Modeling and Experimental Study on Dynamic Characteristics of Dual-Mass Flywheel Torsional Damper

*eoretical modeling and experimental research are carried out on the dynamic torsional characteristics of a dual-mass flywheel (DMF) in this paper. Firstly, the structure and working principle of the DMF are analyzed. Secondly, the arc spring is analyzed by the discrete element method. For the different frictional torques in the working process, the linear fitting and equivalent energy methods are used to model the frictional torque under the dynamic condition of the arc spring. *e fractional derivative model is used to model the viscous damping of DMF. *en, the parameter identification and model verification are carried out on the model, and the model error is analyzed. Finally, an experimental study on the dynamic torsional characteristics of DMF is carried out. *e results demonstrate that the torsional stiffness of the DMF varies with the excitation amplitude and frequency. *is modeling and test method can be used for structural design and performance prediction analysis of DMF.


Introduction
Torsional vibration of an internal combustion engine vehicle powertrain is another major source of vibration excitation for vehicles in addition to road excitation. e torsional vibration of the powertrain is mainly caused by the cyclical change of the engine cylinder pressure and the inertial force generated by the reciprocating motion of the crank-link mechanism, which causes the engine output torque to fluctuate and brings torsional vibration. e application of the elastic torsional vibration damper mounted on the clutch disc reduces the torsional vibration of the powertrain, but due to the limited space of the clutch torsion damper spring, the torsion angle is small, the torsional stiffness is large, and the effect of vibration reduction is limited [1,2]. Moreover, the clutch torsional vibration damper cannot reduce the resonance speed of the powertrain below the idle speed so that the transmission system still has the possibility of resonance in the common speed range of the automobile. Although the promotion and application of high-power and lightweight engines have achieved the goal of improving vehicle power and reducing emissions, this has also led to more severe torsional vibration of automobile transmission systems, which has become an urgent problem for traditional internal combustion engine vehicles to improve NVH performance of complete vehicles. As a new type of the torsional vibration damper, the DMF has a large torsion angle and a small torsional stiffness compared to the clutch torsional vibration damper so that the speed corresponding to the natural frequency of the powertrain is much lower than the idle speed. erefore, the powertrain does not resonate within the normal speed range of the engine, reducing the torsional vibration of the powertrain and improving the ride comfort of the vehicle [3].
At present, the DMF plays a very important role in the vibration reduction, noise reduction of the powertrain, solving the problem of engine knocking, improving the smoothness of the shift, and so on. At the same time, there are many academic researches on the dual-mass flywheel. Kim used the discrete element method to analyze the effect of the friction on the arc spring on torsional performance of the DMF [4]; Tom conducted theoretical and experimental research on the torsional characteristics of long arc spring DMF and estimated the engine output torque using DMF [5].
rough the experimental study on the torsional characteristics of the DMF, Chen established a nonlinear dynamic model of the DMF based on the viscous damping [6]; Song proposed a DMF with continuous variable stiffness by using shape constraints and conducted theoretical and experimental studies on its torsional characteristics [7,8]. Zu et al. proposed an intelligent magnetorheological DMF, which can realize the real-time adjustment of the damping coefficient to achieve semiactive control of the torsional vibration of the powertrain [9]; Tang et al. studied the effect of dual-mass flywheels on the torsional vibration performance of hybrid electric vehicles and proposed a motor torque control methods to suppress the torsional vibration [10][11][12]. e existing literature has carried out a lot of research on the structure of the DMF, torsional characteristics, and vibration damping performance, but it can be found that there are few studies on the dynamic torsional characteristics (dynamic stiffness and lag angle) of the DMF. e dynamic torsional characteristics of the DMF directly determine its damping performance, and the test results also show that the performance will change greatly under dynamic torsion. Based on the existing literature, the dynamic torsion characteristics of DMF are modeled and tested in this paper.
is paper is organized as follows: Section 1 is devoted to describing the composition and working principle of the DMF; Section 2, the dynamic characteristics of DMF are modeled. In Section 3, the parameter identification and experimental verification of the dynamic characteristic model built in Section 2 are carried out, and the model error is analyzed. In Section 4, the dynamic characteristics of DMF are experimentally studied. Section 5 summarizes the conclusions of this paper.

