The dynamic performance of a belt drive system is composed of many factors, such as the efficiency, the vibration, and the optimal parameters. The conventional design only considers the basic performance of the belt drive system, while ignoring its overall performance. To address all these challenges, the study on vibration characteristics and optimization strategies could be a feasible way. This paper proposes a new optimization strategy and takes a belt drive design optimization as a case study based on the multidisciplinary design optimization (MDO). The MDO of the belt drive system is established and the corresponding sub-systems are analyzed. The multidisciplinary optimization is performed by using an improved genetic algorithm. Based on the optimal results obtained from the MDO, the three-dimension (3D) model of the belt drive system is established for dynamics simulation by virtual prototyping. From the comparison of the results with respect to different velocities and loads, the MDO method can effectively reduce the transverse vibration amplitude. The law of the vibration displacement, the vibration frequency, and the influence of velocities on the transverse vibrations has been obtained. Results show that the MDO method is of great help to obtain the optimal structural parameters. Furthermore, the kinematics principle of the belt drive has been obtained. The belt drive design case indicates that the proposed method in this paper can also be used to solve other engineering optimization problems efficiently.

A belt drive system, as a drive device, is extensively used to transmit power between different rotational machine elements in many engineering applications.1 The belt drive system is widely used in machinery products, such as lathes, automotive engines, washing machines, excavators, and agricultural equipment. As a complex drive system, the vibration of the belt drive system mainly includes transverse vibrations and longitudinal vibrations of the continuum belt spans.2,3 The longitudinal vibration is mainly constrained by the constitutive relation of the belt. The transverse vibration of the belt drive, however, is usually affected by the structural parameters.

The conventional design does not take the vibration of the belt drive into account. The vibration analysis of the belt drive is usually performed after the structure design. For the lateral vibration performance, many literatures have focused on this issue. For example, Zhang and Zu4–6 calculated eigenvalues and transverse vibrations of a belt drive system. Ghayesh and Ahmadian7,8 studied the transverse vibrations of the belt drive system based on the multiple scales method. Pan9 proposed an efficient method to evaluate the natural vibration characteristics of the belt drive system. Kong10 used the Galerkin method to obtain the natural vibration characteristics in the space domain and time domain. Shangguan11 obtained the natural frequency of transverse vibrations of the belt drive system based on a single moving beam model. It is observed that all the above transverse vibration analyses only focus on the existing structure rather than the optimization issues.

By the surrogate model technology and the genetic optimization algorithms, the belt vibration has therefore been an area of active study.12 Nurudeen A. Raji2 used the response surface technology to carry out the optimization design of the structural parameters for the belt drive system. Nouri and Zu13 chose the belt tension as the objective function to optimize the analysis based on sequential quadratic programming (SQP) and Kuhn-Tucker method, but they ignored the impact of structural parameters on the transverse vibration. Hou14 proposed a method to search for the optimum design, but it is only in the optimization of the algorithm. All the researchers above focused on new methods or new algorithms, but ignored the coupled effects of different disciplines. In this paper, the MDO framework is adopted, which takes into account the performance of the transverse vibrations in the design of the belt drive system.

The reminder of the paper is organized as follows. Section II develops mathematical model of the transverse vibration for the belt drive. The MDO framework is established in Section III, including the system optimization objective, sub-systems analyses and restrictions. Section IV establishes the three-dimension (3D) belt drive system. The transverse vibration performance is discussed based on MDO results in section V. Section VI gives the concluding remarks.

The belt drive system is usually designed for some specific functions and used for the power transmission from one driving pulley to the other. The system relies on the friction between the pulley and the belt for power transmission. There are many advantages using the belt system, such as the vibration absorption and the long-distance transmission. However, many disadvantages also exist, for example the transverse vibrations. To study the dynamic performance of the belt, the theoretical model and kinematic equations of the belt drive are established.

