Sage Journals: Discover world-class research

Abstract

Multi-source drive system with the feature of compact, small size and other advantages is widely used in large engineering machinery, such as shield machine, wind turbines, and shearer. In this article, a reasonable power transmission form is designed and the electromechanical-hydraulic coupling model of the multi-source drive system including the hydraulic pump-motor lumped parameter model and gear system dynamics model is established based on the co-simulation of MATLAB and AMEsim. Taking the pump flow pulsation and the time-varying meshing stiffness as the external and internal excitation of the multi-source drive system, respectively, the vibration and the dynamic characteristics of the multi-source drive system and the transfer characteristics of the dynamic excitation are analyzed. Results show that the flow-speed pulsation and the pressure-torque pulsation are gradually reduced along the direction of the transmission chain. As the external and internal excitation, the flow pulsation and the time-varying meshing stiffness will cause complex influences on the vibration and the dynamic characteristics of the multi-source transmission system. The findings provide a reference basis for the design of the multi-source drive system.

Keywords

Multi-source drive system electromechanical-hydraulic short transmission flow pulsation time-varying meshing stiffness vibration characteristics dynamic characteristics

Introduction

Multi-source drive system with the feature of compact, small size and other advantages is widely used in shield machine, wind turbines, shearer, and so on. At present, research on multi-source drive system mainly focuses on the load sharing, synchronous control, and multi-stage separation of transmission system. Instead, research on the interaction of the coupling for multi-stage in the system is still relatively rare. For multi-source drive system, Jeftenić et al.¹ researched multi-motor synchronous control to improve the load sharing of system. Yang et al. established the coupled mechanical and electrical coupling dynamics model using the centralized parameter method and analyzed its dynamic load sharing characteristics. The results showed that the system non-uniform load characteristics can be improved by reducing the asynchrony of speed or increasing the load.² W Sun et al.³ researched the inherent characteristics of a multi-source drive system. The flow and pressure pulsation of piston pump are the main causes of vibration and noise in the hydraulic system. Edge et al.⁴ studied the effects of the inlet pressure on the flow fluctuation and the pressure changes in piston chamber. Yang et al.⁵ established a piston pump flow pulsation model on complex conditions and analyzed the effects of the outlet pressure of the flow pulsation in steady-state and transient conditions. A Kahraman⁶ analyzed the inherent and dynamic characteristics of planetary gear, and Zhao and Lan⁷ established a dynamics model of the cutting transmission system and revealed the periodic vibration characteristics of the gear system caused by the stiffness excitation and the meshing impact. Guo⁸ established a comprehensive model of shearer cutting unit based on the vibration theory, multibody dynamics, nonlinear finite element, and virtual prototyping technology, as well as revealed the response characteristic of the cutting unit in constant load and varying load conditions.

The electromechanical coupling effect is general in many fields, and there are some other scholars who have studied this phenomenon, especially for the motor-driven system, such as robots and space manipulator with harmonic drive. Liu et al.⁹ proposed a coupled dynamic model of the motor-driven flexible manipulator system (MDFMS) and studied the vibration responses of the flexible manipulator under different velocities, accelerations, and structure parameters, as well as the effect mechanism of the driving motor on the vibration responses. Zhao et al. proposed a new method to analyze the dynamic reliability and parameter sensitivity of space manipulator with harmonic drive. The proposed method extends the previous response surface method (RSM) and focuses on the characteristics of the output dynamical property of the space manipulator.¹⁰

However, many scholars only do a detailed analysis on the pump or gear transmission system and seldom analyze the effects of pump flow pulsation or time-varying mesh stiffness and both of them on the electromechanical-hydraulic coupling system. When the pump and multi-stage gear system are coupled together, new dynamic characteristics may be induced. As a result, most current dynamics studies of multi-source drive system are difficult to guide the design of this system. Based on the patents previously developed by the research group,¹¹ a coupling dynamic model of multi-source drive system, which is based on the driving of hydraulic pump, is established by conducting co-simulation of the MATLAB and AMEsim. This article analyzed the vibration and dynamic characteristics of the transmission system under the periodic excitation with respect to the pump pulsation and time-varying mesh stiffness, based on the electromechanical-hydraulic coupling model.

