# Optimization of Dual-Mass Flywheel Parameters Based on a Multicondition Simulation Test

Authors: Maoqing Xie; Shangrui Wang; Zhengfeng Yan; Leigang Wang; Guanhua Tan
Journal: Shock and Vibration
 (2022)

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## Abstract

A dual-mass flywheel is an important part of an automobile transmission system whose function is to ensure the smooth transmission of engine power to a gearbox. Vehicle vibration and noise are major indicators of vehicle comfort, while dual-mass flywheel technology can preferably solve the comfort problem. Based on the created torsional vibration model, this paper describes the testing of the torsional characteristics under various conditions with an LMS AMESim simulation. The results show that the noise in the cab is fundamentally below 60 dB. The computational formula for the angular stiffness of an arc coil spring is derived, and then the spring angular stiffness is optimized with arithmetic averaging, which reveals an adjusted angular stiffness of 12.8 Nm/°. Additionally, the requirement for the angular acceleration of the WOT input shaft (i.e., being less than 500 rad/s2) is satisfied for various gears. The test results show that the double-mass flywheel can attain a vibration isolation efficiency of around 85%, which more effectively reduces the transmission vibration and noise and improves the vehicle comfort.


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## Body

## 1. Introduction

The NVH performance of vehicles has always been an important indicator for evaluating vehicle quality. In vehicle optimization design, the choice of test conditions greatly affects the optimization effect of the NVH performance. Most existing studies only perform analysis with idle and normal driving conditions, thus causing many limitations in the parameter optimization process of dual-mass flywheel design. Hence, studying the powertrain torsional vibration characteristics under multiple conditions is of great significance to the optimization of dual-mass flywheel parameters.Lv et al. [1] and Li and Shi [2] summarized the generation mechanisms and control methods of vehicle powertrain NVH performance and demonstrated the principles with which the dual-mass flywheel reduced the powertrain torsional vibration and improved driving comfort. Bhagate et al. [3] created a 1D multibody mathematical model of torsional vibration for a front-drive vehicle powertrain, with which the flywheel inertia, transmission shaft stiffness, and clutch stiffness were optimized. Additionally, the powertrain NVH was reduced through mathematical modeling and optimization. For the case of a six-speed transmission with an overdrive gear, Wu et al. [4] built a multiobjective optimization model by using the vehicle energy utilization efficiency and the standing start continuous shift acceleration time as the objective function. Furthermore, the NSGA-II algorithm was modified to allow its higher accuracy and efficiency in handling the multiobjective optimization problem for vehicle transmission systems. He et al. [5] and Chen et al. [6] developed a 19-DOF equivalent system model for a vehicle powertrain under driving conditions and experimentally verified the model validity. To avoid the excitation speed of engine torsional vibration, the model was simplified into a 6-DOF model with an engine, gearbox, transmission shaft, drive axle, wheel, and body. In research carried out by Kelly et al. [7] and Szpica et al. [8], the inherent properties of a powertrain were analyzed with a multi-DOF torsional vibration model under typical conditions, and the performance of a DMF-CS damper was optimized by virtual prototyping, ultimately forming a complete set of stiffness matching design methods for DMF-CS dampers. In 2009, Schaper et al. [9] created a detailed dynamic model for a dual-mass flywheel to examine two arc springs and their rail friction behaviors, for which both the redirection force and the centripetal effect acted on the springs along the radial direction for friction generation. In the research of Chen et al. [10], the effects of the dual-mass flywheel parameters on the powertrain torsional vibration were analyzed under typical conditions, and the dual-mass flywheel was designed according to the parameters of a certain vehicle and modeled through simulation. Utilizing the simulation model, the performance of the flywheel vibration damping was analyzed, and finally, the damping effect of the involved dual-mass flywheel was verified through a bench test. Chen et al. [11] and Tang et al. [12] investigated the hysteresis, stick-slip, and speed-related properties of a basic dual-mass flywheel through static and dynamic tests. Based on the experimental results, a nonlinear model of torque transmitted by the flywheel was built, which consisted of nonlinear elastic and friction torques. The nonlinear characteristics of the rotational stiffness were explored, and a nonlinear model of the rotational speed-dependent stiffness was built. Additionally, a nonlinear 2-DOF system of the involved dual-mass flywheel was created through a Bouc–Wen simulation of the hysteretic friction torque, followed by friction torque estimation with the Levenberg–Marquardt method. Finally, considering the nonlinear stiffness of the model, the Bouc–Wen model parameters were estimated based on the dynamic test data. He et al. [5] investigated the effects of dual-mass flywheel parameters on the powertrain torsional vibration during the vehicle launch process. Their results showed the flywheel underwent drastic oscillations when the transmission system was at the resonance region, thus verifying the model validity. Furthermore, the appropriate moment of inertia under launch conditions was the design requirement for the dual-mass flywheel. In the research of Chen et al. [13], a dynamic characteristics simulation model was established for dual-mass flywheels. Additionally, a set of adjustable-damping semiactive controlled devices for a magnetorheological fluid dual-mass flywheel was designed by exploiting the controllable viscosity feature of the magnetorheological fluid by the magnetic field intensity, which conformed to the flywheel damping requirements under different powertrain conditions. Furthermore, the start and stop conditions of the designed device were analyzed. In the research of Jiang and Chen [14], Song et al. [15], and Zeng et al. [16], the method for the moment of inertia distribution was explored based on the structural and operational principles of a circumferential arc spring dual-mass flywheel, as well as the torsional vibration model created under idle and normal driving conditions. Additionally, modal analysis and vibration test verification were carried out for the vehicle power system. Zeng et al. [17] investigated the torsional properties of a long circumferential arc spring dual-mass flywheel by constructing static and multi-DOF torsional vibration models and performed simulations and experiments to verify the flywheel damping characteristics. In the research of Wang et al. [18] and Liu et al. [19], a multicondition simulation test was employed to separately explore the vibration properties of confluent and single-stage planetary transmissions, and then the powertrain NVH was analyzed based on the vibration properties under multiple conditions, which provided a reference for the planetary transmission design under complex conditions. Gao et al. [20] analyzed the vehicle ride comfort under multicondition random inputs and concluded that the vehicle velocity was directly influential for the driving comfort. Chen et al. [21] studied the dual-mass flywheel torsional vibration damping characteristics of a shock absorber under many conditions, including the engine, DMF, transmission gear pair, and clutch friction model, a six-degrees of freedom nonlinear model of the transmission system design including the ignition conditions, idle speed condition, startup condition at a constant speed, driving conditions, and many conditions with a simulation strategy. The damping performance of the DMF under different working conditions was studied.Taking a holistic look at the current progress of vehicle powertrain torsional vibration research around the world, increased attention has been paid to the creation of torsional vibration models and the accuracy of simulation conditions. Although many researchers have applied multiple conditions when analyzing powertrain vibration characteristics, few have introduced multiple conditions for investigating the effects of a dual-mass flywheel on powertrain vibration characteristics such as idle, start/stalling, starting, WOT, crawling, and TIP IN/OUT conditions. In this paper, a dynamic model is built for a powertrain with a long-arc spring dual-mass flywheel, and the vibration damping performance of the flywheel is examined under multiple conditions through a simulation, which proves that there is highly efficient flywheel damping under various complex conditions. After the optimization of the arc spring angular stiffness, the optimization effect is verified through field testing. Additionally, this study verifies that the multicondition simulation test allows the more accurate and reasonable optimization of parameters, thereby providing a reference for the matching and optimal design of dual-mass flywheel transmission systems.


