A noncircular face gear (NFG) conjugated with a pinion is a new type of face gear which can transmit variable velocity ratio and in which two time-varying excitations exist, namely the meshing stiffness excitation and instantaneous center excitation. Considering the tooth backlash, static transmission error and multifrequency parametric excitation, a nonlinear dynamic model of the NFG pair is presented. Based on the harmonic balance method and discrete Fourier transformation, a semi-analytic approach for the nonlinear dynamic model is given to analyze the dynamic behaviors of the NFG. Results demonstrate that, with increase in the eccentric ratio, input velocity and error amplitude, the NFG will undergo a non-rattle, unilateral rattle and bilateral rattle state in succession, and a jump phenomenon will appear in the dynamic responses when the rattle state of the gears is transformed from unilateral rattle to bilateral rattle.
Noncircular face gear (NFG) drive is a new type of variable transmission ratio mechanism with the advantages of light weight, high interchangeability and convenient installation (Liu et al., 2017). It is widely applicable in agricultural machinery, robots, automobiles, automatic machines and so on. Generally, irregular rattle vibration often appears in a gear system in light load or no load conditions, which not only has negative effects on the dynamic characteristics of the gears but also causes an unpleasant rattling noise. Due to the variable transmission ratio, the rattle vibration occurs in the noncircular gear transmission system more easily, which is a key problem to be solved urgently for noncircular face gears.
The rattle vibration of gears is a strongly nonlinear dynamic behavior caused by the tooth backlash. The vibration shock model, based on the lumped parameter method, is generally applied in theoretical research. The gear pair is a basic element in gear systems. Correspondingly, the torsional vibration model is the fundamental form of the dynamic model of gears. Comparin et al. presented a single degree of freedom nonlinear model for the purely torsional vibration and found three rattle states of gears (Comparin and Singh, 1989). By considering the stiffness of the bearings and shafts, a coupled translation–rotation vibration model is proposed by Kahraman and Singh (1991) to investigate the amplitude frequency features and chaotic vibration. On the basis of the coupled dynamic model of a single stage gear, Zhang et al. (2003a, b) introduced an impact equation to describe the sudden change in the dynamic behaviors of the gear when tooth impact occurs and indicated the influences of the speed fluctuation and unbalanced mass on the rattle vibration. For helical gears, a lateral–torsional–axial coupled nonlinear dynamic model was proposed to discuss the effects of system parameters on the vibration and the bifurcation behaviors by Wei et al. (2013).
Based on the dynamics of a single-stage gear drive, the dynamic characteristics of planetary gear systems were deeply studied. In the works of Al-shyyab and Kahraman (2007), a semi-analytical approach, based on the harmonic balance method, discrete Fourier transformation and Newton's method, is developed to solve the nonlinear torsional vibration model of the planetary gears. Nikolic-Stanojevic et al. (2013) presented a new fractional order model to describe the planetary gears, with the fractional order mode analyzed. The nonlinear dynamic behaviors and the effects of parameters on the dynamic responses of planetary gears were studied by Bahk and Parker (2011), who found that teeth separation still occurs even if the planetary gears are under heavy load. Wu et al. (2011) applied a harmonic balance method to obtain the steady-state solution of nonlinear dynamic model of the compound planetary gear train and analyzed the influences of the stiffness, clearance and errors on the dynamic features. In addition, since the gear rattle severely influences the NVH (noise, vibration and harshness) quality, the dynamics of an automotive transmission gearbox attracted broad attention all the time. Dong et al. (2004) proposed a rattle vibration model of gears comprising both the teeth clearance and the clutch clearance. Shangguan et al. (2018) studied the influences of the clutch on the rattle vibration of the transmission system and presented a method for reducing the gear rattle by controlling the stiffness and damping of the clutch. The works of Bozca (2018) on the dynamics of the gearbox indicated that the transmission errors can be reduced by the optimization of the module, number of teeth and backlash to decrease the rattle noise.
