Solving for the minimum value of pressure for a Air Cushion Vehicle using the Newton Raphson method.

Objective:

Writing a python script to solve for the minimum value of pressure using the Newton Raphson method.

where p denotes the cushion pressure, h the thickness of the ice field, r the size of the air cushion, σ the tensile strength of the ice, and β is related to the width of the ice wedge.

Taking β = 0.5, r = 40 feet and σ = 150 pounds per square inch (psi), to serve the following purposes:

1. Using the Newton Raphson method for finding out the value of pressure for h = 0.6 ft.
2. Finding the optimal relaxation factor for this problem with the help of a suitable plot.
3. Tabulate the results of p for h = [0.6,1.2,1.8,2.4,3,3.6,4.2] assuming a suitable relaxation factor using python.

Problem Statement:

Air-Cushion Vehicle (ACV), a craft designed to travel close to but above ground or water. It is also called a ground-effect machine or Hovercraft. These vehicles are supported in various ways.

It is characterized by movement in which a significant portion of the weight is supported by forces arising from air pressures developed around the craft, as a result of which they hover near the Earth’s surface. It is this proximity to the surface that chiefly distinguishes such craft from aircraft, which derive their lift from aerodynamic forces created by movement through the air.

Some of them have a specially designed wing that will lift them just off the surface over which they travel when they have reached a sufficient horizontal speed (the ground effect). Others are supported by fans that force air down under the vehicle to create lift. In a plenum chamber vehicle, the rate of leakage of this air from underneath the vehicle is reduced by placing a skirt around the lower edge of the craft. In an annular jet vehicle, the rate of leakage is reduced by directing the air downward and inward from the outer edges of the vehicle. Air propellers, water propellers, or water jets usually provide forward propulsion.

The basic elements of an air-cushion vehicle are a hull, beneath which a skirt system is attached and on which accommodation for passengers, crew, and freight is built; a propulsion system; and a lift system that feeds air into the plenum chamber below the craft to provide a cushion. The propulsion and lift systems can be driven by the same power plant or by separate units. If a common power plant is used, the system is known as an integrated lift-propulsion system. Some early craft had only one airflow generating system, which was used for both lift and propulsion, but optimum efficiency for both requirements was difficult to achieve simultaneously, and separate systems are generally used.

Air-cushion vehicles can attain higher speeds than can either ships or most land vehicles and use much less power than helicopters of the same weight.

Solution Procedure:

Following is the method to determine the minimum cushion pressure needed to break a given thickness of ice using an air cushion vehicle, using the Newton Raphson method:

This equation is known as Muller’s Equation

Where,

p - cushion pressure

h - thickness of the ice field

r - the size of the air cushion

σ - tensile strength of the ice

β -width of the ice wedge.

The differentiation of Muller’s Equation w.r.t Pressure

Newton Raphson Method:

The Newton-Raphson method is one of the most widely used methods for root finding. It can be easily generalized to the problem of finding solutions to a system of non-linear equations, which is referred to as Newton's technique. Moreover, it can be shown that the technique is quadratically convergent as we approach the root.

Unlike the bisection and false position methods, the Newton-Raphson (N-R) technique requires only one initial value x₀, (initial guess for the root). Function f(x) can be rewritten as the following using a Taylor series expansion in (x-x₀):

where f'(x) denotes the first derivative of f(x) with respect to x, f''(x) is the second derivative, and so forth. Now, suppose the initial guess is pretty close to the real root. Then (x-x₀) is small, and only the first few terms in the series are important to get an accurate estimate of the true root, given x₀. By truncating the series at the second term (linear in x), we obtain the N-R iteration formula for getting a better estimate of the true root:

Thus, the N-R method finds the tangent to the function f(x) at x=x₀ and extrapolates it to intersect the x-axis to get x₁. This point of intersection is taken as the new approximation to the root and the procedure is repeated until convergence is obtained whenever possible. Mathematically, given the value of x = x_i at the end of the ith iteration, we obtain x_i+1 as

This is called the Newton Raphson iteration.

Steps to be followed:

1. Initialization of the program by importing required libraries
2. Defining a function by simplifying (breaking into smaller terms) the involved equation
3. Defining the derivative of the equation involved.
4. Defining inputs for the width of ice-wedge, size of air cushion, tensile strength in PSF, the thickness of the ice, an initial value of the first iteration, and relaxation factor.
5. Defining loops
6. Plotting the required table
7. Plotting the graph between Cushion Pressure vs Ice Thickness
8. Re-defining inputs to find optimum relaxation factor.
9. Defining the optimum relaxation factor
10. Plotting the number of iterations and relaxation factor

Code:

Results:

*Value of pressure for a height of 0.6 ft is 4.239063886378783

*The Optimum Relaxation Factor= 0.9950251256281407

*The tabulated result (with relaxation factor =1) is as follows:

Conclusions:

1. As the thickness of ice increases the pressure required to break it increases. Therefore, we observed a direct relationship between the pressure and thickness of the ice. It follows a slightly half-parabolic path.
2. As the relaxation factor increases, the no. of iterations involved in obtaining the required value is decreased. Therefore, the required value will be obtained at a shorter period.
3. The optimum relaxation factor value hence found is less than 1.