## 2. Materials and Methods

### 2.1. Torsional Vibration Model of a Powertrain

The torsional vibration model is the basis for the optimization analysis of torsional damper performance parameters. Given the complexity of a vehicle transmission system, the mass, stiffness, and damping distribution are considerably uneven, which necessitates the creation of a multi-DOF discrete model including mass-stiffness-damping. In this study, multi-DOF models for powertrain torsional vibration are built with idle and driving conditions for a common four-cylinder engine front-drive vehicle based on the vehicle powertrain parameters. Figure1 describes the powertrain system layout, for which the powertrain uses a long-arc spring dual-mass flywheel, and Figure 2 illustrates a long-arc spring dual-mass flywheel structure.Figure 1
Powertrain system layout.Figure 2
Structure diagram of arc spring damper flywheel assembly.Concerning the performance comparison of a DMF and clutch, many simulations and experiments have been carried out around the world. After comparison and analysis, the DMF is found to have a good damping effect under idle conditions and at low velocity and normal driving zones, although the damping effect is not very effective at high-velocity zones. The reason for this is that under idle conditions, the engine speed fluctuates greatly. In view of the good isolation effect of a DMF on the low-frequency torsional vibrations, the torsional vibration control of a powertrain system under idle conditions should be set as the primary goal of DMF design. As is clear from the aforementioned structural and operational principles, the DMF controls the powertrain torsional vibration by assigning the moment of inertia and the multistage torsional stiffness of the primary and secondary flywheels. Thus, the inertia and the multistage stiffness determine the damping performance of the primary and secondary flywheels. In the modeling process, the dual-mass flywheel is divided into two moments of inertia. One part is the moment of inertia of the dual-mass flywheel primary flywheel, and the other part is the equivalent moment of inertia of the dual-mass flywheel secondary flywheel and clutch. Figure3 displays the torsional vibration model under idle conditions [22, 23].Figure 3
Torsional vibration model under idle conditions.The torsional damping of a mechanical powertrain system is normally rather small. At smaller damping values (ζ < 1), the natural frequencies with and without damping differ only by about 5%, so the impact of damping is disregarded during the natural frequency calculation. The equation for powertrain torsion can be derived from the relevant knowledge of vibration mechanics as follows:(1)Jθ¨+Kθ=T,J1θ¨1+K1θ1−θ2=0,J2θ¨2−K1θ1−θ2+K2θ2−θ3=T1,J3θ¨3−K2θ2−θ3+K3θ3−θ4=0,J4θ¨4−K3θ3−θ4=0,where θ=θ1θ2⋯θn represents the torsion angle matrix of various rotating parts, Ji represents the inertia matrix of the powertrain system, Ki represents the stiffness matrix of the powertrain system, and T1 denotes the engine output torque. The system is assumed to be unexcited [24].(2)J=J1J2J3J4,K=K1−K1−K1K1+K2−K2−K2K2+K3−K3−K3K3,T=0T100.The engine does not output torque for the idle conditions. To analyze the idle characteristics of the powertrain natural torsional vibration, the following matrix equation can be solved:(3)Jθ¨+Kθ=0.Most of the multicondition simulations in this study are carried out based on a powertrain torsional vibration model under driving conditions. In order to facilitate calculation, the model should be simplified, in which the main final drive and differential can be regarded as an equivalent moment of inertia, and the axle, wheel, and whole vehicle can be regarded as an equivalent moment of inertia. The gearbox needs to exist as a separate moment of inertia. Considering the fact that some conditions involve the shift of gears, the distinction between gears is required during the torsional vibration modeling, as shown in Figure4.Figure 4
Torsional vibration model under driving conditions.The dynamic equations of the powertrain system are listed for the case of the first gear TIP IN condition, at which point the equivalent moment of inertia of the gearbox in first gear is represented byJ51. According to Formula (1), the following can be derived:(4)J1θ¨1+K1θ1−θ2=0,J2θ¨2−K1θ1−θ2+K2θ2−θ3=T1,J3θ¨3−K2θ2−θ3+K3θ3−θ4=0,J4θ¨4−K3θ3−θ4+K4θ4−θ5=0,J51θ¨5−K4θ4−θ5+K5θ5−θ6=0,J6θ¨6−K5θ5−θ6+K6θ6−θ7=0,J7θ¨7−K6θ6−θ7=0.In this case, Formula (1) has the following:(5)J=J1J2J3J4J51J6J7,K=K1−K1−K1K1+K2−K2−K2K2+K3−K3−K3K3+K4−K4−K4K4+K5−K5−K5K5+K6−K6−K6K6,T=0T100000.Based on the abovementioned torsional vibration models, the initial model can be built for the studied vehicle, and the torsional vibration characteristics of its powertrain can be simulated under multiple conditions. In this study, LMS AMESim software is used to build the initial powertrain model, and the torsional vibration characteristics are obtained through simulation under multiple conditions by inputting different parameters into submodels depending on the conditions.


