects of the applied control algorithms in a variety of different conditions. Furthermore, all the practical design parameters were used in the computations to ensure the reliability and convincingness of the cosimulation results. In the next section, the detailed dynamic modelling method and modal parameters solution will be discussed. According to MA results, an appropriate AVC structure will be concluded. Section3 mainly focuses on the vibration control algorithms, and the keystones of PID and FxLMS control algorithm are illustrated. The cosimulation study for investigating the feasibility of applied control scheme and evaluating the superiority of the two algorithms is carried out in Section 4. At last, the concrete conclusions are discussed after analyzing the relevant cosimulation results.


## 2. MA and Structure Description of MGTS

The dynamic modelling methods of gear transmission system can be mainly divided into two kinds. The first one is the classical modelling methods (CMM), and another one belongs to the modern modelling methods (MMM). The CMM includes two aspects: discretization and mathematicization. In other words, it means a continuous system with infinite DOFs will be discretized into a lumped parameters dynamic model with finite DOFs firstly. Then the dynamic model can be derived into the relevant analytic model by utilizing the mathematicization method. This mentioned discretization method is usually referred to as the lumped parameter method (LPM), which is one of the most commonly used dynamic modelling methods. For instance, Xu [15], Zhang [16], Ericson [17], and Hammami [18] et al., respectively, used LPM to carry out the dynamic modelling of multistage gear transmission systems. On the other hand, the principle representative of MMM is finite element method (FEM), which can integrate the discretization and mathematicization into a whole. To put it in another way, the FEM can achieve the dynamic modelling process and the derivation process of analytic model combining into one. Therefore, the principle methods discussed here are utilizing LPM to establish a dynamic model of the MGTS to resolve its modal parameters. Furthermore, the FEM will also be applied for carrying out the modal analysis of the MGTS to validate the correctness of both dynamic modelling of the MGTS and modal solution. The concrete contents are given in this section.


