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Abstract

An automatic tensioner that consists of a torsional spring and friction damping elements is widely used in belt drive system. The relation of the applied torque versus the imposed angle of tensioner during loading and unloading processes is described as a hysteretic loop. An analytical model is established for estimating the hysteretic behavior of a tensioner, and measurements for hysteretic loop in quasi-static and dynamic excitation are carried out to validate the analytical model. Taking one engine timing belt drive system as a studying example, the method and the procedure for estimating vibration responses of the nonlinear tensioner are given. An iterative algorithm for predicting the accurate equivalent viscous damping of tensioner is carried out in analyzing a belt drive system. The vibration responses of tensioner are validated by the measurement of timing belt drive system. The developed method presented in this article is useful for predicting the hysteretic behavior of tensioner, vibration responses, and the parameters optimization of a belt drive system.

Keywords

Nonlinear tensioner hysteretic behavior analytical model iterative algorithm vibration responses

Introduction

An automatic tensioner is widely used in a belt drive system, such as timing belt drive system (TBDS). A typical TBDS consists of a driving pulley, a teeth belt, an automatic tensioner, and several accessories pulleys, such as pulleys of camshaft, idle, oil pump, and so on. The tensioner is an important component of TBDS for maintaining belt tension constantly, avoiding the jumping in meshed belt teeth and improving the belt service life by reducing vibration.^1–4

Generally, a typical tensioner is mainly composed of a torsional spring, friction elements, and an eccentric tensioner arm.⁵ The spring functions as an energy storage element for tensioning the belt, and the damping element is used for reducing and absorbing the vibration of the belt.⁶ Owing to the friction damping, the tensioner arm behaves a stick–slip motion, and relation of the applied torque and the imposed angle is a hysteretic loop during loading and unloading processes.⁵ The nonlinear behavior (hysteresis) of tensioner has an evident impact on the capacity of applied load, the dynamic performance and the working life of a belt drive system.^6–8

Now some researchers focused on the model of the dynamic characteristics for tensioner hysteresis, and the identification of model parameters of tensioner. Zhao et al.⁹ developed a mathematical model to describe the hysteretic behavior of a spring-load tensioner. The model was validated and model parameters were identified under different excitations. Bastien et al.¹⁰ proposed a modified Dahl and a Masing model to describe the hysteretic behavior of a tensioner. The tensioner was modeled as a torsional spring stiffness, an equivalent friction stiffness, and an equivalent viscous damping. Michon et al.¹¹ presented a general hysteretic model to describe the dynamic behavior of the tensioner. A polynomial expression was used for approximating the hysteresis loop. Chatelet et al.¹² presented a method to choose the most appropriate contact model of tensioner in predicting hysteretic behavior. The nonlinear forces transmitted along contact surfaces should be described as accurately as possible. Zhu et al.⁵ put forward an analytical model and a three-dimensional (3D) finite element model to investigate the hysteretic characteristics of a tensioner. An experimental device was developed to obtain the friction coefficients of contact pairs and the working torque performance of a tensioner.

Other researchers have studied on the application of tensioner in analyzing dynamic responses of belt drive systems. A rotational vibration model for a belt drive system was established by Hwang et al.¹³ The tensioner was modeled as a linear spring and an equivalent viscous damping. The oscillation angle of tensioner arm and the fluctuation of belt tension are predicted. Kraver et al.¹⁴ presented a tensioner model with a coulomb damping and a viscous damping. The frequency characteristics of tensioner arm oscillation in a belt drive system were analyzed. Cheng and Zu¹⁵ investigated the influences of a tensioner with dry friction on dynamic characteristics in a serpentine belt drive system. Two kinds of period responses of nonstick and stick–slip motions for tensioner arm were obtained. Shangguan and Zeng¹⁶ and Feng et al.¹⁷ investigated the stick–slip motion of tensioner arm in a belt drive system, and analyzed the influence of tensioner parameters on dynamic performance of system. Farong and Parker¹⁸ investigated the influence of dry friction of tensioner on dynamic performance in a belt-pulley system, and discussed the energy dissipation caused by the hysteretic behavior of tensioner and the stick–slip phenomena of tensioner arm.

In the previous studies,^5–18 the research works are focused on the parameters identification and the physical model of tensioner. The analytical model stated in Zhu et al.⁵ is suitable for a specific form, and the method for obtaining the hysteretic behavior of tensioner in a dynamic manner is not given. On the other side, the tensioner is simplified as a linear torsional spring and an equivalent viscous damping in the dynamic analysis of a belt drive system.^13–18 The equivalent viscous damping of tensioner is an approximation value in analysis, which can only be obtained by a known vibration amplitude and frequency of tensioner arm.^15,16 Thus, the contributions of this article are:

To establish an analytical model for calculating the hysteretic behavior of a friction type tensioner based on the structure parameters.

To develop a method for calculating vibration responses of tensioner in TBDS.

To carry out a method for estimating the accurate equivalent viscous damping of tensioner in analysis of a belt drive dynamic.

The organization of this article is as follows: In section “Model for calculating the hysteretic behavior of tensioner,” an analytical model for calculating hysteretic behavior of a friction type tensioner is established. In section “Experiment and model validation for vibration responses of tensioner,” the hysteretic behavior of tensioner is measured and compared with calculations by analytic model. In section “Method for estimating the vibration responses for tensioner in TBDS,” a method for estimating vibration responses of a tensioner in TBDS is presented, and a method for obtaining the accuracy equivalent viscous damping of tensioner is given. In section “Validation for the vibration responses of tensioner,” the vibration responses of tensioner are analyzed by the presented method and validated by the measurement of TBDS.

Model for calculating the hysteretic behavior of tensioner

In this section, the structure of a TBDS tensioner is introduced, and the modeling and calculating methods of the hysteretic behavior for the tensioner are presented.

Structure of tensioner and friction torques

The structure of a tensioner is shown in Figure 1. The tensioner consists of a base (1), a torsional spring (2), a tensioner arm (3), and a friction disk (6), and so on. The description for each component and the mechanical fit among components of tensioner are shown in Tables 1 and 2.

Figure 1.

The structure analysis of the tensioner. (1) Base, (2) torsional spring, (3) tensioner arm, (4) bushing, (5) pivot shaft, (6) friciton disk, (7) pulley, (8) bearing, and (9) bolt. (a) Structure diagram of tensioner, (b) section view and location of the friction zone, (c) mechanical fit among components of tensioner, and (d) functional scheme of tensioner.

Table 1.