### 2.2. Full Vehicle Parameters

At the initial stage of DMF development, simulation calculations are required to determine various performance indicators. The input of the vehicle parameters is very critical since the simulation results are directly affected by the integrality and accuracy of the parameters. Tables1 and 2 list the engine parameters and the gearbox parameters used for the studied vehicle, respectively. The torsional stiffness parameters involved in the simulation and test in this paper, shown in Table 3, are derived from manufacturer data and CAE analysis.Table 1
Engine parameters.
ModelDisplacement/(L)Number of cylindersMaximum torque/(Nm)Driving speed/(rpm)Idle speed/(rpm)GA161.59442801500–4500750Table 2
Gearbox parameters.
MF628 gearbox1st gear2nd gear3rd gear4th gear5th gear6th gearMain retarderSpeed ratio3.7272.0531.371.030.8160.6894.158The moment of inertia/kg·m20.145960.044260.019730.011150.006990.00499—Table 3
The torsional stiffness parameters.
ParameterTorsional stiffness/(Nm/rad)K1126575.7K2121476K3171.9 (the first stage), 733.44 (the second stage)K45042.4 (3rd gear), 6589.5 (4th gear), 7449 (5th gear), 8308.5 (6th gear)K584372.62K67047.9 (left), 5328.9 (right)K724982.8–39995.4Figure5 illustrates the engine cylinder pressure diagram obtained after importing the client-provided data for cylinder pressures into the computational software.Figure 5
Engine cylinder pressure data.Figure6 plots the changes in the engine torque.Figure 6
Engine torque variations.


### 2.3. Initial Model Building

Under different conditions, commonality exists in the computational process of inherent vehicle characteristics, and frequent change of the parameters is required for calculation during the DMF development. Using LMS AMESim software, the initial model of the vehicle powertrain is built based on its structure and the dynamic model developed with basic components. The model allows simulation of the powertrain torsional vibration characteristics under multiple conditions. Figure7 depicts the initial model, which is composed of a four-cylinder engine and its front attachment model, a DMF model, and a gearbox model. The full vehicle model is composed of a final drive, a differential, an axle shaft, and a wheel.Figure 7
Initial model.During model building, the following parameters should be considered: the stiffness, damping, and moment of inertia of the front rotor parts in the front attachment model; the number of engine cylinders; the stiffness, damping, and moment of inertia of the rotor parts such as crankshaft pistons; the cylinder pressure characteristics and throttle opening signal model of the engine in the engine model; the damping spring stiffness, angular damping, and primary and secondary flywheel moments of inertia (including the moments of inertia of clutch pressure and driven plates) in the DMF model; the stiffness, damping, moment of inertia, and speed ratio of specific gears in the gearbox model; and the stiffness and moment of inertia of the final drive, differential and axle shaft tire, the full vehicle mass, frontal area, road damping, in the full vehicle model. Figure8 displays the parameter assignment and commissioning interface for the DMF model.Figure 8
Parameter assignment and commissioning interface of the DMF model.


