Comparative analysis of dynamic load characteristics of artillery steering gear based on different MBD methods

The multi-body dynamics analysis model of the steering gear of a certain type of artillery was established. The dynamic load characteristics of the output gear of the steering gear motor and its driven gear under the given speed and load conditions were calculated based on the three different MBD analysis methods, such as the speed, meshing force, and contact stress. Through the comparative analysis, it was concluded that the calculation results of rigid-flexible coupling analysis were closer to the theoretical value and the fluctuation range was smaller, which provides reference and basis for the prediction of gear fatigue life and the optimization design under dynamic load conditions.


Introduction
The artillery relies on the steering machine to transmit power, drive the turret, and complete the gun aiming action.The structure of the steering gear is compact, the transmission efficiency is high, and the working load is large.The inertia load of the turret is also large during the gun adjustment process.The steering gear is prone to fatigue damage during repeated gun adjustment [1][2][3][4].Therefore, it is necessary to study the fatigue reliability of the steering gear in the actual working process.In the past, the design and research of gear systems mainly focused on static design.With the development of gear transmission systems towards high speed, heavy load, and high precision, static design can no longer meet the demand, and multi-body dynamic design has become an inevitable trend in gear design.Multi-body system dynamics is usually divided into three categories: multi-rigid-body system dynamics, rigid-flexible coupling system dynamics, and multi-flexible system dynamics.The multirigid-body system cannot obtain the stress and strain without considering the structural deformation, which is mainly used to analyze the kinematic characteristics of the mechanical structure.A flexible multibody system can not only carry out kinematic analysis but also analyze the structural strength and fatigue life [5][6][7][8][9][10].RecurDyn is a new generation of multibody dynamics simulation software, which adopts the theory of motion equations in relative coordinate systems and completes recursive algorithms [11][12][13].In this paper, the above three methods are used to establish a multibody dynamic analysis model of a steering gear transmission based on RecurDyn, and the simulation results are compared.The influence of different modeling methods on the calculation results is analyzed, to predict the fatigue life and dynamic of the steering gear.

Modelling theory of multibody dynamics
The dynamics of multi-rigid-body systems is to establish a mathematical model suitable for solving computer programs for kinematics and dynamics analysis of complex systems composed of multiple rigid bodies and to seek efficient and stable numerical solutions.Two different mathematical modeling methods, namely the Lagrange method and the Cartesian method, have been formed in the fields of aerospace and machinery.The Lagrange method is a relative coordinate method, which takes one rigid body as the reference object, and the position of another rigid body relative to the rigid body is described by the generalized coordinate of the hinge.The generalized coordinate is usually the relative rotation angle or displacement between the connecting rigid bodies.In this way, the position of the open-loop system can be completely determined by the Lagrange coordinate matrix q of all hinges.Its dynamic equation is the second-order differential equation group of the Lagrange coordinate matrix [14]: To make the equations stylized and universal, Matrices A and B often contain information describing the topology of the system, which is quite complex in form and requires human intervention in the selection of generalized coordinates, which is not conducive to automatic computer modeling.
The Cartesian method is an absolute coordinate method that takes each object in the system as a unit and establishes a coordinate system consolidated on a rigid body.For a system composed of N rigid bodies, the number of coordinates in the position coordinate array q is 3N (two-dimensional), 6N, or 7N (three-dimensional), and these position coordinates are not independent due to the existence of hinge constraints.The general form of a system dynamics model can be expressed as: where Φ is the constraint equation for the position array q, is the Jacobian matrix of the constraint equation, and λ is the Lagrange multiplier.This type of mathematical model is the DAEs-Differential Algebraic Equations, also known as the Euler-Lagrange Equations, which have a large number of equations.However, the coefficient matrix is sparse, which is suitable for computers to automatically establish a unified model for processing [15][16].

Dynamics modeling of multibody gear system
The gear pair composed of the output gear of a steering motor and the driven gear is taken as the research object, as shown in Figure 1.The steering gear is mainly composed of a motor, electromagnetic clutch, gear transmission chain, planetary differential mechanism, turbine pair, elastic tooth assembly, and so on.It is the rotational drive device of artillery and turret, which is used to change the firing direction of artillery and to accurately aim and maintain the aiming position [5][6][7].The gear geometry model is established by parametric modeling.The multi-rigid body model, rigidflexible coupling model, and multi-flexible body model of a gear transmission system are constructed for multi-body dynamics analysis.The simulation results are comprehensively compared to obtain the influence of different modeling methods on the analysis results.

