Simulation of vibro-impact gear model considering the lubricant influence with a new computational algorithm

  • Fuhao Liu
  • Liang Zhang
  • Hanjun Jiang
  • Xuehua Yu
Technical Paper


In this paper, a vibro-impact gear model incorporating the influences of the lubricant and backlash is formulated. Then, a new computational algorithm validated in comparison with the “stiff” solvers, by defining a transition area and adopting the double-changed time step, is proposed to identify the influences of the lubricant on the dynamic system. The results obtained in this paper indicate that the proposed numerical algorithm not only guarantees the precision of solutions, but also reduces the calculation speed of the whole system. The lubricant can potentially reduce the vibrations in the gear system, and the boundaries for double-sided impacts, single-sided impact and no impact are mainly dependent on the fluctuating driving torque and the stiffness of the lubricant. These results could provide a good source of information on the utilization of vibro-impact modeling and simulation for the study of spur gears dynamic performance, and quantification of the factors such as gear backlash, input power or torque fluctuations, lubrication, rattle, etc. In addition, the proposed numerical method could be used as a basic program of vibro-impact in Matlab environment.


Gear system Computational algorithm Nonlinear vibration Gear backlash 



Dynamic transmission errors


Relative speed


Viscous damping


Total backlash


Time in seconds


Time of one excited period cycle

\(\Delta t\)

Time interval

\(F_{n} \left( t \right)\)

Nonlinear elastic contact force


Rotational inertia of the pinion and gear


Equivalent mass

\(\dot{\theta }_{p,g}\)

Rotational velocity of the pinion and gear

\(\ddot{\theta }_{p,g}\)

Rotational acceleration of the pinion and gear


Pitch radius of the pinion and gear


Driving and driven torque


Mean part of the drag torque


Amplitude of vibratory part of the \(k{\text{th}}\) harmonic


Initial phase of \(i{\text{th}}\) harmonic


Critical viscous damping ratio of lubricant and solid


Stiffness of lubricant and solid


Initial resolution of the numerical solution


Number of the period


Fundamental frequency

Profile contact ratio of the gears


Small value defining transition area

\(\nabla t_{ - , + }^{max}\)

Maximum time step for lubricant and solid contact


Rotational displacements of the pinion and gear

1 Introduction

Multi-mesh gear systems are generally used in all kinds of industrial machinery and the tooth impacts generated by input and output torque fluctuations are often caused by the presence of gear backlash. Such impacts could result in intense vibration problem and heavy dynamic loads that could affect the product quality and reliability of the gear drive.

Dynamic modeling of gear vibration can be used to promote our understanding of the vibration generation mechanisms in gear transmissions as well as the dynamic behaviors in the presence of various types of gear tooth damage [1, 2, 3, 4]. Therefore, a lot of research has been undertaken to study the dynamic modeling of the gear pairs. The first model about gear system can be attributed to the paper [5]. And a comprehensive review of the mathematical models used in gear dynamics can be found in the paper [6]. Basically, the studied model could be classified into two groups: the first one is the dry impact model. Singh et al. developed a five-speed manual transmission of a front wheel drive automobile model to understand, quantify and control the vibro-impacts problem inducted by gear backlash [7]. Kahraman and Singh constructed a spur gear model with backlash to examine the nonlinear frequency response characteristics for both external and internal excitations [8, 9, 10]. Sheng et al. proposed a vibro-impact dynamic model of a spur gear train by incorporating the constant meshing stiffness, the linear time-invariant viscous damping values and the gear backlash non-linearity [11]. Liu et al. presented an analytical model of gear system to demonstrate that the sliding friction could not only introduce vibration but also decrease the dynamic transmission errors (DTE) under proper operation condition [12]. Latter, Jiang and Liu developed an analytical helical gear pair model by considering the mass eccentric and mesh stiffness to show the sliding friction play a certain role in inhibiting the amplitude of the frequency [13, 14].

Another one is lubricant impact model, in which the lubricant cannot be negligible because it influences drag torque applied to unloaded gears. Since there is always a formed film of lubricant in the gear backlash under low load condition, thus there is no direct “impacts” with metal-to-metal. However, there are strong or smooth fluctuations of the film thickness instead, which can change the force transmitted between the teeth flanks [15]. Therefore, the gear-impacting surface can be treated as lubricated connections rather than the usually reported solid-to-solid collision. Most researchers mainly described the lubricant effect as the damping: Brancati et al. used the squeeze model to study the gear rattle by assuming that the damping force is proportional to the oil viscosity [16, 17]. Theodossiades et al. introduced a new approach for understanding the interactions between the transmission gears during engine idle conditions by taking into account the effect of driving-side lubrication [18, 19, 20]. Latter, Guilbault et al. calculated the nonlinear damping at different mesh frequencies and torque amplitudes to illustrate the squeeze film damping had main contribution to the global mesh damping [4]. Recently, Liu et al. extended the developments of Theodossiades to characterize the damping and stiffness by considering the double-sided lubricant film, they verified that the effects of the lubricant film on the coast side is significant under low loading condition [21, 22] and the hydrodynamic flank friction has almost no influence on the gear system [23].