Principle and design of the electromechanical-hydraulic short cutting transmission system

Principle and design of the hydraulic drive circuit

An electromechanical-hydraulic multi-source drive system is proposed based on the transmission type of multi-pump and multi-motor in the shield machine hydraulic system.^12,13 The proposed model offers many advantages over the conventional design. First, the deformation damage of the rocker arm due to overloading and strong shock can be avoided by substituting the long transmission chain system with the multi-pump and multi-motor hydraulic system. Second, the speed of the cutting roller can be adjusted by the volume speed-modulating circuit of variable displacement pump and constant displacement motor. Furthermore, the cutting roller can cut coal with reasonable cutting thickness and achieve the maximum cutting effectiveness. Overall, compared to the traditional long transmission chain system, the hydraulic system has a more desirable impact performance. Figure 1 shows the proposed multi-source drive/transmission system.

Figure 1.

The multi-source drive system.

Figure 2 shows the hydraulic drive circuit schematic of short cutting transmission system. The speed of the constant displacement motor can be adjusted by changing the flow of pump 1 and pump 2. The spill valve is installed in the high-pressure oil circuit to avoid overload. The low-pressure oil circuit is equipped with a small flow supplement oil pump to supplement for the leaks of the variable pump and the hydraulic motor.

Figure 2.

Schematic diagram of the hydraulic drive circuit.

Principle and design of the gear reduction system

The gear reduction system is the most important portion of the cutting unit and its system structure is shown in Figure 3. The coupling is used to connect the hydraulic motors and the pinions. The power of the large gear is transmitted to the sun gear and then from the ring gear to the roller after the second reduction. According to the parameters of the multi-source drive system, the total gear ratio of the gear reduction system is about 50, the coupling gear system ratio is set to be 9 initially, and the second planetary gear system ratio is set to be 5.56.

Figure 3.

Structure of the gear reduction system.

As shown in Figure 3, the power is transmitted from the sun gear to the ring gear in the planetary gear system. The external and internal meshing are the two structural types of the coupling gear system, as shown in Figure 4. According to the external structure of the oblique shaft motor and the internal space limitations of the cutting unit, the structure type a is used in this article. The parameters of the coupling gear system and the planetary gear system are shown in Table 1.

Figure 4.

Structure type of the coupling gear system.

Table 1.

Parameters of the gear system.

Parameter	Coupling gear system		Planetary gear system
Parameter	Large gear	Pinion	Sun gear	Planetary gear	Ring gear
Number	1	3	1	3	1
Teeth number	144	16	19	35	89
Tooth width (mm)	144		109
Engagement angle (°)	20		20
Modulus (mm)	4		8
Equivalent brace stiffness and damping	k_xpi = k_ypi = 10⁸ N/m, k_xb = k_yb = 10⁸ N/m $c_{h} = ξ \sqrt{k_{h} m_{h}}, ξ = 0.1$ k_h = equivalent brace stiffness, m_h = component mass

Modeling of the electromechanical-hydraulic short cutting transmission system

Dynamics model of the coupling gear system

A dynamics model is the basis of dynamics equations. The equivalent dynamics model of the coupling gear system is shown in Figure 5.^2,14,15

Figure 5.

Dynamics model of the coupling gear system.

Three pinions are distributed in the circumferential direction uniformly and each gear unit is regarded as a rigid body of rotation with 3 degrees of freedom along the x direction, the y direction, and the rotational direction. The brace stiffness of the gear transmission shaft and the bearing, as well as the box body, is uniformly equivalent to the bearing stiffness. Here, r_pi (i = 1, 2, 3) and r_b are the base radii of the three pinions and the large gear, respectively; θ_pi and θ_b are the corresponding angular displacements; x_pi, x_b, y_pi, and y_b are the vibration displacements along x and y directions. Also, k_xpi, k_xb, k_ypi, and k_yb are the equivalent brace stiffness of the pinions and the large gear in x and y directions; c_xpi, c_xb, c_ypi, and c_yb are the equivalent damping of the pinions and the large gear in x and y directions; k_pi and c_pi are the meshing stiffness and the damping between the pinions and the large gear. φ_i are the angle distribution of the pinions, φ_i = 2π(i – 1)/3; α_n is the pressure angle.

