# Modeling and Dynamics Analysis of a Dual-Mass Flywheel with the Conformal Contact Action of Friction Damping Ring and Pressure Plate

Authors: Liping Zeng; Jie Huang; Yukun Xu; Liquan Song
Journal: Shock and Vibration
 (2021)

---

## Abstract

To study the nonlinear dynamic characteristics of the dual-mass flywheel (DMF) under the conformal contact action between the friction damping ring and primary flywheel pressure plate, the contact action model is established and analyzed based on Winkler model. Through analysis and calculation, the contact deformation, contact pressure at different contact positions, and equivalent torsional contact stiffness are obtained. The nonlinear dynamic analysis model of three-degree-of-freedom (3DOF) which takes the conformal contact into account and two-degree-of-freedom (2DOF) without considering conformal contact is established. The approximate analytical solution of the nonlinear frequency characteristics of the system at steady state is derived. By comparing with the results obtained from numerical method, the theoretical analysis process is proved to be valid. And it is found that the overall amplitude and angular displacement transmissibility of the 3DOF model are smaller than the 2DOF model, especially at resonance frequency. The effects of the friction damping ring moment inertia, stiffness of DMF, and axial friction torque on the frequency characteristics of system and angular displacement transmissibility are analyzed. The forced vibration response analysis of the 3DOF model is conducted, through which the torsional angle variations of the primary flywheel, friction damping ring, and secondary flywheel with time are obtained. The results show that the amplitude of the secondary flywheel is much smaller than that of the primary flywheel, indicating that the DMF has prominent damping performance.


---

## Body

## 1. Introduction

Dual-mass flywheel (DMF), which was first proposed by companies represented by LuK, can efficiently reduce the torsional vibration and noise, relieve impact, realize the protection of engine and gearbox against overload, and enable the engine to operate at lower speeds to reduce fuel consumption [1–4]. Generally speaking, through reasonable parameter matching, DMF can make the natural frequency of the automobile transmission system isolated outside the frequency corresponding to the normal speed range of the engine [5]. Owing to these significant advantages, now DMF is wildly applied in automotive transmission system [6, 7], and there are many academic researches on the DMF. Berbyuk [8] studied the feasibility of the application of a DMF for heavy-duty truck drivetrain system. And the optimized mass inertia, stiffness, and damping parameters of a DMF were proved to provide the better attenuation of the torque oscillations. To make a DMF with three-stage stiffness have continuous variable stiffness, Song et al. [9] proposed a new structure improvement by using shape constraints and compensation principle. References [2, 10] focused on the measure method of the DMF angular displacement. Tang et al. [11] investigated the effect of a DMF on the torsional vibration characteristics of a power-split hybrid powertrain system. Theodossiades et al. [4] investigated the effect of DMF on impact-induced clonk noise and its severity through experimentation. Chen et al. [12] analyzed the dynamic torsional characteristics of a circumferential arc spring DMF from theoretical study and experiments. And it was found that the torsional stiffness of the DMF varies with the excitation amplitude and frequency. Güllü et al. [13] proved that DMF can make the clutching process perform more smoothly than single-mass flywheel. Li et al. [14] proposed a radial spring DMF with good nonlinear torsional stiffness characteristics owing to the moment inertia of the torsional vibration damper allocated reasonably and the torsional stiffness of the torsional vibration spring designed properly. Mao et al. [15] and Zu et al. [16] established the simulation model of magnetorheological DMF in Amesim software and analyzed the attenuation of torsional vibration of transmission system under various working conditions. Chen et al. [17] built the multidegree powertrain torsional vibration model which contains magnetorheological fluid DMF and semiactive fuzzy control model and researched the torsional vibration damping characteristic of the DMF and the control system in vehicle powertrain system.To meet the requirements of small torsional angle and stiffness at low speed and low torque, and large stiffness and output torque at high speed and high input torque, DMFs generally possess nonlinear stiffness characteristics, i.e., its stiffness will change with the relative angular displacement of the primary flywheel and the secondary flywheel. Song et al. [18] proposed the design theory of a two-stage piecewise variable stiffness DMF, and the torque characteristics and frequency characteristics are analyzed by theoretical study and experiments. Wang et al. [19] investigate the kinetic parameters matching and designing method for a multistage torsional stiffness DMF based on torsional vibration control by using the numerical analysis methods and proved by tests on an actual vehicle. Shi et al. [20] proposed an innovative arrangement of elastic structure applied in DMF to make this vibration damper have multipiece step stiffness and conducted the natural characteristic analysis of the transmission system at driving and idling conditions to evaluate this DMF vibration control effects. For the DMF matching under start-up condition, He et al. [21] investigated the influence of a five-stage variable stiffness DMF kinetic parameter on driveline torsional vibration in engine start-up process.In Ref [18], a friction damping ring structure is introduced into DMF to further reduce torsional vibration based on the equivalent damping effect of the axial friction torque produced by axial preload. According to the DMF structure, high conformal contact is formed between the introduced friction damping ring and the primary flywheel pressure plate. Usually, contact problem is a complex process involving elastic deformation and other factors, and usually, multibody system dynamics problems involving contact have nonlinear characteristics [22]. The existing literature has carried out a lot of research on the structure of the DMF, torsional characteristics, and vibration damping performance. But few studies consider the influence of inner part contact on the dynamic characteristics of DMF in the previous works.The present work is devoted to study the nonlinear dynamic characteristics of DMF with considering the conformal contact action between the friction damping ring and the primary flywheel pressure plate. Through establishing the contact analysis model based on Winkler model, the contact deformation and contact pressure at different contact positions, and the equivalent torsional contact stiffness generated by the conformal contact can be obtained. Then, the nonlinear dynamic analysis model of the DMF is established, and the approximate analytical solution of the nonlinear frequency characteristics of the system at steady state is solved by using averaging method. To verify the validity of the solution process, the results are compared with the system steady-state response amplitude obtained by applying the fourth-order Runge–Kutta numerical method with fixed step. The steady-state frequency characteristics and angular displacement transmissibility of the three-degree-of-freedom (3DOF) model which takes conformal contact into account are compared with the two-degree-of-freedom (2DOF) model without considering conformal contact. The effects of moment inertia ratio, DMF stiffness, and axial friction torque on the frequency characteristics of system dynamic response and angular displacement transmissibility are demonstrated.


## 2. DMF Structure

Figure1 shows the schematic diagram of the DMF structure, which is mainly composed of six parts. The spring 3 used for vibration reduction is fixed between the two flanges of the primary flywheel 1 through two spring seats 2, and the friction damping ring 5 is placed between the primary flywheel pressure plate 6 and the secondary flywheel 4. In the working process of the DMF, the primary flywheel connected with the engine crankshaft is driven to rotate. The primary flywheel compresses the damping springs installed between the flanges, and the spring seats move the secondary flywheel. In addition, the axial friction torque between the primary flywheel and the secondary flywheel is generated by the preload applied in the axial direction of the friction damping ring.Figure 1
Structural diagram of the DMF: 1: primary flywheel; 2: spring seat; 3: spring; 4: secondary flywheel; 5: friction damping ring; 6: primary flywheel pressure plate.


