Mathematical Model Development Using Matlab Code

In this method, the recoil brake and dashpot forces are calculated by evaluating the pressure drop across the orifices and dashpot during the piston movement. The pressure drops when multiplied with the effective area over which it occurs gives the resistive force offered. The total resistive force is calculated by summing the forces across all the orifices

The following assumptions are considered for the purpose of simplifying the modelling procedure

a) Since, major pressure drop and damping is expected to occur across the orifices, the flow in other regions is considered to be laminar, to simplify the governing equations.

b) The flow is considered as fully developed flow.

c) The friction factor is assumed to be zero. Wall friction losses are neglected under the assumption that pressure drop across the orifices are significant. This simplification is appropriate for short flow paths or configurations

d) The value of discharge coefficient of the orifices is constant. The fixed discharge coefficient value assumption provides a reasonable first approximation since experimental calibration data are unavailable.

e) For different positions of piston during recoil and recuperation, only those orifices contributing to significant damping are considered. The damping due to remaining orifices, at that instant, is considered to be negligible.

Initial conditions & displacement, velocity and acceleration relations

The system is stationary before firing. Therefore, the two initial conditions i.e. initial displacement and initial velocity of the system, for solving the second order differential equation Eq. (4), are considered as zero.

x(t = 0) = 0 (5)

v(t = 0) = 0 (6)

The instantaneous displacement (x), velocity (v) and acceleration (a) of the piston at any time step t and subsequent time step (t + Δt), are related by following Eqs. (7) and (8)

x (t + Δt) = (t) + x(t) Δt (7)

v(t + Δt) = v(t) + v(t) Δt (8)

The sign convention for the displacement is considered to be positive when the piston moves in the recoil direction as shown Figure 2.

Calculation of Effective Damping Force by Recoil Brake and Dashpot

The damping force is offered by the fluid resistance when it flows across the orifices. This force is evaluated by calculating the pressure drop occurring across each orifice and multiplying it with the corresponding effective area, over which the pressure drop occurs. The effective damping force of the system is the sum of the damping forces across the individual orifices at that particular instant of time. The flow path of the fluid is different during recoil phase and recuperation phase. Therefore, the damping force is calculated by individually considering the recoil and recuperation phase separately.

Damping Force during Recoil Phase: During recoil phase, the movement of the piston forces the fluid to flow from the outer region, through orifices in the floating shuttle and inclined orifices, into the inner region. The flow then splits into two paths. One flows into the annular orifice, whose orifice area varies with the piston stroke and fills the cylinder volume on the left side of piston. The remaining fluid passes through an annular gap acting as a varying area long annular orifice, formed between the control rod and the inner wall of the piston. Further, this part of the fluid passes through another set of orifices, a micron annular orifice and finally fills the dashpot. The flow direction during recoil inside the cylinder is marked in red arrows as shown in Figure 2. The recoil phase is considered whenever the velocity of the piston is positive i.e. v(t) > 0

Figure 2: Fluid flow direction inside the cylinder during recoil phase and recuperation phase

When recoil starts, the fluid in the outer region flows to the inner region through number of semicircular orifices in the floating shuttle/ spring seating area. The pressure drop across these orifices is calculated by Eq. (B1) in appendix and damping force is calculated by Eq. (9)

Then, the fluid passes through inclined orifices attached to the piston. The pressure drops across these orifices is calculated by Eq. (B2) in appendix and damping force is calculated by Eq. (10).

These two orifices contribute to the damping all the time, irrespective of the position of the piston.

After passing through the inclined orifices, a portion of fluid enters into the long varying area annular orifice. The length of the region of the control rod where diameter is varying is l0. Only when the displacement of the piston is less than l0, the pressure drop of this region will contribute to the damping. Assuming the radius of the control rod varies linearly from start to end, the taper angle is given by α. The radius of control rod as function of distance, rc1x is evaluated. The pressure drops occurring due to the long variable area orifice is calculated using Hagen Poiseuille’s equation given by Eq. (B3) in appendix and the corresponding damping force is given by Eq. (11).