### 2.1. Dynamic Model of MGTS

Considering the torsional-bending vibrations of the transmission shafts and the bending vibrations of the supporting bearings, the equivalent dynamic model of the MGTS with 26 degrees of freedom (DOF) was built as shown in Figure1. Oixiyizii=1,2,3,4 are the fixed coordinate systems for meshing gears, and xbiybizbii=1,2,3,4,5,6 are the fixed coordinate systems for the supporting bearings. The figure in the green dashed block diagram on the upper left of Figure 1 denotes the lumped mass model of the meshing gears. Kmi,Cmi(i=1,2) represent the gear meshing stiffness and damping, respectively, and eii=1,2 are the gear meshing error. Kbxi,Kbyi,Cbxi,Cbyii=1,2,3,4,5,6 denote the radial supporting stiffness and damping of the bearings. θi(i=1,2,3,4) represent the angular displacement of the meshing gears. Iin and Iout severally denote the moment of inertia of the motor and load.Figure 1
The dynamic model of MGTS.In fact, the gear meshing vibrations will change the position of the gear theoretical rotation center, and the dynamic transformation schematic diagram of the gear theoretical center is given in Figure2(a). The blue solid circles (Oixiyi, i=1,3), (Ojxjyj, j=2,4) denote the original position of gears, and the red dashed dot circles (Gixiyi, i=1,3), (Gjxjyj, j=2,4) denote the dynamic position of the gear affected by the meshing vibration. More than this, the meshing vibration will also arouse the transmission shafts generating translational and bending deformations. As shown in Figure 2(b), the blue line B1O1B2 is the original position of the driving transmission shaft in the steady state condition, and the red dashed line Bo1G1Bo2 is the position after deformation owing to the vibrations propagation, projecting Bo1G1Bo2 to x- and y-axis directions. Bo1x,Bo1y and Bo2x,Bo2y are the projective coordinate corresponding to Bo1 and Bo2 in x- and y-axis directions, and O1x,O1y is projective coordinate of point O1. The distances between the projective points and the coordinate origin can be expressed as follows: B1Bo1x=xb1, B1Bo1y=yb1, B2Bo2x=xb2, B2Bo2y=yb2, O1O1x=x1, O1O1y=y1. The line segment Bo1xBo2x intersects with line segment O1O1x at point O1bx. Therefore, the line segment O1O1bx just denotes the translational deformation without considering the bending deformation. Thus, O1xO1bx can be used to represent the bending deformation δ1 of the driving shaft at the location of driving gear rotation center. And δ1 can be divided into x and y direction components δ1x,δ1y. Analogously, the bending deformations in the x and y directions of the transmission shafts at the other meshing gear rotation centers can be expressed as follows:(1)δ1x=x1-τ1xb2-τ2xb1δ1y=y1-τ1yb2-τ2yb1δ2x=x2-τ3xb4-τ4xb3δ2y=y2-τ3yb4-τ4yb3δ3x=x3-τ5xb4-τ6xb4δ3y=y3-τ5yb4-τ6yb3δ4x=x4-τ7xb6-τ8xb5δ4y=y4-τ7yb6-τ8yb5where τj=lij/li(i=1,2,3,4;j=1,2,3,4,5,6,7,8), lij are the distances between the gear rotation centers and the bearing theoretical centers, and li are the lengths between the two bearings theoretical centers located on one shaft. In addition, the deformations δm1, δm2 along the meshing lines can be obtained from Figure 2(a):(2)δm1=x1-x2sin⁡α1+y1-y2cos⁡α1+θ1r1-θ2r2-e1δm2=x3-x4sin⁡α2+y3-y4cos⁡α2+θ3r3-θ4r4-e2where rii=1,2,3,4 is the radius of basic circle and αi(i=1,2) are the angle of the working pressure angles.Figure 2
The dynamic transformation schematic diagram of the internal parts of the MGTS.
(a) (b)Assumingmi(i=1,2,3,4) are the lumped mass of the transmitting gears, mbi(i=1,2,3,4,5,6) are the lumped mass of the bearings. Ctii=1,2,3 and Ktii=1,2,3 denote the torsional damping and torsional stiffness of the transmission shafts, respectively. Csi(i=1,2,3,4) and Ksi(i=1,2,3,4) denote the bending damping and stiffness of the transmission shafts at gear theoretical rotation centers, respectively. And Ii(i=1,2,3,4) are the moments of inertial of the transmitting gears, and Tin,Tout denote the torques of the motor and load.The differential equations of motion (DEOM) can be given by analyzing the relationships among the components of MGTS:(3)Iinθ¨in+Ct1θ˙in-θ˙1+Kt1θin-θ1=TinEquation (3) is the DEOM of the motor, and the DEOM of the transmitting gears can be expressed from (4) to (7):(4)m1x¨1+Cs1δ˙1x+Ks1δ1x+Cm1δ˙m1sin⁡α1+Km1δm1sin⁡α1=0m1y¨1+Cs1δ˙1y+Ks1δ1y+Cm1δ˙m1cos⁡α1+Km1δm1cos⁡α1=0I1θ¨1+Ct1θ˙1-θ˙in+Kt1θ1-θin+r1Cm1δ˙m1+r1Km1δm1=0(5)m2x¨2+Cs2δ˙2x+Ks2δ2x-Cm1δ˙m1sin⁡α1-Km1δm1sin⁡α1=0m2y¨2+Cs2δ˙2y+Ks2δ2y-Cm1δ˙m1cos⁡α1-Km1δm1cos⁡α1=0I2θ¨2+Ct2θ˙2-θ˙3+Kt2θ2-θ3-r1Cm1δ˙m1-r2Km1δm1=0(6)m3x¨3+Cs3δ˙3x+Ks3δ3x+Cm2δ˙m2sin⁡α2+Km2δm2sin⁡α2=0m3y¨3+Cs3δ˙3y+Ks3δ3y+Cm2δ˙m2cos⁡α2+Km2δm2cos⁡α2=0I3θ¨3+Ct2θ˙3-θ˙2+Kt2θ3-θ2+r3Cm2δ˙m2+r3Km2δm2=0(7)m4x¨4+Cs4δ˙4x+Ks4δ4x-Cm2δ˙m2sin⁡α2-Km2δm2sin⁡α2=0m4y¨4+Cs4δ˙4y+Ks4δ4y-Cm2δ˙m2cos⁡α2-Km2δm2cos⁡α2=0I4θ¨4+Ct3θ˙4-θ˙out+Kt3θ4-θout-r4Cm2δ˙m2-r4Km2δm2=0The DEOM of the load can be given by(8)Ioutθ¨out+Ct3θ˙out-θ˙4+Kt3θout-θ4=-ToutThe DEOM of the supporting bearings can be written as(9)mbix¨bi+Cbxix˙bi+Kbxixbi=Fbximbiy¨bi+Cbyiy˙bi+Kbyiybi=Fbyiwhere Cbxi, Kbxi(i=1,2,3,4,5,6) are the supporting damping and stiffness of the supporting bearings. Because of the deformations of the transmission shafts at the theoretical rotation center of the gears, the supporting reactions of the transmission shaft at the supporting bearings are induced, and it can be expressed as below:(10)Fbnii=1,2,3,4,5,6Fbn1=τ2Cs1δ˙1m+Ks1δ1mFbn2=τ1Cs1δ˙1m+Ks1δ1mFbn3=τ4Cs2δ˙2m+Ks2δ2m+τ6Cs3δ˙3m+Ks3δ3mFbn4=τ3Cs2δ˙2m+Ks2δ2m+τ5Cs3δ˙3m+Ks3δ3mFbn5=τ8Cs4δ˙4m+Ks4δ4mFbn6=τ7Cs4δ˙4m+Ks4δ4mn=x,yEquations (1) and (2) are substituted into (3) to (8), and (10) is substituted into (9), and the equations of motion of the MGTS can be written as(11)Mq¨+Cb+Cmq˙+Kb+Kmq=T+F+E=PThe generalized coordinates array is(12)q=θinx1y1θ1x2y2θ2x3y3θ3x4y4θ4xb1yb1xb2yb2xb3yb3xb4yb4xb5yb5xb6yb6θoutTThe mass matrix M can be given as(13)M=diag⁡Iinm1m1I1m2m2I2m3m3I3m4m4I4mb1mb1mb2mb2mb3mb3mb4mb4mb5mb5mb6mb6IoutThe stiffness matrix K=Kb+Km can be expressed as follows:(14)K=ADBCwhere A is a matrix that consists of four different submatrices A1, A2, A3, A4:(15)A=A1A2A3A4where A1 is given as follows:(16)A1=Kt1-Kt1Ks1+Km1sin2⁡α1Km1sin⁡α1cos⁡α1Km1r1sin⁡α1-Km1sin2α1-Km1sinα1cosα1-Km1r2sinα1Km1sin⁡α1cos⁡α1Ks1+Km1cos2⁡α1Km1r1cos⁡α1-Km1sinα1cosα1-Km1cos2α1-Km1r2cosα1-Kt1Km1r1sin⁡α1Km1r1cos⁡α1Kt1+Km1r12-Km1r1sin⁡α1-Km1r1cosα1-Km1r1r2-Km1sin2⁡α1-Km1sin⁡α1cos⁡α1-Km1r1sin⁡α1Ks2+Km1sin2α1Km1sinα1cosα1Km1r2sinα1-Km1sin⁡α1cos⁡α1-Km1cos2⁡α1-Km1r1cos⁡α1Km1sinα1cosα1Ks2+Km1cos2α1Km1r2cosα1-Km1r2sin⁡α1-Km1r2cos⁡α1-Km1r1r2Km1r2sinα1Km1r2cosα1Kt2+Km1r22The nonzero element of A2 is A27,3=-Kt2, and the nonzero element of A3 is A3(3,7)=-Kt2, as for A4, it can be expressed as below:(17)A4=Ks3+Km2sin2⁡α2Km2sin⁡α2cos⁡α2Km2r3sin⁡α2-Km2sin2⁡α2-Km2sin⁡α2cos⁡α2-Km2r4sin⁡α2Km2sin⁡α2cos⁡α2Ks3+Km2cos2⁡α2Km2r3cos⁡α2-Km2sin⁡α2cos⁡α2-Km2cos2⁡α2-Km2r4cos⁡α2Km2r3sin⁡α2Km2r3cos⁡α2Kt2+Km2r32-Km2r3sin⁡α2-Km2r3cos⁡α2-Km2r3r4-Km2sin2⁡α2-Km2sin⁡α2cos⁡α2-Km2r3sin⁡α2Ks4+Km2sin2⁡α2Km2sin⁡α2cos⁡α2Km2r4sin⁡α2-Km2sin⁡α2cos⁡α2-Km2cos2⁡α2-Km2r3cos⁡α2Km2sin⁡α2cos⁡α2Ks4+Km2cos2⁡α2Km2r4cos⁡α2-Km2r4sin⁡α2-Km2r4cos⁡α2-Km2r3r4Km2r4sin⁡α2Km2r4cos⁡α2Kt3+Km2r42The nonzero element of B is B(13,13)=-Kt3, and the matrix C can be given as follows:(18)C=diag⁡Kbx1Kby1Kbx2Kby2Kbx3Kby3Kbx4Kby4Kbx5Kby5Kbx6Kby6Kt3The matrix D is expressed as below:(19)D=0⋯0-τ2Ks1-τ1Ks1-τ2Ks1-τ1Ks1⋱0⋯⋮-τ4Ks2-τ3Ks2-τ4Ks2-τ3Ks2⋱⋮⋯⋮-τ6Ks3-τ5Ks3⋱-τ6Ks3-τ5Ks3⋮⋯0⋱-τ8Ks4-τ7Ks4-τ8Ks4-τ7Ks40⋯-Kt3The damping matrix C′=Cb+Cm can be obtained by replacing the stiffness parameters with corresponding damping parameters. The external load matrix is given in (22).


