Nonlinear dynamic analysis of power reflux hydraulic transmission system

3.1. Purely rotational nonlinear dynamic model of planetary gear train

Fig. 2 contains a scheme of the purely rotational nonlinear dynamic model of the planetary gear train [26].

Newton’s law is utilized to obtain the equations of motion of the planetary gear train:

where:

Fig. 2Purely rotational nonlinear dynamic model of planetary gear train

3.2. Dynamic model of torque converter

The dynamic model of the torque converter proposed in this paper is based on four first-order differential equations [27], which are implemented in Matlab software. Eqs. (4-6) are the conservation of momentum equations of pump, turbine, and stator, respectively.

Eq. (7) is the conservation of energy equations of the torque converter:

The parameters that define the state of the models are the pump’s angular speed $n_p$, turbine’s angular speed $n_t$, stator’s angular speed $nst$, and volumetric flow rate $Q$. The pump torque $T_p$ and turbine torque $T_t$ were also integrated into the model.

3.3. Nonlinear dynamic model of PRHTS

Fig. 3 illustrates a nonlinear dynamic model of the PRHTS. By merging the equilibrium equations of the planetary gear train dynamic and torque converter dynamic, the equilibrium equation of the PRHTS dynamics is established.

Fig. 3Nonlinear dynamic model of PRHTS

4.1. Nonlinear dynamic responses

The possible steady motion types of a nonlinear dynamic system can be divided into four types: stationary state, periodic motion, quasi-periodic motion, and chaotic motion. The period of the periodic motion is the integral multiple of the excitation period; however, the quasi-periodic motion contains several frequencies which have no common divisor. Long-term prediction of periodic motion and quasi-periodic motion, which are insensitive to the system parameters and initial conditions, are feasible. Chaotic motion is a type of unusual steady state of motion, which has been discovered in recent years. Chaotic motion is a broadband and continuous spectrum motion, which is similar to a random motion under a single period excitation. Chaotic motion cannot be long-term predicted and is sensitive to system parameters.

Chaos signifies an apparently random irregular motion in a deterministic system; it is an important branch of nonlinear science. The Chaos theory states that there may be a simple, deterministic nonlinear law behind the complex phenomenon. From another perspective, a simple and completely determined nonlinear dynamic system can also produce a similar random irregular motion that cannot be predicted over a long period. The Chaos theory breaks the traditional determinism, and builds a bond between certainty and randomness. It also provides a new way to explain and analyze the complex phenomena in nature.

4.2. Analytical method

In this study, the nonlinear dynamics of the PRHTS, illustrated in Fig. 3, are analyzed by using bifurcation diagrams, dynamic trajectories, phase plane diagram, and Poincaré maps. The basic principles of each analytical method are reviewed in the following sub-sections.

This study creates the first analytical method to identify the onset of chaotic motion with the use of dynamic trajectories of the PRHTS. The dynamic trajectories of the PRHTS provide a fundamental indication as to whether the system behavior is periodicor non-periodic. The dynamic trajectories of chaotic motion display randomness and aperiodicity. However, it is not feasible to identify the onset of chaotic motions using them because the dynamic trajectories of quasi-periodic motion also display similar randomness and aperiodicity. Therefore, it is challenging to distinguish between quasi-periodic motion and chaotic motion with only dynamic trajectories. Accordingly, another form of analytical method is required.

Displacement and speed can indicate an object’s kinestate at any time. Therefore, displacement and speed are called state variables. Phase plane diagrams, in which $x$- and $y$-coordinates are displacement and speed, respectively, display the variation of state variables with time. In general, the phase plane diagram of a periodic motion is a closed curve. The phase plane diagram of a quasi-periodic motion is not a closed curve; however, it has certain regularity. The phase plane diagram of chaotic motion comprises lines of reciprocal chiasmata and twine.

In nonlinear dynamic systems, Poincaré maps are composed of state variable points of a time series corresponding to a constant time interval, which is the period of the excitation force. Poincaré maps of quasi-periodic motion comprise several closed curves. For a chaotic motion, a Poincaré map is a fractal structure comprising a number of irregularly-distributed points. Finally, for a periodic motion, a Poincaré map consists of several discrete points. Therefore, Poincaré maps are an effective analytical method to distinguish between quasi-periodic motion and chaotic motion.