## 3. Conformal Contact Action of the Friction Damping Ring and Pressure Plate

### 3.1. Winkler Model

Contact between parts in multibody system is a common phenomenon. The analysis and solution of the contact problems generally involve many factors such as friction theory, dynamics, elastic deformation, and calculation method. And now it has been a topic of interest to researchers. As illustrated by Corral in [23], now there are many contact models, such as Hertz model, Hooke’s model, and Kelvin–Voigt model, but they generally have specific applicable characteristics. The Winkler model can be well applied to the contact pressure calculation of high conformal contact problems and wear simulation, which can avoid the complexity of contact pressure calculation [24, 25]. For example, Su et al. [26] used Winkler model to calculate joint reaction and contact pressure, and the result shows that the contact pressure calculated by this model has a better accuracy than the result computed by Hertz contact theory. Mukras et al. [27] found that Winkler contact model can consider the influence of the shape and size of the hinge joint better than other models when analyzing the wear of hinged joints with clearance. Zhu et al. [25] proposed a NLCP model with considering the nonlinear relationship between contact force and contact deformation based on Winkler contact model when analyzing the wear of hinged joints with clearance.The Winkler model ignores the shearing action on the contact surface, and the contact interface is modeled as a series of springs. As shown in Figure2, the relationship between contact pressure and contact deformation (penetration depth) is described as follows:(1)p=E∗δL,where E∗ is the synthesis elastic modulus of the two contacting objects, δ is the deformation, and L is the length before deformation.Figure 2
Winkler model.


### 3.2. Contact Deformation

Figure3(a) shows the initial state before the deformation; the primary flywheel pressure plate 6 and the friction damping ring 5, and the friction damping ring 5 and the secondary flywheel 4 are face-to-face contacts. We establish a rectangular coordinate system Oxy with the rotation center O of the DMF as the origin point. The contact surface of the pressure plate and the friction damping ring is an arc with O1 as the center, and the eccentric distance lOO1 = e. The contact surface of the friction damping ring and the secondary flywheel is an arc with O point as the center, and the interaction between the two can be approximated as no deformation.Figure 3
Contact deformation of the friction damping ring. (a) Initial state. (b) Contact deformation after a relative angleθ.
(a)(b)It is assumed that pointC is a contact point on the contact surface between the friction damping ring and the primary flywheel pressure plate when the DMF works as shown in Figure 3(b). Under torque action, the angular displacement of the pressure plate relative to the friction damping ring is θ, and the interaction line between the pressure plate and the friction damping ring moves to the arc with O2 as the center (solid line II) from the arc with O1 as the center (dotted line I). At this time, the elastic deformation of the point C at the angle of α with the y-axis is lCD = δα.The arc I takesO1 as the center of the circle, and the coordinates before the deformation of the contact point C are (rsinα, e + rcosα). The equation of the straight line O1C is y = xcotα+e. Thus, the equations of arc I and II are as follows:(2)Ι:x2+y−e2=r2,ΙΙ:x−esinθ2+y−ecosθ2=r2.The coordinates of pointD can be obtained by substituting the equation of line O1C into the equation of arc II. Then, the length of CD is obtained:(3)δα=r2−e2sinα−θ−sinα2+ecosα−θ−cosα−r.In order to simplify the analysis and calculation, equation (3) is regarded as a function of the independent variable α, and the first three terms are expanded by Taylor series at α = 0 for approximate calculation:(4)δα≈θesinα−θ22ecosα+er≈θesinα.For the actual structure, in equations (3) and (4), e = 16 mm, r = 60 mm. When θ = 1° and the angle of contact area αmax is 50°, the approximate deformation and the actual deformation are shown in Figure 4. The small difference between the approximate deformation and the actual deformation indicates that this approximate equivalence is reasonable.Figure 4
Contact deformation at different positions.The outer arc equation of the friction damping ring is as follows:x2 + y2 = R2. H is the intersection point of line O1C and the outer circular arc of the friction damping ring. The length of O1H can be calculated and expressed as(5)lO1H=xHsinα=R2−e2sin2α−ecosα.Normal thickness of the friction damping ring at pointC before deformation is as follows:(6)Lα=lCH=lO1H−r=R2−e2sin2α−r+ecosα.


### 3.3. Torque Generated by Conformal Contact Action

As shown in Figure5(a), suppose the contact pressure at point D on the contact surface is pα and the unit contact area is A, the contact force Fα at the contact point can be expressed as(7)Fα=pαA=E∗δαLαrhdα,where dα is a small arc degree and h is axial contact length.Figure 5
Contact force analysis. (a) Contact force. (b) Contact stiffness with different contact angleαmax.
(a)(b)The torque generated by forceFα is as follows:(8)Tα=FαlOE=Fαesinα.According to equations (1)–(4) and (6)–(8), the total torque produced by the contact action of the friction damping ring and the primary flywheel pressure plate can be obtained:(9)Td=∫Tα=Kcθ,where Kc is the conformal contact stiffness between the friction damping ring and the primary flywheel pressure plate, and it is expressed as(10)Kc=E∗rhe2∫sin2αR2−e2sin2α−r+ecosαdα.The relationship between the contact stiffness and the angle of the contact regionαmax is shown in Figure 5(b). It can be seen that Kc increases nonlinearly with the increase of the contact angle αmax, and Kc is about 20000N·m/rad when αmax = 50°.


## 3.1. Winkler Model

Contact between parts in multibody system is a common phenomenon. The analysis and solution of the contact problems generally involve many factors such as friction theory, dynamics, elastic deformation, and calculation method. And now it has been a topic of interest to researchers. As illustrated by Corral in [23], now there are many contact models, such as Hertz model, Hooke’s model, and Kelvin–Voigt model, but they generally have specific applicable characteristics. The Winkler model can be well applied to the contact pressure calculation of high conformal contact problems and wear simulation, which can avoid the complexity of contact pressure calculation [24, 25]. For example, Su et al. [26] used Winkler model to calculate joint reaction and contact pressure, and the result shows that the contact pressure calculated by this model has a better accuracy than the result computed by Hertz contact theory. Mukras et al. [27] found that Winkler contact model can consider the influence of the shape and size of the hinge joint better than other models when analyzing the wear of hinged joints with clearance. Zhu et al. [25] proposed a NLCP model with considering the nonlinear relationship between contact force and contact deformation based on Winkler contact model when analyzing the wear of hinged joints with clearance.The Winkler model ignores the shearing action on the contact surface, and the contact interface is modeled as a series of springs. As shown in Figure2, the relationship between contact pressure and contact deformation (penetration depth) is described as follows:(1)p=E∗δL,where E∗ is the synthesis elastic modulus of the two contacting objects, δ is the deformation, and L is the length before deformation.Figure 2
Winkler model.


## 3.2. Contact Deformation

Figure3(a) shows the initial state before the deformation; the primary flywheel pressure plate 6 and the friction damping ring 5, and the friction damping ring 5 and the secondary flywheel 4 are face-to-face contacts. We establish a rectangular coordinate system Oxy with the rotation center O of the DMF as the origin point. The contact surface of the pressure plate and the friction damping ring is an arc with O1 as the center, and the eccentric distance lOO1 = e. The contact surface of the friction damping ring and the secondary flywheel is an arc with O point as the center, and the interaction between the two can be approximated as no deformation.Figure 3
Contact deformation of the friction damping ring. (a) Initial state. (b) Contact deformation after a relative angleθ.
(a)(b)It is assumed that pointC is a contact point on the contact surface between the friction damping ring and the primary flywheel pressure plate when the DMF works as shown in Figure 3(b). Under torque action, the angular displacement of the pressure plate relative to the friction damping ring is θ, and the interaction line between the pressure plate and the friction damping ring moves to the arc with O2 as the center (solid line II) from the arc with O1 as the center (dotted line I). At this time, the elastic deformation of the point C at the angle of α with the y-axis is lCD = δα.The arc I takesO1 as the center of the circle, and the coordinates before the deformation of the contact point C are (rsinα, e + rcosα). The equation of the straight line O1C is y = xcotα+e. Thus, the equations of arc I and II are as follows:(2)Ι:x2+y−e2=r2,ΙΙ:x−esinθ2+y−ecosθ2=r2.The coordinates of pointD can be obtained by substituting the equation of line O1C into the equation of arc II. Then, the length of CD is obtained:(3)δα=r2−e2sinα−θ−sinα2+ecosα−θ−cosα−r.In order to simplify the analysis and calculation, equation (3) is regarded as a function of the independent variable α, and the first three terms are expanded by Taylor series at α = 0 for approximate calculation:(4)δα≈θesinα−θ22ecosα+er≈θesinα.For the actual structure, in equations (3) and (4), e = 16 mm, r = 60 mm. When θ = 1° and the angle of contact area αmax is 50°, the approximate deformation and the actual deformation are shown in Figure 4. The small difference between the approximate deformation and the actual deformation indicates that this approximate equivalence is reasonable.Figure 4
Contact deformation at different positions.The outer arc equation of the friction damping ring is as follows:x2 + y2 = R2. H is the intersection point of line O1C and the outer circular arc of the friction damping ring. The length of O1H can be calculated and expressed as(5)lO1H=xHsinα=R2−e2sin2α−ecosα.Normal thickness of the friction damping ring at pointC before deformation is as follows:(6)Lα=lCH=lO1H−r=R2−e2sin2α−r+ecosα.