### 2.2. Modal Analysis of MGTS Based on FEM

In this paper, determining an appropriate AVC structure of the MGTS is a primary task for reducing the vibration magnitude of the MGTS, and ascertaining an accurate actuating position for the PZT is a key objective. Furthermore, another noteworthy problem is the excitation frequencies that should be selected carefully to avoid the entire system generating resonances. Therefore, a MA was carried out to solve the above problems.For the undamped free vibration system, the vibration equations can be obtained by simplifying (11), as shown in (20):(20)Mq¨+Kq=0The essence of modal solving process is a method for decoupling the differential equations, converting the physical coordinates of the differential equations to modal coordinates. Thus, (17) can be written as follows:(21)K-wi2Mφi=0(22)P=TinCm1e˙1sin⁡α1+Km1e1sin⁡α1Cm1e˙1cos⁡α1+Km1e1cos⁡α1r1Cm1e˙1+r1Km1e1-Cm1e˙1sin⁡α1-Km1e1sin⁡α1-Cm1e˙1cos⁡α1-Km1e1cos⁡α1-r2Cm1e˙1-r2Km1e1Cm2e˙2sin⁡α2+Km2e2sin⁡α2Cm2e˙2cos⁡α2+Km2e2cos⁡α2r3Cm2e˙2+r3Km2e2-Cm2e˙2sin⁡α2-Km2e2sin⁡α2-Cm2e˙2cos⁡α2-Km2e2cos⁡α2-r4Cm2e˙2-r4Km2e20⋮0-ToutIn (22), wi,φii=1,2,…,n denote the ith order natural frequency and the ith order modal shape of the MGTS. For the free vibration system, lower order vibration modals have more prominent contribution rate and determine the dynamic response characteristics of the system [19], and forasmuch, the 1st- to 6th-order modal parameters are extracted by calculation method and FEM, respectively. And the main parameters for calculating the modal parameters of the MGTS are given in Table 1.Table 1
Parameters of MGTS.
Variable Numerical value Unit m 1/m2 0.53/1.264 Kg m 3/m4 0.56/1.526 Kg m b i 0.25 Kg r 1/r2 0.19/0.37 M r 3/r4 0.23/0.35 M α 1/α2 20/20 ( ∘ ) K t 1 8.0024 × 1 0 4 N / m K t 2 8.0425 × 1 0 4 N / m K t 3 5.056 × 1 0 4 N / m K s 1 2.4972 × 1 0 7 N / m K s 2 2.3162 × 1 0 7 N / m K s 3 2.7134 × 1 0 7 N / m K s 4 1.9721 × 1 0 7 N / m K b 5.5 × 1 0 7 N / m I i n 0.0614 kg⋅m2 I 1 9.416 × 1 0 - 4 kg⋅m2 I 2 8.05 × 1 0 - 4 kg⋅m2 I 3 9.042 × 1 0 - 4 kg⋅m2 I 4 9.416 × 1 0 - 4 kg⋅m2 I o u t 0.0144 kg⋅m2 Poisson ratio 0.3 Density 7.85 × 1 0 3 Kg/m3 Young’s Modulus 2 × 1 0 5 MPaAs for the modal parameters solved by FEM, firstly, the three-dimensional model of MGTS was built in the UG NX software.Then the model was exported as a Parasolid file into ANSYS Workbench platform. Mechanic properties of materials used for the modal analysis are elastic modulus, Poisson's ratio, and material density; these parameters are given in Table1. Using automatic mesh generation, then the preceding six-order modal parameter was extracted. The three-dimension model and the meshed model of MGTS are given in Figures 3(a) and 3(b), respectively. And the comparative analysis of the natural frequencies with the two methods is given in Table 2.Table 2
The comparisons of natural frequency by two different methods.
Order Calculation method FEM Error rate 1st 1547.2 1587 2.5% 2nd 1889.82 1938.2 2.49% 3rd 2261.2 2328.3 2.88% 4 t h 2588.4 2656.5 2.56% 5th 3268.94 3288 0.57% 6th 3299.51 3324.3 0.74%Figure 3
The meshed modal of MGTS.
(a) (b)From Table2, we can find that the general relative error rate between the two solutions is less than 5%, which is theoretically permissible. To some extent, the dynamic model of MGTS established in forward has been validated. Furthermore, the preceding six-order mode shapes are arrayed in proper sequence as shown in Figures 4(a)–4(f). From Figure 4, we can find that the 1st-, 4th-, and 6th-order modal shapes are mainly taking on the bending vibration, and the 2nd-, 3rd-, and 5th-order modal shapes are mainly taking on bending-torsional coupling vibration. In general, the system will generate some modal vibrations due to the influence of the work load. And the total system response at work is a linear superposition of the response of each excited mode. Therefore, it is necessary to analyze the modals of MGTS for determining a AVC structure and a position where the PZTs should act on. According to Figure 4, it is apparent to know that the max deformation of the gears mainly appeared at gear teeth and its inside. And it is too hard to fix the PZTs on the gear teeth firmly during work. However, the max and main deformation for the transmission shafts mainly appeared at the neighbourhood of the gears; thus it is a good choice to apply the PZT working on the transmission shafts. Therefore, this paper applied a pair of PZTs acting on the high speed and low speed shafts. This active shaft control concept can be verified by a series of cosimulations; furthermore, a better actuating position for the PZTs can also be obtained by the numerical simulations, all of these will be discussed in detail in Section 4. And the details of the MGTS are discussed in the next section.Figure 4
The preceding six-order modal shapes of MGTS.
(a) (b) (c) (d) (e) (f)


### 2.3. Structure Description of MGTS

The mesh excitations interact with the MGTS dynamic characteristics to generate the time-varying mesh forces, which will bring about the complex vibrations. Then the vibration energy is transmitted through the transmission shafts and supporting parts to the host housing, but just because the existence and transmitting of the vibration will have a severe effect on the working performance of the MGTS. Hence, a AVC structure for reducing the MGTS vibrations was developed based on the previous MA in this paper. The three-dimensional view of the MGTS and the cross-sectional view expounding the inner details of the proposed outfit are shown in Figures5(a) and 5(b). The main structural configuration includes a motor, a load, a pair of PZTs, and the MGTS. The motor and load are connected with MGTS through a pair of couplings. As for the PZTs, they can be mounted on the auxiliary supports, and the auxiliary supports can be mounted on both side walls of the gearbox housing.Figure 5
The three-dimension and cross-sectional view of MGTS.
(a) (b)Furthermore, because the transmission shafts may generate vibration deformation, the PZTs may have some lateral motions that may affect the PZTs. Hence, the setup must be carefully designed such that the PZTs should not be directly connected to the additional bearing raceway. Just because of this, a decompression unit is immobilized between the PZT and the addition bearing for cutting down the longitudinal force to avoid PZT generating unnecessary damage, and the detailed view is given in Figure5(a).