After long varying area annular orifice, the fluid flows across peripheral semicircular orifices before the dashpot. From the initial piston position, till the piston head displaces up to l0, the flow enters through the varying area annular orifice. The pressure drop is given by Eq. (B4) in appendix and damping force during this condition is given by Eq. (12).

When the piston displaces beyond l0, the flow directly goes into these orifices. In this condition, the pressure drop is given by Eq. (B5) in appendix and damping force is given by Eq. (13).

The micron orifice’s annular region is formed by the overlap of dashpot’s piston and inner cylindrical hole in the piston. This overlap happens from the initial position of the piston till it displaces by lpd which is the length of the extended cylindrical rod. Therefore, the pressure drop contributed by this orifice will be applicable only during the phase when the displacement is less than lpd. As the piston displaces beyond lpd, the fluid flows directly into the dashpot. The pressure drop is calculated using Hagen Poiseuille’s equation given by Eq. (B6) in appendix and the damping force is given by Eq. (14).

Damping Force During Recuperation Phase: At the end of recoil phase, the compressed spring expands forcing the piston to the initial equilibrium position. Due to the movement of the piston, the fluid stored in the chamber volume is forced to flow across the annular orifice formed by the piston head and the control rod. Then the fluid flows from the inner region to the outer region through inclined orifices and peripheral orifices in the floating shuttle/spring seating area. At the end of recuperation phase, the fluid in the dashpot flows out through the micron orifice providing a cushioning effect. Then it flows to outer region through the long varying area orifice. The flow direction during recuperation inside the cylinder is marked in black arrows as shown in Fig. 2. The recoil phase is considered whenever the velocity of the piston is negative i.e., v(t) < 0

When the recuperation starts, the fluid is forced to flow through the annular orifice formed by the piston head and control rod. To account for the varying diameter of the control rod, mean diameter of its’s ends is considered. The pressure drop is given by Eq. (B7) in appendix and damping across this orifice is given by Eq. (15).

Then the fluid enters into the outer region through the inclined peripheral orifices. The pressure drop is given by Eq. (B8) in appendix and damping across this orifice is given by Eq. (16).

This part of fluid also passes through the orifices in the floating shuttle/spring seating area. The pressure drop is given by Eq. (B9) in appendix and damping across this orifice is given by Eq. (17).

These above orifices contribute to the damping all the time, irrespective of the position of the piston

Due to movement of the piston, fluid in the dashpot exits and passes through the peripheral orifices adjacent to the dashpot. The pressure drop in these orifices is significant only when the position of the piston is in such a way that there is no overlap of dashpot piston cylinder. This condition occurs when the displacement of the piston is greater than the lpd. The pressure drop is given by Eq. (B10) in appendix and damping across this orifice is given by Eq. (18) respectively.

At the end of recuperation, complete fluid in the dashpot is forced out through the micron level orifice. As discussed earlier, the pressure drop contributed by the micron level orifice will be applicable only during the phase when the piston comes back to a position where the displacement is less than lpd. The drop in pressure in this orifice is given by Eq. (B11) in appendix and damping force is given by Eq. (19).

Total Effective Damping Force Due to Recoil Brake and Dashpot: The sum of damping forces due to all the orifices at a particular instant of time gives the total damping. The total damping force is given by Eq. (20)

Calculation of Piston Displacement and Transmitted Force

From the governing equation of motion Eq.(4), acceleration can be calculated by

Using the initial conditions Eqs. (5) and (6), Eq. (21) is solved to get the instantaneous acceleration at time step t. The velocity and displacement values corresponding to next time increment (t + Δt), is calculated by using the Eqs. (5,6).

The force transmitted to the structure during recoil and recuperation is given by the sum of forces due to the recuperator, recoil brake and dashpot given as shown in Eq. (22).