## 2.1. Torsional Vibration Model of a Powertrain

The torsional vibration model is the basis for the optimization analysis of torsional damper performance parameters. Given the complexity of a vehicle transmission system, the mass, stiffness, and damping distribution are considerably uneven, which necessitates the creation of a multi-DOF discrete model including mass-stiffness-damping. In this study, multi-DOF models for powertrain torsional vibration are built with idle and driving conditions for a common four-cylinder engine front-drive vehicle based on the vehicle powertrain parameters. Figure1 describes the powertrain system layout, for which the powertrain uses a long-arc spring dual-mass flywheel, and Figure 2 illustrates a long-arc spring dual-mass flywheel structure.Figure 1
Powertrain system layout.Figure 2
Structure diagram of arc spring damper flywheel assembly.Concerning the performance comparison of a DMF and clutch, many simulations and experiments have been carried out around the world. After comparison and analysis, the DMF is found to have a good damping effect under idle conditions and at low velocity and normal driving zones, although the damping effect is not very effective at high-velocity zones. The reason for this is that under idle conditions, the engine speed fluctuates greatly. In view of the good isolation effect of a DMF on the low-frequency torsional vibrations, the torsional vibration control of a powertrain system under idle conditions should be set as the primary goal of DMF design. As is clear from the aforementioned structural and operational principles, the DMF controls the powertrain torsional vibration by assigning the moment of inertia and the multistage torsional stiffness of the primary and secondary flywheels. Thus, the inertia and the multistage stiffness determine the damping performance of the primary and secondary flywheels. In the modeling process, the dual-mass flywheel is divided into two moments of inertia. One part is the moment of inertia of the dual-mass flywheel primary flywheel, and the other part is the equivalent moment of inertia of the dual-mass flywheel secondary flywheel and clutch. Figure3 displays the torsional vibration model under idle conditions [22, 23].Figure 3
Torsional vibration model under idle conditions.The torsional damping of a mechanical powertrain system is normally rather small. At smaller damping values (ζ < 1), the natural frequencies with and without damping differ only by about 5%, so the impact of damping is disregarded during the natural frequency calculation. The equation for powertrain torsion can be derived from the relevant knowledge of vibration mechanics as follows:(1)Jθ¨+Kθ=T,J1θ¨1+K1θ1−θ2=0,J2θ¨2−K1θ1−θ2+K2θ2−θ3=T1,J3θ¨3−K2θ2−θ3+K3θ3−θ4=0,J4θ¨4−K3θ3−θ4=0,where θ=θ1θ2⋯θn represents the torsion angle matrix of various rotating parts, Ji represents the inertia matrix of the powertrain system, Ki represents the stiffness matrix of the powertrain system, and T1 denotes the engine output torque. The system is assumed to be unexcited [24].(2)J=J1J2J3J4,K=K1−K1−K1K1+K2−K2−K2K2+K3−K3−K3K3,T=0T100.The engine does not output torque for the idle conditions. To analyze the idle characteristics of the powertrain natural torsional vibration, the following matrix equation can be solved:(3)Jθ¨+Kθ=0.Most of the multicondition simulations in this study are carried out based on a powertrain torsional vibration model under driving conditions. In order to facilitate calculation, the model should be simplified, in which the main final drive and differential can be regarded as an equivalent moment of inertia, and the axle, wheel, and whole vehicle can be regarded as an equivalent moment of inertia. The gearbox needs to exist as a separate moment of inertia. Considering the fact that some conditions involve the shift of gears, the distinction between gears is required during the torsional vibration modeling, as shown in Figure4.Figure 4
Torsional vibration model under driving conditions.The dynamic equations of the powertrain system are listed for the case of the first gear TIP IN condition, at which point the equivalent moment of inertia of the gearbox in first gear is represented byJ51. According to Formula (1), the following can be derived:(4)J1θ¨1+K1θ1−θ2=0,J2θ¨2−K1θ1−θ2+K2θ2−θ3=T1,J3θ¨3−K2θ2−θ3+K3θ3−θ4=0,J4θ¨4−K3θ3−θ4+K4θ4−θ5=0,J51θ¨5−K4θ4−θ5+K5θ5−θ6=0,J6θ¨6−K5θ5−θ6+K6θ6−θ7=0,J7θ¨7−K6θ6−θ7=0.In this case, Formula (1) has the following:(5)J=J1J2J3J4J51J6J7,K=K1−K1−K1K1+K2−K2−K2K2+K3−K3−K3K3+K4−K4−K4K4+K5−K5−K5K5+K6−K6−K6K6,T=0T100000.Based on the abovementioned torsional vibration models, the initial model can be built for the studied vehicle, and the torsional vibration characteristics of its powertrain can be simulated under multiple conditions. In this study, LMS AMESim software is used to build the initial powertrain model, and the torsional vibration characteristics are obtained through simulation under multiple conditions by inputting different parameters into submodels depending on the conditions.