The parameter vector of the coupling gear system is set to be: X =[x_p₁, x_p₂, x_p₃, x_b, y_p₁, y_p₂, y_p₃, y_b, θ_p₁, θ_p₂, θ_p₃, θ_b] and then the differential equation of the motion for the coupling gear system is established

{\begin{matrix} m_{pi} {\overset{\cdot\cdot}{x}}_{pi} + c_{xpi} {\overset{\cdot}{x}}_{pi} + k_{xpi} x_{pi} = - F_{pi} \sin (α_{n} + φ_{i}) \\ m_{pi} {\overset{\cdot\cdot}{y}}_{pi} + c_{ypi} {\overset{\cdot}{y}}_{pi} + k_{ypi} y_{pi} = - F_{pi} \cos (α_{n} + φ_{i}) \\ I_{pi} {\overset{\cdot\cdot}{θ}}_{pi} = T_{pi} - F_{pi} r_{pi} \\ m_{b} {\overset{\cdot\cdot}{x}}_{b} + c_{xb} {\overset{\cdot}{x}}_{b} + k_{xb} x_{b} = \sum_{i = 1}^{3} F_{pi} \sin (α_{n} + φ_{i}) \\ m_{b} {\overset{\cdot\cdot}{y}}_{b} + c_{yb} {\overset{\cdot}{y}}_{b} + k_{yb} y_{b} = \sum_{i = 1}^{3} F_{pi} \cos (α_{n} + φ_{i}) \\ I_{b} {\overset{\cdot\cdot}{θ}}_{b} = \sum_{i = 1}^{3} F_{pi} r_{b} - T_{b} \end{matrix}

(1)

In equation (1), m_pi (i = 1, 2, 3) and m_b are the mass of the three pinions and the large gear, respectively; I_pi and I_b are the inertia of the pinions and the large gear, respectively; F_pi is the meshing force between the pinions and the large gear. The meshing force can be calculated directly by the following formula without considering the meshing error

\begin{matrix} F_{pi} = c_{pi} [({\overset{\cdot}{x}}_{pi} - {\overset{\cdot}{x}}_{b}) \sin λ_{n} - ({\overset{\cdot}{y}}_{pi} - {\overset{\cdot}{y}}_{b}) \cos λ_{n} + r_{pi} {\overset{\cdot}{θ}}_{pi} + r_{b} {\overset{\cdot}{θ}}_{b}] \\ + k_{pi} [(x_{pi} - x_{b}) \sin λ_{n} - (y_{pi} - y_{b}) \cos λ_{n} + r_{pi} θ_{pi} + r_{b} θ_{b}] \end{matrix}

(2)

In equation (2), λ_n = α_n + φ_i. Combining equations (1) and (2), the differential equations of the system can be written by a standard matrix equation

[M] {\overset{\cdot\cdot}{X}} + [C] {\overset{\cdot}{X}} + [K] {X} = {P}

(3)

Dynamics model of the planetary gear system

The dynamics model of the planetary gear system with 6 degrees of freedom is shown in Figure 6.¹⁶

Figure 6.

Dynamics model of the planetary gear system.

The dynamics equations of the planetary gear system can be written as

{\begin{matrix} I_{s} {\overset{\cdot\cdot}{θ}}_{s} + c_{sg} {\overset{\cdot}{θ}}_{s} + k_{sg} θ_{s} = - \sum_{j = 1}^{n} r_{s} F_{spj} \\ I_{p} {\overset{\cdot\cdot}{θ}}_{pj} = r_{p} (F_{rpj} - F_{spj}) \\ {\bar{I}}_{c} {\overset{\cdot\cdot}{θ}}_{c} + \sum_{j = 1}^{n} I_{p} ({\overset{\cdot\cdot}{θ}}_{pj} - {\overset{\cdot\cdot}{θ}}_{c}) + c_{cg} {\overset{\cdot}{θ}}_{c} + k_{cg} θ_{c} \\ = \sum_{j = 1}^{n} r_{c} (F_{rpj} - F_{spj}) \\ I_{r} {\overset{\cdot\cdot}{θ}}_{r} + c_{rg} {\overset{\cdot}{θ}}_{r} + k_{rg} θ_{r} = - \sum_{j = 1}^{n} r_{r} F_{rpj} \end{matrix}

(4)

In equation (4), θ_s, θ_pj (j = 1, 2, 3), θ_c, and θ_r are the angular displacements of the sun gear, the planetary gear, the planet carrier, and the ring gear, respectively; θ_c = 0; r_s, r_p, and r_r are the base radii of the sun gear, the planetary gear, and the ring gear, respectively; r_c is the rotation radii of the planet carrier center; I_s, I_p, and I_r are the inertia of the sun gear, the planetary gear, and the ring gear, respectively; ${\bar{I}}_{c}$ is the centralized inertia of planetary gear and the planet carrier, ${\bar{I}}_{c} = I_{c} + n I_{p} + n m_{p} r_{c}^{2}$ . Furthermore, m_p is the mass of the planetary gear, I_c is the inertia of the planet carrier; c_sg and k_sg are the equivalent brace stiffness and the damping of the sun gear, respectively; c_cg and k_cg are the equivalent brace stiffness and the damping of the planet carrier, respectively; and c_rg and k_rg are the equivalent brace stiffness and the damping of the ring gear, respectively. F_spj is the dynamic meshing force between the sun gear and the planetary gear and F_rpj is the dynamic meshing force between the planetary gear and the ring gear