In the above studies, the nonlinear dynamic modeling of the gear pairs has achieved successful results and the effects of the lubricant damping on the gear system is general well understood. However, some developments are still needed to ensure a rapid and precise representation of the gear mesh model with a new computational algorithm (NCA). This mainly includes two respects: For one hand, many factors such as gear backlash [8], fluctuation of the driving/driven torque or speed [24] and the lubricant [25, 26] synchronously exert a complicated influence on the gear rattle. Therefore, development of a more accurate mathematical model considered more factors to describe the dynamic behaviors has become an important issue. However, to the authors’ knowledge, constructing a vibro-impact model considered the influence of the lubricant is novel. For another hand, the high value of the contact solid stiffness and the nonlinear backlash impact will cause the ill-conditioning and numerical “stiff” [27], which could present particular difficulties at the simulation stage. In order to overcome this numerical problem, various solution techniques, including analog/digital simulation [28], numerical integration [29, 30], (multi-term) harmonic balance method [31, 32, 33, 34, 35], multi-scale method [36] and finite element method [3, 37, 38, 39] are used to solve the dynamic model in order to obtain the gear dynamic response. Lately, Martin even used the periodic Green’s function in the form of truncated Fourier series to find the analytical periodic solution in a steady state [40]. However, the above methods require for a long time and special concern. Actually, most of the gear dynamic researchers had recognized this problem implicitly, but a few researchers can avoid this obstacle for solving. Actually, most of the solution algorithms available in the literature can not be immediately applied to examine this problem even used the “stiff” solvers, because the Jacobian matrix of the linearizing gear system is discontinuities and varying around different operation point [41]. Hence, it is vital important to develop a NCA which does not use the Jacobian matrix to avoid this “stubborn” as suggested in the paper [7, 42].

Therefore, the objective of this work is to extend the methodology in reference paper [21] and focus on the above-mentioned issues in order to comprehensively understand the gear rattle dynamics. The rest of this paper is organized as follows: in the next section, a vibro-impact model of spur gear is presented by considering the torsional fluctuations resident on both input pinion and drag torque, and lubricant in the gear backlash is considered as the linear element. Then by defining a transition area between solid contact and lubricant contact, a proposed NCA with the double-changed time step is proposed in Sect. 3. Subsequently, the NCA is validated by comparison to the “stiff” solvers in Matlab to illustrate its advantages, and the behaviors of lubricant at different condition and its influences on the dynamic respond of gear system are also discussed. In the last section, concluding remarks are presented.

2 Dynamic model formulation

The system in this paper consists of two gears rotating with the absolute angular velocity \(\dot{\theta }_{p,g}\) as shown in Fig. 1, where \(I_{p,g}\) are rotary inertias, \(R_{p,g}\) are the pitch radius and \(T_{p,g}\) are torques. Here, the subscripts \(p\) and \(g\) refer to the pinion and the wheel. \(K\) and \(C\) represents the nonlinear contact stiffness and damping, respectively. In this paper, the spur gear teeth are assumed to be perfectly involute along their meshing region and no static transmission errors are accounted for. The driving torque \(\left( {T_{p} } \right)\) can be decomposed into mean and perturbation parts [12]
$$T_{P} (t) = T_{pm} + \sum\limits_{i = 1}^{n} {T_{pp}^{i} } \cos \left( {i\omega t + \varphi_{p}^{i} } \right)$$
here, \(t\) is the time in second. \(T_{pm}\) is the mean part of the drag torque, \(T_{pp}^{i}\) and \(\varphi_{p}^{i}\) are the corresponding amplitude of vibratory part and the initial phase of the \(i{\text{th}}\) harmonic, respectively. \(\omega\) is the excitation frequency. Under no power loss condition, the drag torque \(\left( {T_{g} } \right)\) should be equal to the driving torque multiplied by the transmission ratio, that is \(T_{g} = T_{p} R_{g} /R_{p}\).
Fig. 1

Dynamic model of a pair of spur gears

The gear pairs in Fig. 1 can be represented by a rigid disk coupled through nonlinear elastic contact force \(F_{n}\). There will be two types of motion that take place in this gear system: (1) slipping of gear tooth with lubricant contact, (2) impacting of the engage tooth with solid contact. Thus, the nonlinear elastic contact force \(F_{n} \left( {\text{t}} \right)\) including nonlinear stiffness/damping force can be written as [43]
$$F_{n} (t) = C\dot{x}(t) + \left( {Kx(t) + W} \right)$$
here, \(x = R_{p} \theta_{p} - R_{g} \theta_{g}\) is the DTE. \(C\) is the viscous damping, which can be generally estimated as
$$C = 2\varsigma \sqrt {I_{\text{eq}} K}$$
where \(I_{\text{eq}}\) is the gear pair’s equivalent mass. \(\zeta\), \(K\) and \(W\) are the parameters whose values are depended on the state of the gear contact, when \(\left| x \right| < L/2\)
$$\zeta = \zeta_{1} ,\;K = K_{1} ,\;W = 0,$$
and when \(\left| x \right| \ge L/2\)
$$\zeta = \zeta_{2} ,\,K = K_{2} ,\,W = L(K_{1} - K_{2} ){\text{sign}}(x)/2,$$
here, \(\zeta_{1,2}\) and \(K_{1,2}\) are the critical viscous damping ratios and stiffnesses of lubricant/solid contact, respectively.The lubricant stiffness and critical viscous damping can be obtained using the trace method [21]. The solid stiffness is mainly depended on the both state of gear engagements and geometrical dimension, which can be defined as
$$K_{2} = K_{d} \left( {\varepsilon - 1} \right) - K_{s} \left( {\varepsilon - 2} \right)$$
here, \(\varepsilon\) is the profile contact ratio, \(K_{d,s}\) are the stiffnesses when the gear pairs in double-tooth and single-tooth engagements, which can be obtained using analytical model or a finite element method [3, 44].
The nonlinear elastic contact force versus DTE is described in Fig. 2, in which the solid line is the trace of the hysteresis loop. The hysteresis loop, demonstrated by a closed cycle, consists of two dashed lines: one is the line for \(\dot{x} > 0\) and another is the line for \(\dot{x} < 0\). When the wheel and pinion move towards each other, \(\dot{x} > 0\) and the nonlinear elastic contact force \(F_{n} \left( t \right)\) follows the lower locus. Likewise, when \(\dot{x} < 0\), the wheel and pinion rebound and the nonlinear elastic contact force \(F_{n} \left( t \right)\) follows the upper locus. As shown in Fig. 2a–c, a thin lubricant film between the backlashes generates the first damping \(\zeta_{1}\) and stiffness \(K_{1}\), within the initial displacement of \(\left| x \right| < L/2\). Otherwise, the nonlinear elastic contact force \(F_{n} \left( t \right)\) is governed by parameters \(\zeta_{2}\) and \(K_{2}\). The shape of the hysteresis mainly depends on the critical viscous damping ratio \(\zeta_{1,2}\). One can see that a “jump” of the elastic contact force happens at \(\left| x \right| = L/2\) when \(\zeta_{1} \ne \zeta_{2}\) as shown in Fig. 2a–b. And this “jump” could generate a higher acceleration of the DTE, which will lead to excessive vibration and noise. Under the case of \(\zeta_{1} = \zeta_{2}\), the “jump” disappears. Finally, consider a special condition of the above case in Fig. 2d, when the lubricant in backlash is neglected, i.e. \(K_{1} = 0\). As described by reference papers [8, 32], this condition can simulate a purely vibro-impact case with nonlinear stiffness and linear damping.
Fig. 2

