EN222: Mechanics of Solids
Homework 4: The Dreaded Polar Coordinates!
Due Thursday February 3, 2003
Division of Engineering
Brown University
1.
Derive results corresponding to those developed in class for the Spherical coordinate system for
the Cylindrical system.
Your answer should include:
1.
Formulas that convert from ,,
rz
q
to
123
,,
xxx
and back again
2.
The position vector in the {
i,j,k
} basis in terms of ,,
q
3.
The components of the basis vectors
{,,}
q
eee
in the {
i,j,k
} basis, and corresponding
expressions for {
i,j,k
} vectors in terms of
q
4.
Basis change matrices that map vector and tensor components between {
i,j,k
} and
q
5.
Expressions for the derivatives of the basis vectors
q
in terms of ,,
q
6.
A derivation of the gradient operator in cylindrical coordinates
7.
The matrix of components of the gradients of a vector field
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Fig 1. shows a test designed to measure the response of a polymer to large shear strains.
The
sample is a hollow cylinder with internal radius
0
a
and external radius
1
a
.
The inside diameter
is bonded to a fixed rigid cylinder.
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 Spring '09
 HUAJIANGAO
 Coordinate system, Spherical coordinate system, Polar coordinate system, Vector field, basis vectors

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