To date, plenty of studies have reported on the backlash nonlinear dynamics of different kinds of gear systems, which state that many complex nonlinear dynamic phenomena in gear systems exist, such as the multifrequency response, jump phenomenon, multivalue response, bifurcation and chaos. These results not only establish a theoretical basis for the dynamic design of high quality gears but also provide a useful reference for the dynamic research of noncircular gears. A comparison of internal excitations between circular and noncircular gears was made in the literature (Liu et al., 2016), in which a torsional vibration model was presented to investigate the parametric vibration characteristics of planar noncircular gears. Through an experimental method, Liu et al. (2012a, b) tested the vibrational performance of elliptical gears under different rotational velocities and torques. Then they proposed a multi-degree-of-freedom torsional vibration model of the experimental prototype of elliptical gears, with a consideration of the tooth clearance, and calculated the dynamic responses by the numerical method (Liu et al., 2013). The curve face gear is a new kind of spatial gear mechanism with a variable transmission ratio. Lin et al. (2015) constructed the nonlinear torsional vibration model of the gears and pointed out that multi-periodic, quasi-periodic and chaotic vibration phenomena appear under different mesh frequencies. Furthermore, they used the bond graph theory to establish a coupled dynamic model of the spatial noncircular gear system, with dynamic efficiency (Lin et al., 2016) and nonlinear dynamic features (Cai and Lin, 2017) being analyzed. The dynamics of the noncircular planetary gear train was studied in the literature (Yuan et al., 2018), including the torsional vibration model and dynamic behaviors under different loads and speeds.
It can be seen from the above studies on the dynamics of noncircular gears that the time-varying instantaneous center excitation is a particular internal excitation for noncircular gears, which leads to more complex vibration behaviors. Compared with the existing noncircular gears, the noncircular face gear drive has different time-varying instantaneous center excitation and dynamic characteristics. To reveal its dynamic behaviors, the nonlinear dynamic model of the noncircular face gear is presented, with the consideration of the tooth backlash, the time-varying meshing stiffness and the instantaneous center in this article. A semi-analytical approach, based on the harmonic balance method, discrete Fourier transformation and Newton's method, is utilized to obtain the periodic steady-state responses of the gear. The effects of the input speed, static transmission error, eccentric ratio, meshing stiffness and load torque of the gears on their rattle vibration are analyzed in detailed. The experimental results verify the correctness of the theoretical analysis on the dynamic behaviors of NFG.
Combining the transmission features of noncircular gears and face gears, a new gear mechanism, comprising a pinion and a noncircular face gear with orthogonal axes, is presented as shown in Fig. 1.
Noncircular face gear drive.
In the engagement of the NFG pair, the pitch cylinder of the pinion is tangent to the pitch surface of the NFG, as illustrated in Fig. 2. On the pitch surface of NFG, there is a noncircular closed curve called a pitch curve, which keeps pure rolling on the surface of the pitch cylinder of pinion; thus, the transmission ratio of the gear pair can be written as follows:
Pitch surfaces of NFG pair.
In conventional machines, a serial mechanism composed of a pair of cylinder gears and a pair of noncircular gears is usually used to both reduce the rotational velocity of the motor and achieve a variable output speed, as shown in Fig. 3. The NFG pair in Fig. 1 can implement the function of the gear train set in Fig. 3. Using the NFG pair to replace the serial gear mechanism in mechanical equipment could reduce half of the weight and space and improve the efficiency of the transmission system (Liu et al., 2017).
Gear train composed of circular and noncircular gears.
The mathematical models of the pitch curve and the tooth surface of the NFG were given in the literature (Liu et al., 2019) and are not repeated here. The main topic of this paper is the nonlinear dynamic behavior of the NFG under multifrequency periodic excitation.
The NFG is a special kind of noncircular gear. However, it still transmits power by tooth meshing like circular gears. So, there are three typical kinds of internal excitations in the NFG, namely the meshing stiffness, the static transmission error and the meshing shock excitation. In addition, the position of the relative instantaneous center varies with the rotation of the gears; hence, a special internal excitation appears in gears with the changeable transmission ratio, which is named the instantaneous center activation (Liu et al., 2019).
The NFG pair is simplified to the model shown in Fig. 4. The pinion is modeled as a cylindrical rigid body, with constant rotational radius
Simplified model of NFG pair.
We define the rotational angle of the NFG pair yielding to the theoretical
transmission ratio as the rigid angle. The rotational angle caused by the
elastic formation of teeth is specified as the elastic angle. In the
engagement, the real rotational angle of the gears is the sum of the rigid
and the elastic angle. When the NFG pair rotates in the theoretical transmission ratio
Then, the relative displacement of the spring in Fig. 4 can be written as follows:
Consider a lumped parameter model consisting of the meshing stiffness, the
meshing damping, the static transmission error and the tooth backlash of the
NFG pair in Fig. 5.
Dynamic model of the NFG pair.