Gear backlash nonlinearity. a \(\zeta_{1} > \zeta_{2}\); b \(\zeta_{1} < \zeta_{2}\); c \(\zeta_{1} = \zeta_{2}\); d \(K_{1} = 0\)

Using Newton’s second law, the corresponding equations of motion can be arranged as:
$$I_{P} \ddot{\theta }_{p} (t) + R_{p} F_{n} (t) = T_{P} (t)$$
$$I_{g} \ddot{\theta }_{g} (t) - R_{g} F_{n} (t) = T_{g} (t)$$

3 Proposed numerical method

With the governing equations for the gear pairs in Fig. 1, the motion of the system is simulated in discrete time steps. In order to decrease the whole calculation time, the two-maximum time-steps are used in the whole process. For solid contact, extremely small integration time step is required because the interval of the impacts is negligibly small, and the maximum time step \(\Delta t_{ - }^{\hbox{max} }\) used for the solid contact can be given by
$$\Delta t_{ - }^{\hbox{max} } = {{t_{f} } \mathord{\left/ {\vphantom {{t_{f} } N}} \right. \kern-0pt} N} = {{2\pi } \mathord{\left/ {\vphantom {{2\pi } {\left( {\omega N} \right)}}} \right. \kern-0pt} {\left( {\omega N} \right)}}$$
here, \(t_{f}\) is the excitation period, \(N\) is the initial resolution of the numerical solution. Because the stiffness of the solid contact is very high, thus there is no remarkable change of the DTE at two adjacent time-steps. Therefore, the initial resolution \(N\) should be enough lager to ensure accurate simulation. However, for the case of the lubricant contact, it is totally different because the stiffness of the lubricant film is much smaller than that of solid contact. Thus, a bigger time step \(\Delta t_{ + }^{ \hbox{max} }\) is adequate to simulate the lubricant contact case, and a maximum step \(\Delta t_{ + }^{ \hbox{max} }\) for the lubricant contact is defined by
$$\Delta t_{ + }^{ \hbox{max} } = C_{0} \left( {K_{1} ,K_{2} ,T_{p} ,T_{g} } \right)\Delta t_{ - }^{ \hbox{max} }$$
here, \(C_{0}\) is a variable number, whose value is mainly depended on the system parameters, such as \(K_{1}\), \(K_{2}\), \(T_{p}\) and \(T_{g}\). In this paper, \(C_{0} = 100\) is used for simplicity, and the relationship between \(C_{0}\) and system parameters is the next scope we will study.
For convenience, the vibro-impact gear system Eqs. (7) and (8) can be simplified as
$$\ddot{x}(t) + c\dot{x}\left( t \right) + kx\left( t \right) + w = f_{m} + \sum\limits_{i = 1}^{n} {f_{p}^{i} \cos \left( {i\omega t} \right)}$$
where \(c = C/I_{\text{eq}}\), \(k = K/I_{\text{eq}}\), \(w = W/I_{\text{eq}}\) and \(f_{m,p} = T_{pm,pp} R_{p} /I_{p} + T_{gm,gp} R_{g} /I_{g}\).
Assuming the conditions for any specify time \(t_{0}\) is given, the relative velocity \(\left( {\dot{x}} \right)\) and relative displacement \(\left( x \right)\) of the gear over the time interval \(\Delta t = t - t_{0}\) can be obtained by integrating Eq. (11)
$$\begin{aligned} \dot{x}\left( t \right) & = \dot{x}\left( {t_{0} } \right) + f_{m} \Delta t - w\Delta t - c\left( {x\left( t \right) - x\left( {t_{0} } \right)} \right) \\ & \quad + \sum\nolimits_{{i - 1}}^{n} {\frac{{f_{p}^{i} \left( {\sin \left( {i\omega t_{0} } \right)} \right)}}{{i\omega }}} \\ & \quad - k\left( {x\left( {t_{0} } \right)\Delta t + \dot{x}\left( {t_{0} } \right)\frac{{\Delta t^{2} }}{2} + \ddot{x}\left( {t_{0} } \right)\frac{{\Delta t3}}{6}} \right) \\ \end{aligned}$$
$$\begin{aligned} x\left( t \right) & = x\left( {t_{0} } \right) + \dot{x}\left( {t_{0} } \right)\Delta t + f_{m} \frac{{\Delta t^{2} }}{2} \\ & \quad - \sum\nolimits_{{i = 1}}^{n} {f_{p}^{i} \left( {\frac{{\cos \left( {i\omega t} \right) - \cos \left( {i\omega t_{0} } \right)}}{{\left( {i\omega } \right)^{2} }} + \frac{{\sin \left( {i\omega t_{0} } \right)\Delta t}}{{i\omega }}} \right)} \\ & \quad - w\frac{{\Delta t^{2} }}{2} - c\left( {\dot{x}\left( {t_{0} } \right)\frac{{\Delta t^{2} }}{2} + \ddot{x}\left( {t_{0} } \right)\frac{{\Delta t3}}{6}} \right) \\ & \quad - k\left( {x\left( {t_{0} } \right)\frac{{\Delta t^{2} }}{2} + \dot{x}(t_{0} )\frac{{\Delta t^{3} }}{6} + \ddot{x}\left( {t_{0} } \right)\frac{{\Delta t^{4} }}{{24}}} \right) \\ \end{aligned}$$