## 3.3. Torque Generated by Conformal Contact Action

As shown in Figure5(a), suppose the contact pressure at point D on the contact surface is pα and the unit contact area is A, the contact force Fα at the contact point can be expressed as(7)Fα=pαA=E∗δαLαrhdα,where dα is a small arc degree and h is axial contact length.Figure 5
Contact force analysis. (a) Contact force. (b) Contact stiffness with different contact angleαmax.
(a)(b)The torque generated by forceFα is as follows:(8)Tα=FαlOE=Fαesinα.According to equations (1)–(4) and (6)–(8), the total torque produced by the contact action of the friction damping ring and the primary flywheel pressure plate can be obtained:(9)Td=∫Tα=Kcθ,where Kc is the conformal contact stiffness between the friction damping ring and the primary flywheel pressure plate, and it is expressed as(10)Kc=E∗rhe2∫sin2αR2−e2sin2α−r+ecosαdα.The relationship between the contact stiffness and the angle of the contact regionαmax is shown in Figure 5(b). It can be seen that Kc increases nonlinearly with the increase of the contact angle αmax, and Kc is about 20000N·m/rad when αmax = 50°.


## 4. Dynamic Analysis Model of DMF

### 4.1. Dynamic Model

For the cyclical change of the engine cylinder pressure and the inertial force generated by the reciprocating motion of the crank link mechanism in internal combustion engines, the output torque from the engine changes continuously, which can be generally expressed as [19, 28]:(11)Te=T0+∑γ=0.5∞Tγsinγωt+ψr,where T0 is the average torque caused by cylinder pressure in a cycle, γ represents the harmonic number, γ = 1/2, 1, 3/2, 2, …, Tγ is the amplitude of torque at harmonic number γ, ω is the angular velocity of crankshaft, and ψγ is the initial phase at harmonic number γ. Assume the external excitation on the primary flywheel is in a harmonic change form: Te = Tpsinωt.Compared with the DMF, the moment inertia of vehicle subsystem behind clutch-driven plate and wheel is very large. Therefore, the wheel and vehicle subsystem is assumed to be rigid [29, 30]. As shown in Figure 6(a), 3DOF torsional vibration analysis model is established to study the DMF dynamic characteristics according to vehicle powertrain system, where θ1 is the angular displacement of the primary flywheel, θ2 is the angular displacement of the friction damping ring, θ3 is the angular displacement of the secondary flywheel, J1 is the equivalent moment inertia of the primary flywheel and its front parts, J2 is the moment inertia of the friction damping ring around the rotation center O, J3 is the equivalent moment inertia of the secondary flywheel and clutch-driven plate, cs is the damping between the primary flywheel and the secondary flywheel, kg and cg are the equivalent stiffness and damping associated with the automotive driveline, Ts is the torque acting through the springs between the primary flywheel and the secondary flywheel, Td(θ1, θ2) is the contact torque of the friction damping ring and the primary flywheel pressure plate, and Mf is the axial friction torque between the single friction damping ring (there are three uniformly arranged friction damping rings in total) and second flywheel, which is produced by the axial preload. In addition, the 2DOF model shown in Figure 6(b) is also established without considering the contact influence between the friction damping ring and the primary flywheel pressure plate.Figure 6
DMF analysis model. (a) 3DOF model. (b) 2DOF model. (c) Piecewise variable stiffness of the damping spring.
(a)(b)(c)As shown in Figure6(c), the stiffness of the DMF is divided into two stages: k1 at small angular displacements and k2 at large angular displacements, namely(12)k=k1,if−β≤θ≤β,k2,else,where θ = θ1 − θ3, β is the relative angular displacement when the stiffness begins to change.Then, the torqueTs produced by the damping springs between the primary flywheel and the secondary flywheel is expressed as(13)Ts=k1θ,if−β≤θ≤β,k2θ−β+k1β,ifθ>β,k2θ+β−k1β,else..Supposeθ has the form θ = AcosΘ under the engine sinusoidal torque excitation, where A is amplitude, Θ = ωt + ψ, ψ is the phase angle.WhenA≥β, Ts is expanded by Fourier series as(14)Ts=k1+1πk2−k12φ0−sin2φ0cosΘ+1πk2−k1×∑m=1∞1msin2mΘ0−1m+1sin2m+2Θ0cos2m+1Θ2m+1A,where Θ0=arccosβ/A.LetTs=k1+1/πk2−k12Θ0−sin2Θ0AcosΘ and ke=k1+1/πk2−k12Θ0−sin2Θ0.The dynamic equations of the 3DOF system are illustrated as(15)J1θ¨1+3Kcθ1−θ2+keθ1−θ3+csθ˙1−θ˙3=TpsinωtJ2θ¨2+Mfsgnθ˙2−θ˙3=Kcθ1−θ2J3θ¨3+kgθ3+cgθ˙3=keθ1−θ3+csθ˙1−θ˙3+3Mfsgnθ˙2−θ˙3,where(16)ke=k1,ifA<β,k1+1πk2−k12Θ0−sin2Θ0,ifA≥β,The dynamic equations of the 2DOF system are given as(17)Jθ¨1+keθ1−θ3+csθ˙1−θ˙3=Tpsinωt−3Mfsgnθ˙1−θ˙3,J3θ¨3+kgθ3+cgθ˙3=keθ1−θ3+csθ˙1−θ˙3+3Mfsgnθ˙1−θ˙3,where J = J1 + 3J2.