Bifurcation is an important characteristic of nonlinear dynamic systems. Bifurcation represents the response of nonlinear dynamic system with varying control parameters. A bifurcation diagram summarizes the essential dynamics of nonlinear dynamic systems and is an effective method for observing its nonlinear dynamic response. Bifurcation diagrams of periodic motion consist of several curves. For a chaotic motion, a bifurcation diagram is a cloudy point set within a control parameter region. Bifurcation diagram is necessary for a comprehensive understanding of nonlinear dynamic systems. However, the use of only bifurcation diagrams is not enough to distinguish between quasi-periodic motion and chaotic motion. Accordingly, various analytical methods are required to analyze nonlinear dynamics. In this study, the bifurcation diagrams are generated using two control parameters – the internal excitation frequency, $\mathrm{\Omega }$, and meshing damping ratio, $\zeta$, both of which are dimensionless.

4.3. Effect of dimensionless internal excitation frequency

The nonlinear dynamic equations of the PRHTS are solved using the fourth-order Runge-Kutta method. The sampled data are used to generate the bifurcation diagrams, dynamic trajectories, phase plane diagram, and Poincaré maps of the PRHTS to study its nonlinear dynamic behavior [28]. The dimensionless internal excitation frequency ($\mathrm{\Omega }$) of the PRHTS is one of the key parameters that affects the dynamic behavior of the mechanical transmission system. Fig. 4 presents the bifurcation characteristic of the PRHTS in terms of the dimensionless displacement, using $\mathrm{\Omega }$ as a bifurcation parameter. It can be observed that when the dimensionless internal excitation frequency is varied, various types of motion forms such as periodic motion, quasi-periodic motion, and chaotic motion are generated. Specifically, bifurcation diagram is a line at low values of the $\mathrm{\Omega }$ (< 1.13). According to the theory of bifurcation diagram, the PRHTS exhibits period-1 motion. However, when $\mathrm{\Omega }$ is between 1.13 and 1.25, two disjoint lines are shown in the bifurcation diagram. Therefore, the period-1motion of the PRHTS is replaced by the period-2 motion. As the value $\mathrm{\Omega }$ is further increased, many scattered points appear in the bifurcation diagram. Combined with Poincaré map, it can be determined that the period-2 motion turns into the quasi-periodic motion when the value $\mathrm{\Omega }$ is further increased from 1.25 to 1.36. Finally, for $\mathrm{\Omega }\mathrm{}$> 1.36, the PRHTS performs chaotic motion. The motion form, dynamic trajectories, phase plane diagram, and Poincaré maps at $\mathrm{\Omega }\mathrm{}$= 1.1, 1.2, 1.3, and 1.5 are presented in Figs. 5-8, respectively, to illustrate the dynamic characteristics of the system [29].

Fig. 5 presents the dynamic trajectories and phase plane diagram when $\mathrm{\Omega }$ is 1.1. The dynamic trajectories have the form of a sine wave, and their phase plane diagram is a closed circle. This implies that the system response is period-1 motion, which is consistent with the bifurcation diagram.

When $\mathrm{\Omega }$ is 1.2, the dynamic trajectories display a periodic motion, and the phase plane diagram comprises two closed circles (Fig. 6). This implies that the system motion state is transformed from periodic-1 motion to periodic-2 motion, which is consistent with the bifurcation diagram.

Fig. 7 and Fig. 8 present dynamic trajectories and phase plane diagrams when $\mathrm{\Omega }$ is 1.3 and 1.5, respectively. In Fig. 7 and Fig. 8, it can be observed that dynamic trajectories and phase plane diagrams at $\mathrm{\Omega }\mathrm{}$= 1.3 and 1.5 are similar. Accordingly, the Poincaré maps presented in Fig. 9 enable one a clear view to distinguish between the quasi-periodic motion and chaotic motion. When $\mathrm{\Omega }$ is 1.3, the Poincaré map comprises two closed circles, as illustrated in Fig. 9(a). When $\mathrm{\Omega }$ is 1.5, the Poincaré map forms a fractal structure comprising a number of irregularly distributed points, as illustrated in Fig. 9(b). Therefore, the system motion state is quasi-periodic motion when $\mathrm{\Omega }$ is 1.3, and chaotic motion when $\mathrm{\Omega }$ is 1.5.