Multi-rigid-body modeling
The three-dimensional geometric model is imported into the RecurDyn analysis module.Firstly, the local coordinate system of the parts is established to determine the relative position of the space, and the geodetic coordinate system is added to determine the position relationship of each part relative to the whole.Then we add constraints between the parts, drive, and load of the transmission system.Since the gear rotates in a fixed axis, the RevJoint is defined between the two gears and the rack (GROUND).
The force transfer between the parts in the machinery is realized through the contact of the two parts, and the two rigid bodies in contact with each other in the rigid body mechanical system cannot invade each other.That is, contact is a unilateral constraint.Contact and collision are often linked together.For the phenomenon of contact and collision, the continuous contact force method (that is, based on the penalty function) can be adopted.The contact calculation method based on the penalty function is widely used.The contact and collision process can be simulated more realistically by dealing with continuous dynamic problems.Surface contact can be used for rigid body contact in RecurDyn.Surface contact is the use of key surfaces to simplify complex solid contact, which can calculate complex and arbitrary contact problems.Surface to Surface and Extended Surface to Surface can be selected, and the difference is that the discrete methods of surfaces as Action are different [16].In this paper, a face-to-face contact method is used to pre-set the geometric surfaces that two gears may contact as a face set (Face Surface), as shown in Figure 2.

Gear rigid-flexible coupling model
In the multi-rigid-body analysis, since each component in the model is rigid, the elastic deformation of the component is ignored, and the dynamic stress of the component cannot be obtained from the analysis results.To reflect the dynamic characteristics of the gear more truly, the elastic deformation of the gear must be considered to establish a flexible gear that can reflect the elastic deformation [17][18].When the modal flexibility (RFLEX) method is used for analysis, the flexibility of key components is first processed.Flexibility processing needs to apply professional finite element flexibility to the modal analysis of rigid gear, generate the modal flexible body file required, and import this file into RecurDyn to replace the original rigid body.The rotation pair constraints are added between the two gears and the ground, and the position of the flexible body is on the main node of the rigid area, as shown in Figure 3.
At this time, the parts are independent and need to define contact between the gear contact surface.There are many types of flexible body contact, such as flexible surface-flexible surface contact, flexible surface-rigid surface contact, ball-flexible surface contact, flexible line-flexible line contact, and flexible line-surface contact.In this paper, flexible surface-rigid surface contact is used in rigidflexible coupling.The flexible gear contact surface is predefined as a patch set, and the contact pair is established between the predefined rigid body contact surface (Face-Surface).

3.3Multi-flexible body model
The multi-flex analysis is based on the multi-rigid body model and the rigid-flexible coupling model, and the motor gear is also flexible to form a multi-flexible coupling simulation model.The motor gears are flexibly treated in the same way as the driven gears, and the multi-flex simulation model is finally built, as shown in Figure 4.The gear pair contact setting adopts the flexible body-specific FSurface to FSurface contact, that is, the flexible body surface contacts with the flexible body surface.The tooth surface of the two gears is predefined as the Patch sets.When setting the contact, the two faces are directly selected as the contact surface, which can greatly facilitate the operation.The speed defined as STEP (TIME, 0.1, 0, 0.2, 2*PI) is applied to the motor rotation gear.The load is defined as STEP (TIME, 0.1, 0, 0.2, 100, 000) on the rotational pair of the driven gear, and the load torque increases from 0 to 1, 000 N‧m in 0.1 s-0.2 s.

4.1Comparative analysis of the instantaneous speed of driven gear
The speed applied to the motor gear join is the driving load, the driving speed is consistent in all three cases, and the variable speed of the driven gear changes with time under the three analysis methods, as shown in Figure 5.Comparison of instantaneous speeds of driven gears.In Figure 5, the speed curves all show a random fluctuation trend, and the fluctuation average and amplitude of the stable section are obtained for comparison, as shown in Table 1.Combined with Figure 5 and Table 1, it can be seen that the average fluctuation of the instantaneous speed obtained by the three simulation methods as a whole is close to coincident, the average speed is -2.1 rad/s, the transmission ratio is 2.99, and the error is very small compared with the theoretical value 3. Through the standard deviation and other indicators to analyze, the results of the rigid-flexible coupling and multi-flexible analysis are very close, and the standard deviation of the multi-rigid body analysis is the largest, which means that the fluctuation is more intense.

Comparative analysis of instantaneous response torque of motor gears
Figure 6 shows the comparison of the instantaneous response torque of the motor gear, and Table 2 is the statistical analysis table of the instantaneous response torque of the motor gear.It can be seen from the chart that the rigid-flexible coupling and multi-flexible coupling simulation results are almost the same, the simulation mean deviation of the multi-rigid body results is slightly larger, and the fluctuation amplitude of the instantaneous response torque during the multi-rigid body simulation is also larger than that of the other two analysis methods.