## 2.1. Dynamic Model of MGTS

Considering the torsional-bending vibrations of the transmission shafts and the bending vibrations of the supporting bearings, the equivalent dynamic model of the MGTS with 26 degrees of freedom (DOF) was built as shown in Figure1. Oixiyizii=1,2,3,4 are the fixed coordinate systems for meshing gears, and xbiybizbii=1,2,3,4,5,6 are the fixed coordinate systems for the supporting bearings. The figure in the green dashed block diagram on the upper left of Figure 1 denotes the lumped mass model of the meshing gears. Kmi,Cmi(i=1,2) represent the gear meshing stiffness and damping, respectively, and eii=1,2 are the gear meshing error. Kbxi,Kbyi,Cbxi,Cbyii=1,2,3,4,5,6 denote the radial supporting stiffness and damping of the bearings. θi(i=1,2,3,4) represent the angular displacement of the meshing gears. Iin and Iout severally denote the moment of inertia of the motor and load.Figure 1
The dynamic model of MGTS.In fact, the gear meshing vibrations will change the position of the gear theoretical rotation center, and the dynamic transformation schematic diagram of the gear theoretical center is given in Figure2(a). The blue solid circles (Oixiyi, i=1,3), (Ojxjyj, j=2,4) denote the original position of gears, and the red dashed dot circles (Gixiyi, i=1,3), (Gjxjyj, j=2,4) denote the dynamic position of the gear affected by the meshing vibration. More than this, the meshing vibration will also arouse the transmission shafts generating translational and bending deformations. As shown in Figure 2(b), the blue line B1O1B2 is the original position of the driving transmission shaft in the steady state condition, and the red dashed line Bo1G1Bo2 is the position after deformation owing to the vibrations propagation, projecting Bo1G1Bo2 to x- and y-axis directions. Bo1x,Bo1y and Bo2x,Bo2y are the projective coordinate corresponding to Bo1 and Bo2 in x- and y-axis directions, and O1x,O1y is projective coordinate of point O1. The distances between the projective points and the coordinate origin can be expressed as follows: B1Bo1x=xb1, B1Bo1y=yb1, B2Bo2x=xb2, B2Bo2y=yb2, O1O1x=x1, O1O1y=y1. The line segment Bo1xBo2x intersects with line segment O1O1x at point O1bx. Therefore, the line segment O1O1bx just denotes the translational deformation without considering the bending deformation. Thus, O1xO1bx can be used to represent the bending deformation δ1 of the driving shaft at the location of driving gear rotation center. And δ1 can be divided into x and y direction components δ1x,δ1y. Analogously, the bending deformations in the x and y directions of the transmission shafts at the other meshing gear rotation centers can be expressed as follows:(1)δ1x=x1-τ1xb2-τ2xb1δ1y=y1-τ1yb2-τ2yb1δ2x=x2-τ3xb4-τ4xb3δ2y=y2-τ3yb4-τ4yb3δ3x=x3-τ5xb4-τ6xb4δ3y=y3-τ5yb4-τ6yb3δ4x=x4-τ7xb6-τ8xb5δ4y=y4-τ7yb6-τ8yb5where τj=lij/li(i=1,2,3,4;j=1,2,3,4,5,6,7,8), lij are the distances between the gear rotation centers and the bearing theoretical centers, and li are the lengths between the two bearings theoretical centers located on one shaft. In addition, the deformations δm1, δm2 along the meshing lines can be obtained from Figure 2(a):(2)δm1=x1-x2sin⁡α1+y1-y2cos⁡α1+θ1r1-θ2r2-e1δm2=x3-x4sin⁡α2+y3-y4cos⁡α2+θ3r3-θ4r4-e2where rii=1,2,3,4 is the radius of basic circle and αi(i=1,2) are the angle of the working pressure angles.Figure 2
The dynamic transformation schematic diagram of the internal parts of the MGTS.
(a) (b)Assumingmi(i=1,2,3,4) are the lumped mass of the transmitting gears, mbi(i=1,2,3,4,5,6) are the lumped mass of the bearings. Ctii=1,2,3 and Ktii=1,2,3 denote the torsional damping and torsional stiffness of the transmission shafts, respectively. Csi(i=1,2,3,4) and Ksi(i=1,2,3,4) denote the bending damping and stiffness of the transmission shafts at gear theoretical rotation centers, respectively. And Ii(i=1,2,3,4) are the moments of inertial of the transmitting gears, and Tin,Tout denote the torques of the motor and load.The differential equations of motion (DEOM) can be given by analyzing the relationships among the components of MGTS:(3)Iinθ¨in+Ct1θ˙in-θ˙1+Kt1θin-θ1=TinEquation (3) is the DEOM of the motor, and the DEOM of the transmitting gears can be expressed from (4) to (7):(4)m1x¨1+Cs1δ˙1x+Ks1δ1x+Cm1δ˙m1sin⁡α1+Km1δm1sin⁡α1=0m1y¨1+Cs1δ˙1y+Ks1δ1y+Cm1δ˙m1cos⁡α1+Km1δm1cos⁡α1=0I1θ¨1+Ct1θ˙1-θ˙in+Kt1θ1-θin+r1Cm1δ˙m1+r1Km1δm1=0(5)m2x¨2+Cs2δ˙2x+Ks2δ2x-Cm1δ˙m1sin⁡α1-Km1δm1sin⁡α1=0m2y¨2+Cs2δ˙2y+Ks2δ2y-Cm1δ˙m1cos⁡α1-Km1δm1cos⁡α1=0I2θ¨2+Ct2θ˙2-θ˙3+Kt2θ2-θ3-r1Cm1δ˙m1-r2Km1δm1=0(6)m3x¨3+Cs3δ˙3x+Ks3δ3x+Cm2δ˙m2sin⁡α2+Km2δm2sin⁡α2=0m3y¨3+Cs3δ˙3y+Ks3δ3y+Cm2δ˙m2cos⁡α2+Km2δm2cos⁡α2=0I3θ¨3+Ct2θ˙3-θ˙2+Kt2θ3-θ2+r3Cm2δ˙m2+r3Km2δm2=0(7)m4x¨4+Cs4δ˙4x+Ks4δ4x-Cm2δ˙m2sin⁡α2-Km2δm2sin⁡α2=0m4y¨4+Cs4δ˙4y+Ks4δ4y-Cm2δ˙m2cos⁡α2-Km2δm2cos⁡α2=0I4θ¨4+Ct3θ˙4-θ˙out+Kt3θ4-θout-r4Cm2δ˙m2-r4Km2δm2=0The DEOM of the load can be given by(8)Ioutθ¨out+Ct3θ˙out-θ˙4+Kt3θout-θ4=-ToutThe DEOM of the supporting bearings can be written as(9)mbix¨bi+Cbxix˙bi+Kbxixbi=Fbximbiy¨bi+Cbyiy˙bi+Kbyiybi=Fbyiwhere Cbxi, Kbxi(i=1,2,3,4,5,6) are the supporting damping and stiffness of the supporting bearings. Because of the deformations of the transmission shafts at the theoretical rotation center of the gears, the supporting reactions of the transmission shaft at the supporting bearings are induced, and it can be expressed as below:(10)Fbnii=1,2,3,4,5,6Fbn1=τ2Cs1δ˙1m+Ks1δ1mFbn2=τ1Cs1δ˙1m+Ks1δ1mFbn3=τ4Cs2δ˙2m+Ks2δ2m+τ6Cs3δ˙3m+Ks3δ3mFbn4=τ3Cs2δ˙2m+Ks2δ2m+τ5Cs3δ˙3m+Ks3δ3mFbn5=τ8Cs4δ˙4m+Ks4δ4mFbn6=τ7Cs4δ˙4m+Ks4δ4mn=x,yEquations (1) and (2) are substituted into (3) to (8), and (10) is substituted into (9), and the equations of motion of the MGTS can be written as(11)Mq¨+Cb+Cmq˙+Kb+Kmq=T+F+E=PThe generalized coordinates array is(12)q=θinx1y1θ1x2y2θ2x3y3θ3x4y4θ4xb1yb1xb2yb2xb3yb3xb4yb4xb5yb5xb6yb6θoutTThe mass matrix M can be given as(13)M=diag⁡Iinm1m1I1m2m2I2m3m3I3m4m4I4mb1mb1mb2mb2mb3mb3mb4mb4mb5mb5mb6mb6IoutThe stiffness matrix K=Kb+Km can be expressed as follows:(14)K=ADBCwhere A is a matrix that consists of four different submatrices A1, A2, A3, A4:(15)A=A1A2A3A4where A1 is given as follows:(16)A1=Kt1-Kt1Ks1+Km1sin2⁡α1Km1sin⁡α1cos⁡α1Km1r1sin⁡α1-Km1sin2α1-Km1sinα1cosα1-Km1r2sinα1Km1sin⁡α1cos⁡α1Ks1+Km1cos2⁡α1Km1r1cos⁡α1-Km1sinα1cosα1-Km1cos2α1-Km1r2cosα1-Kt1Km1r1sin⁡α1Km1r1cos⁡α1Kt1+Km1r12-Km1r1sin⁡α1-Km1r1cosα1-Km1r1r2-Km1sin2⁡α1-Km1sin⁡α1cos⁡α1-Km1r1sin⁡α1Ks2+Km1sin2α1Km1sinα1cosα1Km1r2sinα1-Km1sin⁡α1cos⁡α1-Km1cos2⁡α1-Km1r1cos⁡α1Km1sinα1cosα1Ks2+Km1cos2α1Km1r2cosα1-Km1r2sin⁡α1-Km1r2cos⁡α1-Km1r1r2Km1r2sinα1Km1r2cosα1Kt2+Km1r22The nonzero element of A2 is A27,3=-Kt2, and the nonzero element of A3 is A3(3,7)=-Kt2, as for A4, it can be expressed as below:(17)A4=Ks3+Km2sin2⁡α2Km2sin⁡α2cos⁡α2Km2r3sin⁡α2-Km2sin2⁡α2-Km2sin⁡α2cos⁡α2-Km2r4sin⁡α2Km2sin⁡α2cos⁡α2Ks3+Km2cos2⁡α2Km2r3cos⁡α2-Km2sin⁡α2cos⁡α2-Km2cos2⁡α2-Km2r4cos⁡α2Km2r3sin⁡α2Km2r3cos⁡α2Kt2+Km2r32-Km2r3sin⁡α2-Km2r3cos⁡α2-Km2r3r4-Km2sin2⁡α2-Km2sin⁡α2cos⁡α2-Km2r3sin⁡α2Ks4+Km2sin2⁡α2Km2sin⁡α2cos⁡α2Km2r4sin⁡α2-Km2sin⁡α2cos⁡α2-Km2cos2⁡α2-Km2r3cos⁡α2Km2sin⁡α2cos⁡α2Ks4+Km2cos2⁡α2Km2r4cos⁡α2-Km2r4sin⁡α2-Km2r4cos⁡α2-Km2r3r4Km2r4sin⁡α2Km2r4cos⁡α2Kt3+Km2r42The nonzero element of B is B(13,13)=-Kt3, and the matrix C can be given as follows:(18)C=diag⁡Kbx1Kby1Kbx2Kby2Kbx3Kby3Kbx4Kby4Kbx5Kby5Kbx6Kby6Kt3The matrix D is expressed as below:(19)D=0⋯0-τ2Ks1-τ1Ks1-τ2Ks1-τ1Ks1⋱0⋯⋮-τ4Ks2-τ3Ks2-τ4Ks2-τ3Ks2⋱⋮⋯⋮-τ6Ks3-τ5Ks3⋱-τ6Ks3-τ5Ks3⋮⋯0⋱-τ8Ks4-τ7Ks4-τ8Ks4-τ7Ks40⋯-Kt3The damping matrix C′=Cb+Cm can be obtained by replacing the stiffness parameters with corresponding damping parameters. The external load matrix is given in (22).