## 2.2. Full Vehicle Parameters

At the initial stage of DMF development, simulation calculations are required to determine various performance indicators. The input of the vehicle parameters is very critical since the simulation results are directly affected by the integrality and accuracy of the parameters. Tables1 and 2 list the engine parameters and the gearbox parameters used for the studied vehicle, respectively. The torsional stiffness parameters involved in the simulation and test in this paper, shown in Table 3, are derived from manufacturer data and CAE analysis.Table 1
Engine parameters.
ModelDisplacement/(L)Number of cylindersMaximum torque/(Nm)Driving speed/(rpm)Idle speed/(rpm)GA161.59442801500–4500750Table 2
Gearbox parameters.
MF628 gearbox1st gear2nd gear3rd gear4th gear5th gear6th gearMain retarderSpeed ratio3.7272.0531.371.030.8160.6894.158The moment of inertia/kg·m20.145960.044260.019730.011150.006990.00499—Table 3
The torsional stiffness parameters.
ParameterTorsional stiffness/(Nm/rad)K1126575.7K2121476K3171.9 (the first stage), 733.44 (the second stage)K45042.4 (3rd gear), 6589.5 (4th gear), 7449 (5th gear), 8308.5 (6th gear)K584372.62K67047.9 (left), 5328.9 (right)K724982.8–39995.4Figure5 illustrates the engine cylinder pressure diagram obtained after importing the client-provided data for cylinder pressures into the computational software.Figure 5
Engine cylinder pressure data.Figure6 plots the changes in the engine torque.Figure 6
Engine torque variations.


## 2.3. Initial Model Building

Under different conditions, commonality exists in the computational process of inherent vehicle characteristics, and frequent change of the parameters is required for calculation during the DMF development. Using LMS AMESim software, the initial model of the vehicle powertrain is built based on its structure and the dynamic model developed with basic components. The model allows simulation of the powertrain torsional vibration characteristics under multiple conditions. Figure7 depicts the initial model, which is composed of a four-cylinder engine and its front attachment model, a DMF model, and a gearbox model. The full vehicle model is composed of a final drive, a differential, an axle shaft, and a wheel.Figure 7
Initial model.During model building, the following parameters should be considered: the stiffness, damping, and moment of inertia of the front rotor parts in the front attachment model; the number of engine cylinders; the stiffness, damping, and moment of inertia of the rotor parts such as crankshaft pistons; the cylinder pressure characteristics and throttle opening signal model of the engine in the engine model; the damping spring stiffness, angular damping, and primary and secondary flywheel moments of inertia (including the moments of inertia of clutch pressure and driven plates) in the DMF model; the stiffness, damping, moment of inertia, and speed ratio of specific gears in the gearbox model; and the stiffness and moment of inertia of the final drive, differential and axle shaft tire, the full vehicle mass, frontal area, road damping, in the full vehicle model. Figure8 displays the parameter assignment and commissioning interface for the DMF model.Figure 8
Parameter assignment and commissioning interface of the DMF model.


## 3. Results and Discussion

### 3.1. Simulation Result

Simulations are performed after the model is built. To ensure that the angular accelerations of the engine and the gearbox input shaft conform to the full vehicle requirements, the following parameters are adjusted: the damping spring stiffness, damping angle, and damping of the DMF, as well as the moments of inertia of the primary and secondary flywheels. As shown in Figure9, the angular accelerations at the gearbox input shaft are considerably smoother than those at the engine end, and all of the angular accelerations are below 500 rad/s2, thus proving the good damping effect of the DMF.Figure 9
Angular acceleration frequency spectra.Based on the created dynamic and initial models, the characteristics of the idle (Figure10), start/stalling (Figure 11), starting (Figure 12), WOT (Figure 13), crawling (Figure 14), and TIP IN/OUT (Figure 15) conditions can be obtained, which is shown further in this study.Figure 10
Idle conditions. (a) Cold idle clutch. (b) Hot idle load.
(a)(b)Figure 11
Start/stalling conditions. (a) Start. (b) Stalling.
(a)(b)Figure 12
Starting condition. (a) 1st gear start. (b) 1st gear loaded start.
(a)(b)Figure 13
Crawling condition.Figure 14
WOT condition.Figure 15
TIP IN/OUT condition. (a) 2nd gear POT. (b) 3rd gear POT.
(a)(b)


### 3.2. Experimental Verification

For the test experiment described in this section, the abovementioned conditions are regarded as the test conditions. Additionally, to ensure the accuracy of the test results, the test vehicle is required to travel normally without dysfunction; the ECU and suspension should be preliminarily calibrated; and the air conditioner, headlight, heater, cigarette lighter, and battery should operate normally. This paper will use professional testing equipment to research the vibration of each component of the vehicle. BBM PAK vibration and noise testing equipment and an analysis system, MK II portable data acquisition front end, and PAK analysis software are used in the experiment. After the engine heats up and the clutch is connected, the vehicle enters the driving mode, so the engine speed rises quickly to about 1,500 rpm while keeping the engine stable. The acquired signals include the speed signal of the engine flywheel starter ring gear, the speed signal of the gearbox first shaft constant meshing gear, the vibration signal of the driver seat slide rail, the vibration signal of bearing vibration, and lubrication conditions in the gearbox system [25, 26]. Figure 16 illustrates the noise spectra in the cab, according to which the cab interior noise is basically below 60 dB.Figure 16
Noise spectra in the cab.Figure17 plots the experimental versus simulation comparison curves. As is clear, the simulated angular acceleration at the gearbox input satisfies the practical requirement, and the test results are very close to the simulations. Thus, the simulation results are instructive.Figure 17
Simulated versus experimental DMF performance.