Comparative analysis of instantaneous meshing force
Figure 7 shows the comparison curve of the dynamic meshing force of the gear, and Table 3 is the statistical analysis table of the gear dynamic meshing force.As can be seen from the figure and table, the results of the rigid-flexible coupling analysis are between the results of the other two analysis methods, and the fluctuation amplitude of the rigid-flexible coupling analysis results is the smallest from the standard deviation and variance and other indicators.The gear meshing force obtained by theoretical calculation is about Fn = 13, 963 N, which is close to the results of rigid-flexible coupling analysis.The above comparative analysis is close to the theoretical value from the perspective of transmission ratio and meshing force, which verifies the accuracy of the simulation model.Overall, the results of the rigid-flexible coupling analysis method are closer to the theoretical values and the fluctuation amplitude is smaller.
From the perspective of operational efficiency, multi-rigid body analysis is the fastest.However, it cannot analyze structural stress-strain without considering structural deformation, so it is only suitable for kinematic analysis.The computational efficiency of the rigid-flexible coupling model is higher than that of the multi-flex coupling simulation.In addition to the algorithmic reasons, the number of meshes of the multi-flex analysis model is much higher than that of the rigid-flexible coupling model, and the computer computing power requirements are also higher.When analyzing practical problems, if you only care about the structural strength and fatigue reliability of some key components, you can use the rigid-flexible coupling method to only flex the key parts, which can improve the calculation efficiency without reducing the accuracy.

Conclusion
In this paper, a pair of meshing gears of an artillery steering machine is selected, and three different multibody dynamic analysis methods are used for modeling and simulation calculations.The rigidflexible coupling analysis method can be obtained by comparing with the theoretical values to ensure the accuracy of the results and have high computational efficiency, which has certain advantages over multi-rigid body analysis and multi-flexible analysis.It should be noted that this paper only selects a pair of meshing gears for preliminary analysis.The number of gear pairs in the gear transmission system of the entire directional machine is greater, the structure is more complex, and the amount of calculation is increased.Only the flexible body model of the heavy part can be established in the multibody dynamic analysis, and the remaining gears can be rigidized.The research conclusions of this paper can guide the subsequent modeling of multibody dynamics of steering machine gear transmission systems and lay the foundation for fatigue life prediction based on multibody dynamics and optimal design under dynamic load conditions.

Figure 1 .
Figure 1.Geometric model of a steering gear and driven gear.

Figure 3 .
Figure 3. Rigid-flexible coupling gear model.Figure 4. Flexible-flexible coupling gear model.To facilitate the comparison of the analysis results, the same boundary conditions are applied to the three MBD models, namely motor gear speed and the driven gear load torque.The STEP function is used to add motion to the rotation pair.The speed defined as STEP (TIME, 0.1, 0, 0.2, 2*PI) is applied to the motor rotation gear.The load is defined as STEP (TIME, 0.1, 0, 0.2, 100, 000) on the rotational pair of the driven gear, and the load torque increases from 0 to 1, 000 N‧m in 0.1 s-0.2 s.

Figure 4 .
Figure 3. Rigid-flexible coupling gear model.Figure 4. Flexible-flexible coupling gear model.To facilitate the comparison of the analysis results, the same boundary conditions are applied to the three MBD models, namely motor gear speed and the driven gear load torque.The STEP function is used to add motion to the rotation pair.The speed defined as STEP (TIME, 0.1, 0, 0.2, 2*PI) is applied to the motor rotation gear.The load is defined as STEP (TIME, 0.1, 0, 0.2, 100, 000) on the rotational pair of the driven gear, and the load torque increases from 0 to 1, 000 N‧m in 0.1 s-0.2 s.

Figure 5 .
Figure 5.Comparison of instantaneous speeds of driven gears.In Figure5, the speed curves all show a random fluctuation trend, and the fluctuation average and amplitude of the stable section are obtained for comparison, as shown in Table1.Combined with Figure5and Table1, it can be seen that the average fluctuation of the instantaneous speed obtained by the three simulation methods as a whole is close to coincident, the average speed is -2.1 rad/s, the transmission ratio is 2.99, and the error is very small compared with the theoretical value 3. Through the standard deviation and other indicators to analyze, the results of the rigid-flexible coupling and multi-flexible analysis are very close, and the standard deviation of the multi-rigid body analysis is the largest, which means that the fluctuation is more intense.Table 1.Statistical analysis table of instantaneous speed of driven gears.

Figure 6 .
Figure 6.Comparison of instantaneous response torques of motor gears.Table2.Statistical analysis of instantaneous response torque.

Figure 7 .
Figure 7.The comparison curve of the dynamic meshing force of the gear.

Table 1 .
Statistical analysis table of instantaneous speed of driven gears.

Table 2 .
Statistical analysis of instantaneous response torque.

Table 3 .
Statistical analysis table of dynamic gear meshing force.