In this paper, Eqs. (12) and (13) are valid for all intermediate time values, but the parameters (\(k, c,w\)) are changed according to the state of the gear system.

According to the Fig. 3, the motion for lubricant and solid contact can be easily determined. Although the transition area is very small (\(\varepsilon = 10^{ - 15}\) m), the two types of motion (solid and lubricant contact) still can happen in this critical area. Therefore, it is very difficult to determine the exact contact state. In this paper, the criterion for differentiating the two types of motion in the transition area is defined as:for lubricant contact
$${\text{sign}}\left( {x\left( {t_{0} } \right)} \right) \ne {\text{sign}}\left( {\dot{x}\left( {t_{0} } \right)} \right)$$
and for solid contact
$${\text{sign}}\left( {x\left( {t_{0} } \right)} \right) \ne {\text{sign}}\left( {\dot{x}\left( {t_{0} } \right)} \right)$$
Fig. 3

Transition area between solid contact and lubricant contact

In the first place, the current state of the motion (lubricant or solid contact) needs to be estimated in advantage. For the motion in lubricant contact, the time interval from lubricant contact to the next possible solid contact should to be determined. A possible solid contact after lubricant contact will occur only when
$$x(t_{0} + \Delta t_{ + i} ) = {{L{\text{sign}}\left( {\dot{x}(t_{0} )} \right)} \mathord{\left/ {\vphantom {{L{\text{sign}}\left( {\dot{x}(t_{0} )} \right)} 2}} \right. \kern-0pt} 2}$$
here, \(\Delta t_{ + i} = t_{ + i} - t_{0}\) is the time interval from lubricant contact to the solid contact. If \({\text{sign}}\left( {\dot{x}\left( {t_{0} } \right)} \right) > 0\), the impact happens at the mating surface and the wheel rotated with the action of the pinion; and if \({\text{sign}}\left( {\dot{x}\left( {t_{0} } \right)} \right) < 0\), the impact happens at the non-mating surface and the pinion rotated with the action of the wheel.
Combining Eqs. (16) and (13) gives
$$\begin{aligned} \frac{L}{2}{\text{sign}}\left( {\dot{x}(t_{0} )} \right) = x(t_{0} ) + \dot{x}(t_{0} )\Delta t_{ + i} + f_{m} \frac{{\Delta t_{ + i}^{2} }}{2} \hfill \\ - \sum\limits_{i = 1}^{n} {f_{p}^{i} \,\left( {\frac{{\cos \,(i\omega t) - \cos (i\omega t_{0} )}}{{\left( {i\omega } \right)^{2} }} + \frac{{\sin (i\omega t_{0} )\Delta t_{ + i} }}{i\omega }} \right)} \hfill \\ - c\left( {\dot{x}(t_{0} )\frac{{\Delta t_{ + i}^{2} }}{2} + \mathop x\limits^{..} (t_{0} )\frac{{\Delta t_{ + i}^{3} }}{6}} \right) - k\left( {x(t_{0} )\frac{{\Delta t_{ + i}^{2} }}{2} + \dot{x}(t_{0} )\frac{{\Delta t_{ + i}^{3} }}{6} + \ddot{x}(t_{0} )\frac{{\Delta t_{ + i}^{4} }}{24}} \right) \hfill \\ \end{aligned}$$
Because \(\Delta t_{ + i}\) is infinitesimally small, the trigonometric function in Eq. (17) could be simplified by using Taylor’s formula in the region near the point \(\Delta t_{ + i}\) and neglecting the truncation error. So, Eq. (17) can be rearranged as
$$\frac{L}{2}{\text{sign}}\left( {\dot{x}(t_{0} )} \right) \approx x(t_{0} ) + \dot{x}(t_{0} )\Delta t_{{ + i}} + \frac{{\Delta t_{{ + i}}^{2} }}{2}{\kern 1pt} f_{m} + \sum\limits_{{i = 1}}^{n} {f_{p}^{i} \cos (i\omega t_{0} ) - c} \dot{x}(t_{0} ) - kx(t_{0} )$$
Then, the quadratic formula is used to estimate the time until solid contact happens
$$\Delta t_{ + i} = {{( - b_{ + } \pm \sqrt {b_{ + }^{2} - 4a_{ + } c_{ + } } )} \mathord{\left/ {\vphantom {{( - b_{ + } \pm \sqrt {b_{ + }^{2} - 4a_{ + } c_{ + } } )} {(2a_{ + } )}}} \right. \kern-0pt} {(2a_{ + } )}}$$
$$a_{ + } = {{\left( {f_{m} + \sum\limits_{i = 1}^{n} {f_{p}^{i} \cos (i\omega t_{0} ) - c} \dot{x}(t_{0} ) - kx(t_{0} )} \right)} \mathord{\left/ {\vphantom {{\left( {f_{m} + \sum\limits_{i = 1}^{n} {f_{p}^{i} \cos (i\omega t_{0} ) - c} \dot{x}(t_{0} ) - kx(t_{0} )} \right)} 2}} \right. \kern-0pt} 2}$$
$$b_{ + } = \dot{x}(t_{0} )$$
$$c_{ + } = {{x(t_{0} ) - L{\text{sign}}\dot{x}(t_{0} )} \mathord{\left/ {\vphantom {{x(t_{0} ) - L{\text{sign}}\dot{x}(t_{0} )} 2}} \right. \kern-0pt} 2}$$
Next, the time step can be defined by comparing with the \(\Delta t_{ + i}\) and \(\Delta t_{ + }^{ \hbox{max} }\). If \(\Delta t_{ + }^{ \hbox{max} }\) is small than \(\Delta t_{{ + {\text{i}}}}\), then Eq. (13) is used to determine whether the solid contact happens
$$\left| {x(n + 1)} \right| > L/2 + \varepsilon$$
while the value of \(t(n + 1)\) is reset to
$$t(n + 1) = t(n) + \Delta t_{ + }^{\hbox{max} }$$
If Eq. (23) is not satisfied, Eq. (13) is re-evaluated using Eq. (24). And if Eq. (23) is satisfied, only one possibility could happen for the value of \(\Delta t_{ + }^{ \hbox{max} }\): the value is out of range so that the solid contact happens and the value of \(\Delta t_{ + }^{ \hbox{max} }\) should be reduced using the formula
$$\Delta t_{ + }^{\hbox{max} } = \Delta t_{ + }^{\hbox{max} } - (\Delta t_{ + }^{\hbox{max} } )^{2}$$
and \(\Delta t_{ + }^{ \hbox{max} }\) is re-evaluated using Eq. (25) until Eq. (23) is not satisfied.
If \(\Delta t_{ + }^{ \hbox{max} }\) is bigger than \(\Delta t_{ + i}\), Eq. (13) is used to determine whether the value of the DTE is located in allowable range, as shown in Fig. 3. The modified solutions after some iteration should yield closer values to the real solution. The error criteria are defined as
$$\left\| {x\left( {n + 1} \right)\left| { - L/2} \right|} \right\| < \varepsilon$$