### 3.3. Optimization and Verification of DMF Parameters

The torsional curve of a DMF is mainly dependent on parameters such as the angular stiffness and damping. In this study, the torsional angular stiffness of a DMF is investigated. This torsional angular stiffness is dependent on the arc spring, which is an elastic element. There are various design schemes involving one-stage stiffness or multistage stiffness. Optimization of the arc spring angular stiffness is conducive to alleviating the powertrain vibration problem and improving driving comfort. The angular stiffness of an arc spring is optimized herein with the constructed LMS AMESim simulation model of the powertrain system. The angular stiffness of the DMF arc spring is selected as the design variable for optimization analysis, and the input shaft angular accelerations for various WOT conditions are used as the objective function to make the parameter below 500 rad/s2. To achieve a reasonable angular stiffness, first deriving the characteristic formula for the long-arc spiral spring is necessary. Figure 18 presents the schematic of the arc spring.Figure 18
Schematic diagram of arc spring.A DMF with two sets of inner and outer arc springs is studied herein. This DMF features two-stage torsional angular stiffness. The torsional angular stiffnessK1,2I provided by the outer arc spring acts under idle conditions, while the torsional angular stiffness K1,2R provided jointly by both sets of arc springs acts under driving conditions, and K1,2I ≤ K1,2R. According to the structural characteristics of the DMF, it is clear that its torsional angular stiffness does not exceed that of the driven disc torsional damper.Under idle conditions, the torque is small, the compression of the inner springs is limited, only the outer arc springs are active, and the ultimate working angle isθ1. Under driving conditions, the torque is large, the inner and outer arc springs work together, and the ultimate working angle is θ2. The total torsional angle of the DMF arc springs is θ = θ1+θ2, and its value is usually around 60. Considering the size of the DMF and the installation space of the arc springs, θ is set to 60. Combining the working characteristics and the stress requirements for the inner and outer arc springs, the ultimate working angles θ1 and θ2 are set separately.The ultimate working torqueMj of the DMF depends on the maximum torque Temax of the engine. To endow a certain buffer capacity, the Mj value is generally slightly larger than the Temax value:(6)Mj=ξTemax.In Formula (6), Temax denotes the maximum engine torque, whose value is 280 Nm based on the abovementioned description, and ξ denotes the torque reserve coefficient, which equals 1.5 for the vehicle studied herein [27].Accordingly, the ultimate working torque Mj of the DMF is as follows:(7)K1,2Iθ1+K1,2Rθ2≥Mj=ξTemax.Taking the abovementioned factors collectively, the constraint on DMF torsional stiffnessK1,2R under driving conditions can be obtained, which is converted into a degree measurement as follows [28]:(8)7.32≤K1,2R≤13.62Nm/°.Table4 lists the actual test results of the vehicle loading. The entire test process requires an outdoor environment with sunny weather, dry air, and no wind or breeze with a speed greater than 3 m/s. The test track should be located on a clean, dry, and flat first-class highway, and the pavement slope must be less than 0.1%.Table 4
Actual test results of sample loading.
Working condition2nd angular acceleration of the engine/(rad/s2)2nd angular acceleration of input shaft/(rad/s2)Vibration isolation rate (%)Note (rpm)1st gear WOT224436983.5630192nd gear WOT293543885.0824613rd gear WOT306145885.0425064th gear WOT326654583.3116455th gear WOT364660083.5416496th gear WOT375771980.861559According to practical requirements, the input shaft angular acceleration under various WOT conditions must be less than 500 rad/s2. The DMF damping stiffness is optimized and adjusted with the arithmetic averaging of the elastic element stiffness. As shown in Figure 19, the load when the inner arc springs start to work is equal to the mean of the spring loads at zero load and full load:(9)TK=0.5T0+TW.Figure 19
Optimization schematic of torsional characteristic parameters.Additionally, the frequency corresponding to the mean load betweenT0 and TK is equal to the frequency corresponding to the mean load between TK and TW, where Km denotes the angular stiffness of the outer-arc spring, and Ka denotes the angular stiffness of the inner arc spring:(10)T0+TK2Km=TW+TK2Km+Ka.Substituting Formula (9) into (10) yields:(11)KaKm=2TW−2T03T0+TW.Letting(12)TWT0=λ.Through simplification, the proportional relationship between the angular stiffness values of the inner and outer arc springs can be obtained [29]:(13)KaKm=2λ−23+λ.According to the constraint on the DMF torsional stiffnessK1,2R under driving conditions in Formula (13), and the proportional relationship between the inner and outer arc spring stiffness in Formula (13), the angular stiffness optimization of the arc springs can be achieved.Table5 lists the adjustment results for the arc spring parameters. The adjusted angular stiffness of the arc springs satisfies the constraint. Figure 20 depicts the DMF damping stiffness before and after adjustment, while Figure 21 presents the comparison between the preadjustment and postadjustment simulations. As is clear from the figure, all the angular accelerations satisfy the practical requirements after adjusting the spring stiffness.Table 5
Arc spring parameter adjustment.
Arc springThe wire diameter/(mm)Effective number of turnsFree length/(mm)Angle stiffness/(Nm/°)Torque/(Nm)Outer arc springNot adjust3.8441603182Inner arc springBefore the adjustment3.55611512.7190.5After the adjustment3.2441219.8205.8Figure 20
Vibration damping stiffness of DMF.Figure 21
Comparison of simulations before and after adjustment.Table6 lists the actual loading test results after adjustment. According to the table, the input shaft accelerations in various gears are all below 500 rad/s2, and the vibration isolation rates are also markedly improved, which proves that the adjustments are effective and in line with the simulations as well.Table 6
Actual test results of loading after adjustment.
Working condition2nd angular acceleration of the engine/(rad/s2)2nd angular acceleration of input shaft/(rad/s2)Vibration isolation rate (%)Note (rpm)1st gear WOT223137983.0131512nd gear WOT280636187.1321973rd gear WOT297940586.4019374th gear WOT290744184.8316285th gear WOT302645684.9315726th gear WOT305245984.961535


