CHEN Xuehua，F(xiàn)ENG Jian

1.Chongqing Radio ＆ TV University，Chongqing 400052，China;2.Technical center of CN Gpower Gearbox Co，Ltd，Chongqing 402263，China

1.Introduction

A large-size reduction gearbox test system is one of key experimental equipment used for test the performance of a large-size reduction gearbox in the industry.Its vibration and noise characteristics are very important for predicting the performance of the test gearbox.The current design method of the gearbox test system is based on a static design method，which only considers the experimental functions and cannot describe its dynamic characteristics.When a static design method is used，its vibration problem may be found when the test system is completed，and it is difficult to analyze the reasons of the vibration problem for a whole test system.In addition，it is difficult to control the vibration and noise of the large-size reduction gearbox using an experimental method because its structure is very complex.Hence it is very important to investigate the dynamic characteristics of the large-size reduction gearbox，which can provide some guidance for its vibration and noise control.A lot of research works have been done in this area.

For instance，Richards and Pines［1］proposed a passive method to reduce transmitted vibration caused by the gear mesh contact dynamics base on the dynamic characteristics of the periodic structural components.Guan et al.［2］developed a comparative analysis method for the dynamic performances of four potentially feasible actuation concepts used to control the housing vibration in a gearbox test system.Cheon et al.［3］presented a new approach to control the vibration of a simple spur gear pair with phasing，which is based on controlling the vibration in mesh stiffness by adding another pair of gears with half-pith phasing.Zhang et al.［4］proposed a dynamic model coupling bending and torsional vibration of a gear pair system to study the vibration of the gear pair system caused by the mass unbalance.Li et al.［5］developed a three degree of freedom numerical model of a spur gear pair system using a lumped mass method.They provided an active control method to control the gear meshing point vibration caused by the transmission error based on the numerical results.Some other researchers also studied the different vibration control method for various gearboxes as shown in Refs.［6-8］.However，the above research works only focused on the vibration control of a simple gear pair system.There are few research works on the large-size reduction gearbox test system.Hence it is necessary to investigate the dynamic characteristics of the large-size reduction gearbox test system for its vibration and noise control.

In this paper，a new simulation method is developed to analyze the vibration of a large-size reduction gearbox test system，which consists of a FE method and a torsional analysis method.The natural frequencies，the mode shapes and the harmonic response of the gearbox test system are studied using the FE method.The torsional vibration of the gearbox test system is analyzed by a proposed torsional model.The simulation results show that the proposed method may be used to the vibration and noise analysis and control of a large-size test gearbox test system.

2.Finite Element analysis

2.1.Analysis of natural frequencies and mode shapes

A large-size reduction gearbox test system is studied in this section，which consists of two motors，shaft couplings，a speed-increasing gearbox，a dynamometer，and other support saddles as shown in Fig.1.Since the whole test system has a complex structure，it is difficult to obtain the natural frequencies and the model shapes of the large-size reduction gearbox test system using an experimental method.Hence a FE method is utilized to calculate its natural frequencies and mode shapes using commercial FE software.Here，because the technical know-how of the new large-size reduction gearbox test system，the results from the speed-increasing gearbox is given in this study.But it has slight effects on the quality of this paper because the analysis method of the other component of the gearbox test system is similar.The first ten natural frequencies of the speed-increasing gearbox is show in Tab.1，and one typical mode at the eighth natural frequency of the speed-increasing gearbox is shown in Fig.2.

Fig.1 Schematic of a large-size reduction gearbox test system

Tab.1 First ten natural frequencies of the speedincreasing gearbox with its support saddles

Tab.2 plots the rotational frequencies of the input shaft of the speed-increasing gearbox.Comparisons analysis between the natural frequencies of the speed-increasing gearbox and the rotational frequencies of the input shaft of the speed-increasing gearbox show that the resonance vibration may be caused when the input shaft speed is at 1 020 r/min(4fsand 5fs)and 1 060 r/min(3fs，4fsand 5fs).The mode shape at the eighth natural frequency shows the resonance vibration at the bottom of the speed-increasing box when the input shaft speed is at 1 060 r/min(5fs=90 Hz).The simulation results show that the vibration problem can be predicted at the beginning of the design stage of the large-size reduction gearbox test system using the proposed method，which cannot be predicted using the traditional static design method.

