#### Abstract

This paper describes a research work on driveline modeling, torsional vibration analysis, and clutch damper parameters optimization for reducing transmission gear rattle on the vehicle creeping condition. Firstly, major driveline components, including quasi-transient engine, multistage stiffness clutch damper, detailed manual transmission and differential mechanism, and LuGre tire, are modeled, respectively. Secondly, powertrain system modeling adopting a two-stage stiffness clutch damper is constructed and analyzed. Transient responses predicted by the model show that the driveline undergoes severe torsional vibration and transmission gear rattle phenomenon. By analysis, it is concluded that the clutch damper works jumping between the first- and second-stage stiffness, which results in this problem for the creeping condition. Then, a three-stage stiffness clutch damper is proposed innovatively to solve this problem. It is shown that severe driveline vibration and gear rattle phenomenon are inhibited effectively. Finally, it draws a conclusion that clutch damper parameters could have a great effect on driveline vibration and gear rattle phenomenon and a three-stage stiffness clutch damper could be utilized to solve gear rattle phenomenon efficiently on the vehicle creeping condition.

#### 1. Introduction

Vibroimpacts in manual transmission (MT) are of critical concern to vehicle manufacturers based on noise, vibration, and reliability consideration. Gear rattle is a typical gear noise that is generated under the existence of torsional fluctuations, which, in turn, leads to gear teeth impact of unloaded gears fluctuating within tooth lash. The impact collision is transmitted to the transmission housing via shafts and bearings and then converted into an audible rattle noise, which is broadband in the frequency spectrum. Rattle noise has a distinct sound quality that differentiates it from other noises produced by other sources in the vehicle, which makes passengers usually annoyed by this noise and attribute it to some vehicle companies. So a better understanding of the dynamic behavior of drivelines and transmission gear rattle mechanism is in urgent need and has drawn many scholars’ attention.

Gear rattle phenomenon is a comprehensive problem of the driveline that includes many nonlinearities of multistage clutch damper, gear meshing stiffness, gear backlash, drag torque, and so on. These nonlinearities make it difficult to analyze the mechanism for this phenomenon. Some attempts on numerical simulation and experiment studies are conducted in some literature.

In terms of numerical simulation, initial research on gear rattle focused on one gear pair. Nakamura firstly modeled one straight spur gear pair in which the time-varying meshing stiffness was equivalent to square wave function and static transmission error was the sum of harmonic Fourier series. It gave the moment of gear rattling clearly through the numerical simulation method [1]. Since then, many research scholars paid more attention to solving algorithms of mathematical models. Comparin and Singh utilized harmonic balance method to solve one gear pair rattling model, which arrived at the fact that there was two-side impact, one-side impact, or no impact with some parameters changing [2]. Kahraman and Singh found that one gear pair nonlinear property involved subharmonic response and chaos response by the numerical simulation method and harmonic balance method [3]. As the study continued, research object was transferred from one simple gear pair to complicated gear transmission system. Based on the four-degree-of-freedom model of one gear pair, Bozca et al. proposed empirical model and torsional vibration model based optimization of a 5-speed gearbox design parameters to reduce rattle noise in an automotive transmission. Despite the geometric parameters optimization, overall rattle noise level was reduced and all optimized geometric design parameters also satisfied all constraints [4, 5]. Besides, gear rattle problem is regarded as a comprehensive problem of the driveline system. Most of the driveline models used for the driveline torsional vibration analysis are lumped discrete models with a few degrees of freedom. Wang et al. described a model most early for torsional vibration of automotive manual transmission (MT) to analyze and predict gear rattle of all speeds. Accordingly, a rattle index was used to compare the rattle levels produced by different gear pairs. But in that model gear meshing stiffness was constant and self-excited vibration of time-varying stiffness was ignored [6]. Wu and Luan paid attention to the impact of gear meshing stiffness on the vehicle driveline torsional vibration and gave a comparison of simulation between variable meshing stiffness and averaged stiffness of loaded gear pairs based on overall powertrain system [7]. Robinette et al. developed a representative model for a front wheel drive (FWD) vehicle with MT by lumped parameter analysis and presented functional relations for torque losses associated with shafts, gears, seals, lubricating oil flow, and bearing clearance as a function of basic design parameters [8]. Drag torque including bearing friction torque, oil shearing torque, or oil churning torque was then validated by experimental results [9]. De La Cruz et al. considered different lubrication states influence of gear pairs on the rattle phenomenon and proposed a rattle index in consideration of lubrication state [10]. Fietkau and Bertsche proposed a simulation approach for loaded and unloaded gear contacts, which include oil films and elastic deformations. This approach is validated with experiment measurement and it is concluded that lubricant condition could not be ignored [11]. Theodossiades et al. took into consideration the effect of lubrication during engine idle conditions and examined the influence of lubrication in torsional vibration. It is shown that the lubricant film behaved as a time-varying nonlinear spring-damper element and could have a great influence on the gear rattle problem [12, 13]. Crowther et al. put forward 6-degree-of-freedom (DOF) model using a frequency sweep with engine excitation derived from measured data with two-stage gear meshing and an unloaded gear pair. It is found that the gear rattling is more severe when the engine speed passes the resonation frequency region of the system natural model. It concluded that an effective dynamic engine model is needed in order to get transient driveline component motion and then rattle phenomenon actually [14]. Bhagate et al. put forward a 6-DOF mathematical model for the torsional vibrations of front wheel drive automotive drivetrain and developed the optimization of sensitive system parameters for reducing the driveline rattle [15]. So as for transmission modeling, various factors such as gear pair time-varying stiffness, gear friction, bearing friction, and gear oil churning loss are in urgent need in the future modeling work.