## 2.2. Modal Analysis of MGTS Based on FEM

In this paper, determining an appropriate AVC structure of the MGTS is a primary task for reducing the vibration magnitude of the MGTS, and ascertaining an accurate actuating position for the PZT is a key objective. Furthermore, another noteworthy problem is the excitation frequencies that should be selected carefully to avoid the entire system generating resonances. Therefore, a MA was carried out to solve the above problems.For the undamped free vibration system, the vibration equations can be obtained by simplifying (11), as shown in (20):(20)Mq¨+Kq=0The essence of modal solving process is a method for decoupling the differential equations, converting the physical coordinates of the differential equations to modal coordinates. Thus, (17) can be written as follows:(21)K-wi2Mφi=0(22)P=TinCm1e˙1sin⁡α1+Km1e1sin⁡α1Cm1e˙1cos⁡α1+Km1e1cos⁡α1r1Cm1e˙1+r1Km1e1-Cm1e˙1sin⁡α1-Km1e1sin⁡α1-Cm1e˙1cos⁡α1-Km1e1cos⁡α1-r2Cm1e˙1-r2Km1e1Cm2e˙2sin⁡α2+Km2e2sin⁡α2Cm2e˙2cos⁡α2+Km2e2cos⁡α2r3Cm2e˙2+r3Km2e2-Cm2e˙2sin⁡α2-Km2e2sin⁡α2-Cm2e˙2cos⁡α2-Km2e2cos⁡α2-r4Cm2e˙2-r4Km2e20⋮0-ToutIn (22), wi,φii=1,2,…,n denote the ith order natural frequency and the ith order modal shape of the MGTS. For the free vibration system, lower order vibration modals have more prominent contribution rate and determine the dynamic response characteristics of the system [19], and forasmuch, the 1st- to 6th-order modal parameters are extracted by calculation method and FEM, respectively. And the main parameters for calculating the modal parameters of the MGTS are given in Table 1.Table 1
Parameters of MGTS.
Variable Numerical value Unit m 1/m2 0.53/1.264 Kg m 3/m4 0.56/1.526 Kg m b i 0.25 Kg r 1/r2 0.19/0.37 M r 3/r4 0.23/0.35 M α 1/α2 20/20 ( ∘ ) K t 1 8.0024 × 1 0 4 N / m K t 2 8.0425 × 1 0 4 N / m K t 3 5.056 × 1 0 4 N / m K s 1 2.4972 × 1 0 7 N / m K s 2 2.3162 × 1 0 7 N / m K s 3 2.7134 × 1 0 7 N / m K s 4 1.9721 × 1 0 7 N / m K b 5.5 × 1 0 7 N / m I i n 0.0614 kg⋅m2 I 1 9.416 × 1 0 - 4 kg⋅m2 I 2 8.05 × 1 0 - 4 kg⋅m2 I 3 9.042 × 1 0 - 4 kg⋅m2 I 4 9.416 × 1 0 - 4 kg⋅m2 I o u t 0.0144 kg⋅m2 Poisson ratio 0.3 Density 7.85 × 1 0 3 Kg/m3 Young’s Modulus 2 × 1 0 5 MPaAs for the modal parameters solved by FEM, firstly, the three-dimensional model of MGTS was built in the UG NX software.Then the model was exported as a Parasolid file into ANSYS Workbench platform. Mechanic properties of materials used for the modal analysis are elastic modulus, Poisson's ratio, and material density; these parameters are given in Table1. Using automatic mesh generation, then the preceding six-order modal parameter was extracted. The three-dimension model and the meshed model of MGTS are given in Figures 3(a) and 3(b), respectively. And the comparative analysis of the natural frequencies with the two methods is given in Table 2.Table 2
The comparisons of natural frequency by two different methods.
Order Calculation method FEM Error rate 1st 1547.2 1587 2.5% 2nd 1889.82 1938.2 2.49% 3rd 2261.2 2328.3 2.88% 4 t h 2588.4 2656.5 2.56% 5th 3268.94 3288 0.57% 6th 3299.51 3324.3 0.74%Figure 3
The meshed modal of MGTS.
(a) (b)From Table2, we can find that the general relative error rate between the two solutions is less than 5%, which is theoretically permissible. To some extent, the dynamic model of MGTS established in forward has been validated. Furthermore, the preceding six-order mode shapes are arrayed in proper sequence as shown in Figures 4(a)–4(f). From Figure 4, we can find that the 1st-, 4th-, and 6th-order modal shapes are mainly taking on the bending vibration, and the 2nd-, 3rd-, and 5th-order modal shapes are mainly taking on bending-torsional coupling vibration. In general, the system will generate some modal vibrations due to the influence of the work load. And the total system response at work is a linear superposition of the response of each excited mode. Therefore, it is necessary to analyze the modals of MGTS for determining a AVC structure and a position where the PZTs should act on. According to Figure 4, it is apparent to know that the max deformation of the gears mainly appeared at gear teeth and its inside. And it is too hard to fix the PZTs on the gear teeth firmly during work. However, the max and main deformation for the transmission shafts mainly appeared at the neighbourhood of the gears; thus it is a good choice to apply the PZT working on the transmission shafts. Therefore, this paper applied a pair of PZTs acting on the high speed and low speed shafts. This active shaft control concept can be verified by a series of cosimulations; furthermore, a better actuating position for the PZTs can also be obtained by the numerical simulations, all of these will be discussed in detail in Section 4. And the details of the MGTS are discussed in the next section.Figure 4
The preceding six-order modal shapes of MGTS.
(a) (b) (c) (d) (e) (f)


