Abstract
Customers call for better performance, such as miniaturization, low noise, and higher load capacity of gear system. The microsegment gear is a new tooth form whose tooth profile curve is composed of many micro segments. This paper investigates the bifurcation and chaos characteristics of spur microsegment gear pair. To improve the authenticity of the solution, the timevarying mesh stiffness is expressed in term of piecewise function, and the normal profile deviation is expressed in the form of Fourier series. The influence of damping coefficient, excitation frequency, internal and external excitation on bifurcation and chaos properties of the system are analyzed. The numerical results show that the dynamic system is very sensitive to damping coefficient. With the increasing of damping coefficient, the number of bifurcation and impressive jump tends to decrease; and the intervals of dimensionless frequency leading to unstable or chaotic motion have a trend to concentration. The influence of external and internal excitation on bifurcation characteristics are also investigated. Comparison results show that, internal excitation has a greater effect than external excitation based on corresponding amplitudefrequency diagrams and bifurcation diagrams.
1. Introduction
Gears are the indispensable components of industrial machinery, automobile, ship, locomotive, airplane, and other machines. The customers require higher load capacity, rotating speed, powerweight ratio, lower vibration and noise of gearing.
Involute gears have been investigated systematically and comprehensively since 1694. The involute gear driving thus has developed to be the most important mechanical transmission gradually. Modern industry requires gears to implement miniaturization, as well as to improve powerweight ratio. Consequently, the shortages, such as relative sliding, convexconvex pattern of contact and minimum tooth number for gears without undercut, make involute gear difficult to meet the needs of modern industry equipment.
Relative sliding leads to friction loss between the contact teeth directly. The convexconvex contact pattern reduces contact strength. This effect is more significant as the rotating speed and load increases. The minimum tooth number for gears without undercut prevents the miniaturization of the system.
To overcome these shortages, Komori [13] proposed a new type of high strength gear profile named LogiX gear in 1980’s whose tooth profile is composed of many micro involute segments. The LogiX tooth profile has been known as that having concaveconvex pattern of contact like WN (WildhaberNovikov) tooth profile and the improvements are LogiX tooth profile can be applied to a spur/helical gear and have a feature of linecontact. While, Komori didn’t point out any actual application of this type of gearing could be limited by the processing problem.
Actually, in NC (numerical control) machining, computerized interpolation is employed in producing plane, cylinder or complex curved surface. A final tooth profile is composed of a large number of segments and the final profile will be closer to the designed position with an increase in the quantity of micro segment. Inspired by the NC machining technology and the design strategies of LogiX gear, Han proposed a new tooth profile named microsegment tooth profile. By employing micro segments to replace micro involutes, a microsegment gear can be easier processed in NC machining centers [4]. The author has carried out theoretical and experimental researches of microsegment in his PhD dissertation. The basic characteristics such as strength problem were researched; the effect of primary parameters on tooth profile shape was discussed; and the hob cutter was designed [5]. The studies on the temperature rise comparison [6], transmission efficiency calculation [7] and strength [5] show that microsegment gear has better static performance than involute gear.
Dynamic analysis is an effective method of predicting the system’s performance. Over the past few decades, a lot of researches have been conducted on gear dynamics [822], while most of them are centered on involute gears. Several of these studies are associated with noninvolute gear according to available literature. [23] developed a dynamic model of a singlestage cycloid drive and considered the dynamic behavior of cycloid planetary gear trains. [24] investigated the approach on parametric modeling and dynamic contact analysis of noninvolute beveloid gears which have advantages of relieving the high dynamic stress and contact shock problem of intersecting axes beveloid gear pairs. Recently, we employed the singledegreeoffreedom (SDOF) model for microsegment gear transmission with a small change [25]; the profile deviation between microsegment and involute gear is regarding as the displacement excitation. Timevarying mesh stiffness is calculated by finite element method. In reference [25], three working conditions were assumed to explore the difference between two kinds of tooth profile, and the contrast analysis result was shown that the microsegment gear has a better dynamic performance in highspeed and heavy load condition.
In this paper, the bifurcation and chaos characteristic will be explored. The influence of mesh frequency, internal and external excitation will be investigated mainly. We have great expectations to lay a foundation for microsegment gear dynamic theoretical system.
2. Dynamic model for microsegment gear
The constructing principle, presented in Appendix A1, indicated that, each micro segment involute (simplified as straight line finally) has its own base circle, and they are disciplinary changing from one micro segment to the next one, while the involute tooth profile has an invariable base circle. The feature of changing base circle means changing line of action and direction of mesh force. Thus, it is unacceptable to use the SDOF model directly. Here, an adjustment has been made to let the SODF model suit for microsegment gear, i.e. the tooth profile deviation between microsegment gear and involute gear is considered as another displacement excitation.