If Eq. (26) is not satisfied, there will be two errors happen for \(\Delta t_{ + i}\): its value is too large or too small. If \(\Delta t_{ + i}\) is too large, the value of \(\Delta t_{ + i}\) is reset to
$$\Delta t_{ + i} = \Delta t_{ + i} - \Delta t_{ + i}^{2}$$
and Eq. (27) is re-evaluated until Eq. (23) is not satisfied. If \(\Delta t_{ + i}\) is too small, \(\Delta t_{ + i}\) will be choose as the time step
$$t(n + 1) = t(n) + \Delta t_{ + i}$$
and \(\Delta t_{ + i}\) is re-evaluated using Eq. (19) until Eq. (26) is satisfied.
Due to the fluctuation of the excitation parameters (pinion’s speed, drag torque), it is impossible for that the gear system always has solid contact. Therefore, the changes in time from solid contact to the next possible lubricant contact also need to be determined. And a possible lubricant contact will happen when the meshed tooth begins to secede. At that time, the DTE has the form
$$x\left( {t_{0} + \Delta t_{ - i} } \right) = - L{\text{sign}}\left( {{\dot{x}}\left( {t_{0} } \right)} \right)/2$$
here, \(\Delta t_{ - i} = t_{ - i} - t_{0}\) is the time interval from solid contact to the lubricant contact.
Combining Eqs. (29) and (13) gives
$$\begin{aligned} - \frac{L}{2}{\text{sign}}(\dot{x}\left( {t_{0} } \right)) & = x(t_{0} ) + \dot{x}(t_{0} )\Delta t_{{ - i}} + f_{m} \frac{{\Delta t_{{ - i}}^{2} }}{2} - w\frac{{\Delta t_{{ - i}}^{2} }}{2} \\ & \quad - \sum\limits_{{i = 1}}^{n} {f_{p}^{i} {\kern 1pt} \left( {\frac{{\cos {\kern 1pt} (i\omega t) - \cos (i\omega t_{0} )}}{{\left( {i\omega } \right)^{2} }} + \frac{{\sin (i\omega t_{0} )\Delta t_{{ - i}} }}{{i\omega }}} \right)} \\ & \quad - c\left( {\dot{x}(t_{0} )\frac{{\Delta t_{{ - i}}^{2} }}{2} + \ddot{x}(t_{0} )\frac{{\Delta t_{{ - i}}^{3} }}{6}} \right) \\ & \quad - k\left( {x(t_{0} )\frac{{\Delta t_{{ - i}}^{2} }}{2} + \dot{x}(t_{0} )\frac{{\Delta t_{{ - i}}^{3} }}{6} + \ddot{x}(t_{0} )\frac{{\Delta t_{{ - i}}^{4} }}{{24}}} \right) \\ \end{aligned}$$
here, \(\Delta t_{ - i}\) is very small. Similarly, Eq. (30) can be simplified as
$$- \frac{L}{2}{\text{sign}}\left( {\dot{x}(t_{0} )} \right) \approx x(t_{0} ) + \dot{x}(t_{0} )\Delta t_{ - i} + \left( {\begin{array}{*{20}l} {f_{m} \sum\limits_{i = 1}^{n} {\cos (i\omega t_{0} } )} \\ { - c\mathop x\limits^{.} (t_{0} ) - kx(t_{0} ) - w} \\ \end{array} } \right)\frac{{\Delta t_{ - i}^{2} }}{2}\,$$
then, \(\Delta t_{ - i}\) could be estimated by the quadratic formula
$$\Delta t_{ - i} = {{( - b_{ - } \pm \sqrt {b_{ - }^{2} - 4a_{ - } c_{ - } } )} \mathord{\left/ {\vphantom {{( - b_{ - } \pm \sqrt {b_{ - }^{2} - 4a_{ - } c_{ - } } )} {(2a_{ - } )}}} \right. \kern-0pt} {(2a_{ - } )}}$$
$$a_{ - } = {{\left( {f_{m} + \sum\limits_{i = 1}^{n} {f_{p}^{i} \cos (i\omega t_{0} ) - c} \mathop x\limits^{.} (t_{0} ) - kx(t_{0} ) - w} \right)} \mathord{\left/ {\vphantom {{\left( {f_{m} + \sum\limits_{i = 1}^{n} {f_{p}^{i} \cos (i\omega t_{0} ) - c} \mathop x\limits^{.} (t_{0} ) - kx(t_{0} ) - w} \right)} 2}} \right. \kern-0pt} 2}$$
$$b_{ - } = \dot{x}(t_{0} )$$
$$c_{ - } = {{x(t_{0} ) - L{\text{sign}}\dot{x}(t_{0} )} \mathord{\left/ {\vphantom {{x(t_{0} ) - L{\text{sign}}\dot{x}(t_{0} )} 2}} \right. \kern-0pt} 2}$$
The time step can be defined by comparing with the \(\Delta t_{ - i}\) and \(\Delta t_{ - }^{ \hbox{max} }\). If \(\Delta t_{ - }^{ \hbox{max} }\) is smaller than \(\Delta t_{ - i}\), then Eq. (13) is used to determine whether the lubricant contact happens
$$\left| {x(n + 1)} \right| < L/2 - \varepsilon$$
while the value of \(t(n + 1)\) is reset to
$$t(n + 1) = t(n) + \Delta t_{ - }^{\hbox{max} }$$
If Eq. (36) is satisfied, there is only one possibility for \(\Delta t_{ - }^{ \hbox{max} }\): its value is out of range so that the lubricant contact happens and the value of \(\Delta t_{ - }^{ \hbox{max} }\) should be reduced to
$$\Delta t_{ - }^{\hbox{max} } = \Delta t_{ - }^{\hbox{max} } - \left( {\Delta t_{ - }^{\hbox{max} } } \right)^{2}$$
and Eq. (13) is re-evaluated until Eq. (36) is not satisfied.
And if \(\Delta t_{ - }^{ \hbox{max} }\) is bigger than \(\Delta t_{ - i}\), then Eq. (26) needs to be judged whether the value of the DTE falls into the allowable area. If \(\Delta t_{{ - {\text{i}}}}\) is too large or too small, it is impossible for that Eq. (26) can be satisfied. And more specifically, if \(\Delta t_{{ - {\text{i}}}}\) is out of range, the value of \(\Delta t_{ - i}\) then can be reduced by
$$\Delta t_{ - i} = \Delta t_{ - i} - \Delta t_{ - i}^{2}$$
and Eq. (39) is re-evaluated until Eq. (36) is not satisfied. If \(\Delta t_{{ - {\text{i}}}}\) is too small, then \(\Delta t_{{ - {\text{i}}}}\) will be chosen as the time step
$$t\left( {n + 1} \right) = t(n) + \Delta t_{ - i}$$
and \(\Delta t_{ - i}\) is re-evaluated using Eq. (32).
In the whole process above, the minimal positive real values of \(\Delta t_{ + i}\) and \(\Delta t_{ - i}\) can be determined according to the physical meaning. But one must note that the time \(\Delta t_{{ + {\text{i}}}}\) and \(\Delta t_{{ - {\text{i}}}}\) is depended not only on the physical parameter, but also on the state of the gear system. Thus it is possible for that all the values for Eqs. (19) and (32) are imaginary numbers. Under this condition, the value of \(t(n + 1)\) is then reset to Eqs. (24) and (37), respectively. The whole process can not be stopped until
$$t(n + 1) > Mt_{f}$$
where \({{M}}\) is number of the period. The flowchart of the numerical solution is shown in Fig. 4.
Fig. 4