In the belt drive system, the belt is fitted with a certain initial tension on the two pulleys, and the belt has a certain degree of elasticity, so that the transverse vibration of the belt drive is similar to the string vibration. In order to describe the transverse vibration of the belt drive, the center line of the two pulleys is set to be the X axis, and the center of the driving pulley is set to be the origin of the coordinate system. When the side of the belt is subjected to the transient mechanical action, the belt undergoes free vibration, and the distance from the coordinate system is x, and the micro segment of length is dx, as shown in figure 1.

From Figure 1, it is seen that the belt drive is not steady. The kinematic equations are given by:

Qcosθx,tQ+Qxdxcosθx+dx,t+Tsinθx+dx,tTsinθx,t=ρ2y2tM+MxdxMQ+Qxdx=0
(1)

As the vibration during the drive is very small, the following equations can be obtained.

sinθx,tyx,tx
(2)
sinθx+dx,tyx,tx+2yx,tx2
(3)
cosθx,t=cosθx+dx,t1
(4)

where θ is the angle between the center line and the belt and y(x,t) is the transverse vibration at the instantaneous time t.

The relationship between the bending moment, the bending stiffness and the deformation can be known from the material mechanics, the equation is given by:

EI2yx,t2x=Mx,t
(5)

Combining equations (3), (4), and (5), the kinematic equations can be obtained as:

EI4y4xT2y2x+ρ2y2t=0
(6)

Assuming that there is no eccentricity and both ends of the vibrating belt are simplified as hinged ends, the following boundaries are satisfied when the belt is transversely vibrating.

y0,t=ya,t=02y0,t2x=2ya,t2x=0
(7)

The vibration of the belt can be expressed by the following equation:

yx,t=YxHt
(8)

By substituting equation (7) into equation (6), according to the principle of the superposition of solution, the general solution of equation (6) is given by:

yx,t=n=1Ancosωst+Bnsinωstsinnπxa
(9)

where

ωs2=EIρnπa4+Tρnπa2
(10)

In order to investigate the natural frequency and the natural vibration of the transmission vibration of the belt drive system, the free vibration equation of the system is obtained by ignoring the damping factors in the system:

mÿ+Ky=0
(11)

Usually, equation (11) has the following form of results:

y=usinφt+θ
(12)

where

u=u1u2u3unT
(13)

By substituting equation (12) into equation (11), the following equation can be obtained by the deformation equation:

Kφ2mu=0
(14)

The frequency of the transmission vibration and the vibration mode of the belt drive can be obtained by equation (15):

m1Ku=φ2u
(15)

The MDO’s essence is the method and the tool that integrate multiple disciplines for the analysis and the calculation. In order to obtain the optimal values of the system optimization objectives, MDO utilizes the synergies between disciplines to achieve parallel optimizations.

findxminfx,ux;s.t.  hix,ux=0,i=1,2,,m    gjx,ux0j=1,2,,nSx,ux=S1x,u1xuNxSNx,u1xuNx
(16)

where x is the design variables; u(x) is the state variables of the system; f(x,u(x)) is the overall objective of MDO; hi(x,u(x)) is the equality constraint; gj(x,u(x)) is the inequality constraint; and S(x,u(x)) is the equation of state.

MDO has a variety of solving methods which can be divided into two main categories: single-level formulations and multi-level formulations. This paper uses the CO method to solve MDO problems. As a widely used MDO method, the CO method has a high computational efficiency. As illustrated in figure 2, the CO method can maintain a high self-consistency during the analysis and optimization of sub-systems or sub-disciplines, which makes CO a suitable framework for parallel processing especially when solving a large system.18 

The system level of the CO method is restrained by the residual error. The CO method can be given by:

minfx=f1x+f2x+f3xs.t.J1x=x1y112+x2y122=0  J2x=x1y212+x2y222=0  J3x=x1y312+x2y322=0
(17)

where y11, y12, y21, y22, y31, and y32 are the auxiliary variables of MDO, respectively.