2.1. Mathematical representation of dynamic model for microsegment gear pair
The mechanical model is shown in Fig. 1. Two gears are combined with supporting shafts, which are regarded as rigid bodies with large support stiffness. The pinion and gear are represented by their base circles with radius ${r}_{bp}$ and ${r}_{bg}$, respectively. The equations of motion can be expressed as:
where, ${I}_{p}$ and ${I}_{g}$ are polar mass moments of inertia. ${m}_{p}$ and ${m}_{g}$ are masses. $K\left(t\right)$ is timevarying mesh stiffness. ${c}_{m}=2\zeta \sqrt{{k}_{m}/\left(1/{m}_{p}+1/{m}_{g}\right)}$ is viscous damping. $D\left(t\right)$ is timevarying profile deviation of microsegment gear comparing with involute gear. The static transmission error $e\left(t\right)$ is regarded as internal excitation which can be expressed in a Fourier series in the form:
Fig. 1a) Snap shot of contact pattern (at t= 0) of a spur microsegment gear pair; b) the nonlinear dynamic model
a)
b)
The external excitations ${T}_{P}$ and ${T}_{g}$ are acting on the pinion and gear which can be decomposed into average torque ${T}_{m}$ and fluctuating external torque excitation $T\left(t\right)$. To simplify the calculation, the fluctuation of output torque is neglected, i.e. ${T}_{g}=\mathrm{}{T}_{gm}$ and input torque can be expressed via Fourier series as:
The backlash function $h$ is usually calculated by:
By employing a composite coordinate:
Eq. (1) can be reformed as:
with:
where ${m}_{e}$ is the equivalent mass. ${F}_{m}$ is average force and ${F}_{p}\left(t\right)$ is the fluctuating force related to external torque excitation. ${F}_{e}\left(t\right)$ is the fluctuating force related to internal displacement excitation.
2.2. Two important parameters in dynamic model $\left(\mathit{K}\left(\mathit{t}\right),\mathit{D}\left(\mathit{t}\right)\right)$
The main difference between proposed dynamic model and the traditional SDOF is embodied in two important parameters $K\left(t\right)$ and $D\left(t\right)$.
2.2.1. Timevarying mesh stiffness $\mathit{K}\left(\mathit{t}\right)$
The timevarying mesh stiffness greatly affects the dynamics of gear system. According to the available literature, the mesh stiffness is usually simplified or calculated by the empirical equations. These methods are often based on the classical theory of elasticity and numerical approaches. However, it is considered to be unsuitable to use these methods on the dynamic research of microsegment gear due to the special profile. Therefore, the finite element method is utilized to calculate the mesh stiffness. With the design parameters of microsegment gears listed in table 1, the 3D finite element model for a pair of tooth of microsegment gear is shown in Fig. 2.
Fig. 2The entity model of microsegment gear
The 3D finite element model contains about 9840 elements and the tooth thickness in mesh model is set to 1mm. Four end faces and one cylindrical face are fixed, and the other cylindrical face is defined as the loading surface. Since the gear meshing is a nonlinear process, the contact surfaces are defined as nonlinear contacting to get a more realistic result.
Table 1Common design parameters of the spur gear pairs
Parameter  Pinion/Gear 
Number of teeth  45 
Transverse module (mm)  2 
Width (mm)  20 
Mass (kg)  0.885 
Moments of inertia (kg$\bullet $m^{2})  9.988e004 
Density (kg/m^{3})  9860 
Elasticity modulus (GPa)  205 
Poisson ratio  0.3 
In the meshing process of a gear set with a contact ratio $\epsilon \mathrm{}$lower than 2, we can conceptually consider that there are always two pairs of teeth (pair #1 and pair #2) under contact. In Fig. 1(a), the meshing teeth pair with contact point located between point A and point B is defined as pair #1, and the meshing teeth pair with contact point located between point B and point C is defined as pair #2.
Fig. 3The mesh stiffness of microsegment gear
The mesh stiffness for a pair of teeth from A to C, can be calculated by the FEM mentioned above, and can be expressed in the form of Fourier series expansion as follow:
where ${k}_{m}$ is average mesh stiffness. ${k}_{aj}$, ${k}_{bj}$ are amplitudes of the $j$th order harmonic. ${w}_{k}$ is fitting frequency; then we have $T=2\pi /{w}_{k}$, here $T$ is the period of the mesh stiffness $k\left(t\right)$ for one pair of teeth, rather than the timevarying mesh stiffness for the gear set. A good fit can be achieved with the first eight orders of the Fourier series, as show in Fig. 3 and fitting parameters are listed in Table 2.