## 3.1. Simulation Result

Simulations are performed after the model is built. To ensure that the angular accelerations of the engine and the gearbox input shaft conform to the full vehicle requirements, the following parameters are adjusted: the damping spring stiffness, damping angle, and damping of the DMF, as well as the moments of inertia of the primary and secondary flywheels. As shown in Figure9, the angular accelerations at the gearbox input shaft are considerably smoother than those at the engine end, and all of the angular accelerations are below 500 rad/s2, thus proving the good damping effect of the DMF.Figure 9
Angular acceleration frequency spectra.Based on the created dynamic and initial models, the characteristics of the idle (Figure10), start/stalling (Figure 11), starting (Figure 12), WOT (Figure 13), crawling (Figure 14), and TIP IN/OUT (Figure 15) conditions can be obtained, which is shown further in this study.Figure 10
Idle conditions. (a) Cold idle clutch. (b) Hot idle load.
(a)(b)Figure 11
Start/stalling conditions. (a) Start. (b) Stalling.
(a)(b)Figure 12
Starting condition. (a) 1st gear start. (b) 1st gear loaded start.
(a)(b)Figure 13
Crawling condition.Figure 14
WOT condition.Figure 15
TIP IN/OUT condition. (a) 2nd gear POT. (b) 3rd gear POT.
(a)(b)


## 3.2. Experimental Verification

For the test experiment described in this section, the abovementioned conditions are regarded as the test conditions. Additionally, to ensure the accuracy of the test results, the test vehicle is required to travel normally without dysfunction; the ECU and suspension should be preliminarily calibrated; and the air conditioner, headlight, heater, cigarette lighter, and battery should operate normally. This paper will use professional testing equipment to research the vibration of each component of the vehicle. BBM PAK vibration and noise testing equipment and an analysis system, MK II portable data acquisition front end, and PAK analysis software are used in the experiment. After the engine heats up and the clutch is connected, the vehicle enters the driving mode, so the engine speed rises quickly to about 1,500 rpm while keeping the engine stable. The acquired signals include the speed signal of the engine flywheel starter ring gear, the speed signal of the gearbox first shaft constant meshing gear, the vibration signal of the driver seat slide rail, the vibration signal of bearing vibration, and lubrication conditions in the gearbox system [25, 26]. Figure 16 illustrates the noise spectra in the cab, according to which the cab interior noise is basically below 60 dB.Figure 16
Noise spectra in the cab.Figure17 plots the experimental versus simulation comparison curves. As is clear, the simulated angular acceleration at the gearbox input satisfies the practical requirement, and the test results are very close to the simulations. Thus, the simulation results are instructive.Figure 17
Simulated versus experimental DMF performance.