### 4.2. Approximate Analytical Solution of the Nonlinear Dynamics Equations

It is known that approximate analytical or semianalytical solutions can be formulated by using the methods of either Krylov–Bogoliubov, Galerkin, Ritz, averaging, harmonic balance, or multiple scales. The method of averaging is valid in the first approximation and can be used to examine the stability behavior. The main idea of the method of averaging is that the derivatives of the amplitude and initial phase of the nonlinear vibration system are regarded as a slowly varying function of time and replaced by the average value of one cycle.In this work, the method of averaging is used to solve the nonlinear dynamics equations. According to the nonlinear vibration analysis of multiple degrees-of-freedom systems in reference [31], the damping force, axial friction torque, and input excitation in equation (15) are multiplied by a small parameter ε; then, equation (15) is deformed as follows:(18)θ¨1=−1J13Kc+keθ1+31J1Kcθ2+1J1keθ3+ε1J1Tpsinωt−csθ˙1−θ˙3,θ¨2=1J2Kcθ1−1J2Kcθ2−ε1J2Mfsgnθ˙2−θ˙3,θ¨3=1J3keθ1−1J3ke+kgθ3+ε1J3csθ˙1−θ˙3−cgθ˙3+3Mfsgnθ˙2−θ˙3.Letθ1=x1,θ˙1=x2,θ2=x3,θ˙2=x4,θ3=x5,θ˙3=x6, a1=3Kc+ke/J1, a2=3Kc/J1, a3=ke/J1, a4=Kc/J2, a5=ke/J3, a6=ke+kg/J3, and when ε = 0, equation (18) can be written as(19)x˙1=x2,x˙2=−a1x1+a2x3+a3x5,x˙3=x4,x˙4=a4x1−a4x3,x˙5=x6,x˙6=a5x1−a6x5.From equation (19), the natural frequency ωni(i = 1∼3) can be determined by the following formula(20)a1−ωni2a4−ωni2a6−ωni2−a2a4a6−ωni2−a3a5a4−ωni2=0.Since the parametersa1, a3, a5, and a6 contain ke, the natural frequency ωni is also a nonlinear function of Ai. Taking the following transformation,(21)xs=∑i=13AiGsiφi,s=1,2,3,4,5,6,where Ai is amplitude and φi is phase, and both are time functions, and Gsi is a transformation function.Inserting equation (21) into equation (18), we get(22)∑i=13dAidtGsiφi−∑i=13AiGsi∗φidφidt−ωni=εFs,where(23)F1=0,F2=Tpsinωt−csθ˙1−θ˙3J1,F3=0,F4=−MfJ2sgnθ˙2−θ˙3,F5=0,F6=csθ˙1−θ˙3−cgθ˙3+3Mfsgnθ˙2−θ˙3J3,.A set of fundamental solution of equation (19) is given as(24)G1i=cosωnit,G1i∗=sinωnitG2i=−ωnisinωnit,G2i∗=ωnicosωnitG3i=a4a4−ωni2cosωnit,G3i∗=a4a4−ωni2sinωnitG4i=−a4ωnia4−ωni2sinωnit,G4i∗=a4ωnia4−ωni2cosωnitG5i=a5a6−ωni2cosωnit,G5i∗=a5a6−ωni2sinωnitG6i=−a5ωnia6−ωni2sinωnit,G6i∗=a5ωnia6−ωni2cosωnit.The adjoint equation of equation (19) is as follows:(25)X˙1=a1X2−a4X4−a5X6,X˙2=−X1,X˙3=−a2X2+a4X4,X˙4=−X3,X˙5=−a3X2+a6X6,X˙6=−X5.Also, a set of fundamental solutions of equation (25) can be obtained as follows:(26)H1i=cosωnit,H1i∗=sinλitH2i=−1ωnisinωnit,H2i∗=1ωnicosωnitH3i=a2a4−ωni2cosωnit,H3i∗=a2a4−ωni2sinωnitH4i=−a2a4−ωni21ωnisinωnit,H4i∗=−a2a4−ωni21ωnicosωnitH5i=a3a6−ωni2cosωnit,H5i∗=a3a6−ωni2sinωnitH6i=−a3a6−ωni21ωnisinωnit,H6i∗=−a3a6−ωni21ωnicosωnit.There is a relationship between formulas (24) and (26):(27)∑sG∗siφiHsi∗φi=∑sGsiφiHsiφi=1+a2a4a4−ωni22+a3a5a6−ωni22=Δi.Multiplying equation (22) by Hsi(φi) and Hsi∗φi, respectively, we obtain(28)dAidt=εΔi∑sFsHsiφi=εϕiAi,φi,dφidt=ωni−εΔiAi∑sFsHsi∗φi=ωni−εϕi∗Ai,φi.ForAi and φi, we take the following transformation:(29)Ai=yi+εUit,yi,ϑi,φi=ωt+ϑi+εVit,yi,ϑi.When studying the system main resonance vibration, takeλi–ω as the same order infinitesimal as ε, and require the derivative of new variables yi, ϑi as follows:(30)dyidt=εYiyi,ϑi+ε2Yi∗t,yi,ϑi,ε,dϑidt=εZiyi,ϑi+ε2Zi∗t,yi,ϑi,ε.For equations (29) and (30), Yi and Zi do not contain periodic functions of time t, and Ui, Vi, Yi∗, and Zi∗ are the functions of ϑi and t with period 2π.Substituting equation (30) into equation (29) and letting the first-order coefficient of ε be equal, we get(31)Yi+dUidt=ϕi,Zi+dVidt=−ϕi∗.In order to make the functionsYi and Yi not contain time t, put(32)Yi=12π∫02πϕidψ,Zi=ωni−ω−12π∫02πϕi∗dψ.In equation (30), we take the first-order square coefficient of ε, and let the right end be equal to zero. Because the first-order resonant amplitude is much larger than the second-order and third-order amplitudes, the first-order resonant frequency characteristics are mainly analyzed here. The first-order approximate analytical solution is set as θ1=A1cosωt+φ1, θ2=a4/a4−ωn12A1cosωt+φ1, θ3=a5/a6−ωn12A1cosωt+φ1. After a series of calculation, the following function is obtained:(33)dA1dt=ε−12J1ωn1Δ1Tpcosφ1−12πΔ1S1A1+S2,dφ1dt=ωn1−ω+ε12J1ωn1Δ1A1Tpsinφ1,where(34)S1=a3J3a6−ωn12−1J1a5a6−ωn12−1csπ+a3a5J3a6−ωn122cgπ,S2=3a3J3a6−ωn12ωn1−a2J2a4−ωn12ωn14Mfsgna5a6−ωn12−a4a4−ωn12.The steady-state solution can be found by settingdA/dt=0,dφ/dt=0. The relationship between the system excitation frequency ω and the amplitude A1 and phase φ1 can be obtained.(35)ω=ωn12−2δeA12±TpΔ1J1A12+4δeA12δeA12−ωn12,φ1=arctanω2−ωn12A12δeω.where(36)δe=−12πΔ1S1A1+S2.For the 2DOF model, according to equation (17), the first-order natural frequency of the corresponding linear systems can be given by:(37)ωn1′=12b1+b3−b1−b32+4b1b2,where b1=ke/J,b2=ke/J3,b3=ke+kg/J3.Similarly, the first approximate analytical solution of the 2DOF model system can be written asθ1=A1′sinωt+φ1′, θ3=b1−ωn1′2/b1A1′cosωt+φ1′, and the relationship between ω and amplitude A1′, and phase φ1′ in the first approximate analytical solution are deduced (38)ω=ωn1′2−2δe′A1′2±TpA1′J2+4δe′A1′2δe′A1′2−ωn1′2φ1′=arctanω2−ωn1′2A1′2δe′ω,where(39)δe′=−12πΔ1′Q1A1′+Q2Q1=b1−ωn12′2b1b2J3cg−b1−ωn1′2b2J3−1Jωn1′2b1csπ,Q2=−12ωn1′b1−ωn1′2b2J3−1JMf.Equations (35) and (38) are the nonlinear relationship between the excitation frequency and amplitude and initial phase, which can be solved by numerical method to obtain the corresponding amplitude-frequency characteristics and phase-frequency characteristics.For rotor systems, angular displacement transmissibility is one of the important indexes of vibration reduction performance. The torsional angle amplitude ratio of the secondary flywheel and the primary flywheelTr and Tr′ is the angular displacement transmissibility of the 3DOF model and the 2DOF model, respectively, which can be expressed as(40)Tr=a5a6−ωn12,Tr′=b1−ωn1′2b1.We define the moment inertia ratioλ = J2/J1, and the main parameters of the DMF are as follows: r = 60 mm, R = 92.5 mm, J1 = 0.15 kg·m2, λ = 0.02, J3 = 0.055 kg·m2, l = 70.9 mm, e = 16 mm, Kc = 20000 N·m/rad, h = 10 mm, kg = 501 N·m/°, β = 16°, k1 = 9.22 N·m/°, k2 = 23.67 N·m/°, and Mf = 2 N·m. The synthesis elastic modulus E∗ can be calculated with the material properties of the primary flywheel pressure plate and friction damping ring shown in Table 1.Table 1
Material properties.
Properties/bodiesPrimary flywheel pressure plateFriction damping ringMaterialSteelPa66Elastic modulus (GPa)2068.3Poisson ratio0.30.28To verify the correctness of the solution method in Section3.2, the Runge–Kutta numerical method is applied to obtain the amplitude-frequency characteristics of the primary flywheel angular displacement of the 3DOF model and 2DOF when Tp = 100 N·m. As shown in Figure 7(a), the numerical results agree with the current analytical results well at different frequencies, which indicates that the calculation and analysis of the averaging method are accurate and valid. The commonly used speed of the vehicle engine is 800–4000 rpm, corresponding to the angular frequency range of 83.8–418.9 rad/s. From equation (20), the first-order and second-order resonant frequencies of the system are about 57 rad/s and 728 rad/s, respectively, while the third-order resonant frequency is higher, so the resonant frequency range of the system can avoid in the frequency range of the engine torque variation. In addition, through numerical analysis, it is found that the response amplitude of the second-order and third-order resonant frequencies is also much smaller than that of the first-order resonant frequency. Therefore, we mainly analyze the first-order resonant frequency characteristics without considering the second-order and third-order resonant frequency characteristics in Section 3.2.Figure 7
Amplitude-frequency characteristics of the primary flywheel angular displacement. (a) Comparison of the numerical method result and the analytical result of the 3DOF model and 2DOF model whenTp = 100 N·m. (b) Comparison of the 3DOF model and 2DOF model with different Tp.
(a)(b)The first-order resonance amplitude-frequency characteristics of the 2DOF model and 3DOF model under different excitation torque amplitudesTp are shown in Figure 7(b). It can be seen that the larger the excitation torque, the larger the overall response amplitude and the wider the resonance frequency band. Under the same excitation frequency, the amplitude of the 3DOF model is smaller than that of the 2DOF model, especially nearby the resonant frequency. The maximum value of the 3DOF model is 10.2°, 20.7°, and 28.7° when Tp = 40 N·m, 80 N·m, and 120 N·m, respectively, while the maximum value of the 2DOF model is 10.9°, 21.6°, and 30.2°. Since the 3DOF model takes into account the elastic deformation under the contact action of the structure, it is more closer to the actual situation. Next, the frequency characteristics, angular displacement transmissibility, and forced vibration of the 3DOF model are mainly analyzed.The effects of the moment inertia ratioλ, DMF stiffness ke, and contact stiffness Kc on the natural frequency of the 3DOF model can be analyzed, respectively from equation (20), as shown in Figure 8. When λ changes from 0 to 0.1, as shown in Figure 8(a), the first-order natural frequency changes from 58.8 rad/s to 51.6 rad/s, the third-order natural frequency changes from 36520.3 rad/s to 1316.8 rad/s, while the second-order natural frequency basically maintains at 729.1 rad/s. Therefore, J2 has a great influence on the first-order and third-order natural frequencies but little influence on the second-order natural frequencies. Figure 8(b) demonstrates that when the torsional stiffness ke changes from 0 N·m/° to 30 N·m/°, the first-order natural frequency ωn1 changes from 0 to 100.9 rad/s, and the second-order and third-order natural frequencies are about 729.1 rad/s and 2658.3 rad/s, respectively. Therefore, ke mainly affects the first-order natural frequency and has little influence on the second- and third-order natural frequencies. Besides, from Figure 8(c), it can be concluded that the contact stiffness Kc mainly affects the third-order natural frequency and has little effect on the first-order and second-order natural frequency.Figure 8
Influence ofλ, ke and Kc on the natural frequency of the 3DOF model. (a) Influence of λ. (b) Influence of ke. (c) Influence of Kc.
(a)(b)(c)