In terms of rattle experiments, Couderc et al. designed and built early an experimental setup of a vehicle driveline for the prediction of the dynamic behavior of vehicle drivelines. It is concluded that the simulation model validated by the experimental setup could provide transient response truly [16]. Bellomo et al. analyzed the contribution of individual sound source to the overall rattle noise by means of noise-source analysis and proposed a pareto-optimal solution to reduce the rattle noise emission, utilizing a rattle test bench [17]. This improved test bench reproduced the branched driveline system, rather than the one-string driveline system in [16]. Forcelli et al. set up a virtual engine simulator for automotive transmission and conducted a parametric sensitivity study for amplitude of the torsional vibration. Moreover, a relationship between the vibroacoustics measurements and the human perception was found [18]. Barthod et al. analyzed the rattle threshold and the rattle noise evolution for different multiharmonic excitation parameters and mechanical gearbox parameters through a bench test [19]. Crowther and Rozyn introduced a gear rattle test rig, in which the electric motor drives the transmission at a steady mean speed via a double telescoping Hooke’s joint. By changing the angle of the joints, the amplitude of the vibration could be adjusted [20]. Baumann and Bertsche built one gear pair test rig for rattle research and compared the rattle intensity under different lubricant oil condition. It is found that adopting high viscosity oil could inhibit rattle phenomenon when angular acceleration of the input shaft is larger [21]. Brancati et al. set up a specific test rig for one lightly loaded gear pair, which is able to acquire the relative rotation motion of gears by two high resolution incremental encoders. Based on measurement data from this test rig, a gear rattle metric based on the wavelet multiresolution analysis was proposed [22].

The clutch damper is a component of drivelines that could have significant influence on the torsional dynamic behavior of drivelines. Gear rattle phenomenon can be greatly reduced by opportunely setting some clutch parameters such as the multistage torsional springs. Steinel examined the influence of the twin-mass flywheel on the driveline natural characteristics and transient responses. It was shown that the twin-mass flywheel was the ideal solution for drivetrains of which the vibrations could not be reduced sufficiently if there was no need for the consideration of costs [23]. Prasad et al. found that elimination of gear rattle could be achieved by maximizing the hysteresis of clutch thereby absorbing the energy being transferred through the subjective and objective evaluation in the passenger bus experiment [24]. But it is obvious that maximizing the hysteresis of the clutch damper would reduce the transmission efficiency of the powertrain system. Xu et al. introduced a novel clutch damper with three-stage stiffness and solved the rattle phenomenon effectively in low torque condition compared with the damper with two-stage stiffness by vehicle experiments [25]. Similarly, many research scholars found that the clutch damper property plays an important role in reducing driveline vibration and rattle phenomenon [26, 27].

This paper presents a lumped parameters model capable of predicting the driveline vibration, the onset of loose gear rattle, and the clutch damper optimization for reducing loose gear rattle. Firstly, a description of the driveline and modeling of major components are presented. Then, the driveline model is used to perform transient analysis of current systems and provide a comprehensive understanding of a four-cylinder and four-stroke engine excitation, the strong nonlinearities of the driveline elements (including multistage clutch stiffness and frictional hysteresis), and parameter excitations of loaded gear pair meshing stiffness. The driveline model is divided into the baseline vibration and the rattling vibration. The baseline vibration is taken as the excitation to the rattling vibration and it is neglected that the rattling vibration has an effect on the baseline vibration. A detailed manual transmission modeling could reproduce the onset of rattle phenomenon of unloaded gear pairs. Finally, a comparison of the baseline vibration and the rattling vibration between using a two-stage stiffness clutch damper and using an improved three-stage stiffness clutch damper is studied on the vehicle creeping condition, which shows that it is achievable to optimize clutch damper parameters for reducing driveline vibration and gear rattle.