Using Newton's second law of motion, the differential equations of motion of
the NFG pair are deduced as follows:
Let the pinion rotate at a given speed. The deformations of the contacting
teeth of the two gears only generate the elastic rotational angle of the
NFG. Substituting
Assume that Eq. (9) has a periodic solution
Substituting Eqs. (11)–(12) into Eq. (9), we can obtain a algebraic equation
set,
In the algebraic equation set
First, discretize the Eq. (13) by taking the following:
Set the transmission ratio of the NFG pair as follows:
Since the tooth shapes of the NFG are complicated and different from each
other, the finite element method is applied to compute the mesh stiffness of
the gears. Figure 6a shows a single-tooth model of the NFG in a finite
element software, which is constructed based on the meshing theory in the
literature (Liu et al., 2019). The teeth of the NFG and the pinion contact
at a point. A unit normal force is added at the meshing point of the two
tooth surfaces. Correspondingly, the tooth deformation is calculated by the
finite element software, as shown in Fig. 6b. We extract the deformation data,
Finite element model of a single tooth of the NFG.
Dealing with the contacting teeth of the NFG and the pinion as serial springs, we can obtain the meshing stiffness curves of a pair of teeth of the gears. In an engagement, two pairs of the meshing teeth can be represented by two parallel springs. According to the alternation rule of one and two pairs of teeth, the synthesizing meshing stiffness curves of the gears are obtained, as shown in Fig. 7, where the rotational speed of the pinion is 1 rad per second. Due to the asymmetry of the teeth of the NFG, the stiffness curve in every meshing period is various.
Synthesizing mesh stiffness of the NFG pair.
In most cases, the harmonic of the fundamental frequency of excitations
plays a leading role in the steady-state response of the dynamic system,
which is also given special attention in practice. With the consideration of the convenience of the calculation and actual requirement in engineering, the
expressions of the meshing stiffness and static transmission error are
rewritten on the basis of Eqs. (6)–(7) as follows:
Parameters of the NFG pair.
The dynamic equation in Sect. 3 is utilized for the analysis of the
nonlinear rattle behavior of the NFG. Since the expressions of the meshing
stiffness and the transmission error take the first-order harmonic alone,
the variable
Figure 8 illustrates the time domain diagram, phase diagram and the
frequency–amplitude diagram of the relative displacement between gears. It
can be seen from Fig. 8a that the values of the relative displacement are
negative or zero. At the same time, there exits a sudden change in relative
velocity at
Dynamic response with
Reduce the input velocity
Dynamic response with
When the input speed
Dynamic response with
From the above analysis, it can be seen that, under the combined excitation of the time-varying instantaneous center and meshing stiffness, the vibration behavior of a non-circular face gear is more complex than that of a circular gear. The change in velocity will lead to different rattle vibration behaviors. In order to further reveal the evolution law of the rattle vibration behavior of non-circular face gear, the influences of parameters such as the input speed, eccentricity, static transmission error, meshing stiffness and load on the rattle behavior are studied by the control variable method below. The parameter rsv is introduced to describe the rattle state. When rsv is 0, 1 or 2, it represents the no-rattle, unilateral rattle and bilateral rattle of gears respectively. In addition, ppv is referred to as the peak-to-peak value of the steady-state response that is used to reflect the degree of gear vibration. By calculating the rsv and ppv of NFG drive under different system parameters, the evolution characteristics of the rattle vibration behavior of the gear were analyzed.
Let the rotational velocity of the pinion increase from 100 to 250 rad per second, and let the values of the other parameters in Table 1 be invariable. Figure 11a shows the change in the rattle state along with
Change in vibration with input velocity.
The peak-to-peak value of the vibration of the NFG varying with
Figure 12a depicts the change in the rattle state, along with the amplitude
of the transmission error. As
Change in vibration with the amplitude of the static transmission error.
The variation in the rattle state with the eccentricity of the NFG is
investigated. Figure 13a shows that the NFG also undergoes the three rattle
states with the increase in eccentric ratio. The eccentric ratio is the
essential difference between noncircular and circular face gear. When
The peak-to-peak value of the NFG grows with the increase in
Change in vibration with eccentric ratio.
We define the ratio of
Change in vibration with the amplitude coefficient of meshing stiffness at different input velocities.
Let the load torque,
Change in vibration with load torque.
