Designing A Pirate Cave Door Due Date For This Project

Designing A Pirate Cave Door 1the Due Date For This Project 1augusti

Designing A Pirate Cave Door 1the Due Date For This Project 1augusti

designing a pirate cave door 1 The Due Date for this project 1 August. Introduction Intent. The pedagogical goal of this project is for students to use numerical integration techniques to solve an engineering design question that cannot be tackled through any analytic means. It is meant to be an exercise that requires a bit of creativity, along with the ability to apply “integration” on a problem that you have not fully practiced in homework, asking you to read, understand, and design. Problem description.

You have been hired by Bill Johnson to design the door to an underwater pirates cave that is being constructed. Bill wants the door to be in the shape of a skull and has provided a hand-drawn sketch to give the basic idea. (See Figure 1.) Bill has already decided that the door will be 8 cm thick and constructed from stainless steel. The additional specifications are related to the shape: Figure 1: Client provided sketch of the desired door, with reference line at top indicating the waterline.

• The door will be in the shape of a skull.
• The door must be between 3 m and 4 m in height.
• The door must be at least 2 m wide.
• The top of the door will be 2 m below the surface of the water.
• The door will be mounted on a pivoting point at the center of pressure, so that it can be easily opened even with sea pressure against the door.

In addition to providing a scaled drawing of the door shape, you are also required to determine:

• The total hydrostatic force on the door, so that the door mount can be properly designed.
• The location of the center of pressure, so that the pivot can be placed at the appropriate location.

You should present your findings in a short technical report.

Mathematical and Physics (hydrostatics) background information. In Section 8.3 of the book, we worked on problems of hydrostatic force. So, computing that force is something that you (in principle) already know how to do.

Recall that hydrostatic pressure at a distance x meters below the waterline is given by P(x) = ρgx, where ρ is the density, and g is acceleration due to gravity. If we denote by w(x) the width of the plate at depth x, then we compute the total force on the plate by integration:

F_HS = ∫_a^b P(x)w(x) dx.

The center of pressure is the point where the total sum of a pressure field acts on a body. When pressure varies across the body, the center of pressure is shifted downward compared to the centroid. Calculating the center of pressure involves evaluating the integral of x*P(x)w(x)dx, divided by the total force, as per the formula:

x_{c p} = ∫_a^b x P(x)w(x) dx / ∫_a^b P(x)w(x) dx.

Project tasks include selecting an appropriate shape for the door, using numerical integration (preferably Simpson's Rule) to compute physical quantities, and preparing a technical report that includes your shape, calculations, and analysis. You must organize your report into sections, using complete sentences and proper formatting. You may use external references with proper citations. Equations should be properly typeset. You are encouraged to reflect on errors, accuracy, and limitations of your computations within your report. Use of external help for mathematical calculations is not permitted, other than instructor guidance.

Paper For Above instruction

The design of an underwater pirate cave door in the shape of a skull presents a fascinating engineering challenge that integrates principles of hydrostatics with creative design. The principal goal is to determine the hydrostatic forces acting on the door, as well as the optimal location for the pivot point at the center of pressure, ensuring ease of operation despite the substantial sea pressure exerted on the structure. To achieve these objectives, the project entails modeling the door shape, applying numerical integration to evaluate the hydrostatic forces, and analyzing the pressure distribution to identify the exact point of force application.

First, selecting an appropriate shape for the door is essential. The shape must resemble a skull and conform to the dimensional constraints: a height between 3 to 4 meters, a width of at least 2 meters, and a position such that the top of the door is 2 meters below the water surface. For modeling purposes, one might consider an outline derived from a standard skull profile, scaled appropriately. A typical approach is to digitize or sketch the shape and generate a functional representation w(x), describing the width at different depths. Such a shape can be parameterized using geometric functions, Bezier curves, or even a composite of simple geometric segments that closely approximate the desired form.

With the shape established, the next step involves calculating the total hydrostatic force acting on the door. Hydrostatic pressure increases linearly with depth, described mathematically as P(x) = ρgx. The force on each infinitesimal horizontal strip of the door at depth x is proportional to this pressure and the width w(x). Integrating this differential force across the entire height of the door yields the total force, which is performed numerically using Simpson's Rule for greater accuracy. This requires discretizing the height interval into an even number of segments, calculating pressure and width at each point, and applying Simpson's formula to approximate the integral.

Subsequently, calculating the center of pressure involves evaluating the integral of x P(x)w(x) dx, which weights each horizontal strip by its depth and pressure. Dividing this by the total force provides the vertical position where the net hydrostatic pressure acts. Understanding this position is critical for mounting the pivot point. Placing the pivot at the center of pressure minimizes the torque required for opening the door, compensating for the hydrostatic force exerted at various depths.

The final report should include illustrative diagrams of the shape, detailed numerical calculations, and discussions on the implications of the results. Analyzing potential sources of error, such as discretization errors or shape approximation inaccuracies, enhances the credibility of the findings. Moreover, discussing how finer partitions or more precise shape models could improve the accuracy offers insight into engineering trade-offs. Through this project, students demonstrate an application of theoretical hydrostatics to a creative and practical engineering problem, blending analytical techniques with design considerations.

References

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