Table 2Fitting parameters of stiffness (Nμm1 mm1) and synthetically normal deviation (mm)
$j$  ${k}_{aj}$  ${k}_{bj}$  ${d}_{aj}$  ${d}_{bj}$ 
1  –6.541e1  –2.101e4  –8.088e2  1.733e05 
2  –6.419e1  –2.997e4  2.627e2  –3.418e05 
3  –2.142e1  –1.55e4  6.399e3  5.01e05 
4  –6351e2  –6.556e05  7.992e4  –6.465e05 
5  –4.607e2  –6.824e05  9.012e4  7.744e05 
6  1.491e2  2.454e06  3.851e4  –8.812e05 
7  1.028e2  1.405e05  1.61e4  9.64e05 
8  8.566e3  –7.323e07  6.113e4  –0.000102 
${k}_{m}$  8.622  /  
${d}_{m}$  /  –0.01078 
Then, the timevarying mesh stiffness can be obtained by summing the stiffness of pair #1 and pair #2, as show in Fig. 3. According to Fig. 3, it can be found that the frequency of timevarying mesh stiffness function is $\epsilon {w}_{e}$, and the piecewise stiffness functions can be expressed as:
$=\left\{\begin{array}{ll}{k}_{m}+\sum _{j=1}^{8}\left(\begin{array}{c}{k}_{aj}\mathrm{cos}\left(j{\omega}_{k}\left(\mathrm{m}\mathrm{o}\mathrm{d}\left(t,\frac{T}{\epsilon}\right)+\frac{T}{\epsilon}\right)\right)\\ +{k}_{bj}sin\left(j{\omega}_{k}\left(\mathrm{m}\mathrm{o}\mathrm{d}(t,\frac{T}{\epsilon})+\frac{T}{\epsilon}\right)\right)\end{array}\right),& 0\le \mathrm{m}\mathrm{o}\mathrm{d}\left(t,T\right)<\frac{T\epsilon T}{\epsilon},\\ 0,& \frac{T\epsilon T}{\epsilon}\le \mathrm{m}\mathrm{o}\mathrm{d}\left(t,T\right)<\frac{T}{\epsilon},\end{array}\right.$
$=\left\{\begin{array}{ll}k\left(\mathrm{m}\mathrm{o}\mathrm{d}\left(t,\frac{T}{\epsilon}\right)+\frac{T}{\epsilon}\right)+k\left(\mathrm{m}\mathrm{o}\mathrm{d}\left(t,\frac{T}{\epsilon}\right)\right),& 0\le \mathrm{m}\mathrm{o}\mathrm{d}\left(t,\frac{T}{\epsilon}\right)<\frac{T\epsilon T}{\epsilon},\\ k\left(\mathrm{m}\mathrm{o}\mathrm{d}\left(t,\frac{T}{\epsilon}\right)\right),& \frac{T\epsilon T}{\epsilon}\le \mathrm{m}\mathrm{o}\mathrm{d}\left(t,\frac{T}{\epsilon}\right)<\frac{T}{\epsilon},\end{array}\right.$
$=\left\{\begin{array}{ll}\begin{array}{c}2{k}_{m}+\sum _{j=1}^{8}\left(\begin{array}{c}{k}_{aj}\mathrm{cos}\left(j{\omega}_{k}\left(\mathrm{m}\mathrm{o}\mathrm{d}\left(t,\frac{T}{\epsilon}\right)+\frac{T}{\epsilon}\right)\right)\\ +{k}_{bj}\mathrm{sin}\left(j{\omega}_{k}\left(\mathrm{m}\mathrm{o}\mathrm{d}\left(t,\frac{T}{\epsilon}\right)+\frac{T}{\epsilon}\right)\right)\end{array}\right)\\ +\sum _{j=1}^{8}\left(\begin{array}{c}{k}_{aj}\mathrm{cos}\left(j{\omega}_{k}\mathrm{m}\mathrm{o}\mathrm{d}\left(t,\frac{T}{\epsilon}\right)\right)\\ +{k}_{bj}\mathrm{sin}\left(j{\omega}_{k}\mathrm{m}\mathrm{o}\mathrm{d}\left(t,\frac{T}{\epsilon}\right)\right)\end{array}\right),\end{array}& 0\le \text{mod}\left(t,\frac{T}{\epsilon}\right)<\frac{T\epsilon T}{\epsilon},\\ {k}_{m}+\sum _{j=1}^{8}\left(\begin{array}{c}{k}_{aj}\mathrm{cos}\left(j{\omega}_{k}\mathrm{m}\mathrm{o}\mathrm{d}\left(t,\frac{T}{\epsilon}\right)\right)\\ +{k}_{bj}\mathrm{sin}\left(j{\omega}_{k}\mathrm{m}\mathrm{o}\mathrm{d}\left(t,\frac{T}{\epsilon}\right)\right)\end{array}\right),& \frac{T\epsilon T}{\epsilon}\le \mathrm{m}\mathrm{o}\mathrm{d}\left(t,\frac{T}{\epsilon}\right)<\frac{T}{\epsilon}.\end{array}\right.$
2.2.2. Timevarying profile deviation $\mathit{D}\left(\mathit{t}\right)$
The normal deviation of microsegment gear compared with the standard involute profile will be analyzed in this section. In Fig. 4, the geometric models of the single tooth for both microsegment gear and standard involute gear are established. It is clear that these two profiles are obviously different.