DMF Structural Principle
e structure of the DMF is shown in Figure 1. It is composed of the first mass, the second mass, the starting ring gear, the force transfer plate, and the long arc spring [13]. e starting ring gear is connected to the first mass by an interference fit. e first mass is bolted to the end flange of the engine crankshaft. e flange is connected to the second mass by rivets. e second mass is bolted to the clutch assembly; the first mass assembly and the second mass assembly are connected by a low stiffness arc spring and can rotate relatively. When the engine crankshaft rotates, it drives the first mass to compress the arc spring through the boss. e other end of the arc spring drives the ears on both sides of the transmission plate, thus driving the second mass to rotate and realizing the transmission of power from the engine to the gearbox. e DMF structure solves the shortcomings of the traditional clutch torsional vibration damper spring, such as small torsional angle and large stiffness. Moreover, the moment of inertia of the first mass and the second mass can be flexibly distributed so that the resonance speed of the powertrain is kept away from the operating speed of the engine. Also the torque fluctuation of the input shaft of the transmission and the shifting noise are greatly reduced, and the shift smoothness and the NVH performance of the vehicle can be improved [14,15].

Dynamic Characteristic Modeling
e arc spring is the key component of the DMF structure. e performance of the arc spring determines the torsional vibration reduction performance of the DMF. erefore, the arc spring is first analyzed and modeled in this paper.

Arc Spring Frictional Torque Modeling.
e frictional torque is generated by using the arc spring and the sheath during the working process which is composed of two parts, as shown in Figure 2. e first part is that as the torsion angle θ increases, it produces a radial positive pressure along the radial direction of the distribution, thereby generating a frictional torque during the arc spring is compressed by the torque T; the other part is that the arc spring is subjected to centrifugal force to generate positive pressure on the sheath, which generates frictional torque when the speed of DMF is Ω.

Frictional Torque Generated by Radial Component.
e discrete element method is used to analyze the influence of arc spring on the dynamic characteristics of DMF. e total mass of arc spring is m, which is divided into n units. Each unit has a mass of m i (i � 1, 2, 3, . . . , n) and each element stiffness is k 1 � k 2 � · · · � k n ; c i is the arc spring damping coefficient [4]. Arc spring is shown in Figure 3(a). e equivalent model of the discrete arc spring is shown in Figure 3(b), and the angle between the elements is θ i .
Firstly, the quasi-static compression is analyzed (without considering the frictional torque generated by centrifugal force). e k-th arc spring mass element is used to analyze the stress state of the arc spring, as shown in Figure 4. F k−1 is the force of the last spring mass element, F k is the force of the next spring mass element, and F F is the tangential friction force of the slide way. F 0 is the radial force, F N is the support force of the slide to the arc spring, R is the distribution radius of the arc spring, μ is the friction coefficient, and k i is the stiffness coefficient. e arc spring is loaded clockwise, and the force is decomposed in the normal and tangential directions. According to the force balance, the following equation is obtained.
Tangential direction: Normal direction: where ω k is the angular velocity of the mass element, which can be approximated to zero due to quasi-static loading. According to formula (2), the following equation is obtained: 2 Shock and Vibration e tangential friction force is e angle di erence of any two elements is where α is the central angle of the arc spring's free length and n is the number of arc spring elements. e tangential force formula of the i-th element can be obtained:  Shock and Vibration 3 When unloading, except for the reverse friction force, other forces are the same as those loading. So the tangential force formula of the i-th element can be obtained: According to formulas (6) and (7), the relationship between the friction spring torque and the torsion angle of the arc spring under quasi-static loading conditions (without considering the in uence of centrifugal force) can be obtained, as shown in Figure 5. Figure 6 shows that the frictional torque generated is in a quasi-linear relationship with the torsion angle, and the relationship is obtained by linear tting: where p 1 0.5619 is the tting coe cient. e tting results are shown in Figure 6.

