Submarine Abstract: The Objectives Of Our Experiment Were To

Submarineabstractthe Objectives Of Our Experiment Were To Gain Underst

Abstract The objectives of our experiment were to gain understanding of dynamical systems and modeling experience through MATLAB and SIMULINK simulation and understanding second order system concepts such as damping ratio and natural frequency in dynamic systems. The problem was to simulate a submarine that is towed by ship using a 300 ft cable. The cable manufacturer indicated that the cable has elastic properties so we assumed that the cable is acting like a spring. Therefore, we modeled the problem using SIMULINK in MATLAB after determining the equation of motion of the submarine so that we could get what was asked in the problem. The maximum force in the tow cable which was 8194.1 lb and it occurs at 1.76 seconds after the towing starts.

The maximum force occurs when the submarine goes through its maximum acceleration because force is equal to mass times acceleration, and since the mass is constant, force is at maximum when the acceleration is maximum. Also, the elongation in the tow cable at steady state is 3.59 ft. All of the results we got agree with the theory, which means that we used the theoretical model to validate our simulation. The problem is a simple dynamics problem involving a second-order differential equation. We have a submarine being towed by a ship, with the submarine’s mass denoted as m1, the ship’s mass as m2, and the cable stiffness as k.

The stiffness k can be obtained using the relation: k = (appropriate formula or reference). The wave and viscous drag on the submarine can be assumed to be linearly proportional to its forward velocity under tow, represented by the damping coefficient b. The free body diagram (see figure 1) shows that both the submarine and the ship are moving to the right (positive x1 and x2 directions), with the ship pulling the submarine while opposing forces from waves and viscous drag act against motion. We focus on the submarine’s motion to derive the equation of motion, which is based on the fundamental law ð¹ = ð‘šð‘¥ [2].

Two main forces act on the submarine: the spring force 𑘠ð‘¥! − ð‘¥! in the positive direction, and the viscous drag forces ð‘ð‘¥! in the negative direction. Rewriting the equation, ð‘š!ð‘¥! = 𑘠ð‘¥! − ð‘¥! − ð‘ð‘¥! [3]. Recognizing that ð‘¥! = v, the velocity, integrating to find the displacement x2 = v t, the motion equation becomes ð‘¥! = ! !"!!! !!!! !! [5], and the equation governs the dynamics of the system, allowing simulation in MATLAB’s SIMULINK environment. The length of the cable at steady state can be calculated by: 𑘠= ∠! [7], and the damping ratio ζ = ! ! !" [8].

Results from the simulation show that the system’s behavior aligns closely with theoretical expectations. For example, the maximum elongation in the sample calculation was 3.6 ft, which matches the graph results in the report. Changing the cable length influences the maximum force, velocity, displacement, and elongation, demonstrating the impact of cable properties such as damping ratio. Specifically, increasing the length to 694.34 ft reduces the maximum force to 6397.8 lb, with a velocity of 5.22 ft/s, and a displacement of 116.67 ft, while elongation increases to 8.3 ft. This indicates the importance of cable design considerations in dynamic responses. The quick settlement of the submarine in shorter cables reflects higher restoring forces and damping, showcasing the system’s stability characteristics.

Paper For Above instruction

The following paper explores the modeling, simulation, and analysis of a submarine towed by a ship using MATLAB and SIMULINK, emphasizing second-order dynamic systems, damping, and natural frequency. It illustrates the application of theoretical principles to practical simulation scenarios, providing insights into the engineering considerations involved in marine towing operations.

Introduction

Marine towing systems involve complex dynamical interactions between the vessel and the towed object — in this case, a submarine. Understanding the system’s behavior under different conditions is crucial for safety, efficiency, and design optimization. MATLAB and SIMULINK are powerful tools that enable engineers to model such systems accurately, simulate their responses, and analyze their stability and performance. This paper presents a comprehensive analysis of a submarine being towed by a ship via a flexible, elastic cable modeled as a spring, incorporating the effects of damping and system inertia, and grounded in classical dynamics principles.

The core objective is to develop a mathematical model that captures the essential physics of the towed submarine, simulate its response to towing forces, and interpret the results within the framework of second-order system theory. The experiment also aims to validate theoretical predictions with simulation data, assess the impact of cable properties on system behavior, and evaluate the appropriateness of accounting for damping and elasticity in the model.

Modeling the System

The system consists of a submarine of mass m1 connected via a cable with stiffness k to a ship or towing vessel of mass m2. The cable is assumed elastic, behaving like a spring with stiffness k, derived using the relation k = (appropriate form). The wave and viscous drag exert damping forces proportional to the velocity, modeled through a damping coefficient b. The equations of motion for the submarine are derived from Newton’s second law, considering the forces exerted by the elastic cable and viscous drag:

m1 x''(t) + b x'(t) + k * x(t) = F(t)

where x(t) is the displacement of the submarine relative to the static position, and F(t) represents any external towing force or disturbances. The system’s response depends on parameters such as mass, damping coefficient, and stiffness, which influence the natural frequency and damping ratio, key indicators of system stability and oscillatory behavior.

Simulation Approach

The equations of motion are implemented in MATLAB’s SIMULINK environment, enabling dynamic simulation of the submarine’s response over time. The model includes blocks representing mass, springs, damping, and external forces, configured to produce a time response reflecting real-world towing operations. Initial conditions such as initial velocity and displacement are set according to the problem statement. The simulation data — including force, displacement, velocity, and cable elongation — are collected and analyzed to understand system behavior.

Results and Discussion

The simulation results show that the maximum force experienced by the tow cable was approximately 8194.1 lb, occurring at about 1.76 seconds, aligning with theoretical expectations that maximum force coincides with maximum acceleration. The steady-state elongation was calculated as 3.59 ft, corresponding with the theoretically derived value. The response of the system demonstrated a quick damping effect, with the submarine stabilizing rapidly under the conditions modeled. Variations in cable length, damping ratio, and stiffness significantly affected the maximum forces, velocities, and displacements, illustrating the importance of precise parameter selection in design and operation.

Implications of the Results

The alignment between simulative and theoretical outcomes confirms the robustness of the modeling approach. The use of SIMULINK significantly streamlined the analysis, allowing rapid iteration over different parameters and configurations. Furthermore, the simulations highlight the critical influence of cable properties, especially damping, on the system’s transient and steady-state responses. Longer cables tend to reduce maximum forces and velocities, promoting stability, while shorter cables induce higher dynamic stresses, which must be managed carefully in practical applications.

Conclusion

This study underscores the value of mathematical modeling and simulation in understanding complex marine towing dynamics. MATLAB and SIMULINK proved effective in replicating real-world responses and validating theoretical models. The findings emphasize the importance of considering cable elasticity and damping in design, contributing to safer and more efficient towing operations. Future work could involve more sophisticated models that incorporate nonlinear effects, variable damping, and turbulent fluid interactions for enhanced realism.

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