Fig. 4The geometric models for both microsegment gear and standard involute gear
To research the influence of different profiles on the dynamic characteristics of the gear system, the magnitude of deviation should be accurately expressed. In the process of meshing for a pair of teeth, the contact point of the pinion moves from start point to end point and the corresponding point in the gear moves from end point to start point. This process is divided into many small sections in terms of angle, and each microangle corresponds to a normal deviation, as show in Fig. 4. Due to the cooperative working of two gears, the displacement excitation of profile deviation $d\left(t\right)\mathrm{}$ should be the normal resultant for both of the gear and pinion body in Fig. 5.
Identical to the timevarying mesh stiffness, the synthetically normal deviation of a pair of teeth also can be expressed as following Fourier series expansion, the fitting parameters are also listed in Table 2:
where ${d}_{m}$ is average deviation of synthetically normal deviation. ${d}_{aj}$, ${d}_{bj}$ are amplitudes of the jth order harmonic. The timevarying profile synthetize deviation for the example system is also a piecewise function which can be expressed as follow:
$=\left\{\begin{array}{ll}\begin{array}{c}2{d}_{m}+\sum _{j=1}^{8}\left(\begin{array}{c}{d}_{aj}cos\left(j{\omega}_{k}\left(\mathrm{m}\mathrm{o}\mathrm{d}\left(t,\frac{T}{\epsilon}\right)+\frac{T}{\epsilon}\right)\right)\\ +{d}_{bj}\mathrm{sin}\left(j{\omega}_{k}\left(\mathrm{m}\mathrm{o}\mathrm{d}\left(t,\frac{T}{\epsilon}\right)+\frac{T}{\epsilon}\right)\right)\end{array}\right)\\ +\sum _{j=1}^{8}\left(\begin{array}{c}{d}_{aj}\mathrm{cos}\left(j{\omega}_{k}\mathrm{m}\mathrm{o}\mathrm{d}\left(t,\frac{T}{\epsilon}\right)\right)\\ +{d}_{bj}\mathrm{sin}\left(j{\omega}_{k}\mathrm{m}\mathrm{o}\mathrm{d}\left(t,\frac{T}{\epsilon}\right)\right)\end{array}\right),\end{array}& 0\le \text{mod}\left(t,\frac{T}{\epsilon}\right)<\frac{T\epsilon T}{\epsilon},\\ {d}_{m}+\sum _{j=1}^{8}\left(\begin{array}{c}{d}_{aj}\mathrm{cos}\left(j{\omega}_{k}\mathrm{m}\mathrm{o}\mathrm{d}\left(t,\frac{T}{\epsilon}\right)\right)\\ +{d}_{bj}\mathrm{sin}\left(j{\omega}_{k}\mathrm{m}\mathrm{o}\mathrm{d}(t,\frac{T}{\epsilon})\right)\end{array}\right),& \frac{T\epsilon T}{\epsilon}\le \text{mod}\left(t,\frac{T}{\epsilon}\right)<\frac{T}{\epsilon}.\end{array}\right.$
As shown in Fig. 5, the timevarying deviation curve is not continuous at the critical point of single and double teeth contact area, i.e., point D and point B. However, ${F}_{e}\left(t\right)$ is composed of the second derivative of $e\left(t\right)$ and $D\left(t\right)$ which means $D\left(t\right)$ should be expressed as a continuous differentiable formula. Thus, a further Fourier transform could be more than sufficient.
2.2.3. Nondimensionalization of the dynamic differential equation
As usual, the vibration differential equation of the gear system is processed to be dimensionless to assure the equation don’t depend on the specific physical dimension. Dimensionless equation of motion can be obtained by defining:
Fig. 5The timevarying normal deviation of microsegment gear compared with standard involute gear
Then the original equation of motion Eq. (6) can eventually be put in the normalized form:
$\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}+\sum _{r=1}^{\mathrm{\infty}}{\left(r{\mathrm{\Omega}}_{e}\right)}^{2}{\overline{e}}_{r}\mathrm{cos}\left(r{w}_{e}t+{\phi}_{er}\right)\ddot{\stackrel{}{D}}\left(\tau \right),$
where:
3. Numerical results and discussion
The dynamic performances of a geared system are much effected by the mesh frequency, internal excitation and external excitation. Moreover, these excitations are obviously different in the bifurcation characteristics.
3.1. Influence of excitation frequency
Following parameters are adopted in this section: the amplitudes of internal excitation ${\stackrel{}{e}}_{r}=0$; the external excitation is neglected here and average torque is controlled by input power $P=\mathrm{}$200 KW [26]; the backlash $b=\mathrm{}$20 μm.
The dimensionless nonlinear dynamic Eq. (13) of microsegment gear system with timevarying stiffness, profile deviation and backlash is solved using the fourth order RungeKutta method. The simulations run for 20,000 periods and only the data of the last 1020 periods are plotted to guarantee the data relates to steady state conditions. Then the sampled data is used to generate the bifurcation diagram, times history chart, phase diagram, Poincare map and FFT spectrogram of microsegment gear system in order to obtain an intuitive understanding of the dynamic behaviors.