Tab.2 Rotational frequencies(fs)of the input shaft of the speed-increasing gearbox(in Hz)

2.2.Analysis of the vibration response

To analysis the vibration response of the largesize reduction gearbox test system caused by the offcenter rotational components，a harmonic response method is presented using the FE method.A reference point at the input shaft of the large-size reduction gearbox test system is chosen to study its vibration response as shown in Fig.3.The vibration response along theZdirection is discussed as given in Tab.3 and Fig.4，which is chosen to be tested in practice.Fig.4 plots that peaks occur at the 100 Hz，110 Hz，150 Hz，160 Hz，175 Hz，190 Hz，and 201 Hz.It can be seen that the vibration level of the large-size reduction gearbox test system is large when the rotational frequencies of its shaft are closed to those above frequencies，which should be considered during its design process.

Fig.3 Reference point at the input shaft of the large-size reduction gearbox test system

Tab.3 Amplitude of the vibration response of the reference point 76961 at the input shaft of the largesize reduction gearbox test system

Fig.4 Vibration response of the reference point along the Z direction

3.Torsional Vibration Analysis

3.1.Torsional vibration model

As another key vibration problem of the largesize gearbox，torsional vibration of the rotational componnets should be considered too.Furhermore，the torsional vibration contol is one of the major mthod of the vibration and noise contol methods for the largesize reduction gearbox test system.According toe shematic of the large-size reduction gearbox test system，a seven degree of freedom torsional vibration model of the rotaional component system of the largesize gearbox test sytem is proposed as shown in Fig.5.

Fig.5 A rotational vibration model of the rolation component system of the large-size reduction gearbox test system

The propose mdoel incorparates the following realistic assumptions and considerations.

1)Each rotational component is considered as a rigid plate，which are connected by massless roational springs.

2)The component with a large rotational inertia is assumed as an inertia element，and the component with a small rotational inertia is assumed as a elastic element.

3)All the excitation moments are applied on the inertia elements.

4)The mass of the shafts between two mass elements are shared by the two mass elements connected by the shafts.

5)The rotational inertia and the stiffnesses of the shafts with different rotational speeds are changed into equivalent rotational inertia and equivalent stiffnesses.

6)The deformation of the gears are ignored during their meshing process.The rotational inertias of the gears are applied on the lumped mass on the center line of the master gears based on their transmission ratios.

7)The deformations of the components with large stiffness are ignored.

Based on the above assumptions and considerations，the equations of motion for eache element are derived as follows，

in whichJ1，J2，J3，J4，J5andJ6are the rotational inertia of each inertia element，K1，K2，K3，K4andK5are the stiffnesses of each elastic element，C1，C2andC3are the damping ratios，θ1，θ2，θ3，θ4，θ5and θ6are the rotational displacement vectors，andM1andM2are the driving moment and load，respectively.

3.2.Analysis of the Torsional Vibration

3.2.1.Analysis of the natural Frequencies and the mode shape

The general solution of Eq.(7)is

Then the eigensolutions of Eq.(7)is

Based on Eq.(9)，the natural frequencies and the mode shapes of the rotational component system can be obtained，which are listed in Tab.4.

Tab.4 Natural frequencies and mode shapes of the rotational component system

3.2.2.Forced response

In this study，Matlab/Simulink module in Matlab software is used to solve the Eq.(1)～(6).The loads are based on the experimental date，which are given in Tab.5.

The excitation moment is assumed to be caused by the motor，which is given by

whereM0is the constant moment of the motor;tis the time，andwis the fluctuation frequency of the motor，which is described as

in whichNis the input speed of the motor，and the fluctuation of the moment of the motor ranges from-4%to 4%.

Tab.5 Experimental load parameters

3.2.3.Results and discussions

Fig.6(a)and(b)show the rotational speed of the output shaft of the test gearbox when the input speed of the motor is 565 rpm and the frequency of the load is 9.58 Hz.It can be seen that a beat phenomenon is observed in the time domain rotational speed of the output shaft of the test gearbox.The envelope frequency of the beat phenomenon is closed to 1 Hz and it is a periodic signal.According to the Eq.(10)，the input speed of the motor 1 ranges from 563 rpm to 565 r/min.The simulation results show that a large vibration level of the test gearbox may be caused when the input speed of the large-size reduction gearbox test system is about 565±1 r/min.

Fig.6 Time domain rotational speed of the output shaft of the test gearbox with the input speed of 565 rpm

4.Conclusions

A new simulation method is proposed to control the vibration of a large-size reduction gearbox test system.It consists of a finite element(FE)method and a torsional vibration analysis method.The natural frequencies，the mode shape and dynamic response of the key components of the large-size reduction gearbox test system are calculated by the FE method.Also，a torsional vibration model of the large-size reduction gearbox test system is proposed to investigate the torsional vibration of the gearbox test system.The simulation results show that the vibration analysis and the optimization design of the large-size reduction gearbox test system should be investigated at the beginning of its design stage，which can improve its vibration and noise level.The results also show that the proposed model can be used to analyze and optimize the vibration characteristics of a large-size reduction gearbox test system.

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