## 2.3. Structure Description of MGTS

The mesh excitations interact with the MGTS dynamic characteristics to generate the time-varying mesh forces, which will bring about the complex vibrations. Then the vibration energy is transmitted through the transmission shafts and supporting parts to the host housing, but just because the existence and transmitting of the vibration will have a severe effect on the working performance of the MGTS. Hence, a AVC structure for reducing the MGTS vibrations was developed based on the previous MA in this paper. The three-dimensional view of the MGTS and the cross-sectional view expounding the inner details of the proposed outfit are shown in Figures5(a) and 5(b). The main structural configuration includes a motor, a load, a pair of PZTs, and the MGTS. The motor and load are connected with MGTS through a pair of couplings. As for the PZTs, they can be mounted on the auxiliary supports, and the auxiliary supports can be mounted on both side walls of the gearbox housing.Figure 5
The three-dimension and cross-sectional view of MGTS.
(a) (b)Furthermore, because the transmission shafts may generate vibration deformation, the PZTs may have some lateral motions that may affect the PZTs. Hence, the setup must be carefully designed such that the PZTs should not be directly connected to the additional bearing raceway. Just because of this, a decompression unit is immobilized between the PZT and the addition bearing for cutting down the longitudinal force to avoid PZT generating unnecessary damage, and the detailed view is given in Figure5(a).


## 3. Vibration Control Algorithms

The vibration controller, as the most critical part of the overall vibration control system, has a significant effect on the final control results. Therefore, the control algorithm, as the core of the controller, should be carefully designed for obtaining better control effects. However, most of the traditional control means need an accurate controlled model. But, as for the MGTS, it is pretty hard to build a precise model because its parameters are uncertain. Furthermore, the response characteristic of MGTS is time-variant; therefore, it is not easy to design a constant control system to attain the anticipated requirements using the classical control methods. In this paper, for the purpose of suppressing the vibration of the MGTS, the PID control method and FxLMS adaptive control algorithm were used to establish corresponding control schemes, respectively. The details of these algorithms will be discussed as follows.


### 3.1. PID Control Algorithm

The PID control strategy is the most popular feedback control method used in various industrial controls due to its simple structure, easy implementation, good stability, and strong robustness. As for the classical control method, it is commendable that it has been successfully used for over 50 years. Thus far, PID control is still the basic control method in the process industries. PID control technology and its optimized control technology are accounted for more than 90% in accordance with the survey data [20, 21]. The PID control algorithm mainly includes the proportional (P), integral (I), and derivative (D) three modes. The three modes are also the dominant factor for determining the control effect. The system stability mainly depends on adjusting the P control gain parameter. The steady state error can be controlled by adjusting the I control gain parameter. And the D control has an important effect on increasing the stability of the system and suppressing the overshoot. Thus, determining the three parameters of P, I, and D is the main mission for designing the PID controller [22]. The block diagram of PID control is given in Figure 6(a), and the output of PID controller in time domain is given as below:(23)ut=Kpet+Ki∫t1t2eτdτ+Kddetdt(24)et=xt-ytwhere e(t) is the error signal, u(t) is the output of the controller, and Kp, Ki, and Kd represent the P gain, I gain, and D gain, respectively.Figure 6
The schematic diagram of PID control algorithm and FxLMS algorithm.
(a) (b)


### 3.2. FxLMS Control Algorithm

The FxLMS algorithm is one of the most popular algorithms which has been employed in many other active vibration control studies [23–26]. The FxLMS algorithm control law is utilized for the sake of achieving the high-performance control effect. It does not rely on the prior knowledge and the specific model; moreover, it shows insensitivity when facing the variation of the model parameters. As a matter of fact, the FxLMS algorithm is one of the most effective methods based on the standard LMS algorithm. The existence of the secondary path which is known as one path between the output of adaptive controller and the housing response point may cause the traditional LMS algorithm used for AVC applications to be unstable. As a consequence, a secondary path estimation model is used as a compensator to address this problem. The control schematic diagram of FxLMS algorithm is given in Figure 6(b), and the numerical process of this algorithm is given in Table 3.Table 3
Numerical process of FxLMS algorithm.
Description Algorithm Initialization ω ( 0 ) = 0 Update y ( n ) = ω ( n ) ∗ r ( n ) (iteration) r ′ ( n ) = S ′ ( n ) ∗ r ( n ) e ( n ) = d ( n ) - y ( n ) ∗ S ( n ) ω ( n + 1 ) = ω ( n ) + μ r ′ ( n ) e ( n )The structure of FxLMS algorithm is shown in Figure6(b). H(z) denotes the transfer function between the error excitation signal x(n) and the undealt gear housing vibration d(n). S(z) represents the secondary path transfer function between the output terminal of the controller and the gearbox housing vibration. S′(z) is the estimated model of S(z). The error signal e(n) is equal to acceleration along y direction, the housing vibration response along the gear line-of-action which can be measured through an accelerometer. Moreover, e(n) participates in the FxLMS algorithm iteration for adjusting the adaptive controller’s weight ω(z), μ is the iteration step, and y(n) is the controller output signal.


## 3.1. PID Control Algorithm

The PID control strategy is the most popular feedback control method used in various industrial controls due to its simple structure, easy implementation, good stability, and strong robustness. As for the classical control method, it is commendable that it has been successfully used for over 50 years. Thus far, PID control is still the basic control method in the process industries. PID control technology and its optimized control technology are accounted for more than 90% in accordance with the survey data [20, 21]. The PID control algorithm mainly includes the proportional (P), integral (I), and derivative (D) three modes. The three modes are also the dominant factor for determining the control effect. The system stability mainly depends on adjusting the P control gain parameter. The steady state error can be controlled by adjusting the I control gain parameter. And the D control has an important effect on increasing the stability of the system and suppressing the overshoot. Thus, determining the three parameters of P, I, and D is the main mission for designing the PID controller [22]. The block diagram of PID control is given in Figure 6(a), and the output of PID controller in time domain is given as below:(23)ut=Kpet+Ki∫t1t2eτdτ+Kddetdt(24)et=xt-ytwhere e(t) is the error signal, u(t) is the output of the controller, and Kp, Ki, and Kd represent the P gain, I gain, and D gain, respectively.Figure 6
The schematic diagram of PID control algorithm and FxLMS algorithm.
(a) (b)


## 3.2. FxLMS Control Algorithm

The FxLMS algorithm is one of the most popular algorithms which has been employed in many other active vibration control studies [23–26]. The FxLMS algorithm control law is utilized for the sake of achieving the high-performance control effect. It does not rely on the prior knowledge and the specific model; moreover, it shows insensitivity when facing the variation of the model parameters. As a matter of fact, the FxLMS algorithm is one of the most effective methods based on the standard LMS algorithm. The existence of the secondary path which is known as one path between the output of adaptive controller and the housing response point may cause the traditional LMS algorithm used for AVC applications to be unstable. As a consequence, a secondary path estimation model is used as a compensator to address this problem. The control schematic diagram of FxLMS algorithm is given in Figure 6(b), and the numerical process of this algorithm is given in Table 3.Table 3
Numerical process of FxLMS algorithm.
Description Algorithm Initialization ω ( 0 ) = 0 Update y ( n ) = ω ( n ) ∗ r ( n ) (iteration) r ′ ( n ) = S ′ ( n ) ∗ r ( n ) e ( n ) = d ( n ) - y ( n ) ∗ S ( n ) ω ( n + 1 ) = ω ( n ) + μ r ′ ( n ) e ( n )The structure of FxLMS algorithm is shown in Figure6(b). H(z) denotes the transfer function between the error excitation signal x(n) and the undealt gear housing vibration d(n). S(z) represents the secondary path transfer function between the output terminal of the controller and the gearbox housing vibration. S′(z) is the estimated model of S(z). The error signal e(n) is equal to acceleration along y direction, the housing vibration response along the gear line-of-action which can be measured through an accelerometer. Moreover, e(n) participates in the FxLMS algorithm iteration for adjusting the adaptive controller’s weight ω(z), μ is the iteration step, and y(n) is the controller output signal.