## 4.1. Dynamic Model

For the cyclical change of the engine cylinder pressure and the inertial force generated by the reciprocating motion of the crank link mechanism in internal combustion engines, the output torque from the engine changes continuously, which can be generally expressed as [19, 28]:(11)Te=T0+∑γ=0.5∞Tγsinγωt+ψr,where T0 is the average torque caused by cylinder pressure in a cycle, γ represents the harmonic number, γ = 1/2, 1, 3/2, 2, …, Tγ is the amplitude of torque at harmonic number γ, ω is the angular velocity of crankshaft, and ψγ is the initial phase at harmonic number γ. Assume the external excitation on the primary flywheel is in a harmonic change form: Te = Tpsinωt.Compared with the DMF, the moment inertia of vehicle subsystem behind clutch-driven plate and wheel is very large. Therefore, the wheel and vehicle subsystem is assumed to be rigid [29, 30]. As shown in Figure 6(a), 3DOF torsional vibration analysis model is established to study the DMF dynamic characteristics according to vehicle powertrain system, where θ1 is the angular displacement of the primary flywheel, θ2 is the angular displacement of the friction damping ring, θ3 is the angular displacement of the secondary flywheel, J1 is the equivalent moment inertia of the primary flywheel and its front parts, J2 is the moment inertia of the friction damping ring around the rotation center O, J3 is the equivalent moment inertia of the secondary flywheel and clutch-driven plate, cs is the damping between the primary flywheel and the secondary flywheel, kg and cg are the equivalent stiffness and damping associated with the automotive driveline, Ts is the torque acting through the springs between the primary flywheel and the secondary flywheel, Td(θ1, θ2) is the contact torque of the friction damping ring and the primary flywheel pressure plate, and Mf is the axial friction torque between the single friction damping ring (there are three uniformly arranged friction damping rings in total) and second flywheel, which is produced by the axial preload. In addition, the 2DOF model shown in Figure 6(b) is also established without considering the contact influence between the friction damping ring and the primary flywheel pressure plate.Figure 6
DMF analysis model. (a) 3DOF model. (b) 2DOF model. (c) Piecewise variable stiffness of the damping spring.
(a)(b)(c)As shown in Figure6(c), the stiffness of the DMF is divided into two stages: k1 at small angular displacements and k2 at large angular displacements, namely(12)k=k1,if−β≤θ≤β,k2,else,where θ = θ1 − θ3, β is the relative angular displacement when the stiffness begins to change.Then, the torqueTs produced by the damping springs between the primary flywheel and the secondary flywheel is expressed as(13)Ts=k1θ,if−β≤θ≤β,k2θ−β+k1β,ifθ>β,k2θ+β−k1β,else..Supposeθ has the form θ = AcosΘ under the engine sinusoidal torque excitation, where A is amplitude, Θ = ωt + ψ, ψ is the phase angle.WhenA≥β, Ts is expanded by Fourier series as(14)Ts=k1+1πk2−k12φ0−sin2φ0cosΘ+1πk2−k1×∑m=1∞1msin2mΘ0−1m+1sin2m+2Θ0cos2m+1Θ2m+1A,where Θ0=arccosβ/A.LetTs=k1+1/πk2−k12Θ0−sin2Θ0AcosΘ and ke=k1+1/πk2−k12Θ0−sin2Θ0.The dynamic equations of the 3DOF system are illustrated as(15)J1θ¨1+3Kcθ1−θ2+keθ1−θ3+csθ˙1−θ˙3=TpsinωtJ2θ¨2+Mfsgnθ˙2−θ˙3=Kcθ1−θ2J3θ¨3+kgθ3+cgθ˙3=keθ1−θ3+csθ˙1−θ˙3+3Mfsgnθ˙2−θ˙3,where(16)ke=k1,ifA<β,k1+1πk2−k12Θ0−sin2Θ0,ifA≥β,The dynamic equations of the 2DOF system are given as(17)Jθ¨1+keθ1−θ3+csθ˙1−θ˙3=Tpsinωt−3Mfsgnθ˙1−θ˙3,J3θ¨3+kgθ3+cgθ˙3=keθ1−θ3+csθ˙1−θ˙3+3Mfsgnθ˙1−θ˙3,where J = J1 + 3J2.