#### 2. Description and Modeling of Powertrain System

A classical front wheel drive (FWD) vehicle is a research object. Major components of powertrain system, composed of an inline four-cylinder and four-stroke engine, the clutch damper, a 5-speed MT, the differential mechanism, half axles, and wheels, are as shown in Figure 1.

Effective modeling of powertrain components, which is discussed in this section, is vital to driveline vibration and manual transmission rattle phenomenon analysis. Quasi-transient engine torque is as a power source to the driveline and applicable engine model should consider dynamic output torque rather than steady output torque in order to study transient response. The clutch damper in consideration of elastic torque and hysteresis torque is modeled so that clutch damper parameters affecting the driveline vibration and gear rattle could be analyzed. A detailed 5-speed manual transmission model based on lumped parameters method will also be explained. Simultaneously, the differential mechanism and the tire property are taken into consideration. Furthermore, time-varying meshing stiffness of loaded gear pairs is as an inner excitation in the driveline and accurate and effective calculation method of it could enhance simulation efficiency.

##### 2.1. Quasi-Transient Engine Model

###### 2.1.1. Kinematic Relations of a Single Cylinder

Kinematic diagram of the crank and connecting rod mechanism, which is shown in Figure 2, is calculated by where is the crankshaft angle, is the crankshaft rotation angle speed, is the time, is the crank radius, is the connecting rod length, is the length between the top dead center and the piston center, and , are the translational velocity and acceleration of the piston, respectively.

###### 2.1.2. Force Analysis of a Single Cylinder

Force analysis of the crank and connecting rod mechanism in Figure 3 is derived in where is the cylinder pressure with the change of crank angle, is the piston diameter, is the reciprocating mass including piston, piston ring, piston pin, and connecting rod mass, is the gas pressure force on the piston, is the gas pressure torque, is the reciprocating mass force, and is the reciprocating mass torque.

###### 2.1.3. Transient Engine Friction Model of a Single Cylinder

Engine friction modeling is a key step in the quasi-transient engine model. Transient engine friction model of Rezeka-Henein model is adopted here and engine friction torque is yielded by the following equation [28]:wherewhere are fitting coefficients, is the kinematic viscosity of lubricant oil, is the contact pressure between piston ring and cylinder wall, is the thickness of oil ring, is the inner diameter of cylinder wall, is the number of oil rings, is the number of gas rings, is the thickness of gas ring, is the thickness of lubricating oil film, is the length of piston skirt, is the number of valves, is the force of valve spring, and is the average radius of journal bearing. Some parameters are as shown in Figure 4.

###### 2.1.4. Effective Output Torque of an Inline Four-Cylinder and Four-Stroke Engine

For an inline four-cylinder and four-stroke engine, effective output torque results from the gas torque, reciprocating inertia torque, and friction torque comprehensive in

On the condition of vehicle creeping, engine speed is about 800 rpm and each engine cylinder gas pressure is as seen in Figure 5. Accordingly, effective output torque of four-cylinder and four-stroke engine is as shown in Figure 6.

##### 2.2. The Clutch Model

The clutch plays an important role in driveline vibration, especially in transmission rattle impact. The clutch is composed of two parts or masses when it is engaged. The primary mass is attached to the flywheel rigidly (called the first mass together) and the secondary mass is connected to the input shaft of MT through spline teeth. Multistage springs are placed between the primary mass and the secondary mass.

For an asymmetric two-staged clutch damper in Figure 7(a), the clutch torque is expressed as a function of the relative displacement and the relative velocity and is defined by the sum of elastic torque in Figure 7(b) and hysteresis torque in Figure 7(c) [26]:

**(a)**

**(b)**

**(c)**

**(d)**

The elastic torque is calculated in where is the first-stage stiffness, is the second-stage stiffness of the drive side, is the second-stage stiffness of the coast side, is the third-stage stiffness of the coast side, and , , and are the corresponding transition angles.

The hysteresis torque is defined in where is the first-stage hysteresis torque, is the second-stage hysteresis torque of the drive side, is the second-stage hysteresis torque of the coast side, and is the third-stage hysteresis torque of the coast side.