To overcome the computational challenge and find efficient optimization algorithms, the CO method was adopted in this study. The CO method can optimize multiple disciplines at the same time and has a high computational efficiency. The flow chart of CO is shown in figure 3.

1. The model of sub-systems

The belt drive system is usually designed based on special working conditions. In the traditional design process, the designers usually focus on the function of the product, rather than the overall performance of the product. In order to propose a MDO method for dynamic performance analysis, this paper selected a low-power belt drive system as a case study.

The basic equations of the belt drive are given by:

i=n1/n2d2=id11εv=πd1n160×1000Ld0=2a0+π2d1+d2+d2d124a0a=a0+LdLd02α1=180°d2d1a×57.3°Ki=i21+i5.3Ka=1.2515α/180KL=1+0.45×lgLdL0P=v×K1v0.09K2d1KiK3v2KaKLKi=i×21+i5.31/5.3Z=Pd/P
(18)

where v is the velocity of the driving pulley and Z is the number of belts.

To reduce the power loss caused by bending stress and improve the lifetime of the belt, the diameter of the pulley is set as a design variable. Based on the characteristics of the belt drive system, the design variables of this paper are given by:

x=x1x2T=d1LdT
(19)

As the power increases with the increasing diameter of the pulley, the influence of the pulley diameter should be firstly taken into consideration. The diameter of the pulley is required to meet the overall dimensions. The parameters of the belt drive system are shown in Table I.

Combining the characteristics of the belt drive and the MDO, the system can be divided into three sub-systems, including the structure, the energy consumption, and the vibration. For the structure sub-system, miniaturization is the optimization objective. The structural subsystem can be given by:

findxminf1x=2Ldπd1i+142d1i122s.t.gix0i=1,2,3
(20)

The energy consumption is one of the evaluable criteria for the system performance. Reducing the number of belts can reduce the friction and energy loss, for this purpose, the subsystem can be given by:

findxminf2x=KAPP0KaKL+ΔP0s.t.gix0i=1,2
(21)

The vibration of the belt system is one of the vibration sources of the whole system. In order to reduce the vibration, the system energy is set as one of the sub-disciplines. The system energy can be given by:

findxminf3x=n=1Ancosωst+BnsinωstsinnπxLs.t.gix0i=2,3
(22)

where

ωs2=EIρnπL4+TρnπL2
(23)

2. Formulation of the constraints

The circumference force of the belt drive is proportional to the wrap angle of the pulley. In order to ensure transmission efficiency, the wrap angle of the pulley need to meet the design requirements.

g1x=120°180°i1ax1×60°0
(24)

Besides many advantages, the belt drive also possesses some disadvantages, such as the belt sliding, the inaccurate speed ratio and large volume. Normally, the power of the belt drive is proportional to the belt velocity. As the belt velocity increase, the belt’s power will reduce. To ensure the belt does not slip on the pulley, the velocity of the belt should be kept within a reasonable range. The constraint of the belt velocity is given by:

g2x=πx1n160×100015100
(25)

Increasing the center-to-center distance can not only improve the transmission efficiency, but also reduce the belt fatigue rate. The center-to-center distance is restrained as follows:

g3x=a7000
(26)

3. Results

Crossover probability and mutation probability are the key factors that affect the behavior and performance of the genetic algorithm. Individuals with high fitness are more likely to be protected from entering the next generation than those with low fitness. On the contrary, they are more likely to be eliminated. The improved genetic algorithm is the adjustment of the cross probability and the mutation probability.

Pc=k1fmaxffmaxfavg,ffavgk2,      f<favg
(27)
Pm=k3fmaxffmaxfavg,ffavgk4,        f<favg
(28)

where,

fmax is the maximum fitness value in population;

favg is the average fitness value in population;

f is the larger fitness value in two individuals to cross;

f′ is the fitness value of the individual variation;

ki(i = 1,2,3,4) is a constant in [0,1].