Frictional Torque Generated by Centrifugal
Force. e frictional torque T f2 generated by the centrifugal force of the arc spring is When the speed of DMF is constant, the frictional torque generated by the centrifugal force is constant. e equivalent energy method can be used to calculate the equivalent damping coe cient under periodic excitation [16]. e equivalent damping coe cient can be simpli ed as Finally, the total frictional torque of arc spring is obtained as follows:

Equivalent Model of Arc
Spring. According to Figure 3, the dynamic equation of the i-th element of the curved spring is obtained by Newton's second law: us, the mechanical equation of the whole arc spring can be obtained: Based on the assumptions of discrete elements, m 1 m 2 · · · m n 1 n m, c 1 c 2 · · · c n c, k 1 k 2 · · · k n nk,   Shock and Vibration Simpli cation of equation (13) can be obtained as follows: In order to simplify the calculation, the arc spring is equivalent to a concentrated mass, and formula (15) can be simpli ed as follows: where m e is the equivalent mass, c e is the equivalent damping, k e is the equivalent sti ness, and θ ′ is the equivalent mass torsion angle. Figure 7 is a simpli ed model of DMF, where M 0 is the frictional damping produced by the friction disc. According to the simpli ed model of Figure 7, the mathematical relationship expression of DMF is obtained:

DMF Dynamics Modeling.
where T s is the recovery torque generated by using the arc spring, T fri is the frictional torque, T d is the viscous damping torque, and T is the combined moment of the DMF. e arc spring has complex viscous lubrication in the dynamic working process. In order to explain this complicated dynamic process, the fractional derivative order model is adopted in this paper [17]: where b is the viscoelastic torque coe cient, β is the fractional derivative order, D β (·) is the β-order derivative of (·), and c is the ywheel viscous damping coe cient. e frictional torque generated by using the friction disc is mainly used to reduce the resonance peak, and its expression is as follows: e equivalent damping coe cient is calculated by the equivalent energy method: where ω is the excitation frequency. e elastic recovery torque of DMF can be obtained according to Figure 7 as In summary, the resulting torque of the DMF model is given by the sum of all element torques: e Laplace transform is applied to equation (22), and the dynamic torsion complex sti ness of DMF can be simpli ed:

Parameter Identification and Experimental Verification
In this section, the dynamic characteristics of the DMF are tested, and then the dynamic sti ness curve of the DMF is obtained by the geometric drawing method. e parameters of the model established in the previous section are identi ed and validated.  Figures 8 and 9. Properties of the equipment are as follows: torque range is 0∼1000 N·m, torsion frequency range is 0∼100 Hz, and torsion angle can be 0∼50°. e DMF parameters in this paper are shown in Table 1. e second mass of the DMF is connected to the xed bracket through the clamp. e rst mass is connected with the torsion actuator. e test bench is equipped with an angular displacement sensor and a torque sensor. e angular displacement and the torque signal are collected in real time by using the data acquisition device. e torsion characteristic curve of the DMF is obtained. en the data are processed to obtain a dynamic mass characteristic curve of the DMF.

Test Loading Scheme.
e loading scheme of the DMF dynamic sti ness test is as follows: (i) Dynamic sti ness excitation can be expressed by the following formula: where Θ is the pretwisted angle and θ 0 is the magnitude of the excitation torsion angle.

Shock and Vibration
(ii) e actuator applied static load Θ � 20°to the DMF, and then dynamic load θ 0 sin ωt is applied, where θ 0 � 2.5°and ω � 1 Hz (iii) e angle and torque data were recorded by using a data acquisition device (iv) e dynamic stiffness of the corresponding excitation is calculated by the geometric mapping method (v) Change the excitation frequency and repeat steps 1-4 to get dynamic stiffness at different frequencies (vi) Change θ 0 � 3.5°or θ 0 � 5°and repeat steps 1-5 to get dynamic stiffness at different magnitude of the excitation angle e detailed test combinations are shown in Table 2.