For a three-staged clutch damper in Figure 7(d), the elastic torque and the hysteresis torque are defined in (9) and in (10), respectively. Considerwhere is the second-stage stiffness of the three-staged clutch damper, is the corresponding hysteresis torque, and is the corresponding transition angles.

##### 2.3. Modeling of 5-Speed MT and Loose Gear Drag Torque

###### 2.3.1. MT Mechanism and Equivalent Physical Model

For the transverse 5-speed and two-axis design MT in Figure 8, which includes five forward gear ratios and one reverse gear ratio, input and output shafts are mounted on tapered roller element bearings. The 1st driven gear, 2nd driven gear, 3rd driving gear, 4th driving gear, and 5th driving gear rotate on the input or output shaft through needle bearings. 1st driving and 2nd driving gear are integrated on the input shaft, while 3rd driven, 4th driven, and 5th driven gear are splined on the output shaft. The 1st driven gear and 2nd driven gear utilize the same triple cone synchronizer, which is supported by one hydrodynamic journal bearing, 3rd driving and 4th driving gears utilize one, and 5th driving gear utilizes another one.

Based on lumped parameter modeling method, every gear and synchronizer are equivalent to rotational inertias. The inertia of the segment shaft between two gears or between one gear and one synchronizer is divided into two parts averagely and they will be added on adjacent inertias, respectively. Simultaneously, the segment shaft is equivalent to one rotational stiffness and one rotational damping. Each inertia of one gear pair couples through meshing stiffness, meshing damping, and backlash and drag torques are applied on loose gears. The coupling between the input shaft and the output shaft is obtained by the power transmitting gear pair. The equivalent physical model of 5-speed MT consisting of an arrangement of discrete inertias and stiffness is as shown in Figure 9.

###### 2.3.2. Calculation of Loose Gear Drag Torque

In Figure 9, drag torques , acting on 1st driven gear, 2nd driven gear, 3rd driving gear, 4th driving gear, and 5th driving gear, are generated through bearing friction torque, oil shearing torque, or oil churning torque. Gear windage losses are ignored, since gear speeds are relatively low and loose gears on the input shaft are splash lubricated.

For the 1st speed driven and 2nd speed driven gear rotating on the output shaft, in (11) and in (12) are applied on the gears, respectively:Bearing frictional torque is defined in the following equation [29]:where is the bearing rotation speed, is the bearing average diameter, is a lubrication factor, and is lubrication oil kinematic viscosity.

Oil shearing torque is defined in the following equation [8]:where is the lubrication oil absolute viscosity, is the gear length, is the pitch radius of the gear, is speed differential between the gear and synchronizer or its bounding shaft, and is the radial clearance of the bearing.

Oil churning torque is defined in the following equation [30]:where is the lubrication oil density, is the gear oil churning angle velocity, is the oil-submerged surface area, and is the oil churning coefficient.

For the unloaded 3rd driving gear, 4th driving gear, and 5th driving gear rotating on the output shaft affected by bearing friction, drag torque in (16), drag torque in (17), and drag torque in (18) are applied on the gears, respectively:

##### 2.4. The Differential Model

The bevel gear differential mechanism assembly and kinetic relation of each part are as shown in Figure 10. Rotational angle relation is defined in where is the assembly rotational angle of the final gear, the differential housing, and the planetary-gear pin around the -axis, is the rotational angle of the half axle gear around the -axis, is the rotational angle of the planetary gear around the -axis, and is the speed ratio of the planetary gear to the half axle gear.

Defining and as generalized coordinates, other rotational angles could be presented by these two coordinates:Now, the kinetic energy of the differential assembly is calculated by where is the rotational inertia of the assembly rotational angle of the final gear, the differential housing, and the planetary-gear pin around the -axis, is the rotational inertia of the half axle gear around the -axis, and is the rotational inertia of the planetary gear around the -axis.

##### 2.5. The LuGre Tire Model

For the LuGre tire model, the force analysis and the motion diagram are as shown in Figure 11.

The force analysis of the average lumped LuGre tire model is given by the following equation [31]:where is the average deformation of brush, is the relative velocity between the tire and the ground, is the normalized rubber longitudinal lumped stiffness, is the normalized rubber longitudinal lumped damping, is the normalized viscous relative damping, is the normalized coulomb friction, is the normalized static friction, is the Stribeck relative velocity, is the Stribeck effect index, is the length of the contact patch, is the distribution density function of the longitudinal pressure, is the longitudinal force of the tire, is the vertical force of the tire, is the tire slip rate, is the rotational velocity of the tire, is the rolling radius of the tire, and is the longitudinal road friction coefficient.