After determining the optimization objective and constraints, the MDO problem is programmed. Genetic algorithm is used in the system and subsystem level of the CO method. The results are listed in Table II.

As can be seen, the optimal values of the pulley’s diameter, the length of the belt and the center-to-center distance are 97mm, 1403mm, 618mm, respectively. The CO method not only reduces the size of the belt drive system, but also improves its performance. It is seen that the MDO method can solve the optimal problem between multiple disciplines, and provide the optimal parameters for studying the vibration characteristics of the belt drive system.

By comparing the initial value and corresponding optimized value of the MDO, the diameter of the pulley and belt length have been greatly reduced. To investigate the effect of velocities and loads on transverse vibration, this paper focuses on the numerical simulation of MDO model.

The pulley is usually used to install the belt, which belongs to the hub components. The belt is flexible, so there are several methods to create the belts. The first method is to establish the belt by using the professional FEA software. Second, the belt can be modeled to use ADAMS/Autoflex module. In addition, the flexible beam connection method: one object is discrete into many segments, then this flexible beam is used to connect these rigid bodies together.15–18 

To obtain the accurate model, the model in virtual prototyping environment is established. This method can not only guarantee the accuracy of the pulley, but also easily modify parameters. Assumptions in modeling of a belt are:19 

  1. The belt is uniform;

  2. Vibration amplitude is very small, vibration characteristics are linear;

  3. The bending stiffness of the belt is ignored.

The belt drive system is shown in figure 4.

Set the constraints between the belt and the pulley in the belt drive system. Meanwhile, the direction of the gravity acceleration is along the negative Y-axis, and its value is 9.80665 m/s2. The constraints of the belt drive system are shown in Table III.

where R represents the rotation and C represents the contact.

ADAMS/Solver using multi-step integration methods that contains a predictor and a corrector20–22 is used in this paper to conduct the optimization. In order to reduce the computational cost and get accurate results, this work adopts the Gear Stiff integrator (GSTIFF) and the integrator formulation (SI1), which take into account the constraint derivatives when solving the equations of motion and monitor the integration error on the impulse of the Lagrange Multipliers in the system.22 The integration tolerance in this paper is set as 0.001, the simulation time is 5s and the step size is 0.01s.

To study the transverse vibration characteristics of the belt drive, the belt drive system is designed based on the results of the CO algorithm. The belt drive is vibrating during the operation, the main reason is that the tension on both sides of the pulley is not equal. This work compares the results of the transverse vibration with respect to different velocities. This paper focuses on the vibration of the tight side. The influence of different velocities on the belt drive system are shown in figure 5.

Figure 5 illustrates that the amplitude of the transverse vibration decreases as the velocity increases. The amplitude range on the belt is between -4 mm and 6 mm, the maximum amplitude occurs at 900 r/min. When the velocity of the active pulley increases to 1200 r/min, the vibration is similar to the vibration illustrated in figure 5(a). The amplitude range on the belt is between -4 mm and 4 mm in Fig. 5(b). To obtain the range of vibration frequencies of the belt drive system, figure 4 is obtained by the Fast Fourier Transformation (FFT). The frequency curves are shown in figure 6.

From figure 6, it is seen that the amplitude of the vibration frequency increases as the velocity increases. Figs. 6(a) and 6(b) have many frequencies, including the natural vibration frequency and the multiplier of the belt drive. The maximum amplitudes happen at 89.7 Hz and 204.1 Hz under different velocities, respectively. However, the vibration frequencies are similar to each other, which are all around 20 Hz.

In Figs. 7(a) and (b), it is seen that the velocity does not conduct a smooth movement but with many vibrations. The velocity of the belt in the Y direction is much larger than that in the X direction. The velocity in the Y direction is between 5 mm/s and 45 mm/s. Figs. 7(c) and (d) show that the velocity in the X direction is between 156 mm/s and 164 mm/s and velocity in the Y direction is between 5 mm/s and 45 mm/s. However, the amplitude increases as the velocity increases. To study the influence of different loads on the transverse vibration of the belt drive system, the loads of 300N and 800N are selected for study.

