Consider a square plate that occupies the two-dimensional region = [0,1]2 in the undeformed configuration. The square is rotated (rigid-body rotation) through an angle about the 23-axis (deformed configuration). Let P be an arbitrary material point with coordinates (X₁, X₂) in the undeformed configuration, i.e., the position vector X = Xį€į. Due to the rotation, the material point X is deformed to the point P' with coordinates (x₁, x2), i.e., the position vector x = xiei. Then, answer the following: (i) Express ₁ and 2 in terms of X₁, X₂ and 0, that is obtain the relation x = o(X), where is a vector-valued function and is known (a constant). (ii) Write down the displacement field due to the rotation. Note that the displacement. field is: u(X₁, X₂₁₂) = x - X = (X₁, X₂) - X.
Consider a square plate that occupies the two-dimensional region = [0,1]2 in the undeformed configuration. The square is rotated (rigid-body rotation) through an angle about the 23-axis (deformed configuration). Let P be an arbitrary material point with coordinates (X₁, X₂) in the undeformed configuration, i.e., the position vector X = Xį€į. Due to the rotation, the material point X is deformed to the point P' with coordinates (x₁, x2), i.e., the position vector x = xiei. Then, answer the following: (i) Express ₁ and 2 in terms of X₁, X₂ and 0, that is obtain the relation x = o(X), where is a vector-valued function and is known (a constant). (ii) Write down the displacement field due to the rotation. Note that the displacement. field is: u(X₁, X₂₁₂) = x - X = (X₁, X₂) - X.
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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![Consider a square plate that occupies the two-dimensional region [0, 1]2 in the
undeformed configuration. The square is rotated (rigid-body rotation) through an angle
about the 23-axis (deformed configuration). Let P be an arbitrary material point with
coordinates (X₁, X2) in the undeformed configuration, i.e., the position vector X = Xįei.
Due to the rotation, the material point X is deformed to the point P' with coordinates
(x₁, x2), i.e., the position vector x = x¿eį. Then, answer the following:
=
(i) Express ₁ and 2 in terms of X₁, X₂ and 0, that is obtain the relation x = o(X),
where is a vector-valued function and is known (a constant).
(ii) Write down the displacement field due to the rotation. Note that the displacement
field is: u(X₁, X₂) = x - X = (X₁, X₂) - X.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe5c4e1cc-f8a5-4780-b71d-2bc1a3c6eda4%2F979445b5-23da-404b-889b-cdc6c225966c%2Fs7so3dm_processed.png&w=3840&q=75)
Transcribed Image Text:Consider a square plate that occupies the two-dimensional region [0, 1]2 in the
undeformed configuration. The square is rotated (rigid-body rotation) through an angle
about the 23-axis (deformed configuration). Let P be an arbitrary material point with
coordinates (X₁, X2) in the undeformed configuration, i.e., the position vector X = Xįei.
Due to the rotation, the material point X is deformed to the point P' with coordinates
(x₁, x2), i.e., the position vector x = x¿eį. Then, answer the following:
=
(i) Express ₁ and 2 in terms of X₁, X₂ and 0, that is obtain the relation x = o(X),
where is a vector-valued function and is known (a constant).
(ii) Write down the displacement field due to the rotation. Note that the displacement
field is: u(X₁, X₂) = x - X = (X₁, X₂) - X.
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