## 4.2. Approximate Analytical Solution of the Nonlinear Dynamics Equations

It is known that approximate analytical or semianalytical solutions can be formulated by using the methods of either Krylov–Bogoliubov, Galerkin, Ritz, averaging, harmonic balance, or multiple scales. The method of averaging is valid in the first approximation and can be used to examine the stability behavior. The main idea of the method of averaging is that the derivatives of the amplitude and initial phase of the nonlinear vibration system are regarded as a slowly varying function of time and replaced by the average value of one cycle.In this work, the method of averaging is used to solve the nonlinear dynamics equations. According to the nonlinear vibration analysis of multiple degrees-of-freedom systems in reference [31], the damping force, axial friction torque, and input excitation in equation (15) are multiplied by a small parameter ε; then, equation (15) is deformed as follows:(18)θ¨1=−1J13Kc+keθ1+31J1Kcθ2+1J1keθ3+ε1J1Tpsinωt−csθ˙1−θ˙3,θ¨2=1J2Kcθ1−1J2Kcθ2−ε1J2Mfsgnθ˙2−θ˙3,θ¨3=1J3keθ1−1J3ke+kgθ3+ε1J3csθ˙1−θ˙3−cgθ˙3+3Mfsgnθ˙2−θ˙3.Letθ1=x1,θ˙1=x2,θ2=x3,θ˙2=x4,θ3=x5,θ˙3=x6, a1=3Kc+ke/J1, a2=3Kc/J1, a3=ke/J1, a4=Kc/J2, a5=ke/J3, a6=ke+kg/J3, and when ε = 0, equation (18) can be written as(19)x˙1=x2,x˙2=−a1x1+a2x3+a3x5,x˙3=x4,x˙4=a4x1−a4x3,x˙5=x6,x˙6=a5x1−a6x5.From equation (19), the natural frequency ωni(i = 1∼3) can be determined by the following formula(20)a1−ωni2a4−ωni2a6−ωni2−a2a4a6−ωni2−a3a5a4−ωni2=0.Since the parametersa1, a3, a5, and a6 contain ke, the natural frequency ωni is also a nonlinear function of Ai. Taking the following transformation,(21)xs=∑i=13AiGsiφi,s=1,2,3,4,5,6,where Ai is amplitude and φi is phase, and both are time functions, and Gsi is a transformation function.Inserting equation (21) into equation (18), we get(22)∑i=13dAidtGsiφi−∑i=13AiGsi∗φidφidt−ωni=εFs,where(23)F1=0,F2=Tpsinωt−csθ˙1−θ˙3J1,F3=0,F4=−MfJ2sgnθ˙2−θ˙3,F5=0,F6=csθ˙1−θ˙3−cgθ˙3+3Mfsgnθ˙2−θ˙3J3,.A set of fundamental solution of equation (19) is given as(24)G1i=cosωnit,G1i∗=sinωnitG2i=−ωnisinωnit,G2i∗=ωnicosωnitG3i=a4a4−ωni2cosωnit,G3i∗=a4a4−ωni2sinωnitG4i=−a4ωnia4−ωni2sinωnit,G4i∗=a4ωnia4−ωni2cosωnitG5i=a5a6−ωni2cosωnit,G5i∗=a5a6−ωni2sinωnitG6i=−a5ωnia6−ωni2sinωnit,G6i∗=a5ωnia6−ωni2cosωnit.The adjoint equation of equation (19) is as follows:(25)X˙1=a1X2−a4X4−a5X6,X˙2=−X1,X˙3=−a2X2+a4X4,X˙4=−X3,X˙5=−a3X2+a6X6,X˙6=−X5.Also, a set of fundamental solutions of equation (25) can be obtained as follows:(26)H1i=cosωnit,H1i∗=sinλitH2i=−1ωnisinωnit,H2i∗=1ωnicosωnitH3i=a2a4−ωni2cosωnit,H3i∗=a2a4−ωni2sinωnitH4i=−a2a4−ωni21ωnisinωnit,H4i∗=−a2a4−ωni21ωnicosωnitH5i=a3a6−ωni2cosωnit,H5i∗=a3a6−ωni2sinωnitH6i=−a3a6−ωni21ωnisinωnit,H6i∗=−a3a6−ωni21ωnicosωnit.There is a relationship between formulas (24) and (26):(27)∑sG∗siφiHsi∗φi=∑sGsiφiHsiφi=1+a2a4a4−ωni22+a3a5a6−ωni22=Δi.Multiplying equation (22) by Hsi(φi) and Hsi∗φi, respectively, we obtain(28)dAidt=εΔi∑sFsHsiφi=εϕiAi,φi,dφidt=ωni−εΔiAi∑sFsHsi∗φi=ωni−εϕi∗Ai,φi.ForAi and φi, we take the following transformation:(29)Ai=yi+εUit,yi,ϑi,φi=ωt+ϑi+εVit,yi,ϑi.When studying the system main resonance vibration, takeλi–ω as the same order infinitesimal as ε, and require the derivative of new variables yi, ϑi as follows:(30)dyidt=εYiyi,ϑi+ε2Yi∗t,yi,ϑi,ε,dϑidt=εZiyi,ϑi+ε2Zi∗t,yi,ϑi,ε.For equations (29) and (30), Yi and Zi do not contain periodic functions of time t, and Ui, Vi, Yi∗, and Zi∗ are the functions of ϑi and t with period 2π.Substituting equation (30) into equation (29) and letting the first-order coefficient of ε be equal, we get(31)Yi+dUidt=ϕi,Zi+dVidt=−ϕi∗.In order to make the functionsYi and Yi not contain time t, put(32)Yi=12π∫02πϕidψ,Zi=ωni−ω−12π∫02πϕi∗dψ.In equation (30), we take the first-order square coefficient of ε, and let the right end be equal to zero. Because the first-order resonant amplitude is much larger than the second-order and third-order amplitudes, the first-order resonant frequency characteristics are mainly analyzed here. The first-order approximate analytical solution is set as θ1=A1cosωt+φ1, θ2=a4/a4−ωn12A1cosωt+φ1, θ3=a5/a6−ωn12A1cosωt+φ1. After a series of calculation, the following function is obtained:(33)dA1dt=ε−12J1ωn1Δ1Tpcosφ1−12πΔ1S1A1+S2,dφ1dt=ωn1−ω+ε12J1ωn1Δ1A1Tpsinφ1,where(34)S1=a3J3a6−ωn12−1J1a5a6−ωn12−1csπ+a3a5J3a6−ωn122cgπ,S2=3a3J3a6−ωn12ωn1−a2J2a4−ωn12ωn14Mfsgna5a6−ωn12−a4a4−ωn12.The steady-state solution can be found by settingdA/dt=0,dφ/dt=0. The relationship between the system excitation frequency ω and the amplitude A1 and phase φ1 can be obtained.(35)ω=ωn12−2δeA12±TpΔ1J1A12+4δeA12δeA12−ωn12,φ1=arctanω2−ωn12A12δeω.where(36)δe=−12πΔ1S1A1+S2.For the 2DOF model, according to equation (17), the first-order natural frequency of the corresponding linear systems can be given by:(37)ωn1′=12b1+b3−b1−b32+4b1b2,where b1=ke/J,b2=ke/J3,b3=ke+kg/J3.Similarly, the first approximate analytical solution of the 2DOF model system can be written asθ1=A1′sinωt+φ1′, θ3=b1−ωn1′2/b1A1′cosωt+φ1′, and the relationship between ω and amplitude A1′, and phase φ1′ in the first approximate analytical solution are deduced (38)ω=ωn1′2−2δe′A1′2±TpA1′J2+4δe′A1′2δe′A1′2−ωn1′2φ1′=arctanω2−ωn1′2A1′2δe′ω,where(39)δe′=−12πΔ1′Q1A1′+Q2Q1=b1−ωn12′2b1b2J3cg−b1−ωn1′2b2J3−1Jωn1′2b1csπ,Q2=−12ωn1′b1−ωn1′2b2J3−1JMf.Equations (35) and (38) are the nonlinear relationship between the excitation frequency and amplitude and initial phase, which can be solved by numerical method to obtain the corresponding amplitude-frequency characteristics and phase-frequency characteristics.For rotor systems, angular displacement transmissibility is one of the important indexes of vibration reduction performance. The torsional angle amplitude ratio of the secondary flywheel and the primary flywheelTr and Tr′ is the angular displacement transmissibility of the 3DOF model and the 2DOF model, respectively, which can be expressed as(40)Tr=a5a6−ωn12,Tr′=b1−ωn1′2b1.We define the moment inertia ratioλ = J2/J1, and the main parameters of the DMF are as follows: r = 60 mm, R = 92.5 mm, J1 = 0.15 kg·m2, λ = 0.02, J3 = 0.055 kg·m2, l = 70.9 mm, e = 16 mm, Kc = 20000 N·m/rad, h = 10 mm, kg = 501 N·m/°, β = 16°, k1 = 9.22 N·m/°, k2 = 23.67 N·m/°, and Mf = 2 N·m. The synthesis elastic modulus E∗ can be calculated with the material properties of the primary flywheel pressure plate and friction damping ring shown in Table 1.Table 1
Material properties.
Properties/bodiesPrimary flywheel pressure plateFriction damping ringMaterialSteelPa66Elastic modulus (GPa)2068.3Poisson ratio0.30.28To verify the correctness of the solution method in Section3.2, the Runge–Kutta numerical method is applied to obtain the amplitude-frequency characteristics of the primary flywheel angular displacement of the 3DOF model and 2DOF when Tp = 100 N·m. As shown in Figure 7(a), the numerical results agree with the current analytical results well at different frequencies, which indicates that the calculation and analysis of the averaging method are accurate and valid. The commonly used speed of the vehicle engine is 800–4000 rpm, corresponding to the angular frequency range of 83.8–418.9 rad/s. From equation (20), the first-order and second-order resonant frequencies of the system are about 57 rad/s and 728 rad/s, respectively, while the third-order resonant frequency is higher, so the resonant frequency range of the system can avoid in the frequency range of the engine torque variation. In addition, through numerical analysis, it is found that the response amplitude of the second-order and third-order resonant frequencies is also much smaller than that of the first-order resonant frequency. Therefore, we mainly analyze the first-order resonant frequency characteristics without considering the second-order and third-order resonant frequency characteristics in Section 3.2.Figure 7
Amplitude-frequency characteristics of the primary flywheel angular displacement. (a) Comparison of the numerical method result and the analytical result of the 3DOF model and 2DOF model whenTp = 100 N·m. (b) Comparison of the 3DOF model and 2DOF model with different Tp.
(a)(b)The first-order resonance amplitude-frequency characteristics of the 2DOF model and 3DOF model under different excitation torque amplitudesTp are shown in Figure 7(b). It can be seen that the larger the excitation torque, the larger the overall response amplitude and the wider the resonance frequency band. Under the same excitation frequency, the amplitude of the 3DOF model is smaller than that of the 2DOF model, especially nearby the resonant frequency. The maximum value of the 3DOF model is 10.2°, 20.7°, and 28.7° when Tp = 40 N·m, 80 N·m, and 120 N·m, respectively, while the maximum value of the 2DOF model is 10.9°, 21.6°, and 30.2°. Since the 3DOF model takes into account the elastic deformation under the contact action of the structure, it is more closer to the actual situation. Next, the frequency characteristics, angular displacement transmissibility, and forced vibration of the 3DOF model are mainly analyzed.The effects of the moment inertia ratioλ, DMF stiffness ke, and contact stiffness Kc on the natural frequency of the 3DOF model can be analyzed, respectively from equation (20), as shown in Figure 8. When λ changes from 0 to 0.1, as shown in Figure 8(a), the first-order natural frequency changes from 58.8 rad/s to 51.6 rad/s, the third-order natural frequency changes from 36520.3 rad/s to 1316.8 rad/s, while the second-order natural frequency basically maintains at 729.1 rad/s. Therefore, J2 has a great influence on the first-order and third-order natural frequencies but little influence on the second-order natural frequencies. Figure 8(b) demonstrates that when the torsional stiffness ke changes from 0 N·m/° to 30 N·m/°, the first-order natural frequency ωn1 changes from 0 to 100.9 rad/s, and the second-order and third-order natural frequencies are about 729.1 rad/s and 2658.3 rad/s, respectively. Therefore, ke mainly affects the first-order natural frequency and has little influence on the second- and third-order natural frequencies. Besides, from Figure 8(c), it can be concluded that the contact stiffness Kc mainly affects the third-order natural frequency and has little effect on the first-order and second-order natural frequency.Figure 8
Influence ofλ, ke and Kc on the natural frequency of the 3DOF model. (a) Influence of λ. (b) Influence of ke. (c) Influence of Kc.
(a)(b)(c)