By the LuGre model, the relation between the longitudinal road friction coefficient and the tire slip rate on different ground condition is obtained in Figure 12.

##### 2.6. Calculation of Gear Pair Time-Varying Meshing Stiffness

Finite element analysis (FEA) is the most effective method for helical gear pair time-varying meshing stiffness. The helical gear meshing stiffness is defined as where is the gear pair meshing stiffness, is the normal force of the contact force, is the comprehensive deformation of gear pair, is the bending and shear deformation of one gear on the contact point, is the bending and shear deformation of the other gear on the contact point, and is the contact deformation of the gear pair on the contact point.

Simon got the bending and shear deformation , computational formula of (24), based on large amounts of FEA results through regression analysis [32]. Therefore,where is the elastic modulus, is the normal module, is the coefficient of normal force load point, is the coefficient of the relative radial position between load point and deformation point, is the coefficient of the relative axial position between load point and deformation point, is the teeth number, is the normal pressure angle, is the spiral angle in base on base circle, is the gear modification coefficient, is the addendum, is the dedendum, is the tooth root fillet radius, and is the tooth width.

As for the contact deformation , Cornell derived the following equation [33]:where is the piece length along the tooth width, is the piece force applied on the piece length , is the tooth thickness of one gear, is the tooth thickness of the other gear, is Poisson’s ratio of one gear, is Poisson’s ratio of the other gear, is the elastic modulus of one gear, and is the elastic modulus of the other gear.

Through (23) to (25), the time-varying meshing stiffness of the 1st gear pair (as shown in Figure 9) and the final drive gear pair (as shown in Figure 16) for a two-tooth cycle are shown in Figures 13 and 14.

#### 3. Numerical Modeling and Simulation Algorithm

##### 3.1. Modeling Framework

The 1st shift of MT on the vehicle creeping condition, when gear rattle noise could be perceived clearly by passengers on the researched vehicle, is used as an example. Gear rattle phenomenon is comprehensive results of complex interactions between the baseline vibration for the loaded driveline system and the rattling vibration for unloaded gear pairs in Figure 15. The baseline vibration consists of the engine, the clutch, the 1st gear pair, gears integrated on the input shaft, gears splined on the output shaft, final drive gear pair, the differential, the haft shaft, and the tire, while the rattling vibration concludes lightly loaded gear pairs, namely, the 2nd, the 3rd, the 4th, and the 5th gear pair.

It has been widely recognized in literature that the rattling vibration has little effects on the motion of the baseline vibration [6, 14], which could be utilized to study the overall system behavior more efficiently. The pinion gear motions of lightly loaded gear pairs in the baseline vibration become excitations to loose gear pairs in the rattling vibration. Then, the rattle force of loose gear pairs could be obtained.

##### 3.2. The Baseline Model of Vehicle Driveline System

Dynamic FWD driveline model based on the branched model is described in Figure 16 when the 1st gear pair is engaged. These loaded gear pairs, namely, the 1st gear pair and the final drive gear pair, are considered to be always in contact with a time-varying meshing stiffness, respectively, which is calculated in Section 2.6. Those unloaded gear pairs with lighted load may be driven across the backlash, causing impacts and rattle noise. The driveline model consists of the two-stage stiffness clutch damper model and the detailed MT model, considers the differential property, and utilizes the average lumped parameters LuGre tire model. The input power of driveline system is the effective output torque of the four-cylinder and four-stroke engine. Accordingly, the longitudinal force analysis of the vehicle and the torsional force analysis of the tire are as shown in Figure 17, assuming that vertical left and right tires load of the front or rear axle are equivalent.

In the branched model, the simplified factors include ignoring the oil shearing torque and the oil churning torque applied on the 1st gear pair in the power flow and neglecting dynamic property influence of bearings on the input shaft and the output shaft in Figure 8 and final drive gear bearings.