The description for each component of tensioner.

symbol	Component	Type	Description
1	Base	Fixed	Mounted in the engine block with a limit structure and fixed to the pivot shaft (5)
2	Torsional spring	Revolute pair	One end is fixed to the base (1) and the other end is fixed to the tensioner arm (3)
3	Tensioner arm	Revolute pair	Rotate around the pivot shaft (5) freely with an eccentric structure
4	Bushing	Revolute pair	Installed inside the hole of tensioner arm (3)
5	Pivot shaft	Fixed	Mounted in the engine block by a bolt pin (9)
6	Friction disk	Fixed	Fasten to the pivot shaft (5) by a bolt pin (9)
7/8	Pulley/bearing	Revolute pair	Rotate around the tensioner arm (3)
9	Bolt pin	Fixed	Tighten the pivot shaft (5) to the engine block

Table 2.

The mechanical fit between contact pairs of tensioner.

Symbol	Mechanical fit	Contact pairs
A	Interference	Base/pivot shaft
B	Transition	Bushing/pivot shaft
C	Interference	Bushing/tensioner arm
D	Interference	Friction disk/pivot shaft

As shown in Figure 1(b), owing to the rotation of tensioner arm (3), a counter torque is generated by the torsional spring (2). Two types of friction torques are generated: the friction torque (M_f₁) between the top surface of bushing (4) and friction disk (6), and the friction torque (M_f₂) between the cylindrical contact surfaces of the bushing (4) and the pivot shaft (5). It is seen from Figure 1(d) that the tensioner is modeled as an element with torsional stiffness of spring and an equivalent damping caused by the friction.^10,11

Modeling of friction torque of a tensioner

The working torque of a tensioner (M_t) consists of the spring torque (M_s) caused by the torsional spring, and the friction torques ( M_f₁, M_f₂) caused by the friction pairs. In this section, the method for obtaining the working torque of a tensioner is presented.

The torque (M_s) caused by the torsional spring is calculated by

M_{s} = K_{s} θ_{s} = K_{s} (θ_{t} - θ_{α})

(1)

where $K_{s}$ is the stiffness caused by the torsional spring, $θ_{s}$ is the angular displacements of the spring, $θ_{t}$ and $θ_{α}$ are the angular displacement of tensioner arm and the angular displacement with zero torque of spring in the defined coordinate system, respectively.

The torsional stiffness of spring is calculated by¹⁹

K_{s} = \frac{E_{s} d^{4}}{(64 D n_{s})}

(2)

where $K_{s}$ is the torsional stiffness with a unit Nm/rad, $E_{s}$ is elasticity modulus of the spring, $d$ is the diameter of the spring, $D$ is the mean diameter of spring, and $n_{s}$ is the number of spring coil.

Modeling of $M_{f 1}$

As shown in Figure 2, the top surface of bushing is contacted with the friction disk under a compression spring force (F_N). As the tensioner arm rotates, a friction torque (M_f₁) is generated in the contact area.

Figure 2.

The friction ring between the top of bushing and the friction disk.

Assuming the amount of compression displacement of spring is $S$ , the compression force $F_{N}$ applied to the bushing is

F_{N} = K_{c} \cdot S

(3)

The compression stiffness of the spring $K_{c}$ is calculated by¹⁹

K_{c} = \frac{G d^{4}}{(8 D^{3} n_{s})}

(4)

where $G$ is the shear elasticity of spring.

A tiny ring with an increment dr′ in radius is taken as a studying objective. The force applied to the tiny ring is obtained by

d F_{N} = \frac{F_{N} \cdot 2 r'}{R^{2} - r^{2}} d r'

(5)

where r′ is the radius of the tiny ring, $R$ and $r$ are the outer and inner radius of the friction ring.

Then, the friction torque caused by the compression on the ring is calculated by integration as following

M_{f 1} = \int_{r}^{R} μ_{1} \cdot r' \cdot d F_{N} = \frac{2}{3} μ_{1} F_{N} \frac{R^{3} - r^{3}}{R^{2} - r^{2}}

(6)

where $μ_{1}$ is friction coefficient between the top surface of bushing and the friction disk.

Modeling of $M_{f 2}$

As shown in Figure 3, the friction torque $M_{f 2}$ is generated between the circumferential contact surfaces of the bushing and the pivot shaft. The definitions of parameters in Figure 3 are listed in Table 3.

Figure 3.

The force analysis on the tensioner arm and pivot shaft.

Table 3.

The definitions of parameters in Figure 3.

Symbol	Definition
$T$	Belt tension applied on the tensioner arm
$F_{t}$	Hub load acts on the tensioner arm (on axis O)
$M_{I}$	Overturning moment around the pivot shaft axis P because of an eccentric distance $(Δ_{H})$ between the center of tensioner arm and the center of pivot shaft
$F_{tp}$	Translational force of $F_{t}$ on the pivot shaft (on axis P)
$F_{S}$	Tangential force caused by torsional spring on the tensioner arm
$F_{Sp}$	Translational force of $F_{S}$ on the pivot shaft (on axis P)
$δ$	Angle between the forces $F_{tp}$ and $F_{Sp}$
$γ$	Angle between the hub load $F_{t}$ and the tensioner arm $OP$
$r_{a}$	Radius of pivot shaft

The resultant force applied on the axis $P$ of pivot shaft are calculated by

F_{p} = \sqrt{F_{Sp}^{2} + F_{tp}^{2} - 2 F_{Sp} F_{tp} \cos δ}

(7)

One end of the torsional spring is fixed to the tensioner arm, thus the direction of tangential force (F_S) is determined by the structure of tensioner. The tangential force (F_S) and the translational force (F_Sp) are calculated from the spring torque

F_{Sp} = F_{S} = \frac{M_{s}}{D}

(8)

The working torque (M_t) is generated if the hub load (F_t) is applied on the tensioner arm. It is seen from Figure 3 that the relationship between the hub load (F_t) and the working torque (M_t) is

F_{t} = \frac{M_{t}}{(L_{t} \sin γ)}

(9)

where $L_{t}$ is the length of tensioner arm which is the distance between the center axis $O$ of the tensioner arm and the center axis $P$ of the pivot shaft.

The friction toque generated in cylindrical contact surfaces between the bushing and the pivot shaft is

M_{f 2} = μ_{2} F_{p} r_{a}

(10)

where $μ_{2}$ is the friction coefficient between cylindrical contact surfaces of the bushing and the pivot shaft.

The working torque (M_t) of tensioner is calculated by

M_{t} = M_{s} \pm M_{f 1} \pm M_{f 2}

(11)

where “±” represents the processes of loading and unloading.

According to equations (7), (9), (10), and (11), the friction torque (M_f₂) is coupling with the working torque (M_t). It is difficult to find the explicit expressions of the friction torque (M_f₂) and the working torque (M_t). Thus, an iterative algorithm is proposed in order to obtain these two torques. The iterative method is shown in Figure 4 and the steps are described as follows.