## 5. Steady-State Response Frequency Characteristics

Based on the above analysis and considering that the axial friction torque has the damping equivalent effect to decrease the vibration, the influence of the moment inertia of the friction damping ringJ2, the DMF stiffness (k1 and k2), and the axial friction torque Mf on the nonlinear amplitude-frequency characteristics of the 3DOF model is to study below.When the other parameters remain unchanged and the moment inertia ratioλ is given as 0.01, 0.02, 0.1, and 0.5, the first-order amplitude-frequency characteristic curves are shown in Figure 9(a). With the increase of λ, the first-order resonant frequency becomes small, and the overall amplitude nearby resonant frequency will increase. Therefore, the moment inertia of the friction damping ring J2 increases, the frequency of the resonance region will be significantly decreased, but the amplitude will also be significantly increased.Figure 9
Amplitude-frequency characteristics of the primary flywheel angular displacement. (a) Influence ofλ. (b) Influence of k1. (c) Influence of k2. (d) Influence of Mf.
(a)(b)(c)(d)Whenk1 is taken as 5 N·m/°, 10 N·m/°, 15 N·m/°, and 20 N·m/°, respectively, the amplitude-frequency characteristic curves are shown in Figure 9(b). It can be found that with the increase of k1, the corresponding amplitude-frequency characteristic curve moves to the right, namely the resonance frequency increases, but the response amplitude at the resonance frequency will decrease.In Figure9(c), the amplitude-frequency characteristic curve is given when k2 is 10 N·m/°, 15 N·m/°, 20 N·m/°, and 25 N·m/°, respectively. It can be concluded that the second-stage stiffness k2 of the DMF has a great influence on the amplitude-frequency characteristics of the resonance region. With the increase of k2, the overall amplitude decreases, and the amplitude-frequency characteristic curve moves to the right, namely the resonant frequency increases. The main reason is that when A≥β, with the increase of k2, the equivalent stiffness ke of the system increases, and the first-order natural frequency increases simultaneously, as shown in Figure 8(a).Figure9(d) shows the amplitude-frequency characteristics when the axial friction torque Mf equals 0 N·m, 3 N·m, 6 N·m, and 10 N·m, respectively. The resonance regions corresponding to different axial friction torque Mf are basically the same, but the amplitudes at the same frequency are different. With the increase of Mf, the overall amplitude decreases, especially the peak value at the resonance frequency decreases significantly, and the resonance frequency band becomes narrow. Therefore, increasing the axial friction torque is beneficial to reduce the vibration of the system near the resonant frequency. But, because the friction between the parts will cause energy dissipation, excessive axial friction torque will increase fuel consumption and aggravate the wear and service life of the parts in the normal driving process of the vehicle.