Mathematical Model Development Using Simulink/Simscape Fluids

In this approach, to analyze the dynamic behavior of the tank gun recoil system, a detailed simulation model is developed using Simulink/Simscape Fluids. The working of the recoil system is carefully analyzed and different components such as recoiling mass, recuperator, recoil brake, dashpot are broken down into corresponding blocks to accurately model, simulate their working and interaction with other components. By modelling these components into modular blocks, the Simscape environment enables a structured and scalable approach for simulating the entire system.

The following assumptions, similar to previous approach, are considered for the purpose of simplifying the modelling procedure

1.                  The flows are assumed to be in one dimension, for preliminary analysis.

2.                  The value of discharge coefficient of the orifices is constant. This assumption provides a reasonable first approximation since experimental calibration data are unavailable

3.                  The motion of walls of orifices is not considered.

4.                  The flow friction is assumed to be negligible, considering the pressure drop across the orifices are significant.

 

Overall Modelling Strategy of Recoil System in Simscape Fluids

In the Simulink/Simscape Fluids, individual blocks are created for modelling the mass, recuperator, recoil brake and dashpot. The mass block is connected to the ground structure through two parallel buffer cylinder blocks. Each buffer cylinder block has a recuperator, recoil brake and dashpot blocks representing a mass, spring, damper system. The recuperator block is connected in parallel with the recoil brake and dashpot blocks. The external input force block is connected to provide the firing force to the system. The overall schematic of Recoil system modelled in Simscape Fluids is shown in Fig. 3.

Modelling of External Input Force in Simulink/ Simscape Fluids

The input force is modelled using Ideal force source block which provides the firing force to the model. The firing force Ff, is fed into the model with the help of 1-D lookup Table and Translational Ideal force source blocks as shown in Figure 4. The force vs time values are fed into table data of lookup table and its output is fed into Ideal force source block. The simulation time from the Clock block is fed into lookup table to output the corresponding force value.

Figure 3: Overall schematic of Recoil system modelled in Simscape Fluids

Figure 4: Input force using Ideal force source block

Modelling of mass and recuperator in Simulink/Simscape fluids

The total mass of the recoiling components is modelled by an Ideal mechanical translational mass block. The recuperator i.e. two series springs is modelled by connecting two Ideal mechanical linear spring blocks in series as shown in Figure 3. The stiffness value of the spring, k is entered in the block parameter. To capture the effect of friction between the moving components of the recoil system, a Translational friction block is connected parallel to the cylinders.

Modelling of Recoil Brake in Simulink/Simscape Fluids

The recoil brake is modelled using a Double-acting hydraulic cylinder block. The block has two mechanical translational conserving ports corresponding to the piston rod & cylinder clamping structure and two hydraulic conserving ports. The two hydraulic conserving ports of the double acting cylinder are connected by a hydraulic circuit consisting of all the hydraulic damping elements of the recoil system. The hydraulic damping circuit is responsible for capturing the damping offered by the recoil brake. The different regions in the fluid flow path are modelled using different blocks as shown in Table 1. The hydraulic damping circuit of recoil brake is shown in Figure 5.

Table 1: Blocks used for modelling different regions of recoil brake

Region	Blocks used for modelling the region
Outer Annular region	Annular Orifice
Inclined Orifice region	Fixed Orifices
Gradual Area Change region	Simscape Component (Gradual Area Change)
Variable annular orifice region	Simscape Component (Annular Orifice)
Sudden Area Change region	Simscape Component (Sudden Area Change)

 

Outer Annular Region: As the firing force acts on the piston, it displaces the oil in the region between the buffer cylinder and the piston. The fluid flows between the coils of the spring and enters into the main orifice. This complex flow is simplified and modelled as two annular orifice flow blocks. One annular flow is considered in the gap formed between spring and buffer cylinder and other annular flow is considered between the spring and piston. These two annular orifice blocks are then connected in parallel as shown in Figure 6. Similarly, during recuperation the fluid is forced to flow back in to the outer annular region through the inclined orifice.

Figure 7: Simscape Fixed orifice blocks connected in parallel

Inclined Orifice Region: The oil flows through the inclined orifices during recoil and recuperation phase. This region is represented by a number of Fixed Orifice blocks connected in parallel to each other as shown in Figure7. A subsystem is created selecting the orifices and with the parameters of orifices defined in the Mask of the subsystem as shown in Figure. 5.