## 4. Numerical Simulations

Although the corresponding dynamic analysis can be carried out through the dynamic model established by using LPM, the LPM cannot meet the requirements when it comes to the structural analysis, especially the problem of PZT placement optimization which will be discussed in detail in this paper. Furthermore, the virtual prototype established in ADAMS is more accurate and comprehensive, which can reflect the interior relevant relationships of the components of the MGTS by adding the corresponding constraint relation, such as the spherical constraint, revolute joint, and specific contact constraint, while the lumped parameter model cannot achieve this key point effectively. On the other hand, the cosimulation technique proposed here is suitable for the testing of mechanical structure design and analyzing a whole system behavior and testing of the control system. Therefore, the cosimulations based on Matlab/Simulink and ADAMS will be carried out for testing the feasibility of the AVC scheme adopted in this paper and discussing the control effects dominated by different control algorithms. And the superiority of the two control algorithms used here will be evaluated as well.In view of the analytic results of the above two sections, the AVC analysis of MGTS based on PID and FxLMS algorithms is considered in this section. The diagrammatic sketch of the MGTS with controlled system is given in Figure7.Figure 7
The diagrammatic sketch of the MGTS with controlled system.In Figure7, the No. 1 acceleration sensor is used for acquiring the reference signal for the control algorithm and the No. 2 acceleration sensor is used for picking up the error signal that needs to feed back to the control algorithm. Then the controller outputs the control command to drive the PZTs generating active control force to suppress the vibrations of MGTS. Therefore, to ensure that the analytic results can reflect the realistic situation enough, a cosimulation technique is utilized by two well-known software programmes, ADAMS and Matlab/Simulink. The ADAMS and Matlab/Simulink cosimulation can be mainly divided into two parts. In terms of the first part, first of all, the virtual prototype model of MGTS is established in ADAMS. And the vital mission of this project is to determine the appropriate constraints according to the relationship between all components of the MGTS to guarantee the virtual prototype model up to the actual criterion as far as possible. As for the second important part, the FxLMS adaptive control algorithm is performed by using the user-defined level-2 S function in Matlab. Then the cosimulation control structure based on PID and FxLMS algorithm is established in Simulink.To validate the effectiveness of the proposed method above, the connection between ADAMS and Matlab is established by importing the virtual prototype into Matlab to achieve the interactive simulations. The cosimulation analysis is carried out under the circumstance of the rotational speed and load under steady state condition. That means both of the magnitudes of rotational speed and load remain constant, and the MGTS works under steady state conditions. As discussed by Li et al. [7], for the FGMF, it has been proved that the proposed control method did not perform well and can not achieve significant control at this frequency. Thus, the following study will mainly focus on the single target harmonic vibration control.


### 4.1. PZT Placement Optimization

For testifying the proposed active shaft control concept according to the results of the modal analysis discussed in Section2 and finding an appropriate actuating position for the PZT, a series of cosimulations were carried out. The input shaft was taken as an example for analyzing the possibility of the active shaft control concept. It is clear that the position on the gear’s left is not fit to locate one PZT. Therefore the input shaft is roughly divided into four parts for locating the PZTs, positions 1–4, as shown in Figure 8(a). According to the generation principle of the gear mesh vibrations, the more closer the position to the gear, the greater the amplitude. Thus, the vibration magnitudes for the corresponding positions are given in Figure 8(b). We set the input rotational speed n=1524 r/min and the load torque T=1 N·m. With regard to the four different positions shown in Figure 8(a), the corresponding cosimulation analysis is performed under the same condition. The control variate method can avoid the influences of the other complex factors and ensure the accuracy of the cosimulation results. The cosimulation results are given in Figure 9.Figure 8
The diagrammatic sketch of the actuating position for PZTs and the vibration magnitudes at the corresponding position.
(a) (b)Figure 9
The cosimulation results of the four different positions for fixing the PZTs.As shown in Figure9, with respect to the four different positions for fixing the PZTs, both PID and FxLMS algorithms can suppress the 4x single target harmonic vibration to some extent. For position 1, at the target frequency, about 3 dB vibration reduction can be obtained by PID control method, and about 5 dB vibration reduction can be dominated by FxLMS algorithm. For position 2, the PID control can suppress the vibration approximately 3 dB, and FxLMS can suppress the vibration about 12.5 dB. Similarly, for position 3, the PID control can decrease the vibration about 4 dB, and FxLMS algorithm can reduce the vibration up to 12 dB. However, for position 4, about 7 dB vibration can be reduced by PID, and up to 17 dB vibration reduction can be controlled by FxLMS algorithm. Comparing the vibration reduction results of the four different positions, it can be inferred that the position which has larger vibration magnitude will exert a better control effect; in other words, position 4 is the better actuating position for locating the PZT. Although position division is not accurate enough, at least it can prove the correctness of the conclusion discussed by the modal analysis in Section 2 and provide an applicable method for finding an appropriate actuating position for PZT.


### 4.2. A Single Target Harmonic Vibration Control

In this part, the vibration control of a single gear mesh harmonic up to the fourth order for various rotational speeds was studied. We let the rotational speedn=1524 r/min, 2032 r/min, and 2540 r/min and load torque T=1 N·m and 3 N·m, respectively. In point of these situations, the vibration control simulations of MGTS based on PID and FxLMS algorithm are carried out to analyze the influences of the diversifications of rotational speed and load to the control effect. And the vibration control results of the input shaft at the input gear rotation center are given in Figure 10, and the control effects of the output shaft at the output gear rotation center are given in Figure 11.Figure 10
The vibration control results of the input shaft at the gear rotation center.Figure 11
The vibration control results of the output shaft at the gear rotation center.According to Figures10 and 11, it is obvious to know that, at the single target frequency, the control effect of FxLMS control algorithm is significantly better than PID control method. For instance, when the input shaft rotational speed n=1524 r/min and the load torque T=1 N·m, with regard to the vibration control of input shaft at the gear rotation center, the vibration at 4x mesh target harmonic frequency can be suppressed up to 17 dB by the FxLMS control, and about 7 dB reduction can be obtained by PID control. In addition, for the vibration control of the output shaft at gear rotation center with the same conditions, the FxLMS control algorithm can suppress the vibration up to 19.3 dB, and the PID control can cut down the vibration approximately to 7 dB. As for the control effect of the other conditions, in order to analyze the control results conveniently, the comparisons of the results in Figures 10 and 11 are sorted out in Tables 4 and 5.Table 4
The comparison results of Figure10.
Condition (r/min/1 N⋅m) 1524/1 1524/3 2032/1 2032/3 2540/1 2540/3 Without control/(dB) −1.66384 −0.25402 −0.49957 −0.11854 0.11431 −0.25402 PID/(dB) −8.81372 −6.24047 −7.95936 −4.30991 −8.05673 −4.3829 FxLMS/(dB) −18.5504 −14.0446 −30.4403 −20.6097 −21.7316 −16.5134Table 5
The comparison results of Figure11.
Condition (r/min/1 N⋅m) 1524/1 1524/3 2032/1 2032/3 2540/1 2540/3 Without control/(dB) −15.1778 −1.8346 −0.01693 −0.71803 −5.57155 −2.66723 PID/(dB) −24.6401 −7.47954 −8.8569 −5.78154 −14.8434 −9.01214 FxLMS/(dB) −34.5487 −17.3695 −18.9738 −10.845 −28.5859 −15.8312By analyzing the above contents, for the various target harmonic vibrations at various conditions, it can be found that both PID control and FxLMS control methods can availably suppress the vibrations of the rotation shafts at the gear rotation center. But FxLMS control algorithm is more effective when facing the target harmonic frequency. Furthermore, under the circumstances of the input rotational speeds being the same, with the increase of the load torque, the general control effects of both control methods show a downward trend. Therefore, it has been proved that the load torque has a severe effect on the vibration control. And this result will have a guide for the future experimental study.