Fig. 6 presents four bifurcation diagrams for dimensionless dynamic transmission error of microsegment gear system using dimensionless mesh frequency ${\mathrm{\Omega}}_{k}$ as bifurcation parameter. It can be observed that damping coefficient has a great effect on bifurcation behaviors of the gear system. The jump phenomena and chaotic area change much as damping coefficient increases.
Fig. 6Bifurcation diagrams using Ωk as bifurcation parameter: a) ζ= 0.06; b) ζ= 0.08; c) ζ= 0.1; d) ζ= 0.12
The bifurcation characteristic for the case of $\zeta =\mathrm{}$0.06 is shown in Fig. 6(a). It can be found that the motion is nonharmonicsingleperiodic (NHSperiodic) at low values of dimensionless mesh frequency, i.e. ${\mathrm{\Omega}}_{k}\le \text{0.446}$. The response period $T$ of time history chart equals to excitation period, the phase diagram is a closed noncircle and nonelliptic curve, the resulting trace in Poincare map is a dot and the discrete points locate at the frequencies of $m{f}_{k}$ (where, $m$ is a positive integer, ${f}_{k}={\mathrm{\Omega}}_{k}/2\pi $, similarly hereinafter) in corresponding FFT Spectrogram according to Fig. 7(a). In the meantime, the first distinct amplitude jump happens at ${\mathrm{\Omega}}_{k}=\text{0.41}$.
Fig. 7Simulation results for some important values of Ωk when ζ=0.06: a) NHSperiodic at Ωk=0.25; b) 2Tperiodic at Ωk=0.45; c) NHSperiodic at Ωk=0.552; d) 2Tperiodic at Ωk=0.668; e) chaotic at Ωk=0.72; f) 3Tperiodic at Ωk=0.752; g) 2Tperiodic at Ωk=1.182; h) 3Tpeiodic at Ωk=1.38; i) 2Tperiodic at Ωk=1.67; j) simple harmonic at Ωk=1.74
The motion mutates to be 2Tperiodic as ${\mathrm{\Omega}}_{k}$ reaches 0.448 and this status continues to ${\mathrm{\Omega}}_{k}=\text{0.464}$, as shown in Fig. 7(b), the dominant peaks in FFT Spectrogram are $i{f}_{k}/2$, where, $i$ is an even number. The frequency $j{f}_{k}/2$, where $j$ is an odd number, lead to two deflected and closed cures in phase diagram. After the second jump at ${\mathrm{\Omega}}_{k}=\text{0.464}$, the system comes to be repeatedly transform between NHSperiodic and 2Tperiodic motion until ${\mathrm{\Omega}}_{k}$ gets 0.492. The NHSperiodic motion lasts to ${\mathrm{\Omega}}_{k}=\text{0.66}$ and the third jump occurs at ${\mathrm{\Omega}}_{k}=\text{0.604}$.
As the dimensionless frequency ${\mathrm{\Omega}}_{k}$ further increases, the dynamic behavior of the system reverts to a 2Tperiodic motion again at $\text{0.662}\le {\mathrm{\Omega}}_{k}\le \text{0.682}$. Here, two closed curves appear in the phase diagram and the dominant peaks in FFT Spectrogram are $m{f}_{k}/2$ which is different with the formal 2Tperiodic motion.
After that, the motion evolves to be unstable, even chaotic in the range of $\text{0.684}\le {\mathrm{\Omega}}_{k}\le \text{0.738}$. When the dimensionless frequency ran up to 0.74, the system exhibits 3Tperiodic motion, as shown in Fig. 7(f), the trajectories repeat themselves every three periods, and there are three discrete points on the Poincare map. The corresponding FFT spectrogram has peaks at the points of $m{f}_{k}/3$, but the situation just last for a while, the system turns back to be chaotic.
The state of chaos changes to be stabilized slowly, the system presents obvious 2Tperiodic motion when ${\mathrm{\Omega}}_{k}$ reaches to 1.104, and its stability is getting better and better as ${\mathrm{\Omega}}_{k}$ increases until it comes to 1.186. The chaotic motions appear again in the range of $\text{1.186}\le {\mathrm{\Omega}}_{k}\le \text{1.37}$.
For dimensionless frequency $\text{1.372}\le {\mathrm{\Omega}}_{k}\le \text{1.638}$, the dynamic solutions are 3Tperiodic. What’s more, the system experiences an impressive jump at next moment, and its solution turns to be 2Tperiodic at $\text{1.668}\le {\mathrm{\Omega}}_{k}\le \text{1.696}$ after a brief struggle against unstable state ($\text{1.64}\le {\mathrm{\Omega}}_{k}\le \text{1.644}$ and $\text{1.652}\le {\mathrm{\Omega}}_{k}\le \text{1.666}$) and 2Tperiodic state ($\text{1.646}\le {\mathrm{\Omega}}_{k}\le \text{1.65}$).
Finally, at high values of the dimensionless frequency, i.e. ${\mathrm{\Omega}}_{k}\ge \text{1.698}$, the dynamic behaviors keep simple harmonic, with an ellipse in phase diagram, one point in Poincare map and one frequency in FFT spectrogram, as show in Fig. 7(j).