Methodology used for computation in this paper

4 Results and discussion

The gear parameters used in this paper are given in Table 1. For demonstrating the validation of the NCA, the system is also calculated by employing, variable step Runge–Kutta numerical integration routine (ODE23s and ODE15s) that is suitable for a ‘stiff’ ordinary differential equation. Both the ODE23s and ODE15 solvers can be used to simulate the differential equation with Runge–Kutta method. Nonetheless, ODE15 uses a lower order method than ODE23s. The key distinction for both solvers is the computational mode of the Jacobian matrix: the ODE23s solver renews the Jacobian matrix for every time step. However, The Jacobian matrix of the ODE23s solver can not be updated until the convergence achieved with the current conditions. Thus, ODE23s is more accurate than ODE15s but ODE15s is much faster [45]. Therefore, Fig. 5 and Fig. 6 only show the comparison between NCA and ODE23 s, and the computation time for ODE15 s is still shown in Table 2.
Table 1

Gear data



Inertias (kg m2)

\(I_{p} =\) 7.5886E−5 and \(I_{g} =\) 4.0668E−4

Pitch radius (m)

\(R_{p} =\) 0.034 and \(R_{g} =\) 0.052

Number of teeth

\(Z_{p} =\) 17 and \(Z_{g} =\) 26

Solid stiffness (N/m)

\(K_{s} = 6.20{\text{E}}8\) and \(K_{d} = 1.03{\text{E}}9\)

Lubricant stiffness (N/m)

