6.4
Rotated Coordinates
The previous section addressed translation and this
section addresses rotation. It should be pointed out that the
material developed in this and the previous sections, when
combined, enable you to perform general coordinate
transformations involving both translations and rotations.
Figure 6.4 – 1: Don’t know yet.
Figure 6.4 – 3 shows two coordinate systems that have
been rotated one relative to the other about the z axis. A
differential element dD of the body is shown.
Figure 6.4 – 3: Rotated coordinate systems in a
plane
In the previous section, the origin of the coordinate
system was moved from one point to another without changing
its orientation. The axes of the transformed coordinates were
parallel to the axes of the original coordinates. This section
looks at rotating the coordinates. When the rotation is about one
of the axes of the coordinate system, the rotation is called a
planar rotation. Otherwise, it’s a general rotation. This
section first treats planar rotations and then general rotations. In
both cases, it’s shown how to calculate the integrals of the body
in the rotated coordinate system given the integrals of the body
in the original coordinate system.
The location of the differential element can be written
two ways, as
Planar Rotation of Coordinates
(6.4 – 1)
A coordinate system can be placed anywhere and it can
have any orientation. It’s a two-step process of translation and
rotation, or of rotation and translation, as shown in Fig. 6.4 – 2.
where i, j, and k are unit vectors along the x, y, and z directions,
respectively, and where i’, j’, and k’ are unit vectors along the
x’, y’, and z’ directions, respectively.
r = xi + yj + zk or r = x’i’ + y’j’+ z’k’
Figure 6.4 – 2: In general, the transformation of coordinates consists of a translation and a rotation.
Sub-Section
Planar Rotation of Coordinates
Planar Rotation of Space and Mass
Integrals
General Rotation of Mass Integrals
Section Objectives
Objective
To show how to rotate coordinates in a plane.
To show how to calculate the integrals of a body about a point O in one
orientation given the integrals of the body about point O in another orientation.
The coordinate system is rotated about one of its axes (planar rotation).
To show how to calculate the integrals of a body about a point O in one
orientation given the integrals of the body about point O in another orientation.
The coordinate system is rotated about a general axis.
The x’-y’-z’ coordinate system has been rotated about the z axis
though a positive rotation angle . Since the rotation is about
the z axis, z’ = z and k’ = k in Eq. (6.4 – 1). The transformation
between the coordinates can be found by taking the dot product
of r in Eq. (6.4 – 1) with the unit vectors i, j, i’ and j’
recognizing that (See Fig. 6.4 – 4)
(6.4 – 2)
i  i '  cos ,
i  j'  cos(90   )  sin  ,
j  i'  cos(90   )   sin ,
j  j'  cos .
Planar Rotation of Section and Mass
Integrals
Let’s now express the section integrals Ix, Iy, Ixy, and J
in the original coordinate system in terms of the section
integrals Ix’, Iy’, Ix’y’, and J’ in the rotated coordinate system.
Using Table 6.2 – 2 and Eq. (6.4 – 3), Ix is written out as
(6.4 – 5)
I x   y 2 dA   ( x' sin   y ' cos ) 2 dA 
sin 2   x'2 dA  cos2   y '2 dA 2 cos sin   x' y ' dA
 I x ' cos2   I y ' sin 2   2 I x ' y ' cos sin  .
Equation (6.4 – 5) expresses the section integral Ix in terms of
the section integrals in the rotated coordinate system. Following
similar steps, the other section integrals can be expressed in
terms of the section integrals in the rotated coordinate system.
The planar rotations of the section integrals are
(6.4 – 6)
I x  I x' cos2   I y ' sin 2   2 I x' y ' cos sin  ,
I y  I x' sin 2   I y ' cos2   2 I x' y ' cos sin  ,
Figure 6.4 – 4: The dot product of two unit vectors is the
cosine of the angle between them.
I xy  ( I y '  I x' ) cos sin   I x' y ' (cos 2   sin 2  ),
and
Multiplying r in Eq. (6.4 – 1) by i and j yields
(6.4 – 3)
x  x' cos  y ' sin  ,
y   x' sin   y ' cos ,
z'  z
Equation (6.4 – 3) can be used to find x and y given x’ and y’.
To find x’ and y’ given x and y, multiply r in Eq. (6.4 – 1) by i’
and j’ to get the inverse transformation
(6.4 – 7)
J  J' .
The development of the planar rotations of the mass moments
Ix’x’, Iy’y’, and Iz’z’, and the mass products Ix’y’, Iy’z’, and Iz’x’
follows the same steps as for the planar rotations of the section
integrals. The planar rotations of the mass moments and the
mass products become
(6.4 – 8)
I xx  I x ' x ' cos2   I y ' y ' sin 2   2 I x ' y ' cos sin  ,
I yy  I x ' x ' sin 2   I y ' y ' cos2   2 I x ' y ' cos sin  ,
(6.4 – 4)
x'  x cos  y sin ,
y '  x sin  y cos ,
z  z'
I zz  I z ' z ',
I xy  ( I y ' y '  I x ' x ' ) cos sin   I x ' y ' (cos 2   sin 2  ),
I yz   I z ' x ' sin   I y ' z ' cos ,
I zx  I z ' x ' cos  I y ' z ' sin  .
General Rotation of Mass Integrals
(6.4 – 11)
Section integrals are most often rotated in a plane.
They don’t generally need to be rotated out of the plane. On the
other hand, mass integrals are rotated both in the plane and out
of the plane. The general rotation of a mass integral can be
accomplished by rotating the coordinate system through two
consecutive planar rotations (See Fig. 6.4 – 5).