The frequency response amplitude for the system is shown in Fig. 8.
Fig. 8Frequency response amplitude at various mesh frequency for ζ=0.06
In the case of$\mathrm{}\zeta =\text{0.08}$, it can be seen that the maximum vibration amplitude decreases according to Fig. 6(b). The system performs NHSperiodic motion in the range of ${\mathrm{\Omega}}_{k}\le \text{0.472}$. An impressive jump happens at next moment, and the system presents 2Tperiodic motion at the same time. Before long, the motion turns back to be NHSperiodic start from ${\mathrm{\Omega}}_{k}=\text{0.49}$, with the second jump. As ${\mathrm{\Omega}}_{k}$ increases to 0.6, the motion becomes unstable, and bifurcates into 2Tperiodic motion at ${\mathrm{\Omega}}_{k}=\text{0.69}$ gradually.
As the dimensionless frequency further increases, the system comes into an unstable even chaotic state start from ${\mathrm{\Omega}}_{k}=\text{0.72}$, and then goes back to a stable 2Tperiodic motion slowly at ${\mathrm{\Omega}}_{k}=\text{0.94}$. After a distinct jump at ${\mathrm{\Omega}}_{k}=\text{1.024}$ and a chaotic area $\text{1.026}\le {\mathrm{\Omega}}_{k}\le \text{1.214}$, the 2Tperiodic motion evolves to be 3Tperiodic. What’s more, the3Tperiodic motion further evolves to be 6Tperiodic at ${\mathrm{\Omega}}_{k}=\text{1.33}$.
With another jump happens at ${\mathrm{\Omega}}_{k}=\text{1.39}$, and in the following range of $\text{1.392}\le {\mathrm{\Omega}}_{k}\le \text{1.632}$, the system is mainly under chaotic or unstable state, besides the simple harmonic state happens in $\text{1.542}\le {\mathrm{\Omega}}_{k}\le \text{1.546}$ and $\text{1.594}\le {\mathrm{\Omega}}_{k}\le \text{1.6}$, and 2Tperiodic state happens in $\text{1.616}\le {\mathrm{\Omega}}_{k}\le \text{1.62}$. Then, the system exhibits 2Tperiodic motion again in the range of $\text{1.634}\le {\mathrm{\Omega}}_{k}\le \text{1.692}$. It can be said that the system goes to 2Tperiodic motion after the variation between chaos state, simple harmonic state and 2Tperiodic state.
After that, the system leads to the area of simple harmonic motion.
The diagram of frequency response amplitude for the system is shown in Fig. 9.
Fig. 9Frequency response amplitude at various mesh frequency for ζ=0.08
In Fig. 6(c), the system also exhibits NHSperiodic motion at low value of dimensionless frequency, i.e. ${\mathrm{\Omega}}_{k}\le \text{0.176}$, during this time, the first impressive jump appears at ${\mathrm{\Omega}}_{k}=\text{0.506}$. Then, the motion bifurcates to be 2Tperiodic in the range of $\text{0.718}\le {\mathrm{\Omega}}_{k}\le \text{0.764}$, 4Tperiodic in the range of $\text{0.766}\le {\mathrm{\Omega}}_{k}\le \text{0.858}$, 2Tperiodic in the range of $\text{0.86}\le {\mathrm{\Omega}}_{k}\le \text{0.882}$ and 4Tperiodic in the range of $\text{0.884}\le {\mathrm{\Omega}}_{k}\le \text{0.952}$. The second notable jump happens at ${\mathrm{\Omega}}_{k}=\text{0.938}$. After that, the system tends to be unstable even chaotic until ${\mathrm{\Omega}}_{k}$ gets 1.134. A 3Tperiodic motion appears in the range of $\text{1.134}\le {\mathrm{\Omega}}_{k}\le \text{1.22}$. Subsequently, the dynamic solution goes to be chaotic again as ${\mathrm{\Omega}}_{k}$ rise from 1.222 to 1.472.
Fig. 10Frequency response amplitude at various mesh frequency for ζ=0.1
As the dimensionless frequency further increases, the system exhibits simple harmonic motion before the unstable motion presents again at ${\mathrm{\Omega}}_{k}=\text{1.488}$. Then a 2Tperiodic motion replaces the unstable motion at $\text{1.582}\le {\mathrm{\Omega}}_{k}\le \text{1.648}$. Finally, for the dimensionless frequency $\text{1.65}\le {\mathrm{\Omega}}_{k}\le \text{3}$, the system performs simple harmonic motion.
Fig. 6(d) reveals the bifurcation characteristics for $\zeta =\text{0.12}$ using dimensionless frequency ${\mathrm{\Omega}}_{k}$ as bifurcating parameter. Compared with the cases before, bifurcation characteristics here are much simpler. The system exhibits NHSperiodic motion in the range of ${\mathrm{\Omega}}_{k}\le \text{0.748}$, and then bifurcates to be 2Tperiodic in the range of $\text{0.75}\le {\mathrm{\Omega}}_{k}\le \text{0.906}$. The motion further bifurcates to be 4Tperiodic as ${\mathrm{\Omega}}_{k}$ increases to 0.908, and turn back to be 2Tperiodic momently at $\text{0.918}\le {\mathrm{\Omega}}_{k}\le \text{0.93}$. Subsequently, the system goes to chaotic area start from ${\mathrm{\Omega}}_{k}=\text{0.99}$ after a 4Tperiodic motion in the range of $\text{0.932}\le {\mathrm{\Omega}}_{k}\le \text{0.988}$.