{\begin{matrix} \begin{matrix} F_{spj} = c_{sp} (r_{s} {\overset{\cdot}{θ}}_{s} - r_{cp} {\overset{\cdot}{θ}}_{c} + r_{p} ({\overset{\cdot}{θ}}_{pj} - {\overset{\cdot}{θ}}_{c})) \\ + k_{sp} (r_{s} θ_{s} - r_{cp} θ_{c} + r_{p} (θ_{pj} - θ_{c})) \end{matrix} \\ \begin{matrix} F_{rpj} = c_{rp} (r_{r} {\overset{\cdot}{θ}}_{r} - r_{cp} {\overset{\cdot}{θ}}_{c} - r_{p} ({\overset{\cdot}{θ}}_{pj} - {\overset{\cdot}{θ}}_{c})) \\ + k_{rp} (r_{r} θ_{r} - r_{cp} θ_{c} + r_{p} (θ_{pj} - θ_{c})) \end{matrix} \end{matrix}

(5)

In equation (5), c_sp and k_sp are the meshing damping and the stiffness between the sun gear and the planetary gear; c_rp and k_rp are the meshing damping and the stiffness between the planetary gear and the ring gear.

Calculation of the time-varying meshing stiffness

Stiffness excitation is a dynamic excitation caused by the time variation of composite stiffness in the gear meshing process, and it is also one of the most important dynamic excitation forms in the gear transmission system.¹⁷ The coincidence degree is typically greater than 1 in the gear meshing process, and the alternating meshing phenomenon of the single and double teeth will appear in the gear meshing process with the change of meshing point. As shown in Figure 7, the time-varying meshing stiffness is externalized by a periodic similar rectangular wave function in this article.

Figure 7.

Periodic variation of the meshing stiffness.

In Figure 7, k₁ is the meshing stiffness of the single pair gear teeth, which is also the minimum meshing stiffness; k₂ is the meshing stiffness of the double pair gear teeth, which is also the maximum meshing stiffness. Taking the planetary gear system as an example, the meshing frequency based on the movement of the planetary gear system is calculated as follows when the ring gear is fixed

ω_{s} = \frac{2 π n_{s}}{60} (\frac{z_{s} z_{r}}{z_{s} + z_{r}})

(6)

In equation (6), n_s is the speed of the sun gear; z_s is the number of the sun gear teeth; z_r is the number of the ring gear teeth. The time-varying meshing cycle of the gear is given in equation (7)

T_{s} = \frac{2 π}{ω_{s}}

(7)

The time-varying meshing stiffness based on the Fourier series without the higher-order terms of the meshing frequency is expressed in equation (8)

k (t) = k_{s} + \sum_{m = 1}^{\infty} (b_{m} e^{j ω_{s} (t + t_{0})} + B)

(8)

In equation (8), k_s is the average meshing stiffness, k_s = (2 – ε)k₁ + (ε – 1)k₂; b_m is the mth harmonic amplitude, b_m = 2(k₂ – k₁)sin(mπ(ε – 1))/mπ; ε is the coincidence degree; and t₀ is the initial phase of meshing. B is the conjugate complex of the previous one.

Combining the gear parameters of the planetary gear system in Table 1, the average meshing stiffness k_s of the coupling gear system, the meshing stiffness k₁ of the single pair gear teeth, and the meshing stiffness k₂ of the double pair gear teeth can be calculated. The time-varying meshing stiffness of the coupling gear system and the planetary gear system are shown in Figures 8 –10.

Figure 8.

Time-varying meshing stiffness of the coupling gear.

Figure 9.

Time-varying meshing stiffness between the sun gear and the planetary gear.

Figure 10.

Time-varying meshing stiffness between the planetary gear and the ring gear.