## 6. Angular Displacement Transfer Characteristics

Figure10 shows the angular displacement transmissibility near the first-order resonance frequency region when Tp = 100 N·m. The comparison of the angular displacement transmissibility of the 3DOF model and the 2DOF model is depicted in Figure 10(a). It can be seen that with the increase of frequency, the transmissibility first keeps about 0.018. When the frequency gradually approaches the resonance frequency, it increases to the maximum value and finally decreases to a stable value of about 0.018 with the continuous increase of frequency. Compared to the 2DOF model, the angular displacement transmissibility of the 3DOF model is less in the frequency region of 30–70 rad/s. The maximum transmissibility is about 0.026 and 0.025 for 2DOF model and 3DOF model, respectively.Figure 10
Variation of angular displacement transmissibilityTr with frequency. (a) Comparison of 3DOF model and 2DOF model. (b) Influence of λ. (c) Influence of k1. (d) Influence of (k)2. (e) Influence of Mf.
(a)(b)(c)(d)(e)Figure10(b) shows the angular displacement transmissibility under different moment inertia ratios λ. With the increase of λ, the angular displacement transmissibility curve moves to the left, and the peak value of the transmissibility becomes higher. But the angular displacement transmissibility far from the resonance frequency region is similar.Angular displacement transmissibility with different first-stage stiffnessk1 of the DMF is shown in Figure 10(c). The maximum angular displacement transmissibility is at the resonant frequency. The parameter of k1 has a great influence on transmissibility. The larger k1 is, the higher the transmissibility is, but the smoother the change process of the transmissibility is.Figure10(d) shows the angular displacement transmissibility when the second-stage stiffness k2 of the DMF is given different value. In the frequency region 30–70 rad/s, the angular displacement transmissibility corresponding for different k2 varies greatly. With the increase of k2, the angular displacement transmissibility and the frequency corresponding to the maximum transmissibility become larger. When the frequency is far away from the resonance frequency region, the angular displacement transmissibility with different k2 is basically the same, which is about 0.018, and the larger the k2 is, the faster the transmissibility changes with frequency.Figure10(e) shows the angular displacement transmissibility with different axial friction torques Mf. In the resonance frequency range, the larger the Mf is, the smaller the angular displacement transmissibility is. And the angular displacement transmissibility is almost the same at far away from resonance frequency, which is about 0.018.


## 7. Forced Vibration Response

According to the torsional vibration differential equation (15), the forced vibration response of the system of the 3DOF model is analyzed by using the fourth-order Runge–Kutta numerical method. The first-order nonlinear resonance frequency of the system is about 64.5 rad/s. Let the amplitude of the input excitation torque be 100 N·m, the excitation frequency be 100 rad/s, and all of the initial angular displacements and angular velocities be zeros. The dynamic response of the primary flywheel, friction damping ring, and the second flywheel are shown in Figure 11(a). The amplitude of the angular displacement of the primary flywheel is close to the angular displacement of the friction damping ring, which is about 5° and basically the same as the approximate solution result. Figure 11(b) shows the variation of the relative angular displacement between the friction damping ring and the primary flywheel, which presents a complex variation. The maximum relative angular displacement is about 0.02°, and the corresponding maximum torque caused by conformal contact is 6.98 N·m. Therefore, the contact elastic deformation and stress on the contact region of the friction damping ring and the primary flywheel pressure plate are also changed with time under the variable torque excitation during the working process.Figure 11
Angular displacement time history withTp = 100 N·m and ω = 100 rad/s. (a) Time history of θ1, θ2, and θ3. (b) Time history of θ.
(a)(b)When the excitation frequency equals to the first-order nonlinear resonance frequency 64.5 rad/s, and the initial angular displacement and angular velocity are all zeros, the dynamic responses of the 3DOF are shown in Figure12(a). It can be seen that the primary flywheel, the friction damping ring, and the secondary flywheel have large amplitudes for the system resonates. The change of the angular displacement of the friction damping ring and the primary flywheel is still relatively close, and the amplitudes of both are about 25.2°. The maximum relative angular displacement shown in Figure 12(b) is about 0.04°, and the corresponding maximum torque caused by conformal contact is 13.96 N·m. Compared with the primary flywheel, the amplitude of the secondary flywheel is also smaller, which is basically the same as the results of the averaging method. Figures 11 and 12 show that the angular displacement amplitude of the secondary flywheel is obviously smaller than that of the primary flywheel after the damping effect of the DMF, indicating that the DMF has very excellent damping performance.Figure 12
Angular displacement time history withTp = 100 N·m and ω = 64.5 rad/s. (a) Time history of θ1, θ2, and θ3. (b) Time history of θ.
(a)(b)


## 8. Conclusions

In order to study the nonlinear dynamic characteristics of the DMF under the conformal contact action between the friction damping ring and the primary flywheel pressure plate, a conformal contact analysis model based on the Winkler model is established. The contact deformation, contact pressure at different contact positions, and equivalent torsional contact stiffness are analyzed. Then, two nonlinear dynamic analysis models of the DMF are established, and the averaging method is used to derive the approximate analytical solution of the nonlinear frequency characteristics of the system at steady state. The theoretical analysis process is proved to be valid by comparing with the response amplitude of the system at steady state analyzed by applying Runge–Kutta numerical method. It is found that the structural conformal contact has a certain influence on the nonlinear dynamic characteristics of the system. The overall amplitude and angular displacement transmissibility of the 3DOF model considering the conformal contact effect are smaller than the 2DOF model, especially at resonance frequency. Besides, the 3DOF which considering the conformal contact of the friction ring is more in line with the actual situation. WhenJ2 is small, the result of 2DOF model is close to 3DOF model. And with the increase of J2, the difference between the 2DOF model and 3DOF model results will become larger.The influences of the moment inertia ratioλ, DMF stiffness, and axial friction torque Mf on the dynamic response frequency characteristics and angular displacement transmissibility of the 3DOF model are analyzed. The results indicate that the first-order natural frequency concerned in this study is mainly affected by the friction damping ring moment inertia J2 and DMF stiffness ke. The frequency of the resonance region will be significantly decreased with the increase of the friction damping ring moment inertia J2, but the amplitude of the system vibration will also turn larger. When the first-stage stiffness of the DMF k1 increases, the resonance frequency will increase, but the response amplitude at the resonance frequency decreases. The overall amplitude decreases significantly, but the resonant frequency increases with the increase of the second-stage stiffness of the DMF k2. The increase of the axial friction torque Mf will make the overall amplitude decrease.The angular displacement transmissibility reaches the maximum at the resonance frequency. In the resonance frequency region, the increase ofλ or k2 and the decrease of Mf make the angular displacement transmissibility become larger. At the frequency away from resonance frequency region, the effect of each one of λ, k2, and Mf parameters on transmissibility is small. The parameter of k1 has a great influence on the transmissibility: the larger k1 is, the higher the transmissibility is.For the variation of the relative angular displacement between the friction damping ring and the primary flywheel pressure plate, the contact elastic deformation and stress on the contact region of the friction damping ring and the primary flywheel pressure plate are changing with time under the variable torque excitation during the working process. The angular displacement amplitude of the secondary flywheel is obviously smaller than that of the primary flywheel after the damping effect of the DMF, indicating that the DMF has very excellent damping performance.



---

*Source: 3258356-2021-12-16.xml*