Fig. 11Frequency response amplitude at various mesh frequency for ζ=0.12
The chaotic motion lasts for a long while until the simple harmonic motion replaces it at ${\mathrm{\Omega}}_{k}=\text{1.514}$. In the range of $\text{1.54}\le {\mathrm{\Omega}}_{k}\le \text{1.6}$, the motion shifts back and forth between 2Tperiodic (sometime stable and sometime unstable) and simple harmonic. Finally, the response keeps simple harmonic at high values of ${\mathrm{\Omega}}_{k}$, i.e. ${\mathrm{\Omega}}_{k}\ge \text{1.602}$.
The bifurcation behaviors for the case of $\zeta =\text{0.04}$ and $\zeta =\text{0.14}$ are also investigated using the same method. The plots are not provided here in order to be brief. According to the bifurcation diagrams for different values of the dimensionless frequency, we can find that, impressive jump happens less time as damping coefficient increases.
The unstable and chaotic intervals are much affected as well. According to the discussion above, we can find that the intervals lead to unstable or chaotic responses have a trend to concentration as damping coefficient increases, as shown in Fig. 12.
Fig. 12Intervals of dimensionless frequency lead to unstable or chaotic solution
Moreover, the system is more sensitive to the dimensionless frequency at low value of damping coefficient, i.e., the type of motion changes more frequent as dimensionless frequency increases.
The rotational speed that probably causes an unstable or chaotic should be avoided to ensure stability and reliability of the system according to bifurcation diagrams.
3.2. Influence of external excitation
The fluctuation of torque is an important component of external excitation. To research the influence of fluctuation of torque on the bifurcation characteristics, both the fluctuation amplitude and the fluctuation frequency need significant consideration. Here, the system parameters are given as: ${\stackrel{}{F}}_{m}=\text{0.05}$, $\xi =\text{0.25}$,${\overline{e}}_{r}=\text{0}$, ${\mathrm{\Omega}}_{k}=\text{0.2}$, $b=$20 μm.
Fig. 13 shows the bifurcation process of the system when ${\stackrel{}{F}}_{p1}=\text{0.02}$, ${\mathrm{\Omega}}_{p}={\rho}_{p}{\mathrm{\Omega}}_{k}$, here ${\rho}_{p}$ is used as control parameter.
Fig. 13Bifurcation diagram using ρp as bifurcation parameter, here ρp=Ωp/Ωk
According to Fig. 15, the system presents an interesting bifurcation process. Two excitation sources result in a quasiperiodic motion with the response frequency ${f}_{r}=i{f}_{k}+j{f}_{p}$, here ${f}_{k}={\mathrm{\Omega}}_{k}/2\pi $, ${f}_{p}={\mathrm{\Omega}}_{p}/2\pi $, $i$, $j=$ 0, ±1, ±2,…. So, some special values of $\rho $ will lead to special motions
For example, in the case of ${\rho}_{p}=$0, 1, 2, the system exhibits NHSperiodic motion. In the case of ${\rho}_{p}=$0.5, 1.5, the system performs 2Tperiodic motion. In the case of ${\rho}_{p}=$ 0.25, 0.75, 1.25, 1.75, the virbation response is 4Tperiodic. In the case of ${\rho}_{p}=$0.2, 0.4, 0.6, 0.8, 1.2, 1.4, 1.6, 1.8, the system presents 5Tperiodic motion. The corresponding frequency response amplitude diagram reveals the same situation, as shown in Fig. 14.
Fig. 14Frequency response amplitude at various external excitation frequency Ωp
Compared with the effect of external excitation frequency, the effect of external excitation amplitude is much simpler. The bifurcation diagram presented in Fig. 15 reveals that, at low values of ${\stackrel{}{F}}_{p1}$, the system performs NHS periodic motion. As ${\stackrel{}{F}}_{p1}$ increases, the motion turns to be 4Tperiodic due to the effect of external excitation frequency. What’s more, as ${\stackrel{}{F}}_{p1}$ further increases, the motion keep 4T periodic, the response amplitude is greatly affected.
The corresponding amplitudefrequency characteristic diagram presents the same situation, as shown in Fig. 16.
Fig. 15Bifurcation diagram using Fp1 as bifurcation parameter
Fig. 16Frequency response amplitude at various external excitation amplitude Fp1
It can be found that, the response frequency is composed of nondimensional frequency ${f}_{k}$ and external excitation frequency ${f}_{p}$; as ${\stackrel{}{F}}_{p1}$ increases, the 4Tperiodic motion is performed more obvious, and the amplitudes increase.