Lumped parameter model of the hydraulic pump and motor

A single plunger model is established in AMESim, which is shown in Figure 11. Here, 1 is the oil inlet, 2 is the oil outlet, 3 is the displacement control signal port, and 4 is the plunger speed input port. The lumped parameter model of the pump with nine plungers is established based on the single plunger model, and it is shown in Figure 12.¹⁸

Figure 11.

Single plunger model.

Figure 12.

Multi-plunger hydraulic pump model.

The lumped parameter model of the motor is established by the same method. The pulsation rate δ_A of one variable parameter at the time period of [t₁, t₂] is

δ_{A} = \frac{A_{max} - A_{min}}{A_{t}}

(9)

In equation (9), A_max is the instantaneous maximum value of the variable, A_min is the instantaneous minimum value of the variable, and A_t is the average value of the variable.

Electromechanical-hydraulic coupling model of the short cutting system

As shown in Figure 13, the electromechanical-hydraulic coupling model of the cutting system is established according to the schematic diagram of the system and the various subcomponent models. In Figure 13, q_p₁, q_p₂, q_p, q_m₁, q_m₂, and q_m₃ are the outlet flow of pump 1, the outlet flow of pump 2 outlet, the total outlet flow, the inlet flow of motor 1, the inlet flow of motor 2, and the inlet flow of motor 3, respectively; p_p, p_m₁, p_m₂, p_m₃, and p_r are the pressure of the pumps, the inlet pressure of motor 1, the inlet pressure of motor 2, the inlet pressure of motor 3, and the pressure of the low-pressure oil circuit, respectively. θ_m₁, θ_m₂, θ_m₃, θ_p₁, θ_p₂, θ_p₃, θ_b, θ_s, θ_r, and θ_d are the angular displacements of motor 1, motor 2, motor 3, pinion 1, pinion 2, pinion 3, the large gear, the sun gear, the ring gear, and the roller, respectively. I_m₁, I_m₂, I_m₃, I_p₁, I_p₂, I_p₃, I_b, I_s, I_r, and I_d are the rotational inertia of motor 1, motor 2, motor 3, pinion 1, pinion 2, pinion 3, the large gear, the sun gear, the ring gear, and the roller, respectively. Similarly, k_m₁, k_m₂, k_m₃, k_b, and k_d are the torsional stiffness between motor 1 and pinion 1, motor 2 and pinion 2, motor 3 and pinion 3, the sun gear and the large gear, and the ring gear and the roller, respectively; c_m₁, c_m₂, c_m₃, c_b, and c_d are the corresponding torsional damping values.

Figure 13.

The multi-source drive system model.

The flow loss and the pressure loss of the oil circuit are assumed to be Δq₁, Δq₂, Δq₃, Δp₁, Δp₂, and Δp₃, respectively, and the flow equation of the system is as follows¹⁹

{\begin{matrix} q_{p} = q_{p 1} + q_{p 2} \\ q_{p} = q_{m 1} + q_{m 2} + q_{m 3} + (Δ q_{1} + Δ q_{2} + Δ q_{3}) \end{matrix}

(10)

The pressure equation of the system is as follows

{\begin{matrix} p_{m 1} = p_{p} - Δ p_{1} \\ p_{m 2} = p_{p} - Δ p_{2} \\ p_{m 3} = p_{p} - Δ p_{3} \end{matrix}

(11)

The continuity flow equation and the torque balance equation of the motor are as follows

{\begin{matrix} q_{mi} = C_{m} (p_{mi} - p_{r}) + D_{mi} {\overset{\cdot}{θ}}_{mi} + \frac{V_{m}}{β_{e}} {\overset{\cdot}{p}}_{mi} \\ D_{mi} (p_{mi} - p_{r}) = T_{mi} \end{matrix}

(12)

The dynamics equations of the gear transmission system are as follows

{\begin{matrix} I_{mi} {\overset{\cdot\cdot}{θ}}_{mi} + c_{mi} ({\overset{\cdot}{θ}}_{mi} - {\overset{\cdot}{θ}}_{pi}) + k_{mi} (θ_{mi} - θ_{pi}) = T_{mi} \\ c_{mi} ({\overset{\cdot}{θ}}_{mi} - {\overset{\cdot}{θ}}_{pi}) + k_{mi} (θ_{mi} - θ_{pi}) = T_{pi} \\ I_{b} {\overset{\cdot\cdot}{θ}}_{b} + c_{b} ({\overset{\cdot}{θ}}_{b} - {\overset{\cdot}{θ}}_{s}) + k_{b} (θ_{b} - θ_{s}) = T_{s} \\ I_{d} {\overset{\cdot\cdot}{θ}}_{d} + c_{d} ({\overset{\cdot}{θ}}_{d} - {\overset{\cdot}{θ}}_{r}) + k_{d} (θ_{d} - θ_{r}) = - T_{d} \end{matrix}