Step 1 Set the initial value (M_t₀) to working torque.

Step 2 Calculate the friction torque (M_f₂) using equations (7)–(10).

Step 3 Calculate the working torque (M_t) using equation (11).

Step 4 Calculate the defined error index (λ_t)

λ_{t} = | \frac{M_{t 0} - M_{t}}{M_{t 0}} |

(12)

Step 5 Check the error index (λ_t) in step 4 to see if it is less than a given limit $ε$ . The limit is set to 1% in thisarticle.

If it is “yes,” output the friction torque (M_f₂) and the working torque (M_t) and then stop.

If it is “no,” record the new working torque (M_t) and then go to the next step.

Step 6: Update the initial data with the known working torque (M_t), and go to step 2.

Figure 4.

Flow chart for obtaining the friction torque $M_{f 2}$ and the working torque $M_{t}$ of tensioner.

Analytic model for calculating hysteretic behavior of tensioner

Owing to the existing friction torques, the relation between the working torque and imposed angle behaves in a hysteretic manner as shown in Figure 5.^10,11

Figure 5.

Hysteretic model of tensioner.

It is seen from Figure 5 that the hysteretic loop consists of two parts: the loading process and the unloading process. The loading process concludes the sticking motion of arc segment AB with a nonlinear reaction torque (M_AB), and the slipping motion of line segment BC with a linear reaction torque (M_BC). The point A and point C are the start point and the end point of loading process, respectively. The point $B$ is the transition point between the sticking motion and the slipping motion. Similarly, the unloading process is described with two segments $CD$ and $DA$ .

According to the Bastien et al.¹⁰and Michon et al.¹¹, the slipping motions $BC$ and $DA$ are assumed to be linear, and the working torque are expressed by linear functions as follow

{\begin{matrix} M_{BC} = K_{l} θ_{s} + a, sgn {\overset{\cdot}{θ}}_{s} \geq 0 \\ M_{DA} = K_{u} θ_{s} + b, sgn {\overset{\cdot}{θ}}_{s} < 0 \end{matrix}

(13)

where $K_{l}$ and $K_{u}$ are the slopes of the function which are defined as the stiffness of loading and unloading, the $a$ and $b$ are the intercepts of two line segments $BC$ and $DA$ , respectively.

The stiffness of loading (K_l) or unloading (K_u) process at slipping motion varies with frequency excitation. The dynamic stiffness under excitation is estimated by^20,21

{\begin{matrix} K_{l} = \sqrt{{(K_{l 0} - ω^{2} I_{t})}^{2} + {(ω c)}^{2}} \\ K_{u} = \sqrt{{(K_{u 0} - ω^{2} I_{t})}^{2} + {(ω c)}^{2}} \end{matrix}

(14)

where $K_{l 0}$ and $K_{u 0}$ are the static stiffness of loading and unloading at slipping motion, $c$ and $I_{t}$ are the damping and the inertial moment of the tensioner arm, and $ω$ is the frequency of reciprocating motion

ω = 2 π f

(15)

where $f$ is the frequency of excitation for tensioner arm.

The friction torques are steady versus the imposed angle at slipping motion because the friction coefficient between contact pairs is fixed. Thus, the torques (M_BC and M_DA) can be calculated by equation (11). Then, the static stiffness $K_{l 0}$ and $K_{u 0}$ are estimated by

{\begin{matrix} K_{l 0} = \frac{Δ_{Ml}}{Δ_{θ 0}}, θ_{s} \in [θ_{B}, θ_{0} + θ_{m}] \\ K_{u 0} = \frac{Δ_{Mu}}{Δ_{θ 0}}, θ_{s} \in [θ_{0} - θ_{m}, θ_{D}] \end{matrix}

(16)

where ${Δ_{M}}_{l}$ and ${Δ_{M}}_{u}$ are the variation of working torques (M_BC and M_DA) with an oscillation angle $Δ_{θ 0}$ during loading and unloading processes, respectively.

According to Bastien et al.¹⁰and Michon et al.¹¹, a Duhem model is applied for describing the hysteretic behavior of tensioner. A first-order differential equation is used effectively for disclosing the relationship between the working torque (M_t) and the imposed angle $(θ_{s})$ ¹⁰

{\begin{matrix} \frac{d M_{t}}{dt} = β {\overset{\cdot}{θ}}_{s} (M_{BC} - M_{t} sgn {\overset{\cdot}{θ}}_{s}), sgn {\overset{\cdot}{θ}}_{s}^{3} \geq 0 \\ \frac{d M_{t}}{dt} = β {\overset{\cdot}{θ}}_{s} (- M_{DA} + M_{t} sgn {\overset{\cdot}{θ}}_{s}), sgn {\overset{\cdot}{θ}}_{s} < 0 \end{matrix}

(17)

where $β$ is a positive constant for describing the slope at sticking motion.

The integration of equation (17) gives¹⁰

M_{t} = {\begin{matrix} C_{l} e - β [θ_{s} - (θ_{0} - θ_{m})] + K_{l} θ_{s} + a - \frac{K_{l}}{β}, sgn {\overset{\cdot}{θ}}_{s} \geq 0 \\ C_{u} e - β [θ_{s} - (θ_{0} + θ_{m})] + K_{u} θ_{s} + b + \frac{K_{u}}{β}, sgn {\overset{\cdot}{θ}}_{s} < 0 \end{matrix}

(18)

where $C_{l}$ , $C_{u}$ are the coefficients of loading and unloading process, and these are determined by the value of working torque (M_t) at $θ_{0} - θ_{m}$ and $θ_{0} + θ_{m}$ as follows

{\begin{matrix} C_{l} = \frac{K_{l}}{β} - K_{l} (θ_{0} - θ_{m}) - a + M_{t} |_{θ_{s} = θ_{0} - θ_{m}}, sgn {\overset{\cdot}{θ}}_{s} \geq 0 \\ C_{u} = - \frac{K_{u}}{β} - K_{u} (θ_{0} + θ_{m}) - b + M_{t} |_{θ_{s} = θ_{0} + θ_{m}}, sgn {\overset{\cdot}{θ}}_{s} < 0 \end{matrix}

(19)

It is seen from equation (18) the working torque is described with three parts: an exponential function $(C_{l} e^{- β [θ_{s} - (θ_{0} - θ_{m})]})$ , a linear function $(K_{l} θ_{s} + a)$ and a constant value $- (K_{l} / β)$ . The principle diagram for the exponential functions in equation (18) is shown in Figure 6.

Figure 6.

The principle diagram for the exponential functions.

