Dynamic loading of a water jet propulsion drive of amphibious vehicles

The analysis results of dynamic loading of the amphibious vehicle’ water jet drive are presented, the hypothesis about the durability limitation of the drive due to the onset of resonance forced and parametric oscillations is advanced. Solutions of that problem are justified. In designs of transmissions of all-wheel-drive wheeled vehicle and tractors, amphibious vehicles and others the divided mechanical drives with spatially arranged drivelines (cardan drives) are used. Field experience and results of the experimental efforts of vehicles’ prototypes with such drives indicate limited durability of elements while safety factors are sufficiently. Limited durability of drives’ elements become apparent in damages of shafts’ splined connections, angle gearboxes, water jets, impellers, etc. The analysis of methods of design calculations demonstrates that the well-known techniques do not pay due attention to the estimation of specifics of drives dynamic loading. This work considers theoretic and experimental research of drives dynamic loading as well as justification of ways of its decrease. The objective is achieved by dynamical and mathematical models of systems, simulation modeling of dynamics, oscillating processes analysis with variation of parameters that provide system dynamic stability. Research is performed using the amphibious vehicle, its mobility on the water is provided by water jets use. This solution provides also maneuverability – ability to control of movement direction including reverse movement. Alongside with functional requirements for water jet drives of the amphibious vehicles the special reliability requirements are imposed since failure of the one of elements leads to vehicle mobility disturbance on the water. Kinematic scheme of the drive is given in Figure 1. Water jets drive with quadruple threaded screw of impellers, that placed under waterline, is fulfilled by drivelines from Left (L) and Right (R) of transmission output shafts with rotational speed 1.467 times more than rotational speed of engine ?̇?E.Due to the layout, the transmission axis is displaced relative vehicle centerline. In this regard, the installation angles of drivelines on each side are different (Table 1). * Corresponding author: pasteer@mail.ru MATEC Web of Conferences 224, 02042 (2018) https://doi.org/10.1051/matecconf/201822402042 ICMTMTE 2018 © The Authors, published by EDP Sciences. This is an open access article distributed under the terms of the Creative Commons Attribution License 4.0 (http://creativecommons.org/licenses/by/4.0/). Fig. 1. Kinematic scheme of the drive. Table 1. Values of angular coordinates that determine the spatial arrangement of drivelines Left side Right side shaft 1 shaft 2 shaft 1 shaft 2 Angle γ1 (horizontal plane) 342 1643 228 1643 Angle γ2 (vertical plane) 729 1027 415 1027 The gear ratio of bevel gear of angle gearbox is uav = 0.8947. When torque converter is interlocked the overall gear ratio of water jet drive from engine shaft to water jets impellers is udr = 0.711. Dynamical model reflects interconnection of all elements, their input, output characteristics, and consists of mathematical models of functional systems and units of transmission of vehicle including water jets drives. Such model complicates calculation procedure significantly, however, capabilities of modern computers and software allow to develop computation program that is capable without structural decomposition [1] of tested system to solve differential equation systems. MATEC Web of Conferences 224, 02042 (2018) https://doi.org/10.1051/matecconf/201822402042 ICMTMTE 2018