Figure 6.4 – 5: General rotations can be accomplished by
performing two consecutive planar rotations.
x  x"cos z cos y  y"sin  z  z"cos z sin  y ,
y   x"sin  z cos y  y"cos z  z"sin  z sin  y ,
z  x"sin  y  z"cos y .
The general rotation above was carried out by rotating first
about the y” axis and then about the z’ axis. This kind of
rotation is sometimes called a 2-3 Euler rotation. The 2 refers
to the y” axis and the 3 refers to the z’ axis. There are different
kinds of Euler rotations depending on which first axis is rotated
and which second axis is rotated. Note that the order of the
rotation is important. A rotation y about the y” axis followed
by a rotation z about the z’ axis produces a different coordinate
system than a rotation z about the z” axis followed by a
rotation y about the y’ axis. They are the same, however, when
the rotation angles are small. You can verify this for yourself by
holding up this book and rotating it through two angles and you
can prove it from Eqs. (6.4 – 9) through (6.4 – 11) by making
small angle assumptions.
Let’s now determine the rotated mass integrals. First,
let the mass integrals undergo a planar rotation about the y” axis
through the angle y. In Eq. (6.4 – 8), change (x’ y’ z’) into (z”
x” y”), change (x y z) into (z’ x’ y’), and change z into y to get
(6.4 – 12)
As shown in Fig. 6.4 – 5, the x”-y”-z” coordinates are
first rotated about the y” axis through a positive rotation angle
y to get to the x’-y’-z’ coordinates. Then, the x’-y’-z’
coordinates are rotated about the z’ axis through a positive
rotation angle z to get to the x-y-z axes. Using Eq. (6.4 – 3), the
first planar rotation is
x'  z" cos y  x"sin  y
y '  y"
(6.4 – 9)
z '   z"sin  y  x" cos y
The second planar rotation is
x  x' cos z  y ' sin  z
(6.4 – 10)
y   x' sin  z  y ' cos z
z  z'
Substituting Eq. (6.4 – 9) into (6.4 – 10) yields the general
rotation
I z ' z '  I z"z" cos 2  y  I x"x" sin 2  y  2 I z"x" cos y sin  y ,
I x ' x '  I z"z" sin 2  y  I x"x" cos 2  y  2 I z"x" cos y sin  y ,
I y ' y '  I y" y",
I z ' x '  ( I x"x"  I z"z" ) cos y sin  y  I z"x" (cos 2  y  sin 2  y ),
I x ' y '   I y"z" sin  y  I x" y" cos y ,
I y ' z '  I y"z" cos y  I x" y" sin  y .
Next, let the mass integrals undergo a planar rotation about the
z’ axis through the angle  = z. This is exactly what Eq. (6.4 –
8) does, so
(6.4 – 13)
I xx  I x ' x ' cos 2  z  I y ' y ' sin 2  z  2 I x ' y ' cos z sin  z ,
I yy  I x ' x ' sin 2  z  I y ' y ' cos2  z  2 I x ' y ' cos z sin  z ,
I zz  I z ' z ',
I xy  ( I y ' y '  I x ' x ' ) cos z sin  z  I x ' y ' (cos 2  z  sin 2  z ),
I yz   I z ' x ' sin  z  I y ' z ' cos z ,
I zx  I z ' x ' cos z  I y ' z ' sin  z .
Substituting Eq. (6.4 – 12) into (6.4 – 13) yields the 2-3 Euler
rotation of the mass integrals
(6.4 – 14)
I xx  cos2  y cos2  z I x' ' x' '  sin 2  z I y ' ' y ' '  sin 2  y cos2  z I z ' ' z ''
 2 cos y cos z sin  z I x' ' y ''  2 sin  y cos z sin  z I y ' ' z ' '
 2 cos y sin  y cos2  z I z ' ' x' ' ,
In summary, the mass integrals in a rotated coordinate system
can be found from Eq. (6.4 – 14) if a 2-3 Euler rotation is used.
Note that you’re not restricted to rotate first about the y” axis
and then about the z’ axis. You can rotate about any two
consecutive axes. The calculation of mass integrals that have
undergone a general rotation is illustrated in Example 6.4 – 3.
Key Terms
Euler Rotation; General Rotation; Planar Rotation; Rotated
Mass Integrals; Rotated Section Integrals
I yy  cos2  y sin 2  z I x' ' x' '  cos2  z I y ' ' y ' '  sin 2  y sin 2  z I z ' ' z ' '
 2 cos y cos z sin  z I x' ' y ' '  2 sin  y cos z sin  z I y ' ' z ' '
 2 cos y sin  y sin 2  z I z ' ' x'' ,
I zz  I z" z" cos2  y  I x" x" sin 2  y  2 I z" x" cos y sin  y ,
I xy  cos2  y cos z sin  z I x' ' x' '  cos z sin  z I y ' ' y ' '
 sin 2  y cos z sin  z I z ' ' z ' '
 cos y (cos2  z  sin 2  z ) I x' ' y ' '  sin  y (cos2  z  sin 2  z ) I y ' ' z ''
 2 cos y sin  y cos z sin  z I z ' ' x' ' ,
I yz  cos y sin  y sin  z I x' ' x' '  cos y sin  y sin  z I z ' ' z ' '
 sin  y cos z I x' ' y ' '  cos y cos z I y ' ' z ' '  (cos2  y  sin 2  y ) sin  z I z ' ' x' ' ,
I zx   cos y sin  y cos z I x' ' x' '  cos y sin  y cos z I z ' ' z ' '
 sin  y sin  z I x' ' y ' '  cos y sin  z I y ' ' z ' '  (cos2  y  sin 2  y ) cos z I z ' ' x' ' .
Review Questions
1. Define a planar rotation of a coordinate system.
2. Derive Eq. (6.4 – 7).
3. Under what conditions can the order of the planar rotations
that make up a general rotation be switched?