(13)

As shown in Figure 14, the simulation model of the short cutting transmission system is established by combining the schematic diagram and the theoretical model. The model of the hydraulic transmission system is established in AMESim, the model of the coupling gear system and the planetary gear system is established in MATLAB, and then the co-simulation model of the cutting transmission system is established in AMESim. The plunger pump and the piston motor are used instead of the super elements, the hydraulic pipe is modeled as a constant friction damping model, and the effects of valves in the hydraulic transmission system are ignored. The parameters of the simulation model are shown in Table 2.

Figure 14.

Simulation model of the multi-source drive system.

Table 2.

Parameters of the simulation model.

Name of the parameter	Value	Name of the parameter	Value
Power of the cutting motor (kW)	300	Displacement of the piston motor (mL/r)	3 × 160
Transmission ratio of the increasing gear system	0.655	Piston diameter of the motor (mm)	17
Displacement of the plunger pump (mL/r)	2×250	Length of the contact (mm)	31
Piston diameter of the pump (mm)	29.5	Transmission ratio of the coupling system	9
Length of the contact (mm)	30	Transmission ratio of the planetary system	5.5625
Diameter of the pipe (mm)	39	Inertia of the drum (kg m²)	2000
Thickness of the wall (mm)	10

Analysis of the characteristics of the cutting transmission system based on the periodic dynamic excitation

Analysis of the vibration characteristics of the system based on the flow pulsation

In order to analyze the specific vibration characteristics of the cutting transmission system prominently when the flow pulsation is the external excitation, the meshing stiffness of the coupling gear system and planetary gear system are fixed. The load torque of roller is T_d = 50,000 N m, the simulation time is set at 1.5 s, and the simulation step is set at 10⁻⁵ s. Then, the pump flow pulsation, the motor speed pulsation, and the torque pulsation are obtained, and they are shown in Figures 15 –17.

Figure 15.

Pump outlet flow.

Figure 16.

Motor output speed.

Figure 17.

Motor output torque.

It can be seen from the figure that the output flow of the pump fluctuates on 400 L/min periodically, and the average flow pulsation rate is about 17.33%. In addition, the output speed of the hydraulic motor follows the outlet flow of the pump and the speed fluctuation amplitude changes more. The torque fluctuation amplitude of the motor changes non-uniformly and the pulsation rate is 4.8%. Overall, because of the periodic motion of the pump and the motor plungers, its effects on the hydraulic system are seen as the flow pulsation and its effects on the power components are seen as the torque pulsation.

The output torque of the motor is input to the pinions of the coupling gear system by three couplings and the torque pulsation is a kind of external excitation to the coupling gear system. The vibration displacement, the acceleration, and the meshing force between the pinion and the large gear are shown in Figures 18 –20. It can be seen from Figures 18 and 20 that the vibration displacement and dynamic meshing force of the pinion present the same variation tendency as the output torque of the pump in Figure 17.

Figure 18.

Vibration displacement of the pinion.

Figure 19.

Vibration acceleration of the pinion.

Figure 20.

Dynamic meshing force between the pinion and the gear.

The displacement of the pinion along the meshing line vibrates close to 0.79 × 10⁻³ mm, and the vibration trend is similar to the input torque of the motor. It can be seen from Figure 19 that the acceleration vibration of the pinion caused by the input torque pulsation is severe and has a high frequency. The changing trend of the dynamic meshing force is almost the same as the changing trend of the vibration displacement between the pinion and the large gear.

The pump, the hydraulic circuit outlet, the motor, the coupling gear system, and the planetary gear system are selected to be the five monitoring sites in the short cutting transmission system, respectively. T_d = 75,000 N m, T_d = 50,000 N m, and T_d = 25,000 N m are the three different load conditions. The variables q_p₁ and q_m₁ reflect the flow of the pump and the motor; the variables ω_m₁, ω_p₁, and ω_r reflect the speed of the motor, the pinion, and the ring gear. The variables in the different load conditions are shown in Figure 21.

Figure 21.

Flow and speed pulsation in the different load conditions.