The gear ratio of bevel gear of angle gearbox is uav = 0.8947.When torque converter is interlocked the overall gear ratio of water jet drive from engine shaft to water jets impellers is udr = 0.711.
Dynamical model reflects interconnection of all elements, their input, output characteristics, and consists of mathematical models of functional systems and units of transmission of vehicle including water jets drives.Such model complicates calculation procedure significantly, however, capabilities of modern computers and software allow to develop computation program that is capable without structural decomposition [1] of tested system to solve differential equation systems.Application of that mathematical model of dynamical processes in "enginetransmission -transport vehicle" system is realized in sequential study of the specific, most indicative transient running regimes of transmission with the help of computers by specifically developed program from MATLAB package in Simulink Driveline application [2].Block diagram of software implementation is given in Figure 2. The above listed characteristics are determined for following transient process: water jets acceleration ashore and afloat; vehicle's water entering/leaving when simultaneous operation of track assembly and water jets; reverse gear switching, etc., as well as steadystate movement regimes at constant speed.As an example of computation results, the time function of dynamic torque on the drive shaft of the water jet when switching on/off of the reverse gear and in steady-state regime is given in Figure 3.During the simulation a number of assumptions was made, correctness of which, as well as identification of system's separate parameters were determined during experimental studies in the process of the amphibious vehicle movement.During experimental studies the rotational speed of engine shaft and dynamic torque on connecting shafts of both hull sides were recorded.Analysis of the results of experimental studies of dynamic loading at transient processes showed that dynamic factor in all regimes is less than 1.1…1.2except the reverse gear switching-on regime when dynamic factor reaches value 1.9…2.5 (Figure 4).In steady-state regimes, the engine shaft rotational speed was changed discretely in the range from minimum stable 800 to 2370 with 200-rpm intervals.Fragment of the oscillogram of dynamic torque changing and the spectral density are given in Figure 5. Amplitude-rotational characteristic of the dynamic torque is given in Figure 6.

Fig. 6. Amplitude-rotational characteristic of dynamic torque of water jet drive
Above listed data shows that the average value of torque on drive shafts when rotational speed changing from 800 to 2370 rpm increases according to quadratic dependence for left shaft (Curve 1 in Figure 7) from 41.4 to 370 Nm, and for right shaft (Curve 2 in Figure 7) from 50.9 to 440 Nm, i.e. at full engine speed the value of average torque on the right shaft is 1.19 times more than on the left one.At that, oscillating process, which forms dynamic torque, comes with frequency equal to doubled cardan one.Maximum values of dynamic component are observed at engine shaft rotational speed 1300 … 1500 rpm and are 230 Nm (Curve 3 in Figure 6).Maximum value of the dynamic torque that represents the sum of the average torque and variable component's amplitude is changed according to Curve 4. Comparison of results of numerical simulation and experimental studies coincides with sufficient accuracy at confidence probability minimum 95 %.Nature of dynamic torque changing, as well as analysis of its amplitude and frequency characteristic in all speed range of engine indicates about excitation of parametric oscillations in the system that limit the elements durability.
Parametric oscillations and resonances are dangerous phenomena, as they take place in wide range of disturbing frequency with exponentially increasing amplitudes of the dynamic torque.In the case of the parametric oscillations, the drive design is subjected to dangerous cyclic loading that can lead to fatigue breakdown of the drive elements.Therefore, the main task of the design dynamic analysis in which parametric resonances are excited is determination of the boundaries of the areas of dynamic instability in order to take measures for parametric resonance elimination when reworking.
In the mechanical system under consideration, the parameters formed by drivelines with asynchronous joints are periodically changed.The oscillation amplitude of dynamic torque in the drive is limited in this case and significant resistance moment when the vehicle is on the water does not allow to expand the backlash in the bevel gear of angle gearbox.
Therefore when analyzing the parametric oscillations excited by drivelines the drive is considered as linear system.Coupling transmission and angle gearboxes of the water jet drives is performed by drivelines with asynchronous joints.In the vertical plane the left shaft axis is located relative to the horizontal at angle   = 7°29′ ', and the right one -  = 4°15′.Gear ratio of the asynchronous joints is variable due to periodic changing in angular velocity φ̇1 of the driven joint fork from maximum φ̇1/ cos γ to minimum value φ̇1 • cos γ twice per one rotation of joints relative to the constant speed of the driving joint fork φ̇1 .Hence, driven joint fork and input shaft of angle gearbox rotate at speed φ̇2 = φ̇1 + ∆φ̇1 • cos(2φ̇1 • t) , where ∆ = (