\(K_{1} = 1.9{\text{E}}6\)

Damping ratio

\(\zeta_{1} =\) 0.1 and \(\zeta_{2} =\) 0.04

Backlash (m)

\(L = 1{\text{E}} - 5\)

Pressure angle (°)


Contact ratio


Fig. 5

Time histories without considering the lubricant stiffness when \(T_{pp} = 1\) Nm. a DTE, b speed and c acceleration of DTE. Black line: ODE23s; red line: NCA

Fig. 6

Time histories with considering the lubricant stiffness when \(T_{pp} = 1\) Nm. a DTE, b speed and c acceleration of DTE. Black line: ODE23s; red line: NCA

Table 2

Computing time


\(K_{1} = 0\) N/m

\(K_{1} = 1.9{\text{E}}6\) N/m














The whole simulation is programmed in Matlab R2015a (The Math Works, Inc., Natick, Massachusetts, USA). From the perspective of comparability, the methods (NCA, ODE15s and ODE23s) are running with the same max time-step (\(1 \times 10^{ - 5} {\text{s}}\)) on an iMac with 3.4 GHz Intel Core i5. Because the steady state responses of the gear system are the focus of this paper, it is necessary to run the numerical program sufficiently long until the difference of DTE between the last two periods is less than \(1 \times 10^{ - 11} {\text{m}}\).

According to the vibration theory [46, 47], it can be known that the equilibrium position only depended on the mean part of the exciting force, and the vibration characteristics mainly subject to the fluctuation part of the exciting force. Therefore, in order to facilitate the analysis of the gear vibration, only the fluctuation part of the driving and driven torques could be considered. Figure 5 shows the steady state responses of the time histories obtained from NCA and ODE23 s by neglecting the lubricant stiffness. The horizontal axis represents the dimensionless time normalized by one mesh circle \(\left( {t_{f} } \right)\) and the dimensionless DTE is normalized by the half of the backlash (\(L/2\)). After a close examination of steady state responses in the time domain, it can be easily found that both time histories obtained by NCA and ODE23 s solver yield virtually the same characteristic even in the nonlinear stiffness transition regions. However, the computation times are very different as shown in Table 2: The required calculation times are 1.11551, 19.39326 and 27.06675 s for NCA, ODE15s and ODE23s, respectively. For NCA case, it is even about 1/20 of the calculation time required with ODE15s case. Figure 6 clearly illustrates time histories obtained from NCA and ODE23s with considering the lubricant stiffness, and the value of the lubricant is choose according to our previous research [21]. It is worth to note that the results for both match precisely in peak-to-peak amplitudes. The required times for both methods are also shown in Table 2. Simulation results indicate that the proposed NCA is useful for accelerating calculation speed of the whole system with a high accuracy of the solution.

More specifically, Fig. 7 illustrates the comparisons of the time histories with/without considering the lubricant stiffness. When \(K_{1} = 0\) N/m, the teeth of one wheel firstly engage with those of the other on the driving side. And then with the action of excitation drag torque, the motion on the driving side would suddenly transfer to the coast side. In this process, the gears suffer intensive vibration and there are many impacts happen on the coast side. However, when \(K_{1} = 1.9{\text{E}}6\) N/m, the times of impact happens on driving\coast side is obviously decreased, and the speed and acceleration of the DTE in Fig. 6 are much smaller than that in Fig. 5. Therefore, the lubricant plays a damping role that dramatically alters the character of the gear motion, and consequently reduces the number of impacts.
Fig. 7

Partial enlarged time histories of DTE for Figs. 5 and 6. Dashed line: \(K_{1} = 0\); solid line: \(K_{1} = 1.9{\text{E}}6\) N/m

Figure 8 demonstrates the time evolution of the dimensionless DTE by neglecting the lubricant stiffness. The one with lower frequency and higher magnitude corresponds to the driving cosine torque, and the vibration with higher frequency and lower amplitude is caused by the tooth impact. The locations of the maximum DTE in impact and non-impact state are marked by Ai and Bi (i = 1,2,3), respectively. And the specific values for the maximum DTE are shown in Table 3. As expected, all the maximum DTE increases with the increased drag torque. The two teeth eventually penetrate into each other with an elastic deformation and meanwhile move with a relatively constant acceleration in the non-impact state. However, one must note that two important issues emerge: Firstly, at the start of engaging, impact occurs several times on the driving side before switching over to the coast side. This is mainly because sufficient energy has lost due to viscous damping. Consequently, the normal impact force becomes smaller, and eventually it is not enough to overcome the applied torque. Therefore, the first impact can not drive the tooth to the next contact side, which will require more impacts. Furthermore, the maximum DTE in the state of impact occurs at the second impact, which mainly depended on the excitation frequency. In the state of impact, the acceleration of DTE is very high and the gears suffer tremendous dynamic loads that result in intensive vibration and noise problems. Secondly, the maximum DTE in the state of impact is bigger than that in the state of non-impact when \(T_{gp} = 0\) Nm and the value for the relative location in Table 3 is a minus, which makes clearer that why the gear vibration and noise mostly occurs with unengaged gear pairs.
Fig. 8

The effects of drag torque on DTE without considering the lubricant stiffness when \(T_{pp} = 20\) Nm. Solid line: \(T_{gp} = 0\) Nm; dotted line: \(T_{gp} = 20\) Nm; dashed line: \(T_{gp} = 40\) Nm

Table 3

Values for maximum dimensionless DTE

\(T_{gp}\) (Nm)