The variable p_p₁ reflects the outlet pressure of the pump; the variable p_m₁ reflects the inlet pressure of the motor; the variables T_m₁, T_p₁, and T_r reflect the torque of the motor, the pinion, and the ring gear, respectively. The variables are shown in Figure 22 under a roller load of 75,000 N m. The average pulsation rate of each reference point at the time period of T = 1.1–1.2 s in the different load conditions is shown in Table 3.

Figure 22.

Pressure and torque pulsation of each reference point.

Table 3.

Average pulsation rate of each monitoring site in the three load conditions.

	T_d = 25,000 N m	T_d = 50,000 N m	T_d = 75,000 N m
q_p	15.24%	17.88%	20.92%
p_p	2.33%	2.05%	1.78%
q_m	3.62%	4.26%	4.83%
p_m	2.33%	2.05%	1.78%
ω_m	1.01%	1.24%	1.99%
T_m	7.06%	5.41%	4.7%
ω_p	1.02%	1.22%	1.97%
T_p	2.64%	2.21%	2.21%
ω_r	0.76%	1.28%	2.01%
T_r	1.51%	1.16%	1.31%

It can be seen from Table 3 and Figures 23 and 24 that the greater the roller load, the greater the flow and speed pulsation, but the pressure and torque pulsation will become smaller. In case of the flow and speed, the pulsation rate of each reference point is gradually reduced along the power transmission direction. The torque pulsation is also gradually reduced along the direction of transmission chain. In general, the flow pulsation excitation is gradually reduced along the direction of transmission chain.

Figure 23.

Flow and speed pulsation rate.

Figure 24.

Pressure and torque pulsation rate.

Analysis of the dynamic characteristics of the system based on the periodic excitation

The simulation condition is set as follows: the load of the roller increases from T_d = 50,000 N m to T_d = 85,000 N m at 1.5 s. When only considering the time-varying meshing stiffness excitation or considering both the time-varying meshing stiffness excitation and the flow pulsation excitation, the fluctuations of the flow, the pressure, and the dynamic meshing force between the sun-planet gear pair are shown in Figures 25 –27, respectively. It is observed from the figures that the pump flow pulsation will increase the fluctuations of the system greatly as the external excitation.

Figure 25.

Flow of the system: (a) time-varying meshing stiffness + flow pulsation and (b) time-varying meshing stiffness.

Figure 26.

Pressure of the system: (a) time-varying meshing stiffness + flow pulsation and (b) time-varying meshing stiffness.

Figure 27.

Dynamic meshing force of the sun-planet gear pair: (a) time-varying meshing stiffness + flow pulsation and (b) time-varying meshing stiffness.

In order to analyze the dynamic characteristics of the system further, the flow of the pump, the time-varying meshing stiffness, the torque of the motor, and the dynamic meshing force were analyzed on frequency domain by Fast Fourier Transformation (FFT). The results are shown in Figures 28 –31. It can be seen from Figure 29 that the frequency distribution of the time-varying meshing stiffness is very regular, which is mainly composed of the fundamental frequency and is double the frequency. It can be seen from Figure 28 that the flow pulsation main frequencies are 333, 982, and 2326 Hz. The 982 and 2326 Hz are the triple and the seven times that of this frequency, respectively. Similarly, these frequencies can also be seen in the motor torque. Figure 31 shows that frequency spectrum of the dynamic meshing force is mainly composed of 333, 982, 2326 Hz, and the frequency near them, which are also the same as the main frequency components of the frequency spectrum of the flow. It is observed from the frequency spectrum figures that the pump flow pulsation is the main cause which affects the dynamic characteristics of the system, and as the internal excitation, the time-varying meshing stiffness will cause additional effects on this basis.

Figure 28.

Frequency spectrum of the flow.

Figure 29.

Frequency spectrum of the varying meshing stiffness.

Figure 30.

Frequency spectrum of the motor torque.

Figure 31.

Frequency spectrum of the dynamic meshing force.

Conclusion

This study is based on a new power transmission system with electromechanical-hydraulic multi-source drive, which is proposed by our research group. The electromechanical-hydraulic coupling model of the cutting transmission system is established by the proposed system, and then, the vibration and the dynamic characteristics of the cutting transmission system are analyzed based on the pump flow pulsation and the time-varying meshing stiffness.

The flow pulsation of the plunger pump is transformed into the torque pulsation through the hydraulic motor, which has a significant effect on the vibration characteristics of the coupling gear system. The effects of the flow pulsation on the dynamic characteristics of the gear system are gradually reduced along the power transmission direction. It is also found that the greater the external load, the greater the flow and speed pulsation of the system, and, at the same time, the smaller the pressure and torque pulsation of the system.