Figure 15b shows the ppv curve of the dynamic response of the NFG as the load torque increases. In the bilateral rattle state, the ppv grows slowly with the increase in
A NFG is manufactured by a three axis CNC milling machine, according to the design parameters in Table 1, whose vibration test platform is shown in Fig. 16. A variable frequency motor connects to the pinion with a pair of synchronous pulleys. The NFG is connected to a magnetic powder brake by a couple. Considering that the torsional vibration of the gears would propagate along the supporting bearings, we apply the rectilinear vibration of the bearings to reflect the torsional vibration of the NFG. There are two three-axis acceleration sensors fixed on the bearings of the pinion and the NFG. The one measures the radial vibration of the pinion, while the other measures the radial and axial vibration of the NFG. The vibration signals of the NFG under different working conditions can be tested by adjusting the motor velocity and brake torque.
Vibration test platform of the NFG.
Since the experimental results are the linear vibration data on the bearing seat in Fig. 16, and the theoretical results are the dimensionless torsional vibration data of the NFG, they cannot be compared with each other in the time domain. Figure 17 gives a comparison between the experiment and simulation results in frequency domain in which the rotational velocity of the motor is 864 rpm (revolutions per minute), the reduction ratio of the synchronous belt drive is 1.3, and the load torque is 3 N per meter. It can be seen from Fig. 17 that the theoretical frequency values of instantaneous center excitation and mesh stiffness excitation are 3 and 192 Hz, respectively, while the measured values are 3.2 and 194 Hz, respectively. In addition, the double frequency of the time-varying instantaneous center, the sum and difference frequency of time-varying instantaneous center and the meshing stiffness in Fig. 17a are close to those in Fig. 17b. The frequency components of the theoretical results are in good agreement with those of the experimental results.
Frequency–amplitude diagrams.
To compare the amplitude of the spectrum of the theoretical and experimental
vibration responses, a dimensionless parameter, called the ratio of amplitude, is introduced. Considering that the fundamental frequency plays an important role in the vibration of the system, the ratio of amplitude is expressed by the following:
Furthermore, the theoretical and experimental amplitude ratio data are illustrated in Fig. 18. With the increase in the motor speed, the two curves gradually match. The main reason for the larger error at low speed is that the excitation at low speed is relative small. The meshing vibration of the gear attenuates after it is transmitted along the bearing, which results in a larger error of the measurement data on the bearing seat. When the motor velocity exceeds 616 rpm, the experimental results are found to be in agreement with the calculated ones, which could verify the correctness and validity of the presented theoretical model and solution.
Comparison of theoretical and experimental results.
Comparison of the ratio of amplitude.
The internal excitations of a new gear pair comprised of a pinion and a
noncircular face gear are investigated. Considering the multifrequency
parametric excitation and backlash nonlinearity, the dynamic model of the
NFG pair is established, which is solved by harmonic balance method and
discrete Fourier transformation. The effects of the main system parameters
on the vibrational state of the gears are analyzed in detail. The results
show the following:
The instantaneous center excitation in the NFG pair is low frequency and time varying, which belongs to a parametric type of excitation. It compounds with the time-varying meshing stiffness of the gears to form a complex multifrequency parametric activation. The eccentric ratio, input speed and error amplitude have a great influence on the rattle state of the NFG pair. As the three parameters increase, the NFG undergoes non-rattle, unilateral rattle and bilateral rattle vibration in succession. The effect of the fluctuation of the meshing stiffness on the rattle state is smaller. Under some critical condition, the rattle state of the gear will change with the increase in the amplitude coefficient of the mesh stiffness. In the state of non-rattle and unilateral rattle, the peak-to-peak values of the dynamic response of the NFG grow gradually with the increase in the eccentric ratio, input speed and error amplitude. At the start of bilateral rattle vibration, a jump phenomenon occurs on the ppv curve, which means that the vibration of the gears is suddenly enhanced at this instant. In addition, ppv rises greatly with the increase in the amplitude coefficient of the mesh stiffness, which has a big influence on the vibration amplitude of the gears.
All data included in this study are available upon request from the corresponding author.
DL conceived the idea, developed the theory and acted as corresponding author. GZ performed experiments and simulations and analyzed the results. ZL wrote the paper.
The authors declare that they have no conflict of interest.
This article is part of the special issue “Robotics and advanced manufacturing”. It is not associated with a conference.
The authors would like to thank the National Natural Science Foundation of China (grant no. 51705444) and the University Science and Technology Research Project of Hebei Province (grant no. QN2020266) for financially supporting this work.
This research has been supported by the National Natural Science Foundation of China (grant no. 51705444) and the University Science and Technology Research Project of Hebei Province (grant no. QN2020266).
This paper was edited by Haiyang Li and reviewed by Chao Lin and two anonymous referees.