Fig. 2 .
Fig. 2. Block diagram of the water jet's drive dynamic loading program(MATLAB package, Simulink Driveline application) Developed mathematical model allows to:  determine the characteristics of dynamic loading of water jet drives elements used for their loading spectra development and durability evaluation of relevant elements;  determine the slipping characteristics of friction elements used for evaluation of their operability and durability;  determine the processes of time variations of rotational speed and accelerations of the separate elements of transmission used for the evaluation of quality of transient running regimes.The above listed characteristics are determined for following transient process: water jets acceleration ashore and afloat; vehicle's water entering/leaving when simultaneous operation of track assembly and water jets; reverse gear switching, etc., as well as steadystate movement regimes at constant speed.As an example of computation results, the time function of dynamic torque on the drive shaft of the water jet when switching on/off of the reverse gear and in steady-state regime is given in Figure3.

MATECFig. 3 .
Fig. 3. Time function of dynamic torque on the drive shaft of the water jet when switching on/off of the reverse gear and in steady-state regime Represented results of simulation demonstrate the possibility of emergence of specific forms of oscillating processes in the system (runouts, parametric oscillations) in steadystate regime.In this case, the oscillation frequency corresponds shaft's double rotational speed.In transient processes of the reverse gear switching on/off the dynamic factor increases to 1.9…3.0.During the simulation a number of assumptions was made, correctness of which, as well as identification of system's separate parameters were determined during experimental studies in the process of the amphibious vehicle movement.During experimental studies the rotational speed of engine shaft and dynamic torque on connecting shafts of both hull sides were recorded.Analysis of the results of experimental studies of dynamic loading at transient processes showed that dynamic factor in all regimes is less than 1.1…1.2except the reverse gear switching-on regime when dynamic factor reaches value 1.9…2.5 (Figure4).

Fig. 4 .
Fig. 4. Fragment of the oscillogram of dynamic torque changing on connecting shafts when water jet operates in the reverse gear switching-on regime and steady-state regime.

Fig. 5 .
Fig. 5. Fragment of the oscillogram of dynamic torque changing and the spectral density of the process when water jet operates in steady-state regimeExperimental data analysis indicates that oscillating process has "runouts nature" that caused by summing of periodic components of the torque with near frequency.This is defined by gear ratio value of angle gearbox (19/17); therefore frequencies of periodic components of two drive shafts of the one hull side in all engine speed regimes are distinct from each other by 11.7 %.The fact that torque sensor mounted on the one shaft records cos ) = 1 • sin 2  / cos .Without considering the influence of oscillations on movement of the vehicle inertia mass the differential equation of the system's relative movement is transformed to the form    +  )  =   −   .This equation differs from the common one in that the elastic moment contains periodically changed parameter, i.e. this equation corresponds to the form of the Mathieu differential equation of parametric oscillations [3].To analyze the stability of the parametric oscillations the Mathieu equation is transformed to the form of parameters of the Ince-Strutt diagram + 0 2 [ + 2ℎ cos(2)] = 0 , where abscissa, ℎdiagram ordinate.Abscissa is determined by ratio of frequencies, natural and exciting ,  = (2 0 /) 2 ,  0 2 = /  ,  = 2̇, 2 = 2̇.Diagram ordinateℎ = , where parameter of the modulation depth  = ∆/  ,   = (  −   )/.Ince-Strutt diagram is given in Figure 7, stability region is shaded.

Fig. 7 .
Fig. 7. Ince-Strutt diagram.Analysis of the stability of parametric oscillations is made according to function ℎ =  of the considered system.In the diagram Line 1 corresponds to the parameters of the left drive (  = 7°29′ , с=7780 Nm/rad.Line ℎ =  crosses alternating regions of stability and instability.Parametric resonances are possible in the instability region.Graph 1 of the diagram shows that Line ℎ =  crosses a wide region of instability.This means higher probability of dynamic stability loss, practically at any technically possible value of

Table 1 .
Values of angular coordinates that determine the spatial arrangement of drivelines