The exponential function $(C_{l} e^{- β [θ_{s} - (θ_{0} - θ_{m})]})$ includes a nonlinear region which describes the sticking motion $AB$ and a linear region with zero value which describes the slipping motion $BC$ . Similarly, the exponential function $(C_{u} e^{- β [θ_{s} - (θ_{0} + θ_{m})]})$ is used for describing the motions during unloading process. The definitions for the exponential functions in Figure 6 are given as follow

{\begin{matrix} {C_{l}}^{e - β [θ_{s} - (θ_{0} - θ_{m})]} = 0, sgn {\overset{\cdot}{θ}}_{s} \geq 0, θ_{s} \in [θ_{B}, θ_{0} + θ_{m}] \\ {C_{l}}^{e - β [θ_{s} - (θ_{0} - θ_{m})]} |_{θ_{s} = θ_{0} - θ_{m}} = c_{a} \end{matrix}

(20a)

{\begin{matrix} C_{u} e^{- β [θ_{s} - (θ_{0} + θ_{m})]} = 0, sgn {\overset{\cdot}{θ}}_{s} < 0, θ_{s} \in [θ_{0} - θ_{m}, θ_{D}] \\ C_{u} e^{- β [θ_{s} - (θ_{0} + θ_{m})]} |_{θ_{s} = θ_{0} + θ_{m}} = c_{b} \end{matrix}

(20b)

where $c_{a}$ and $c_{b}$ are the constant values.

It is seen from Figure 5 that the lost energy (E_d) which corresponds to the difference of loading and unloading torques is caused by the existing friction torques. It is calculated by

\begin{matrix} E_{d} = \int_{θ_{0} - θ_{m}}^{θ_{0} + θ_{m}} (M_{l} - M_{u}) d θ_{s} = \\ \frac{1}{β} [e^{- 2 β θ_{m}} (C_{u} - C_{l}) + (C_{l} - C_{u}) + 2 β θ_{0} θ_{m} (K_{l} - K_{u}) \\ + 2 β θ_{m} (a - b) - 2 θ_{m} (K_{l} + K_{u})] \end{matrix}

(21)

Equivalent viscous damping (C_t) of tensioner is a key parameter in analyzing dynamic responses of a belt drive system. According to the hysteretic loop, the viscous damping is calculated by¹⁶

C_{t} = \frac{E_{d}}{(π \cdot ω_{c} \cdot θ_{m}^{2})}

(22)

where the $ω_{c}$ is the amplitude of circular frequency of excitation.

Based on the established analytic model, the procedure for estimating the hysteretic torque of a tensioner is shown in Figure 7.

Figure 7.

The flow chart for estimating the hysteretic torque of tensioner.

Experiment and model validation for vibration responses of tensioner

In this section, test rigs and procedures for measuring the hysteretic behavior of tensioner are presented. The working torque is measured and compared with the calculation.

The test rig for measuring the working torque of tensioner is shown in Figure 8. The tensioner pulley (3) is mounted inside a hook (2), and the hook (2) is connected to an actuator (1). The base of tensioner (3) is installed in a connecting support (4) which is fixed to the fixed jaw (5). The force sensor is installed inside the actuator (1).

Figure 8.

Test rig for measuring working torque of the tensioner. (1) Actuator, (2) hook, (3) tensioner, (4) connecting support, and (5) fixed jaw. (a) Layout of a tensioner test rig, (b) schematic diagram for obtaining the working torque, and (c) measurement spot.

It is seen from Figure 8(b) that the actuator (1) applies a linear displacement $Δ$ , and the reaction force is $F_{e}$ . The tensioner pulley rotates from an initial position $O$ to a new position $O'$ with an angle $Δ_{θ}$ . The angle between the tensioner arm $PO$ and the vertical direction is defined as $α_{0}$ . The oscillation angle of tensioner arm $Δ_{θ}$ is calculated by

Δ_{θ} = \arccos (\frac{L_{t} \cos α_{0} - Δ}{L_{t}}) - α_{0}

(23)

The measured working torque of the tensioner is calculated by

M_{t} = F_{e} L_{t} \sin (Δ_{θ} + α_{0})

(24)

Validation of hysteretic model in a quasi-static excitation

Based on the parameters of tensioner listed in Tables 6 of Appendix, and the model developed in Section “Model for calculating the hysteretic behavior of tensioner,” the working torques of tensioner are calculated and measured from an initial spring position 63.5° to a limit position 90.5° in a quasi-static excitation. The results are shown in Figure 9.

Figure 9.

Hysteretic torque versus angle in a quasi-static excitation (f = 0.01 Hz). (a) The torque versus angle with different coefficients β and (b) the comparison of the measured and calculated torque versus angle.

Based on equations (18) and (20), the positive constant $β$ is the coefficient of exponential function. The value of $β$ determines the behavior of transition region from sticking motion to slipping motion which is shown in Figure 9(a). The slope of sticking motion becomes steeper with a larger value of $β$ .

According to the research results of Bastien et al.¹⁰, the effective value of coefficient (β) is $β = 2$ for a friction type tensioner which is determined by the measured hysteretic loop. It is seen from Figure 9(b) that the relation of calculated torque versus angle has a good correlation with the measurements. The maximum error appears at transition region during loading process. The comparisons of stiffness are shown in Table 4.

Table 4.

Comparisons of stiffness for tensioner.

Symbol	Parameter	Measurement (Nm/°)	Analytic model (Nm/°)	Error (%)
$K_{l}$	Stiffness of loading	0.0218	0.0203	6.88
$K_{u}$	Stiffness of unloading	0.0133	0.0129	3.01
$K_{s}$	Stiffness of spring	0.0163	0.0161	1.23

Validation for hysteretic model in a dynamic excitation

In this section, a measurement for obtaining the dynamic stiffness (K_l and K_u) of tensioner is presented by the test rig in Figure 6. The tensioner arm is excited with a reciprocating motion. The excitation amplitude of tensioner (θ_m) is set as 2.85°, and the excitation frequency (f) are swept from 1 to 35 Hz, respectively. The excitation amplitude of tensioner (θ_m) should be larger than the angle of sticking motion as shown in Figure 5, and it cannot be too large because of the limitation of test rig in a high frequency with large displacement excitation.

The hysteretic loops of tensioner versus different excitation frequencies (from 1 to 35 Hz) are measured one by one by the existing test rig stated in Figure 8. Based on the measured hysteretic loops, the dynamic stiffness of tensioner during loading and unloading processes are obtained by equation (16). The input parameters for tensioner are given in Table 7 of Appendix 1. The measured dynamic stiffness of tensioner versus exciting frequency is given in Figure 10. Based on the measurement, the damping of the tensioner arm $c$ is recognized by equation (14) and the value is shown in Table 7 of Appendix 1. Then, the dynamic stiffness is estimated by the recognized damping of tensioner arm by equation (14).