Dynamic Torsion Performance of DMF.
e dynamic torsion characteristics of DMF are mainly decided by using the complex stiffness k T , dynamic stiffness k d , and lag angle ψ. Referring to the standard of SAE (1085a), the complex stiffness k T is the ratio between the transfer torque and the rotation angle of the DMF, which is the vector sum of the elastic component (dynamic stiffness k d ) and the damping component (C ω ). k d is the projection of k T in the direction of rotation angle. ψ is the angle between complex stiffness k T and the direction rotation angle. Figure 10 is a geometric drawing method for calculating the transfer complex stiffness, dynamic stiffness, and damping lag angle [18][19][20]. e specific calculation method is as follows: where T 0 is the torque amplitude delivered to the second mass and θ 0 is the magnitude of the excitation torsion angle: where S 1 is the area occupied by the ellipse in Figure 11 and S 2 is one-half of the area of the rectangle ABCD in Figure 11.

Model Parameter Identification and
Verification. e parameters of the DMF dynamic characteristics model in this paper can be divided into two categories. One has clear physical meaning and can be given directly. ey are shown  Figure 8: Structure diagram of the test bench. Figure 9: DMF test bench.

DMF
in Table 3. e other kind of parameters should be identified by the model parameter identification method because it has no clear physical meaning.
It should be noted that the effect of centrifugal force on DMF is not taken into account in the dynamic torsion test of DMF. It is because that the test equipment cannot be dynamically loaded under the high-speed rotation condition. Only the dynamic loading at 0 rpm can be performed, so the parameter c ef2 � 0.
In the process of parameter identification, the optimization objective is set as the sum of the relative error values of the dynamic characteristic curve before and after optimization. e objective function is where x is the optimization variable, k de and k ds are the dynamic stiffness values obtained by experiment and simulation, respectively, and ψ de and ψ ds are the lag angle values obtained by experiment and simulation, respectively. Optimization variable is defined as In this paper, the Matlab genetic algorithm toolbox is used for parameter identification. e results are shown in Table 4.
e experimental results are compared as shown in Figures 12-14. e comparison of dynamic stiffness and lag angle results of the model with the results of the test is shown in Figures 12-14. e dynamic stiffness and lag angle results of the model and test match well. us, the reliability of the model can be verified. Figures 12-14 show that the resonance of DMF occurs near 10 Hz. e dynamic stiffness of the DMF is the smallest, and the damping angle is 90°. e idle speed of an automobile engine is generally 750 rpm, and the corresponding ignition frequency of the engine is 25 Hz, which far exceeds the resonance frequency of the DMF, so the powertrain does not resonate during the normal working speed of the vehicle. But the engine must pass through the resonance speed of the DMF in the ignition process, and resonance will occur at this time; due to the short ignition time, the problem of NVH will not be serious. At the same time, the frictional damping generated by using the friction disc will greatly attenuate the torsional vibration of the DMF resonance.

Model Error Analysis.
In this section, the error is reflected by the difference between the calculated value and the experimental value of the model. e curve of the model error with frequency is shown in Figures 15-17.
As shown in Figures 15-17, the errors of dynamic stiffness and lag angle of the model increase first and then decrease with the increase of frequency. Besides, the maximum error occurs near the resonance frequency. When the excitation frequency is greater than the natural frequency, the model error decreases rapidly as the frequency increases. e average error percentages of dynamic stiffness and lag angle are 18.8% and 10.3%, respectively. It is found that when the DMF is near the resonance frequency point, the minimum dynamic stiffness value is only 1.3 N·m/°, while the calculation result of the model is 2.08 N·m/°, and the error is as high as 60%. is is the reason for the larger average error. e resonance state is considered as abnormal working state, and the normal working frequency of the DMF is generally greater than 20 Hz. erefore, only the percentage of the model error above 20 Hz is calculated, which is 4.97% and 1.19%, respectively. e accuracy of the model meets the engineering requirements. e reason for the large error of resonance frequency may be that the arc spring is replaced by equivalent mass.

DMF Dynamic Characteristics Test Results
In this section, frequency dependency characteristics and amplitude dependency characteristics of DMF are experimentally studied by comparing different pretwisted angles and excitation angles amplitude (Table 5).