By the Lagrange equation, the baseline system vibration dynamics is placed in the matrix form: wherewhere diag expresses the diagonal matrix, is the angular displacement (AD) of engine (namely, the flywheel and clutch), is the AD of clutch hub, is the pinion gear AD of the 1st gear pair, and is the wheel gear of the 1st gear pair and the corresponding synchronizer AD. and are the AD of the 2nd gear pair, and are the AD of the 3rd gear pair, and are the AD of the 4th gear pair, and are the AD of the 5th gear pair, is the AD of the 3rd and 4th gear pair synchronizer, is the AD of the 5th gear pair synchronizer, and are the AD of the final drive gear pair, is the AD of a half axle gear about its own rotational axis, and are AD of left and right half axle, and are the AD of left and right tire, is the vehicle longitudinal displacement, is the inertia of flywheel and clutch, is the inertia of clutch hub, is the pinion gear inertia of the 1st gear pair, and is the sum of the wheel gear inertia of the 1st gear pair and the corresponding synchronizer. and are the inertia of the 2nd gear pair, and are the inertia of the 3rd gear pair, and are the inertia of the 4th gear pair, and are the inertia of the 5th gear pair, is the inertia of the 3rd and 4th gear pair synchronizer, is the inertia of the 5th gear pair synchronizer, is the pinion gear inertia of the final drive gear pair, is the sum inertia of differential ring gear, differential shell, planetary gear, and axis pin, and are the inertia of a planetary gear about its own rotational axis, and are the inertia of a half axle gear about its own rotational axis, is the sum inertia of the left half axle, wheel hub, wheel rim, and brake disc, is the sum inertia of the right half axle, wheel hub, wheel rim, and brake disc, is the inertia of the left-front tire, is the inertia of the right-front tire, is the vehicle mass, is the meshing stiffness of gear pairs, is the meshing damping of gear pairs, and are the backlash of unloaded gear pairs. Other and are torsional stiffness and torsional damping, respectively.

Here, in these matrices of , , , **,** and , some parameters are formulated: where is the helical angle on base circle of the pinion gear on the 1st gear, is the helical angle on base circle of the pinion gear on the final drive gear, is the tire dynamic radius, is the rolling resistance coefficient, is the tire bristle average deformation in LuGre tire model, is the tire bristle average deformation rate in LuGre tire model, is the distance from the mass center to the front axle, is the distance from the mass center to the rear axle, is the mass center height, is the vehicle longitudinal acceleration, is the vehicle longitudinal velocity, is the air resistance coefficient, is the vehicle frontal area, and is the air density.

##### 3.3. The Rattling Vibration Model of Unloaded Gear Pairs

The rattling impact is the source of rattle noise. The impact collisions through their gear backlash are transmitted to the transmission housing via shafts and bearings. The vibrations are then converted into an audible rattle. So rattling force is the focus of dynamic study of each gear pair.

For one rattling gear pair, the mechanical model is as shown in Figure 18. Each gear is equivalent to a lumped inertia. As the motion of the pinion gear , which is obtained in the baseline model, is taken as an excitation to the system, for 1st shift, pinion gears include the 2nd driving gear, the 3rd driven gear, the 4th driven gear, and the 5th driven gear in Figure 8. So rattling force of unloaded gear pair is deduced: Here, denotes the relative displacement along the line of conjugate action of unloaded gear pairs. Therefore, each rattling gear pair is then reduced to a single degree of freedom system. and are the backlash function, as shown in Figure 19, and its derivative function, respectively, which are defined aswhere is the driving gear AD, is the driven gear AD, is the base circle radius of the driving gear, is the base circle radius of the driven gear, is the inertia of the driven gear, is the drag torque applied on the driven gear, is the rattling force, is the average meshing stiffness of the gear pair, is the average meshing damping of the gear pair, and is the gear backlash.

##### 3.4. Simulation Method and Numerical Algorithm

As equations of the baseline vibration and the rattling vibration are derived, the driveline vibration includes highly nonlinear factors and the condition number of the system matrix, which is the ratio of its maximum to minimum eigenvalue, is very high. As MATLAB is taken as our numerical simulation tool, a “stiff” problem for ordinary differential equation (ODE) is usually difficult to solve on hand.

MATLAB provides kinds of solvers for stiff ODE, which consist of ODE15s, ODE23s, ODE23t, and ODE23tb. Among them, ODE15s is a variable order solver based on the numerical differentiation formulas. Optionally, it uses the backward differentiation formulas, and it is also known as Gear’s method, which are usually less efficient. ODE23s is based on a modified Rosenbrock formula of order 2. Because it is a one-step solver, it is more efficient than ODE15s at crude tolerances and it could solve some kinds of stiff problems for which ODE15s is not effective [34, 35]. ODE23s is used for the stiff problem on hand and it is found that the efficiency is acceptable.