Figure 8: Simscape Gradual Area Change block

Gradual Area Change Region: The conical region, near the inclined orifice has gradually varying area and it is modelled with a custom Simscape component based on Gradual Area Change (GAC) Block. The GAC block is based on the Local Resistance block. The GAC block is bidirectional and computes pressure loss for both the direct flow (gradual enlargement) and return flow (gradual contraction). As mentioned earlier, the diameter of the control rod varies with respect to length. As a result, the cross-sectional area at the inlet and outlet of the gradual area change region also varies as the piston moves. To calculate the cross-section area at the inlet and outlet of the gradual area change region at a particular instant, the instantaneous diameters corresponding to displacement output of the piston at is calculated. These instantaneous diameters are sent to Gradual Area Change block as shown in Figure 8.

Variable Area Annular Orifice Region: The annular orifice dimensions are decided by the hole diameter of the piston head (constant) and diameter of the control rod (varies with piston displacement). The control rod diameter varies as the piston moves. Therefore, a component block is used to calculate the instantaneous radius of the annular orifice.

The block calculates the instantaneous value of radius of control rod based on the position of piston and sends it to variable area annular orifice block to calculate the instantaneous pressure drop as shown in Figure 9.

Figure 9: Simscape Variable area annular orifice block

Figure 10: Simscape Sudden Area Change block

Sudden Area Change region: Similar to Gradual Area Change block, the Sudden Area Change block is modelled to account the expansion and contraction of oil flowing out from the annular orifice. Similar to GAC block, the calculated pressure loss co-efficient is then passed on to the local resistance block equation. Since the area change depends on the annular orifice dimensions, the instantaneous control rod diameters are calculated and fed to the Sudden Area Change region block as shown in Figure 10.

Modelling of Dashpot in Simulink/Simscape Fluids

To model the damping effect offered by the dashpot, an Ideal Force Source block is used. It is modelled in such a way that it provides an opposing force against the piston motion at the end of the recuperation stroke to provide a cushioning effect. The force acts only after the piston crosses below lpd and is modelled using a MATLAB function block with appropriate condition statements. The dashpot damping force is calculated using Eq. (23).

Measurement Of Recoil System Parameters in Military Vehicle

The primary objective of the past experiment was to estimate the recoil distance, recoil velocity and transmitted force from the recoil system in selected number of military vehicles during live firing at the Field Firing Range. As the firing occurs, the recoil piston rod, which is attached to the gun breech moves backwards and the entire reciprocating mass translates over the fixed cradle, which in turn is assembled with the trunnion block. The cradle is assembled with a deflector guard in order to protect the crew from the cartridge case, which is ejected out from the breech after firing. The Fig. 11 below shows the schematic of a gun and recoil assembly, which is integrated inside the military vehicle.

Figure 11: Armament system assembled in the military vehicle

The data such as recoil length versus time and recoil velocity versus time (using tachometer attached rigidly to the cradle) and pressure of the hydraulic fluid in two buffers versus time (using hydraulic pressure transducer connected to a data acquisition system) were captured during the experiment. The sensors have been positioned at the locations which have been indicated in the Figure 11. During each round of firing, the time variant parameters have been captured.

The experiments were conducted on various military vehicles for checking its repeatability with every round of firing and at different gun angles. The transmitted force to the structure is calculated from the above pressure, which is developed in the buffers. The variation of transmitted force has been captured as a function of stroke and has been used as a benchmark of comparison for the present models, since these estimations have been carried out on the actual military vehicle itself.