Same Pretwisted Angle Θ, Different Excitation
Amplitudes θ 0 . Figures 18 and 19 show the influence of excitation frequency on dynamic stiffness k d and lag angle ψ of DMF under different excitation amplitudes when the pretwisted angle Θ is constant. It can be seen that the dynamic stiffness k d and lag angle ψ of the DMF are greatly affected by the  e dynamic sti ness k d tends to decrease rst and then increase with the increase of the excitation frequency and reaches a minimum near 9 Hz. Because the resonance frequency of the DMF is about 10 Hz, the dynamic sti ness reaches the minimum at this time. When the pretwist is the same, the change of the excitation amplitude θ 0 causes the dynamic sti ness k d characteristic curve shift to the left. erefore, when the excitation frequency is less than 10 Hz, the dynamic sti ness k d decreases with the increase of excitation amplitude θ 0 . When the excitation frequency is greater than 10 Hz, the dynamic sti ness k d increases with the increase of the excitation amplitude θ 0 . But in general, the change in the excitation amplitude θ 0 has less e ect on the dynamic sti ness k d .    Shock and Vibration e lag angle ψ increases as the excitation frequency increases, reaching 90 degrees near 10 Hz. When the excitation frequency exceeds 20 Hz, the lag angle ψ tends to a stable value. When the excitation amplitude θ 0 increases, the characteristic curve of the lag angle ψ shifts to the left. e frequency value corresponding to the 90°value becomes smaller, and the stability value of the lag angle ψ also becomes larger. Shock and Vibration

Same Excitation Amplitude θ 0 , Di erent Pretwisted
Θ. Figures 20-22 show the in uence of the excitation frequency on the dynamic sti ness k d and the lag angle ψ of the DMF. At di erent pretwisted angles Θ when the excitation amplitude θ 0 is constant. It can be seen from the gure that the dynamic sti ness k d of the DMF decreases with the increase of the pretwisted angle Θ under the same excitation amplitude θ 0 , especially when the excitation frequency is greater than 10 Hz. Under the same excitation amplitude θ 0 , the lag angle ψ also increases as the pretwisted angle Θ increases, and the stable value of the lag angle ψ decreases as the pretwisted angle Θ increases. e torsional characteristics test results of the DMF show that the dynamic torsional characteristics have obvious nonlinear characteristics, mainly in the aspects of frequency correlation and amplitude correlation. e torsional characteristics of the DMF determine its damping performance. erefore, frequency variation and variation characteristics need to be fully considered in the process of structure design and parameter matching of the DMF.

Conclusion
In this paper, theoretical modeling and experimental verication of dynamic torsion characteristics of DMF are carried out: (1) Based on the in uence of the spring frictional torque and other factors on the dynamic torsion characteristics of DMF, the nonlinear dynamic torsional vibration model of the DMF is established by using the discrete element method, equivalent energy method, and fractional derivative. e model is veri ed by experiments, and the results show that the   accuracy of the model meets the engineering requirements.
(2) rough the error analysis of the nonlinear dynamic characteristic model of the DMF, the maximum error occurs near the resonance frequency. However, the result is not important, considering that the resonance state is abnormal and the working frequency of the DMF is more than 20 Hz. e model error of the common working frequency is less than 5%, which meets the engineering requirements. (3) rough the experimental study on the dynamic torsional characteristics of the DMF, it is found that the dynamic sti ness and the lag angle have signi cant nonlinear characteristics, and its performance is related to the excitation frequency and pretwisted angle. It is necessary to consider the e ect of nonlinear torsional characteristics on the damping performance in the process of structure design and parameter matching. Frictional torque generated by radial component T f2 : Frictional torque generated by centrifugal force T f : Total frictional torque T f1 and T f2 of arc spring T fri : Frictional torque of the friction disc c, c e1 , c eq : e equivalent damping coe cient R: Arc spring distribution radius θ 0 : Excitation amplitude ω: Excitation frequency T s : e recovery torque of the spring T d : e viscous damping torque b: e viscoelastic torque coe cient β: Order derivative k d : Dynamic sti ness ψ: Lag angle Θ: Pretwist.
Data Availability e data used to support the ndings of this study are included within the article.

Conflicts of Interest
e authors declare that there are no con icts of interest regarding the publication of this paper.