Figure 10.

The dynamic stiffness during loading and unloading processes versus exciting frequency (θ_m = 2.85°).

It is seen from Figure 10 that the trend of measurements and calculations is the same. The stiffness of loading or unloading decreases first with an increasing excitation frequency which is caused by the inertia moment of tensioner arm. Then, the stiffness increases because of the rotational damping of tensioner arm.

Based on the method for estimating hysteretic loop stated in Figure 5 of Section “Analytic model for calculating hysteretic behavior of tensioner,” the working torque of the tensioner versus angle is calculated and compared with the measurement. The static stiffness of tensioner is the stiffness under a zero excitation frequency.

It is seen from Figure 11(a) and (b) that the hysteretic loops of calculation are close to the measurements. The maximum error occurs in the unloading process, because the unloading torque calculated by analytic model is larger than that of measurement. With a low excitation frequency 1 Hz, the loop calculated by the dynamic stiffness of tensioner is almost the same with that of the loop calculated by static stiffness.

Figure 11.

Comparison between the measurement and the calculation by the dynamic analytic model. (a) Hysteretic loop of torque versus angle with an excitation amplitude 2.85° and excitation frequency 1 Hz, (b) hysteretic loop of torque versus angle with an excitation amplitude 5.71° and excitation frequency 1 Hz, (c) hysteretic loop of torque versus angle with an excitation amplitude 2.85° and excitation frequency 15 Hz, and (d) hysteretic loop of torque versus angle with an excitation amplitude 2.85° and excitation frequency 30 Hz.

The dynamic stiffness of tensioner with an excitation frequency 15 Hz is smaller than the static one as stated in Figure 10. Thus, the hysteretic loop calculated by dynamic stiffness moves down comparing with that of the loop calculated by static stiffness as shown in Figure 11(c). With an excitation frequency 30 Hz, the dynamic stiffness of tensioner is close to the static one as stated in Figure 11(d). It is seen from Figure 11(d) that the hysteretic loop calculated by static stiffness is near to that of the loop by dynamic stiffness.

Method for estimating the vibration responses for tensioner in TBDS

As shown in Figure 12, a four-pulley engine TBDS is taken as an example. The system consists of a crankshaft (CRK), two camshafts (CAM1 and CAM2), and a tensioner (TEN). The pulleys and the belt spans are numbered counterclockwise. The driving pulley CRK is numbered as the first pulley. A coordinate system $O - XY$ is established where the origin is located at the center of the CRK. There are $n$ pulleys named as pulley 1, …, pulley $i$ , …, pulley $n$ . The center, radius, angular displacement, moment of inertia for pulley $i$ are defined as $O_{i} (x_{i}, y_{i})$ , $R_{i}$ , $θ_{i}$ , $I_{i}$ , respectively. The belt between pulley $i$ and pulley $i + 1$ is defined as $B_{i}$ with a belt length $L_{i}$ .

Figure 12.

Schematic of a four-pulley timing belt drive system.

Modeling of TBDS

The TBDS includes four parts: the rotational pulley, the automatic tensioner, the meshed teeth region, and the belt tension. In this section, the modeling process for these parts is given.

Equation motion for driven pulley

In Figure 13, a transmitted torque $M_{i}$ applied to the fixed driven pulley $i$ . The moment caused by inertial $(I_{i} {\overset{\cdot\cdot}{θ}}_{i})$ , the moment caused by damping $(C_{i} {\overset{\cdot}{θ}}_{i})$ , and the transmitted torque (M_i) are balanced with the toque caused by belt tensions (T_i_–1, T_i). Thus, the equation of motion for the driven pulley is¹⁷

I_{i} {\overset{\cdot\cdot}{θ}}_{i} + C_{i} {\overset{\cdot}{θ}}_{i} + M_{i} = R_{i} (T_{i - 1} - T_{i}), i = 2, 3, \dots, n

(25)

where $C_{i}$ is the viscous damping of bearing of the pulley.

Figure 13.

The moment applied to the fixed driven pulley.

Equation motion for automatic tensioner

The moment applied to a tensioner arm is shown in Figure 14. The reset direction of the tensioner arm is assumed to be counterclockwise. As the arm rotates clockwise around the pivot $P$ , the spring and the damping element generate resistance moments. Utilizing the theorem of moment of momentum, the equation for tensioner arm is

(I_{i} + I_{t}) {\overset{\cdot\cdot}{θ}}_{t} + C_{t} {\overset{\cdot}{θ}}_{t} + M_{i} = M_{b} + M_{G}

(26)

where $I_{i}$ , $I_{t}$ are moment of inertia for the tensioner pulley rotating around its center and moment of inertia for the tensioner arm rotating around the pivot $P$ , respectively, $C_{t}$ is the equivalent viscous damping of tensioner, $M_{b}$ , $M_{G}$ are moment caused by tensions of the two belt spans adjacent to the tensioner pulley and moment caused by gravity.

Figure 14.

The moment applied to a tensioner arm.

As shown in Figure 14, $T_{i - 1}$ and $T_{i}$ are the tensions of belt with unit vectors $e_{i - 1}$ and $e_{i}$ . The $L_{t}$ is the length of tensioner arm. The moment $M_{b}$ caused by belt tensions in equation (26) is calculated by¹⁷

M_{b} = L_{t} \times e_{i - 1} \cdot T_{i - 1} + L_{t} \times e_{i} \cdot T_{i}

(27)

Modeling of belt length variation on the meshed teeth region

The belt length variation on the meshed teeth region is used for calculating belt tension. Based on the Karolev and Gold,²² the belt length variation on the meshed region depends on belt tensions around the pulley, the number of meshed teeth and the parameters of belt.

Assuming that the gap between each tooth of belt and pulley is assumed to be zero. Figure 15(a) shows the model of a meshed belt and pulley. The teeth of the pulley are assumed to be rigid. The timing belt is modeled as a soft spring $k_{t}$ caused by the belt tooth and a hard spring $k_{c}$ caused by the belt cord along the circumferential direction.

Figure 15.

The meshed model of timing belt. (a) The model of a meshed timing belt and pulley and (b) the forces applied to the timing belt as the pulley drives the belt.

The forces applied to the timing belt are shown in Figure 15(b). The friction force $f_{t}$ on the belt is balanced with the belt tensions $S_{t}$ and ${S_{t}}^{'}$ . The tooth load $Q_{t}$ is balanced with the belt tensions $S_{t - 1}$ and ${S_{t}}^{'}$ . As the pulley drives the belt, the equilibrium equations on the $t th$ belt tooth are described as

S_{t - 1} - {S_{t}}^{'} = Q_{t} \cdot \cos β

(28)

where $β$ is the angle of the tooth profile.