Fig. 4Bifurcation diagram of dimensionless displacement vs. dimensionless internal excitation frequency

Fig. 5Dynamic characteristic curve of PRHTS at Ω = 1.1

a) Dynamic trajectories

b) Phase plane diagram

Fig. 6Dynamic characteristic curve of PRHTS at Ω = 1.2

a) Dynamic trajectories

b) Phase plane diagram

Fig. 7Dynamic characteristic curve of PRHTS at Ω = 1.3

a) Dynamic trajectories

b) Phase plane diagram

Fig. 8Dynamic characteristic curve of PRHTS at Ω = 1.5

a) Dynamic trajectories

b) Phase plane diagram

Fig. 9Poincaré maps of PRHTS at Ω = 1.3 and 1.5

a)$\mathrm{\Omega }\mathrm{}$= 1.3

b)$\mathrm{\Omega }\mathrm{}$= 1.5

4.4. Effect of dimensionless meshing damping ratio

The dimensionless meshing damping ratio of the PRHTS is another key parameter that affects the dynamic behavior of the mechanical transmission system. Fig. 10 presents the bifurcation characteristic of the PRHTS in terms of dimensionless displacement using the dimensionless meshing damping ratio ($\zeta$) as a bifurcation parameter. It can be observed that variation of $\zeta$ generates various kinds of motion forms such as chaotic motion, quasi-periodic motion, and periodic motion. Specifically, it can be observed that many scattered points appear in the bifurcation diagram at low values of $\zeta$. Combined with Poincaré map, it can be determined that the PRHTS exhibits chaotic motion at low values of $\zeta$ (< 0.034). However, when $\zeta$ is between 0.034 and 0.039, the chaotic motion is replaced by quasi-periodic motion. As the value $\zeta$ is further increased, two disjoint lines are shown in the bifurcation diagram. According to the theory of bifurcation diagram, the quasi-periodic motion of the PRHTS transits to period-2 motion when $\zeta$ is further increased from 0.039 to 0.061. Finally, at $\zeta$> 0.061, bifurcation diagram is a line, so the PRHTS performs period-1 motion. The dynamic trajectories, phase plane diagram, and Poincaré maps at $\zeta$= 0.03, 0.035, 0.05, and 0.07 are presented in Fig. 11, Fig. 12, Fig. 14, Fig. 15, respectively, to illustrate the dynamic characteristics of the system.

Fig. 11 and Fig. 12 present the dynamic trajectories and phase plane diagrams when $\zeta$ is 0.03 and 0.035, respectively. In Fig. 11 and Fig. 12, it can be observed from the dynamic trajectories and phase plane diagrams at $\zeta$= 0.03 and 0.035 that it is challenging to distinguish between quasi-periodic motion and chaotic motion. Accordingly, Poincaré maps are illustrated in Fig. 13. When $\zeta$ is 0.03, the Poincaré map is a fractal structure comprising a number of irregularly-distributed points, as illustrated in Fig. 13(a). When $\zeta$ is 0.035, the Poincaré map comprises two closed circles, as illustrated in Fig. 13(b). This implies that the system motion state is chaotic when $\zeta$ is 0.03 and quasi-periodic when $\zeta$ is 0.035.

Fig. 14 presents the dynamic trajectories and phase plane diagram when $\zeta$ is 0.05. The dynamic trajectories display a periodic motion, and their phase plane diagram comprises two closed circles. This implies that the system response is period-2 motion, which is consistent with the bifurcation diagram.

When $\zeta$ is 0.07, the dynamic trajectories display a sine wave and phase plane diagram comprises a closed circle, as illustrated in Fig. 15. This implies that the system motion state is transformed from period-2to period-1 motion, which is consistent with the bifurcation diagram.

Fig. 10Bifurcation diagram of dimensionless displacement with dimensionless meshing damping ratio variation

Fig. 11Dynamic characteristic curve of PRHTS at ζ = 0.03

a) Dynamic trajectories

b) Phase plane diagram

Fig. 12Dynamic characteristic curve of PRHTS at ζ = 0.035

a) Dynamic trajectories

b) Phase plane diagram

Fig. 13Poincaré maps of PRHTS at ζ = 0.03 and 0.035

a)$\zeta$= 0.03

b)$\zeta$= 0.035

Fig. 14Dynamic characteristic curve of PRHTS at ζ = 0.05

a) Dynamic trajectories

b) Phase plane diagram

Fig. 15Dynamic characteristic curve of PRHTS at ζ = 0.07

a) Dynamic trajectories

b) Phase plane diagram