\(K_{1} = 0\) N/m

\(K_{1} = 1.9{\text{E}}6\) N/m

Impact state

Non-impact state

Relative location

Impact state

Non-impact state

Relative location




− 0.017


















With considering the lubricant stiffness, the effects of the drag torque on DTE are shown in Fig. 9. It is can be easily seen that all the maximum DTE in the impact state is much smaller than that in non-impact state, even for non-load case. The lubricant’s kinetic energy, dissipated in the form of noise and heat, could result in reducing the amplitude of DTE. Thus, all the maximum DTE is smaller in comparison with that in Fig. 8, which also can be seen from the Table 3.
Fig. 9

The effects of drag torque on DTE with considering the lubricant stiffness when \(T_{pp} = 20\) Nm. Solid line: \(T_{gp} = 0\) Nm; dotted line: \(T_{gp} = 20\) Nm; dashed line: \(T_{gp} = 40\) Nm

As anticipated, the lubricant with a higher viscosity could generate a larger stiffness and damping. Depending on the lubricant parameters and the fluctuated input torque, there are three cases at which different solutions can be obtained: (1) single-sided impact, which is more likely happen when the engaged gears is in low speed or the drag torque with the unloaded gear pairs; (2) double-sided impact, which demonstrated multiple impact characteristics: several impacts firstly would occur on the driving side and then the driven gear is pushed back to the coast side; (3) without impact, where the tooth separation is not observed. In order to quantify this contribution, the effects of the pinion’s torque and its variation with different lubricant stiffness on the dynamical motion must be ascertained.

Figure 10 represents the boundaries of different impact cases with different lubricant stiffness. And in this section, only the first-order harmonic component in Eq. (11) is taken to simplify the analysis. The horizontal axis represents the input torque (\(T_{p}\)) and the vertical axis tells the torque assign ratio, which is defined as \(\alpha = T_{pp} /T_{pm}\). If a smaller increment is used for the pinion’s torque and its variation, a smoother curve will be obtained. It is clearly show in Fig. 10 that the double sides impact and no impact region are approximately symmetrical about the single side region, and the increased pinion’s torque shrinks the region of the single side impact. On the other hand, the area of the single side impact is broader with the higher lubricant stiffness, which have two effects on the gears system: one is that when \(\alpha > 1\), the higher values of the lubricant stiffness push the boundary move up, thus decreasing the vibration intensity in the gear system. Another is that when \(\alpha < 1\), the higher lubricant stiffness yet pushes the boundary move down, hence reducing the possibility of the vibration. Therefore, the vibration mainly happens at the unloaded gear system with a lower lubricant stiffness, such as neutral rattle.
Fig. 10

Boundaries between a without impact and single side impact, b single side impact and double side impact: line over asterisks: \(K_{1} = 1.0{\text{E}}6\) N/m; line over circles: \(K_{1} = 2.0{\text{E}}6\) N/m; line over squares: \(K_{1} = 3.0{\text{E}}6\) N/m

For the sake of capturing the characteristics of the actual situation, the applications to the gear system in this paper also considers one actual case: the pinion’s torque comprises 1st, 3rd and 5th harmonics with relative amplitudes of 1, 0.5 and 0.25 times of the amplitude of the first harmonic, respectively. From Fig. 11, it can be note that the trends of these boundaries with multi-harmonic are similar with those in Fig. 10. However, multi-harmonic driving torque tends to more easily generate vibration and noise because of their wide-ranging spectral contents as shown in Fig. 11.
Fig. 11

Boundaries between a without impact and single side impact, b single side impact and double side impact. Line over asterisks: \(K_{1} = 1.0{\text{E}}6\) N/m; line over circles: \(K_{1} = 2.0{\text{E}}6\) N/m; line over squares: \(K_{1} = 3.0{\text{E}}6\) N/m

5 Conclusions

In this paper, a vibro-impact gear model considering the influences of the lubricant is developed to strengthen the understanding of gear vibration mechanisms with a transition from high to low and vice versa stiffness of the corresponding ODEs describing the dynamics of the gear system. Due to the numerical simulation of such systems deals with highly ‘stiff’ case, especially for the solid-to-solid regime of contact between the gear teeth, a numerical transition technique utilizing a double-change integration time-step adjustment has been proposed for enhancement of analyzing capability in this study. This method overcomes the limitations of known prior research work, in which the solutions are mainly depended on the iteration of the Jacobian matrix of the linearizing gear system.

Numerical results are presented to illustrate and to quantify the effect of the lubricant, as well as the numerical efficiency and accuracy of the NCA in comparison with standard multistep Matlab solvers such as ODE23s and ODE15s. It is observed that the lubricant played an important role in dynamic system, and the velocity and the acceleration of DTE could be much reduced with the lubricant by comparison with the results obtained from pure vibro-impact model (without lubricant). And two different torque assign ratios are considered against driving torque with different lubricant stiffness in order to systematically illustrate the characteristics of tooth impact.

The NCA used in this paper could provide important suggestions that would help researchers to more easily understand the influences of the lubricant on the impingement mode of the gear system. In the end, this NCA presented in this paper also could be extended to study gear system including the contact ratio, friction and multiple clearances in multi-degree-of-freedom systems, which would be addressed in the future’s article.



The authors acknowledge the financial support from National Natural Science Foundation of China (Grant No. 51305378), Natural Science Foundation of the Jiangsu Higher Education Institutions of China (Grant No. 17KJB460016), China Postdoctoral Science Foundation funded project (2016M590643), Jiangsu Provincial Science and Technology Department (BY2015057-25) and the Research Laboratory of Mechanical Vibration (MVRLAB).


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Copyright information

© The Brazilian Society of Mechanical Sciences and Engineering 2018

Authors and Affiliations

  1. 1.College of Automobile EngineeringYancheng Institute of TechnologyJiangsuChina
  2. 2.Department of Engineering MechanicsShanghai Jiao Tong UniversityShanghaiChina

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