#### 4. Simulation Results Analysis

##### 4.1. The Driveline Vibration Analysis

In the numerical model, required parameters are from a mass production vehicle. A proper and accurate driveline model could insure a practical result. Firstly, a two-stage stiffness clutch damper (see Figure 20) is adopted in the baseline model. And the two-stage stiffness clutch damper characteristics including elastic and hysteresis property, adopted in the original driveline system, are described in Figure 21 in the solid line.

**(a)**

**(b)**

According to (26), in the time domain, the vehicle velocity and the engine speed are obtained in Figures 22 and 23, respectively. From Figure 22, it is found that the vehicle moves forward slowly at the speed between 1.8 m/s and 1.815 m/s, namely, the vehicle creeping speed. In Figure 23, the engine rotates at about 800 rpm and the speed fluctuation amplitude is nearly 80 rpm, while the clutch hub rotates at about 800 rpm and the speed fluctuation amplitude is about 10 rpm. Accordingly, the angular acceleration amplitude of the clutch hub in Figure 25 is much smaller than the acceleration amplitude of the engine in Figure 24. As seen, the clutch damper plays a role in attenuating the fluctuation amplitude of the engine speed in the driveline. But Figure 25 shows that the clutch hub fluctuates remarkably about the mean speed.

**(a)**

**(b)**

On this special condition, it was found that the transmission rattle was severe through the driver subjective evaluation. Now, from the simulation results in Figure 26, it is concluded that the clutch damper works at the angular displacement from 5.7° to 8.6° between the first and second clutch damper mass, namely, the actual working area in the dot-line ellipse in Figure 21. The clutch damper works jumping between the first-stage stiffness and the second-stage stiffness of the drive side and it excites severer driveline torsional vibration, which results in drastic fluctuation vibration of the clutch hub and transmission rattle impact noise that could be perceived by the driver or the passenger.

Besides, in the frequency domain, frequency spectrum of the engine speed (see Figure 27) shows that primary frequencies include 13.43 Hz, 26.86 Hz, and 53.1 Hz, which are one-time frequency, double frequency, and four-time frequency, respectively. Correspondingly, primary frequencies of the clutch hub speed (see Figure 28) include 13.43 Hz, 26.86 Hz, and 53.1 Hz as well. Moreover, amplitudes of eight-time frequency (106.2 Hz), the twelve-time frequency (159.3 Hz), and other frequencies, which are compared with those amplitudes of 13.43 Hz, 26.86 Hz, and 53.1 Hz, are considerable. Through theoretical analysis, amplitudes of higher frequencies are smaller than those of lower frequencies. The two-stage stiffness clutch damper working between the first-stage stiffness and the second-stage stiffness could be explained for the results in Figure 28.

##### 4.2. Rattle Force Analysis of Unloaded Gear Pairs

As explained in Section 3.3, pinion gear motions, which are obtained from the baseline vibration, are excitations to the rattling vibration. The pinion gear motions of the 2nd, 3rd, 4th, and 5th gear pairs are as shown in Figure 29. Accordingly, the 2nd gear pair rattling force , the 3rd gear pair rattling force , the 4th gear pair rattling force , and the 5th gear pair rattling force are as shown in Figure 30.

**(a)**

**(b)**

**(c)**

**(d)**

**(a)**

**(b)**

**(c)**

**(d)**

In Figure 29, pinion gears fluctuate at the mean speed and pinion gear motions of 3rd gear pair, 4th gear pair, and 5th gear pair are nearly consistent with each other. From Figure 30, it is found that two-side rattling impacts happen and larger rattling force is excited in all unloaded gear pairs. Maximum amplitude of the 3rd and 4th gear pair rattling force could be nearly up to 2000 N, while rattle force of the 2nd gear pair is about 1000 N and the 4th gear pair rattle force is about 500 N. So the 3rd and 4th gear pair undergo severe rattle phenomenon. Furthermore, although pinion gear motions of 3rd gear pair, 4th gear pair, and 5th gear pair are nearly consistent, rattling forces of those three gear pairs are completely different, which proves that it is essential to establish a detailed MT model.

#### 5. Clutch Damper Parameters Optimization for Reducing Gear Rattle

##### 5.1. The Driveline Vibration Analysis after Improvement

As concluded in Section 4.1, the two-stage stiffness clutch damper works jumping between the first- and the second-stage stiffness of the drive side, which excites severer driveline vibration and gear rattle phenomenon. So a three-stage stiffness clutch damper for adding one-stage stiffness for low load torque between the first-stage and the second-stage stiffness is proposed innovatively in Figure 31. As seen, other stage property parameters of the two-stage clutch damper are not revised except the added stage property parameters, and this three-stage clutch damper could play a good performance originally for other vehicle driving conditions except the vehicle creeping condition. Nonlinear characteristics of the three-stage clutch damper are as shown in Figure 32 in the solid line.