Taking the belt cord between the tth belt tooth and the $(t + 1) th$ belt tooth as a studying objective, the relationship of tensions on the belt cord are given using Capstan equation²²

S'_{t} = e^{- μ ψ} S_{t}

(29)

where $μ$ is the friction coefficient between the pulley and the belt, $ψ$ is the angle covering the belt piece between two teeth.

The relations among the belt tensions are simplified as a form of difference equation²²

S_{t + 1} - p S_{t} + q S_{t - 1} = 0

(30)

p = 1 + e^{μ ψ} + k_{t} \cos β (e^{μ ψ} - 1) / (k_{c} μ ψ), q = e^{μ ψ}

(31)

where $p$ and $q$ are the coefficients of the difference equation, $k_{t}$ and $k_{c}$ are the stiffness of one tooth and stiffness of one pitch unit of belt, respectively.

The solution of the tension $S_{t}$ is²²

S_{t} = C_{1} λ_{1}^{t} + C_{2} λ_{2}^{t}, t = 1, 2, . . ., m

(32)

λ_{1} = \frac{(q + \sqrt{q^{2} - 4 p})}{2}, λ_{2} = \frac{(q - \sqrt{q^{2} - 4 p})}{2}

(33)

where $m$ is the number of the meshed teeth, the coefficients of $C_{1}$ and $C_{2}$ are deduced from the belt tensions $T_{t}$ and $T_{s}$ of the tight side and the slack side as follow

{\begin{matrix} T_{s} = C_{1} + C_{2} \\ T_{t} = C_{1} λ_{1}^{m} + C_{2} λ_{2}^{m} \end{matrix}

(34)

If the timing belt drives the pulley, the direction of friction force is the opposite as the direction of belt moving. In this case, the frictional coefficient $μ$ in equations (29) and (31) is $- μ$ .

The variation of belt length $Δ_{ti}$ on the meshed region can be calculated by belt tensions $T_{i}$ and $T_{i + 1}$

Δ_{ti} = \sum_{t = 1}^{m} (S_{t}^{'} + S_{t}) \cdot P_{b} / (2 k_{t}) - Δ_{t 0} = F_{ti} (T_{i}, T_{i + 1})

(35)

where $Δ_{t 0}$ is the belt length variation of a meshed teeth region under a belt pretension $T_{0}$ , $P_{b}$ is the pitch of the belt.

Calculation of belt tension

The tension $T_{i}$ of each belt in equations (25) and (27) is the key process variable for modeling the system. Thus, the calculation method of the belt tension is given as following.

1. Formulas for belt tension between two fixed pulleys

In Figure 16, based on the Hook’s law, the belt tension between two fixed pulleys is calculated by¹⁷

T_{i} = T_{0} + K_{i} Δ_{i}

(36)

K_{i} = \frac{E_{b} A}{L_{i}}

(37)

where $T_{0}$ is the pretension of the belt, the $K_{i}$ and the $Δ_{i}$ are the stiffness and the longitudinal elongation of the belt span $B_{i}$ , respectively, $E_{b}$ is the elastic modulus of the belt.

Figure 16.

Belt tension between two fixed pulleys.

The elongation $Δ_{i}$ is determined by angular displacements of two adjacent pulleys, and belt creeping $(Δ_{ci})$ or variation of belt length in meshing region $(Δ_{ti})$ on the pulley wrap arc

Δ_{i} = θ_{i} R_{i} - θ_{i + 1} {R_{i}}_{+ 1} - Δ_{ci} or Δ_{i} = θ_{i} R_{i} - θ_{i + 1} {R_{i}}_{+ 1} - Δ_{ti}

(38)

The variation of belt length $(Δ_{ti})$ in the meshed teeth region is achieved by equation (35), and the creeping elongation $Δ_{ci}$ and the belt stiffness $K_{p (i + 1)}$ of the contact arc area are calculated by¹⁷

Δ_{ci} = \frac{(T_{i + 1} + T_{i} - 2 T_{0})}{(2 K_{p (i + 1)})}

(39)

K_{p (i + 1)} = \frac{E_{b} A}{(α_{i + 1} R_{i + 1})}

(40)

2. Formulas for belt tension between tensioner and fixed pulley

Assuming that the center of tensioner pulley moves from $O_{i + 1}$ to $O'_{i + 1}$ , the contact points move from $W$ and $Q$ to the points $W'$ and $Q'$ . The angular displacement of tensioner arm changes from $θ_{t}$ to ${θ_{t}}^{'}$ and the length of belt span $B_{i}$ changes from $L_{i}$ to $L'_{i}$ as shown in Figure 17.

Figure 17.

Belt tension between tensioner and fixed pulley.

The belt tension is related to the belt elongation between the fixed pulley and the tensioner, the variation of belt length caused by the arm movement, respectively. Thus, the tension is¹⁷

T_{i} = T_{0} + K_{i} (Δ_{i} + Δ L_{i})

(41)

where $Δ L_{i}$ is the length variation of belt span $B_{i}$ by the movement of tensioner arm.

The variation of belt span length $Δ L_{i}$ generated by the tensioner arm movement is represented by

Δ L_{i} = L'_{i} - L_{i} + Δ α \cdot (R_{i} + R_{i + 1})

(42)

where $Δ_{α}$ is the variation of wrap angle of tensioner pulley while tensioner moves from $O_{i + 1}$ to $O'_{i + 1}$ .

If the tensioner arm angles $θ_{t}$ and $θ'_{t}$ are known, the $Δ_{α}$ , $L_{i}$ , and $L'_{i}$ are obtained by the geometric relationship. Thus, the variation of belt span length $Δ L_{i}$ in equation (42) can be rewritten by

Δ L_{i} = g_{i} (θ_{t}, {θ_{t}}^{'})

(43)

Solutions of vibration responses for tensioner

Under the excitations of the driving pulley and the load applied to each driven pulley, the vibration responses of tensioner are obtained by the model developed in section “Modeling of timing belt drive system.”

Method for obtaining the vibration response of tensioner

The excitation that applied to the TBDS consists of a central speed of CRK and fluctuations of angular speed. It is expressed as^16,17

{\overset{\cdot}{θ}}_{1} = 2 π \frac{N}{60} + \sum_{k} A_{k} (N) \cos [k * 2 π \frac{N}{60} t + φ_{k} (N)]

(44)

where $k$ , $A_{k} (N)$ and $φ_{k} (N)$ are the order, the amplitude, and the phase of the excitation, respectively, and $N$ is the average rotational speed of CRK with a unit r/min.

