Variable density structures and components are very common in engineering. For example, one such component is a modern civil aircraft wing. Even if it might be made out of the same material from its connection to the aircraft body to its tip, it has to include several different sub-components at different positions inside its hollow profile. So, if you want to calculate the total mass of the wing approximately at early stages of a design, you can try to find a scalar function that represents this variability in density and then represent the mass as a surface or volume integral. Even in an early stage of design, even a minimalist, but realistic 3D example would have been difficult to solve in an exam situation. We will certainly handle a much simpler case here and it will be in 2D. This introduction was just to give you an insight on what you try to achieve in questions similar to the following. Let's assume that the areal density of our component increases along y-axis and there is no density difference along other axes. So, kg density can be expressed as; p(x, y, 2) = y³ (units : ) Then the total mass could be found by the following surface integral; f p(x, Y, 2) dS Our component's surface could be represented by the following function: z + y3 – z = 0 where 0

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Transcribed Image Text:Variable density structures and components are very common in engineering. For example, one such component is a modern civil aircraft wing. Even if it might be made out of the same material from its connection to the aircraft body to its tip, it has to include several different sub-components at different positions inside its hollow profile. So, if you want to calculate the total mass of the wing approximately at early stages of a design, you can try to find a scalar function that represents this variability in density and then represent the mass as a surface or volume integral. Even in an early stage of design, even a minimalist, but realistic 3D example would have been difficult to solve in an exam situation. We will certainly handle a much simpler case here and it will be in 2D. This introduction was just to give you an insight on what you try to achieve in questions similar to the following. Let's assume that the areal density of our component increases along y-axis and there is no density difference along other axes. So, density can be expressed as; p(x, y, 2) = y³ (units : ) Then the total mass could be found by the following surface integral; l e(x, y, z) dS Our component's surface could be represented by the following function: a + y3 –z = 0 where 0 <x<1 and 0 <y<4 First parameterize this surface into 7 (u, v) form to find dS. Then, express the boundaries for u and v and evaluate the integral to find out the actual mass of this component in kg (hint: you need substitution in evaluating the integral). Write your answer after rounding it to two decimal places (for example: 0.6895 = 0.69 OR 10.494 = 10.49) Answer:

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