## 4.1. PZT Placement Optimization

For testifying the proposed active shaft control concept according to the results of the modal analysis discussed in Section2 and finding an appropriate actuating position for the PZT, a series of cosimulations were carried out. The input shaft was taken as an example for analyzing the possibility of the active shaft control concept. It is clear that the position on the gear’s left is not fit to locate one PZT. Therefore the input shaft is roughly divided into four parts for locating the PZTs, positions 1–4, as shown in Figure 8(a). According to the generation principle of the gear mesh vibrations, the more closer the position to the gear, the greater the amplitude. Thus, the vibration magnitudes for the corresponding positions are given in Figure 8(b). We set the input rotational speed n=1524 r/min and the load torque T=1 N·m. With regard to the four different positions shown in Figure 8(a), the corresponding cosimulation analysis is performed under the same condition. The control variate method can avoid the influences of the other complex factors and ensure the accuracy of the cosimulation results. The cosimulation results are given in Figure 9.Figure 8
The diagrammatic sketch of the actuating position for PZTs and the vibration magnitudes at the corresponding position.
(a) (b)Figure 9
The cosimulation results of the four different positions for fixing the PZTs.As shown in Figure9, with respect to the four different positions for fixing the PZTs, both PID and FxLMS algorithms can suppress the 4x single target harmonic vibration to some extent. For position 1, at the target frequency, about 3 dB vibration reduction can be obtained by PID control method, and about 5 dB vibration reduction can be dominated by FxLMS algorithm. For position 2, the PID control can suppress the vibration approximately 3 dB, and FxLMS can suppress the vibration about 12.5 dB. Similarly, for position 3, the PID control can decrease the vibration about 4 dB, and FxLMS algorithm can reduce the vibration up to 12 dB. However, for position 4, about 7 dB vibration can be reduced by PID, and up to 17 dB vibration reduction can be controlled by FxLMS algorithm. Comparing the vibration reduction results of the four different positions, it can be inferred that the position which has larger vibration magnitude will exert a better control effect; in other words, position 4 is the better actuating position for locating the PZT. Although position division is not accurate enough, at least it can prove the correctness of the conclusion discussed by the modal analysis in Section 2 and provide an applicable method for finding an appropriate actuating position for PZT.


## 4.2. A Single Target Harmonic Vibration Control

In this part, the vibration control of a single gear mesh harmonic up to the fourth order for various rotational speeds was studied. We let the rotational speedn=1524 r/min, 2032 r/min, and 2540 r/min and load torque T=1 N·m and 3 N·m, respectively. In point of these situations, the vibration control simulations of MGTS based on PID and FxLMS algorithm are carried out to analyze the influences of the diversifications of rotational speed and load to the control effect. And the vibration control results of the input shaft at the input gear rotation center are given in Figure 10, and the control effects of the output shaft at the output gear rotation center are given in Figure 11.Figure 10
The vibration control results of the input shaft at the gear rotation center.Figure 11
The vibration control results of the output shaft at the gear rotation center.According to Figures10 and 11, it is obvious to know that, at the single target frequency, the control effect of FxLMS control algorithm is significantly better than PID control method. For instance, when the input shaft rotational speed n=1524 r/min and the load torque T=1 N·m, with regard to the vibration control of input shaft at the gear rotation center, the vibration at 4x mesh target harmonic frequency can be suppressed up to 17 dB by the FxLMS control, and about 7 dB reduction can be obtained by PID control. In addition, for the vibration control of the output shaft at gear rotation center with the same conditions, the FxLMS control algorithm can suppress the vibration up to 19.3 dB, and the PID control can cut down the vibration approximately to 7 dB. As for the control effect of the other conditions, in order to analyze the control results conveniently, the comparisons of the results in Figures 10 and 11 are sorted out in Tables 4 and 5.Table 4
The comparison results of Figure10.
Condition (r/min/1 N⋅m) 1524/1 1524/3 2032/1 2032/3 2540/1 2540/3 Without control/(dB) −1.66384 −0.25402 −0.49957 −0.11854 0.11431 −0.25402 PID/(dB) −8.81372 −6.24047 −7.95936 −4.30991 −8.05673 −4.3829 FxLMS/(dB) −18.5504 −14.0446 −30.4403 −20.6097 −21.7316 −16.5134Table 5
The comparison results of Figure11.
Condition (r/min/1 N⋅m) 1524/1 1524/3 2032/1 2032/3 2540/1 2540/3 Without control/(dB) −15.1778 −1.8346 −0.01693 −0.71803 −5.57155 −2.66723 PID/(dB) −24.6401 −7.47954 −8.8569 −5.78154 −14.8434 −9.01214 FxLMS/(dB) −34.5487 −17.3695 −18.9738 −10.845 −28.5859 −15.8312By analyzing the above contents, for the various target harmonic vibrations at various conditions, it can be found that both PID control and FxLMS control methods can availably suppress the vibrations of the rotation shafts at the gear rotation center. But FxLMS control algorithm is more effective when facing the target harmonic frequency. Furthermore, under the circumstances of the input rotational speeds being the same, with the increase of the load torque, the general control effects of both control methods show a downward trend. Therefore, it has been proved that the load torque has a severe effect on the vibration control. And this result will have a guide for the future experimental study.


## 5. Conclusions

As a guide work for the future experimental study, a dynamic model with 26 DOFs was investigated using LPM in this paper. The modal analysis results solved by using the FEM were compared with the modal solutions resolved by the calculation method to validate the correctness of the dynamic modelling. Then a novel AVC structure of multistage gearbox with built-in PZTs was developed according to the modal results solved by FEM. In addition, in order to examine the feasibility and effectiveness of MGTS vibration control and ascertain the appropriate placement position of PZTs, a series of ADAMS and Matlab cosimulations were carried out based on the PID control and FxLMS adaptive control algorithm. On the basis of simulation results, we can draw several conclusions as follows. First, the compared results of the calculation method and FEM demonstrated the correctness of the modal solving. Second, it is feasible for determining a AVC structure of MGTS in the light of modal analysis. Third, the cosimulation results proved that PID and FxLMS adaptive control algorithms can suppress the gear mesh vibrations of the MGTS effectively. And FxLMS control algorithms are more robust and active than the classic PID control method. At last, the feasibility of the AVC method based on FxLMS algorithm for reducing the MGTS vibrations has been verified by cosimulations, and it will provide a considerable and forceful reference for the future experimental investigation.



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*Source: 7984283-2018-05-23.xml*