According to the established models, the dynamic differential equations of TBDS are derived as follow

{\begin{matrix} I_{i} {\overset{\cdot\cdot}{θ}}_{i} = R_{i} (T_{i - 1} - T_{i}) - M_{i} - C_{i} {\overset{\cdot}{θ}}_{i}, i = 2, 3, 4 \\ (I_{i} + I_{t}) {\overset{\cdot\cdot}{θ}}_{t} = M_{b} + M_{G} - M_{t} - C_{t} {\overset{\cdot}{θ}}_{t}, i = 4 \end{matrix}

(45)

The inputs of the system are the angular speed and the angular displacement of the CRK $({\overset{\cdot}{θ}}_{1}, θ_{1})$ , and the loaded torque of each driven pulley $M_{i}$ . The angular speeds and the angular displacements for each pulley and the tensioner arm $({\overset{\cdot}{θ}}_{i}, θ_{i}, {\overset{\cdot}{θ}}_{t}, θ_{t})$ are unknown values.

It is seen from equation (45) that there are second order components, such as ${\overset{\cdot\cdot}{θ}}_{i}$ . Thus, the equation needs to be rewritten as a first order equation. A state variable are defined as

Y = {[θ_{1}, {\overset{\cdot}{θ}}_{1}, θ_{2}, {\overset{\cdot}{θ}}_{2}, θ_{3}, {\overset{\cdot}{θ}}_{3}, θ_{4}, {\overset{\cdot}{θ}}_{4}, θ_{t}, {\overset{\cdot}{θ}}_{t}]}^{T}

(46)

Substituting equation (46) into equation (45), a first-order equation is obtained as follow

{\begin{matrix} \overset{\cdot}{Y} (2 i - 1) = Y (2 i), i = 1, 2, \dots, 5 & (I) \\ \overset{\cdot}{Y} (2) = - \sum_{k} A_{k} {(N)}^{*} (k^{*} 2 π N {/ 60)}^{*} \sin [k^{*} 2 π Nt / 60 + φ_{k} (N)] & (II) \\ \overset{\cdot}{Y} (2 i) = [R_{i} (T_{i - 1} - T_{i}) - M_{i} - C_{i} Y (2 i)] / I_{i}, i = 2, 3, 4 & (III) \\ \overset{\cdot}{Y} (2 i) = [M_{b} + M_{G} - M_{t} - C_{t} Y (2 i)] / (I_{i - 1} + I_{t}), i = 5 & (IV) \end{matrix}

(47)

It is seen from equation (47) that the formula (I) is obtained by the definition of state variable $Y$ . The acceleration excitation of the CRK $({\overset{\cdot\cdot}{θ}}_{1})$ is given in formula (II) according to equation (44). The acceleration responses of the rotational pulley and the tensioner arm $({\overset{\cdot\cdot}{θ}}_{i}, {\overset{\cdot\cdot}{θ}}_{t})$ are given in formulas (III) and (IV) by rewritting equation (45).

The moment $(R_{i} (T_{i - 1} - T_{i}))$ caused by belt tensions which is applied to the driven pulley $i$ , and the moment (M_b) caused by the tensions of the two belt spans adjacent to the tensioner pulley are the process variables with belt tension $T_{i}$ . Based on equations (38) and (39), the longitudinal elongation of belt span $Δ_{i}$ can be expressed as a function with belt tensions $T_{i}$ and $T_{i + 1}$

Δ_{i} = F_{i} (T_{i}, T_{i + 1})

(48)

According to equations (36) and (41), the belt tension $T_{i}$ is calculated by the longitudinal elongation of belt span $Δ_{i}$ , and can be rewritten by the initial values of angular displacements $(θ_{i} and θ_{t})$ . Then, the moment $(R_{i} (T_{i - 1} - T_{i}))$ and the moment (M_b) are expressed as

R_{i} (T_{i - 1} - T_{i}) = {\begin{matrix} F_{i} (θ_{i - 1}, θ_{i}, θ_{i + 1}), i = 2 \\ G_{i} (θ_{i - 1}, θ_{i}, θ_{i + 1}, θ_{t}), i = 3, 4 \end{matrix}

(49)

M_{b} = F_{b} (θ_{2}, θ_{3}, θ_{4}, θ_{t})

(50)

The initial value of $Y$ is defined as $Y |_{t = 0}$ . The initial angular velocity ${\overset{\cdot}{θ}}_{t} |_{t} = 0$ is acquired by the transmission ratio. Thus, the initial vectors of the system are

{\begin{matrix} θ |_{t = 0} = [0, 0, 0, 0, θ_{t} |_{t = 0}] \\ \overset{\cdot}{θ} |_{t = 0} = {\overset{\cdot}{θ}}_{1} |_{t = 0} \cdot [1, R_{1} / R_{2}, R_{1} / R_{3}, R_{1} / R_{4}, 0] \end{matrix}

(51)

A numerical method Runge–Kutta is used for solving the differential equations as stated in equation (47). The flow chart for calculating the vibration responses of tensioner is given in Figure 18.

Figure 18.

Chart for calculating the vibration responses of tensioner.

The angular displacements $(θ_{i}, θ_{t})$ and the angular velocity $({\overset{\cdot}{θ}}_{i}, {\overset{\cdot}{θ}}_{t})$ versus time of the rotational parts are acquired. Then, the working torque of tensioner $M_{t}$ is obtained by equation (18).

Method for obtaining the equivalent viscous damping of tensioner

It is seen from equation (22) that the equivalent viscous damping $C_{t}$ is relevant to the amplitude of vibration $θ_{m}$ and the circular frequency $ω_{c}$ of tensioner arm. The $ω_{c}$ is determined by the known excitation of CRK, but the amplitude $θ_{m}$ is unknown.

In the existing methods, an approximation value of equivalent viscous damping is used for estimating the oscillation response of tensioner arm.^11–17 Besides, a measured result from the oscillation of tensioner arm in a belt system is added into the input data for obtaining the equivalent viscous damping of tensioner.²³ Here, an iterative algorithm is proposed in Figure 19 in order to estimate the accurate value of amplitude $θ_{m}$ . The error index $λ$ in the flow chart is set as 1%.

Figure 19.

Flow chart for estimating the oscillation angle amplitude of tensioner arm.

Validation for the vibration responses of tensioner

Test rig for measuring the vibration responses of tensioner

The dynamic performance of tensioner is acquired by a test bench rig as shown in Figure 20. A data acquisition and analysis system is used in the measurement.

Figure 20.

Test rig for measuring dynamic performances of a timing belt system. (1) Laser displacement sensor, (2) angle encoder. (a) Schematic diagram of the test rig and (b) layout of the test bench.

It is seen from Figure 20(a), that the angular displacement and angular velocity of the CRK are obtained by the angle encoder, and the laser displacement sensor is used for measuring the oscillation displacement of the tensioner arm.