## 3.3. Optimization and Verification of DMF Parameters

The torsional curve of a DMF is mainly dependent on parameters such as the angular stiffness and damping. In this study, the torsional angular stiffness of a DMF is investigated. This torsional angular stiffness is dependent on the arc spring, which is an elastic element. There are various design schemes involving one-stage stiffness or multistage stiffness. Optimization of the arc spring angular stiffness is conducive to alleviating the powertrain vibration problem and improving driving comfort. The angular stiffness of an arc spring is optimized herein with the constructed LMS AMESim simulation model of the powertrain system. The angular stiffness of the DMF arc spring is selected as the design variable for optimization analysis, and the input shaft angular accelerations for various WOT conditions are used as the objective function to make the parameter below 500 rad/s2. To achieve a reasonable angular stiffness, first deriving the characteristic formula for the long-arc spiral spring is necessary. Figure 18 presents the schematic of the arc spring.Figure 18
Schematic diagram of arc spring.A DMF with two sets of inner and outer arc springs is studied herein. This DMF features two-stage torsional angular stiffness. The torsional angular stiffnessK1,2I provided by the outer arc spring acts under idle conditions, while the torsional angular stiffness K1,2R provided jointly by both sets of arc springs acts under driving conditions, and K1,2I ≤ K1,2R. According to the structural characteristics of the DMF, it is clear that its torsional angular stiffness does not exceed that of the driven disc torsional damper.Under idle conditions, the torque is small, the compression of the inner springs is limited, only the outer arc springs are active, and the ultimate working angle isθ1. Under driving conditions, the torque is large, the inner and outer arc springs work together, and the ultimate working angle is θ2. The total torsional angle of the DMF arc springs is θ = θ1+θ2, and its value is usually around 60. Considering the size of the DMF and the installation space of the arc springs, θ is set to 60. Combining the working characteristics and the stress requirements for the inner and outer arc springs, the ultimate working angles θ1 and θ2 are set separately.The ultimate working torqueMj of the DMF depends on the maximum torque Temax of the engine. To endow a certain buffer capacity, the Mj value is generally slightly larger than the Temax value:(6)Mj=ξTemax.In Formula (6), Temax denotes the maximum engine torque, whose value is 280 Nm based on the abovementioned description, and ξ denotes the torque reserve coefficient, which equals 1.5 for the vehicle studied herein [27].Accordingly, the ultimate working torque Mj of the DMF is as follows:(7)K1,2Iθ1+K1,2Rθ2≥Mj=ξTemax.Taking the abovementioned factors collectively, the constraint on DMF torsional stiffnessK1,2R under driving conditions can be obtained, which is converted into a degree measurement as follows [28]:(8)7.32≤K1,2R≤13.62Nm/°.Table4 lists the actual test results of the vehicle loading. The entire test process requires an outdoor environment with sunny weather, dry air, and no wind or breeze with a speed greater than 3 m/s. The test track should be located on a clean, dry, and flat first-class highway, and the pavement slope must be less than 0.1%.Table 4
Actual test results of sample loading.
Working condition2nd angular acceleration of the engine/(rad/s2)2nd angular acceleration of input shaft/(rad/s2)Vibration isolation rate (%)Note (rpm)1st gear WOT224436983.5630192nd gear WOT293543885.0824613rd gear WOT306145885.0425064th gear WOT326654583.3116455th gear WOT364660083.5416496th gear WOT375771980.861559According to practical requirements, the input shaft angular acceleration under various WOT conditions must be less than 500 rad/s2. The DMF damping stiffness is optimized and adjusted with the arithmetic averaging of the elastic element stiffness. As shown in Figure 19, the load when the inner arc springs start to work is equal to the mean of the spring loads at zero load and full load:(9)TK=0.5T0+TW.Figure 19
Optimization schematic of torsional characteristic parameters.Additionally, the frequency corresponding to the mean load betweenT0 and TK is equal to the frequency corresponding to the mean load between TK and TW, where Km denotes the angular stiffness of the outer-arc spring, and Ka denotes the angular stiffness of the inner arc spring:(10)T0+TK2Km=TW+TK2Km+Ka.Substituting Formula (9) into (10) yields:(11)KaKm=2TW−2T03T0+TW.Letting(12)TWT0=λ.Through simplification, the proportional relationship between the angular stiffness values of the inner and outer arc springs can be obtained [29]:(13)KaKm=2λ−23+λ.According to the constraint on the DMF torsional stiffnessK1,2R under driving conditions in Formula (13), and the proportional relationship between the inner and outer arc spring stiffness in Formula (13), the angular stiffness optimization of the arc springs can be achieved.Table5 lists the adjustment results for the arc spring parameters. The adjusted angular stiffness of the arc springs satisfies the constraint. Figure 20 depicts the DMF damping stiffness before and after adjustment, while Figure 21 presents the comparison between the preadjustment and postadjustment simulations. As is clear from the figure, all the angular accelerations satisfy the practical requirements after adjusting the spring stiffness.Table 5
Arc spring parameter adjustment.
Arc springThe wire diameter/(mm)Effective number of turnsFree length/(mm)Angle stiffness/(Nm/°)Torque/(Nm)Outer arc springNot adjust3.8441603182Inner arc springBefore the adjustment3.55611512.7190.5After the adjustment3.2441219.8205.8Figure 20
Vibration damping stiffness of DMF.Figure 21
Comparison of simulations before and after adjustment.Table6 lists the actual loading test results after adjustment. According to the table, the input shaft accelerations in various gears are all below 500 rad/s2, and the vibration isolation rates are also markedly improved, which proves that the adjustments are effective and in line with the simulations as well.Table 6
Actual test results of loading after adjustment.
Working condition2nd angular acceleration of the engine/(rad/s2)2nd angular acceleration of input shaft/(rad/s2)Vibration isolation rate (%)Note (rpm)1st gear WOT223137983.0131512nd gear WOT280636187.1321973rd gear WOT297940586.4019374th gear WOT290744184.8316285th gear WOT302645684.9315726th gear WOT305245984.961535


## 4. Conclusions

Through this study, the following conclusions can be reached:Torsional vibration models for idle and driving conditions have been developed for which the equivalent moments of inertiaJ5i of various gearbox part assemblies are distinguished by the gears, thereby enabling the model’s applicability to multiple complex driving conditions.Simulation software and testing equipment are utilized to build the initial model, simulate the vehicle powertrain dynamics, and test the entire vehicle, thereby obtaining the NVH characteristics for idle or various driving conditions, which proves the efficient damping performance of the long-arc spring DMF.The computational formula for the angular stiffness of a-long arc spring is derived. Based on this formula and the practical requirements for input shaft angular accelerations for various gears under WOT conditions, optimal design and actual vehicle tests are performed for the spring angular stiffness, ultimately achieving the test results conforming to practical requirements. The correctness of the optimization method is verified, and the NVH performance of the full vehicle is improved. It is also verified that the multicondition simulation test allows more accurate and reasonable results for parameter optimization, which provides a reference for the matching and optimal design of DMF powertrain systems.



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*Source: 2954825-2022-04-14.xml*