Figure 8 illustrates that the amplitude of the transverse vibration decreases as the velocity increases. When the load increases to 800N, the vibration as illustrated from figure 8(b) is only half of that from figure 8(a). The amplitude of the belt drive system under different loads is between -2 mm and 4 mm as illustrated in figure 8. To obtain the range of vibration frequencies of the belt drive system, figure 9 is obtained by the Fast Fourier Transformation (FFT). The frequency curves are shown in figure 9.

From figure 9, it is seen that the amplitude of the vibration frequency increases as the loads increases, which is similar to the influence of the velocity. Figs. 9(a) and 6(b) have many frequencies, including the natural vibration frequency and the multiplier of the belt drive. The maximum amplitudes happen at 169.3 Hz and 106.5 Hz under two different velocities, respectively. Compared with the no-load conditions, the high-order frequency of the belt drive system is more obvious when there is load.

In Figs. 10(a) and (b), it is seen that the velocity does not experience a smooth movement but with many vibrations. The velocity of the belt in the Y direction is much larger than that in the X direction. The velocity in the Y direction is between 5 mm/s and 45 mm/s. Figs. 10(c) and (d) show that the velocity in the Y direction is similar to the velocity illustrated in Figs. 10(a) and (b). From the previous analysis, the transverse vibration of the belt drive system increases as the velocity and loads increases.

In this paper, a multidisciplinary design optimization (MDO) method is adopted to analyze the transverse vibration performance of the belt drive system. This paper proposes a new optimization method and takes a belt drive design as a case. The mathematical model of the belt drive system is established and the transverse vibration analysis is carried out. Meanwhile, the belt drive system is decomposed into three sub-systems according to the disciplines, and each of the sub-systems model is analyzed. The model of the belt drive system is established and carried out for dynamics simulation based on the MDO results. By comparing the results before and after the optimization, the MDO method can effectively reduce the amplitude of the transverse vibration. This paper focuses on the transverse vibration performance of the belt drive MDO model. The results show that the transverse vibration of the tight side decreases as the pulley velocity increases, and the result of slack side is opposite to the tight side. The MDO method can not only greatly reduce the cost, but also achieve the minimum vibration while taking into account the size of the components. The design of the belt drive system is given as an example, which demonstrates that the methodology is helpful to the product design.

This work is supported by National Natural Science Foundation of China (Grant No. 51505061 and No. U1608256).

An, Bn

A constant associated with the order of the fundamental frequency, respectively

a

Center-to-center distance, mm

a0

Normal center-to-center distance, mm

d1, d2

Diameter of the active pulley and the passive pulley, respectively, r/min

EI

The flexural rigidity of the belt, N·mm2

H(t)

Function relationship between the distortion value and the time

i

Transmission ratio

K

The stiffness matrix of the belt drive system;

Ka

Angle correction factor

Ki

Transmission coefficient

KL

Correction factor of the belt

k1, k2, k3

The correlation coefficient of the belt

L0

Specified length of the belt, mm

Ld0

Initial length of the belt, mm

Ld

Standard length of the belt, mm

M

Bending moment of the belt, N·mm

m

The mass matrix of the belt drive system

n

The order of the fundamental frequency, Hz

n1, n2

Velocities of the driving pulley and the driven pulley, respectively; r/min

P

Rated power of the belt, kW

Pd

Rated power, kW

Q

Shearing stress, N

T

Pretightening force, N

t

Time, s

u

The matrix of the mode vector

x

The distance from a point to the y-axis, mm

y

The displacement of transverse vibrations, mm

Y(t)

Spatial state function of belt vibration

Z

Number of the belt

α1

Angle of the pulley, °(degree)

ε

Elastic sliding rate

θ

Initial phase angle, °(degree)

ρ

Mass per unit volume of the belt, kg/m3

φ

The frequency of the simple harmonic motion, Hz

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