The torsional vibration of the CRK is measured first by the angle encoder under an average CRK speed 1000 r/min. The excitation parameters stated in equation (44) are given in Table 5.

Table 5.

The excitation parameters of crankshaft with a speed 1000 r/min.

Order $k$	Amplitude $A_{k} (N)$ /r min⁻¹	Phase $φ_{k} (N)$ /°
2	69.14	45.18
4	9.96	−22.48
6	1.44	−11.56

Figure 21 shows the loaded torque of camshafts versus phase at an average CRK speed 1000 r/min. The camshaft drives the valve with a cyclic varying load, and the phased difference is around 45°.

Figure 21.

The loaded torque versus the phase of camshaft if crankshaft average speed equals to 1000 r/min.

Validation and analysis for vibration responses of tensioner

The coordinate, radius, moment of inertia, and damping for each pulley are shown in Table 8 of Appendix 1, the parameters of timing belt is shown in Table 9 of Appendix 1, respectively. Based on the stated method in section “Solutions of vibration responses for tensioner,” the oscillation angle of tensioner arm is calculated and compared with measurements.

It is seen from Figure 22 that the calculated value is smaller than that of the measurement. The maximum error occurs in a changing from slipping motion to sticking motion of tensioner arm with a value 27.1%. The amplitude of oscillation angle of tensioner arm $(θ_{m})$ is 2.55°, and the excitation frequency (f) is 33.33 Hz which equals to the order 2 excitation under a CRK speed 1000 r/min.

Figure 22.

The oscillation angle of tensioner arm versus time for a crankshaft average speed 1000 r/min.

Using the existing oscillation amplitude $θ_{m}$ and excitation frequency $f$ , the working torque of tensioner is tested again by the test rig as shown in Figure 8. It is seen from Figure 23 that the area (lost energy $E_{d}$ ) of calculated hysteretic loop is little larger than that of the measurement. Thus, the calculated viscous damping of tensioner is larger than the actual value according to equation (22), and as a result, the calculated oscillation angle of tensioner arm is smaller than that of the measurement as shown in Figure 22.

Figure 23.

Working torque versus oscillation angle based on the calculating results (f = 33.33 Hz, θ_m = 2.55°).

According to the calculated oscillation angle of tensioner arm in Figure 22, the spring torque and the working torque of tensioner are calculated by equations (1) and (18), and are shown in Figure 24.

Figure 24.

The working torque of tensioner. (a) The working torque of tensioner versus time and (b) loop of the working torque versus oscillation angle.

It is seen from Figure 24(a) that the working torque changes rapidly when the tensioner arm changes its direction of oscillation. The friction torque reaches maximum with a small angular displacement of tensioner arm, then the working torque of tensioner varies with a small loading or unloading stiffness. Figure 24(b) shows hysteresis loops of the working torque for tensioner. It is seen that the spring provides a basic torque, and the working torque mainly varies with friction torque.

The method stated in Figure 19 of section “Solutions of vibration responses for tensioner” is used for obtaining the equivalent viscous damping of tensioner. The initial amplitude of oscillation angle $θ_{m 0}$ is set as 12.5°. The responses amplitude of oscillation angle $θ_{m}$ and the viscous damping $C_{t}$ under different iterations are shown in Figure 25.

Figure 25.

The viscous damping and amplitude of oscillation under different iterations. (a) The viscous damping and the amplitude of oscillation versus number of iterations and (b) the oscillation angle of tensioner arm versus time.

It is seen from Figure 25(a) that the amplitude of oscillation angle decreases with more iterations because the initial value is larger than the actual value, and the viscous damping increases with a smaller amplitude of oscillation angle $θ_{m}$ based on equation (22). The viscous damping and the amplitude of oscillation angle reach steady values quickly at the first few time of iterations, then distinguish of the iteration results becomes smaller and smaller.

It is seen from Figure 25(b) that the amplitude of the oscillation angle of tensioner arm becomes smaller with more iterations. The vibration frequency of tensioner arm equals to the excitation of CRK. The phase of the oscillation angle delays with fewer iterations because of a smaller equivalent viscous damping.

Conclusion

The established hysteretic model provides a good description for the working torque of a friction type timing belt tensioner. The hysteretic model of tensioner displays a smoothness zone at the sticking motion during loading or unloading process.

The computational accuracy of the hysteretic loop depends on the stiffness of loading or unloading, and the initial torque at the position with maximum amplitude of excitation. The dynamic stiffness of tensioner varies with the frequency of excitation, which has an obvious influence on the hysteretic torque of tensioner.

The iterative algorithm has a good accuracy in estimating the value of oscillation angle of tensioner arm and viscous damping of tensioner in a belt drive analysis. The maximum error occurs in a changing from slip motion to sticking motion of tensioner arm.

Footnotes

Appendix 1

Table 9.

Parameters of the timing belt.

Symbol	Parameter	Value	Unit
$P$	Pitch	9.525	mm
$A$	Cross-sectional area	49.78	mm²
$ψ$	Angle covering the belt piece between two teeth	9	°
$2 β_{t}$	Angle of the tooth profile	32	°
$E_{b}$	Elasticity modulus	5107	MPa
$T_{0}$	Installation tension of equilibrium position	208.6	N
$l_{t}$	Belt length of equilibrium position	1114.42	mm
$k_{t}$	Stiffness of one tooth	252	N/mm
$μ$	Frictional coefficient of belt	0.2	–

Handling Editor: James Baldwin

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The paper is supported by the National Natural Science Foundation of China (No. 51305139).

ORCID iDs

Shangbin Long

Wen-Bin Shangguan

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Methods for estimating hysteretic behavior and vibration responses of a timing belt tensioner

Abstract

Keywords

Introduction

Model for calculating the hysteretic behavior of tensioner

Structure of tensioner and friction torques

Modeling of friction torque of a tensioner

Modeling of M f 1

Modeling of M f 2

Analytic model for calculating hysteretic behavior of tensioner

Experiment and model validation for vibration responses of tensioner

Validation of hysteretic model in a quasi-static excitation

Validation for hysteretic model in a dynamic excitation

Method for estimating the vibration responses for tensioner in TBDS

Modeling of TBDS

Equation motion for driven pulley

Equation motion for automatic tensioner

Modeling of belt length variation on the meshed teeth region

Calculation of belt tension

Solutions of vibration responses for tensioner

Method for obtaining the vibration response of tensioner

Method for obtaining the equivalent viscous damping of tensioner

Validation for the vibration responses of tensioner

Test rig for measuring the vibration responses of tensioner

Validation and analysis for vibration responses of tensioner

Conclusion

Footnotes

Appendix 1

Declaration of conflicting interests

Funding

ORCID iDs

References

Modeling of $M_{f 1}$

Modeling of $M_{f 2}$