3.3. Influence of internal excitation
To investigate the influence of internal excitation on the bifurcation characteristics, the system parameters are assigned as follow: ${\stackrel{}{F}}_{m}=\text{0.05}$, ${\stackrel{}{F}}_{pr}=\text{0}$,$\xi =\text{0.25}$, ${\mathrm{\Omega}}_{k}=\text{0.2}$, ${\mathrm{\Omega}}_{e}={\rho}_{e}{\mathrm{\Omega}}_{k}$, ${\phi}_{er}=$ –0.72, $b=$20 μm. Here, both the influences of ${\mathrm{\Omega}}_{e}$ and ${\overline{e}}_{r}$ on bifucation characteristics are also investigated.
Fig. 17 shows the bifurcation characteristics of dimensionless dynamic transmission error by using ${\rho}_{e}$ as control parameter.
Similar to the bifurcation process using ${\rho}_{p}$ as bifurcation parameter, the response frequency of $\overline{x}$ also composed of the frequencies of two excitation sources, i.e. ${f}_{r}=i{f}_{k}+j{f}_{e}$. Here ${f}_{k}={\mathrm{\Omega}}_{k}/2\pi $, ${f}_{e}={\mathrm{\Omega}}_{e}/2\pi $, $i$, $j=$ 0, ±1, ±2,…. Similarly, special values of ${\rho}_{e}$ will lead to specail motions. In the case of, ${\rho}_{e}=$1, 2, 3, the system performs NHSperiodic motion. In the case of ${\rho}_{e}=$1.25, 1.75, 2.25, 2.75, the system exhibits 4periodic motion. In the case of ${\rho}_{e}=$1.5, 2.5, the system presents 2Tperiodic motion.
The difference is that, the response amplitude $\overline{x}$ has an obviously increase as ${\rho}_{e}$ increases, and when ${\rho}_{e}$ increases to 2.68, the system turns to be unstable. The corresponding amplitudefrequency characteristic diagram is shown in Fig. 18.
Fig. 17Bifurcation diagrams using ρe as bifurcation parameter, here e¯1=0.15
Fig. 18Frequency response amplitude at various internal excitation frequency Ωe
According to figure, it is clear that the response amplitude has a large jumping as ${\rho}_{e}$ increases, and the frequencies resulting in unstable motions are obviously increasing.
Fig. 19Bifurcation diagrams using e¯1 as bifurcation parameter, here Ωk=0.65
The bifurcation characteristics using ${\overline{e}}_{1}$ as control parameter is clearly presented based on Fig. 20. At the low values of ${\overline{e}}_{1}$, the system exhibits 2Tperiodic motion. As ${\overline{e}}_{1}$ increases to 0.04, the motion becomes to be unstable. As ${\overline{e}}_{1}$ further increases to about 0.185, the motion bifurcates to be 4Tperiodic. When ${\overline{e}}_{1}$ increases to 0.23, the system presents chaotic motion. According to the corresponding amplitudefrequency characteristic diagram, the unstable and chaotic phenomenon can be easily found. Another conclusion can be got, the growth of ${\overline{e}}_{1}$ has little influence on response amplitude, but has a great influence on the stability of the motion.
Fig. 20Frequency response amplitude at various internal excitation frequency e¯1
It can also be found that, the response frequency is composed of nondimensional frequency ${f}_{k}$ and internal excitation frequency ${f}_{e}$; as ${\overline{e}}_{1}$ increases, the system bifurcates from 2Tperiodic motion to 4Tperiodic motion. The system performs chaotic motion as ${\rho}_{e}$ increases to 2.68 or as ${\overline{e}}_{1}$ increases to 0.23.
4. Conclusions
The microsegment gear is a new type of gearing and this paper presents a numerical analysis of its bifurcation characteristics. The main conclusions are:
1) The dynamic system is very sensitive to damping coefficient. As damping coefficient increases, the number of bifurcation and impressive jump tend to decrease. In the meantime, the intervals of dimensionless frequency leading to unstable or chaotic motion have a trend to concentration as damping coefficient increases.
2) The motion state of the system often bifurcates though period doubling bifurcation as the dimensionless frequency increases.
3) The influence of external excitation and internal excitation on system’s bifurcation characteristics have similarities and differences. Due to two excitation sources, the responses frequency is composed of nondimensional frequency ${f}_{k}$ and external or internal excitation frequency ${f}_{p}$/${f}_{e}$.
4) The effects of internal excitation on bifurcation characteristics are greater than that of external excitation. As ${\rho}_{e}$ increases to 2.68 or ${\overline{e}}_{1}$ increases to 0.23, the system exhibits chaotic motion. The growth of ${\overline{e}}_{1}$ affects both the response frequency and response amplitude, while the growth of ${\stackrel{}{F}}_{p1}$ only affects the response amplitude.
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This work is supported by the International S&T Cooperation Program of China (2014DFA80440) and the Natural Science Foundation of Anhui Province of China (No. 1408085MKL12).