Results & Discussions

The mathematical model for the tank gun recoil system is developed using MATLAB code and Simscape fluids. The system is modelled as nonlinear 1D, mass, spring and damper system subjected to external force. The firing force acting on the system is evaluated from the experiments. The inertia force is modelled using Newton’s second law and recuperator forces are modelled by using helical spring equations. In the MATLAB code model, recoil brake is modelled using the fluid dynamic equations to evaluate the pressure drop and damping across all the orifices during the piston movement. The governing equation of motion is solved to evaluate the acceleration, velocity and displacement of the piston. The force transmitted to the structure during firing is evaluated by summing the forces exerted by recuperator, recoil brake, dashpot and pressure acting on the piston. In Simscape fluids model, the fluid flow path of the recoil brake is analysed and corresponding hydraulic circuit is modelled using blocks representing different elements of the recoil system. Here the transmitted force is evaluated by using a force sensor block between the system and ground. The results obtained from the models are shown in Figures12, 13, 14.

It is observed that, with the application of firing load, the mass recoils from the initial equilibrium position and its displacement increases. After reaching maximum displacement, the energy stored in the recuperator during recoil phase, forces the mass to the return back to equilibrium position. The mass returns to the initial position without oscillations due to the damping offered by the recoil brake. At the end of recuperation phase, the dashpot provides a cushioning effect by dampening the residual recuperator force available at the end. The transmitted force to the structure increases steeply with the application of firing load and reaches a maximum value. It drops back quickly and finally becomes zero. It is observed that the transmitted force is significantly lesser when compared to the firing force acting on the system proving the fact that the recoil system dissipates the firing loads before transmitting to the structure.

The developed mathematical model has been validated against experimental results, showing strong agreement. The key performance parameters such as maximum recoil displacement, recoil velocity and transmitted force are predicted with acceptable accuracy. The maximum displacement of the recoiling mass during testing was around 349 mm. The maximum displacement predicted by mathematical model developed in MATLAB is 356 mm and that developed using Simscape fluids is 326 mm. The MATLAB and Simscape models show good agreement with experiment with an error of 7 mm and 23 mm in maximum displacement respectively. Similarly, the maximum recoil velocity was found to be 9.32 m/s during testing, while that predicted from the developed mathematical model and Simscape fluids model are 9.24 m/s and 8.92 m/s, respectively and resulting in an error of 0.08 m/s and 0.4 m/s respectively. The maximum transmitted force recorded during testing was 82 tonnes. The force obtained from the developed mathematical model and Simscape fluids model are 82.3 tonnes and 81.8 tonnes repectively, with an error of 0.3 tonnes & 0.2 tonnes respectively. Figure 15 shows the transmitted force variation with respect to recoil distance obtained from the models and experiment. It is clear that both models accurately capture the trend in the variation of transmitted force. This validation confirms that the model is capable of accurately representing the physical system behavior and can be reliably used for further simulation. The developed model will be helpful in understanding of working of existing recoil systems and their effect on the stability of the vehicle. This will serve as foundation in designing and optimizing advanced recoil systems for armoured fighting vehicles.

It can be seen that the results obtained from developed MATLAB model and Simscape fluids model during the recoil phase are very close. The variations observed between the results during the recuperation phase can be attributed to the difference in modelling of dashpot region. In the Simscape fluids model, due to the assumptions considered and based on the standard available blocksets, the actual flow getting in the dashpot could not be modelled accurately, unlike the customization which was possible in the MATLAB code. Hence, the developed mathematical model by using MATLAB code, yields closer results to experimental findings. In this work, laminar, one-dimensional flow with negligible friction and a constant discharge coefficient was assumed to simplify the governing equations and focus on the principal flow mechanisms. While these assumptions do not fully represent the complex, potentially turbulent conditions observed in high-pressure recoil conditions, the resulting model nonetheless produced predictions in close agreement with experimental data. The assumptions, therefore, provide a reasonable trade-off between analytical simplicity and predictive accuracy. Future work will extend the model to include turbulence and transient effects to broaden its applicability.

Acknowledgments

The authors are very thankful to Utkarsh Chitransh, Technical Officer (CEAD) for providing support in creating schematic of armament assembly and Centre for Engineering Analysis & Design (CEAD) division of CVRDE for their support involved in execution of the above work.

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Appendix A: Equations for calculating spring stiffness

Appendix B: Equations for calculating pressure drop across orifices