**(a)**

**(b)**

According to (26), in the time domain, the engine fluctuates at about 800 rpm and speed fluctuation amplitude is nearly 80 rpm in Figure 33, which is similar to the result in Figure 23. But it is obviously found that fluctuation degree of the clutch hub is reduced and the fluctuation amplitude is less than 10 rpm. Similarly, angular acceleration of the clutch hub in Figure 35 is much smaller than that in Figure 25, while the angular acceleration of the engine in Figure 34 is similar to that in Figure 24.

**(a)**

**(b)**

Further analysis of the three-stage clutch damper working AD in Figure 36 shows that it works at the angular displacement from 5° to 8°, namely, the actual working area in the dot-line ellipse in Figure 32. Now, after adopting the three-stage clutch damper, jumping phenomenon between the first-stage stiffness and the second-stage stiffness is eliminated.

Besides, in the frequency domain, frequency spectrum of the engine speed (see Figure 37) is similar to that in Figure 27 and primary frequencies consist of 13.43 Hz, 26.86 Hz, and 53.1 Hz as well. Correspondingly, primary frequencies of the clutch hub speed include 13.43 Hz, 26.86 Hz, and 53.1 Hz as well in Figure 38. But in Figure 38, amplitudes of 106.2 Hz and 159.3 Hz and other frequencies in Figure 28, amplitudes of which could not be neglectful, are reduced to a much lower value. Through comprehensive analysis of those results, jumping phenomenon elimination between the first-stage stiffness and the second-stage stiffness could be explained for the result in Figure 38 after adopting the three-stage stiffness clutch damper.

##### 5.2. Rattle Force Analysis of Unloaded Gear Pairs after Optimization

Through the baseline model, pinion gear motions of the 2nd, 3rd, 4th, and 5th gear pair after optimization are as shown in Figure 39. Compared with the result in Figure 29, speed fluctuations of all pinion gears are apparently much lower, change trends of which are the same with the clutch hub. Then, pinion gear motions are as excitations to unloaded gear pairs and rattle forces of unloaded gear pairs are calculated in Figure 40. Rattle intensities of all unloaded gear pairs are obviously improved and one-side rattle impacts are dominant in all unloaded gear pairs. Maximum rattle force of the 2nd gear pair is less than 150 N and rattle force of the 3rd gear pair is less than 50 N, while rattle force of the 4th and 5th gear pair is similarly less than 100 N. It is concluded that all unloaded gear pairs undergo rattle vibration, but the intensity of rattle impacts is much weaker. So MT rattle phenomenon (or rattle force) is improved after adopting the three-stage stiffness clutch damper on the vehicle creeping condition.

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#### 6. Conclusions

Based on the branched model, including quasi-transient engine model, multistage clutch damper model, detailed MT model, differential model, and LuGre tire model, and considering time-varying stiffness of the 1st speed gear pair and final drive gear pair, 19-DOF model of the baseline vibration is established on the vehicle creeping condition. The rattling vibration is then obtained as the baseline vibration is as an excitation. The baseline vibration and the rattling vibration reproduce a comprehensive study of the driveline system and MT rattle phenomenon. It is concluded that(1)on the creeping condition, the two-stage stiffness clutch damper tends to work jumping between the first- and second-stage stiffness and it causes severer driveline vibration and disturbing rattle noise perceived by passengers. Larger rattling force of two-side impact is excited in all unloaded gear pairs. Maximum rattle force of the 3rd and 4th gear pair is up to about 2000 N, while rattle force of the 2nd gear pair is about 1000 N and rattle force of the 4th gear pair is nearly 500 N;(2)a three-stage stiffness clutch damper is adopted and it could obviously improve the driveline vibration and MT rattle phenomenon on the vehicle creeping condition. One-side impacts are dominant in all unloaded gear pairs. Maximum rattle force of the 4th and 5th gear pair is less than 100 N, while rattle force of the 2nd gear pair is smaller than 150 N and rattle force of the 3rd gear pair is less than 500 N;(3)achievements of numerical simulation developed in this research could be utilized for the design of driveline system and practical strategies for solving MT rattle phenomenon. Currently, all results are mainly obtained from numerical modeling and simulation and they are indispensable to be validated with further experimental results.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgment

The research leading to these results has received funding from the National Natural Science Foundation of China (Grant no. 51175379).