In general, the pump flow pulsation is the main cause that affects the dynamic characteristics of the system, and as the internal excitation, the time-varying meshing stiffness will cause additional effects on this basis.

Footnotes

Handling Editor: Shun-Peng Zhu

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the State Key Development Program of Basic Research of China (2014CB046304) and the Project No. 2018CDJDCD0001 supported by the Fundamental Research Funds for the Central Universities.

ORCID iD

Yang Yang

References

Jeftenić

Bebić

Štatkić

Controlled multi-motor drives. In: Proceedings of the international symposium on power electronics, electrical drives, automation and motion, Taormina, 23–26 May 2006, pp.1392–1398. New York: IEEE.

Yang

et al . Influence of controllable parameters on load sharing behavior of torque coupling gear set. Mech Mach Theory 2018; 121: 286–298.

Sun

Ding

Wei

et al . Hierarchical modeling method and dynamic characteristics of cutter head driving system in tunneling boring machine. Tunn Undergr Sp Tech 2016; 52: 99–110.

Edge

Boston

Xiao

et al . Pressure pulsations in reciprocating pump piping systems part 2: experimental investigations and model validation. Proc IMechE, Part I: J Systems and Control Engineering 1997; 211: 239–250.

Yang

Research status of axial piston pump fluid-borne noise. J Mech Eng 2009; 45: 71–79.

Kahraman

Planetary gear train dynamics. J Mech Des 1994; 116: 714–720.

Zhao

Lan

Dynamic simulation for transmission system of coal winning machine cutting part. J Vib Shock 2014; 33: 106–110.

Guo

Research on dynamic characteristics of drum shearer cutting unit. Xuzhou, China: China University of Mining and Technology, 2014, pp.39–49.

Liu

Yang

et al . Coupled dynamic model and vibration responses characteristic of a motor-driven flexible manipulator system. J Mech Sci 2015; 6: 235–244.

10.

Zhao

Yan

JN.

Analysis of parameter sensitivity of space manipulator with harmonic drive based on the revised response surface method. Acta Astronaut 2014; 98: 86–96.

11.

Sun

Shu

et al . Powertrain of longwall shearer cutting system. China: ZL 2014 1 0728258.3, 2016-08-17.

12.

Xing

Yang

Gong

Drive technique of pumps optimized combination hydraulic system in shield tunneling machine. J Zhejiang Univ 2009; 43: 511–516.

13.

Gong

Yang

Thrust hydraulic system of shield tunnel boring machine with pressure and flow compound control. J Mech Eng 2006; 42: 124–127.

14.

Parker

RG.

A physical explanation for the effectiveness of planet phasing to suppress planetary gear vibration. J Sound Vib 2000; 236: 561–573.

15.

Wei

Sun

et al . Dynamic analysis and load-sharing characteristic of multiple pinion drives in tunnel boring machine. J Mech Sci Technol 2013; 27: 1385–1392.

16.

Song

Zhang

et al . Modified torsional model development and natural characteristics analysis of 2K-H epicyclic gearing. J Mech Eng 2006; 42: 16–21.

17.

Wang

Yang

Wang

Vibration analysis of gear with variable stiffness and transmission error. J Mech Transm 2002; 26: 5–8.

18.

Study on flow ripple and valve plate optimization of axial piston pump. Hangzhou, China: Zhejiang University, 2009, pp.45–46.

19.

Simulation investigation on damping and vibration characteristics of hydraulic pipe. Hangzhou, China: Zhejiang University, 2014, pp.50–59.

Analysis of the characteristics of electromechanical-hydraulic model of multi-source drive/transmission system based on periodic excitation

Abstract

Keywords

Introduction

Principle and design of the electromechanical-hydraulic short cutting transmission system

Principle and design of the hydraulic drive circuit

Principle and design of the gear reduction system

Modeling of the electromechanical-hydraulic short cutting transmission system

Dynamics model of the coupling gear system

Dynamics model of the planetary gear system

Calculation of the time-varying meshing stiffness

Lumped parameter model of the hydraulic pump and motor

Electromechanical-hydraulic coupling model of the short cutting system

Analysis of the characteristics of the cutting transmission system based on the periodic dynamic excitation

Analysis of the vibration characteristics of the system based on the flow pulsation

Analysis of the dynamic characteristics of the system based on the periodic excitation

Conclusion

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References