6.5
Composite Bodies
Figure 6.5 – 1: Metronome
Figure 6.5 – 2: Composite Body CrossSection
The mass moment of the pendulum on a metronome
depends on the location of the counterweight on the pendulum’s
stem. By sliding the counterweight downward, the mass
moment IOzz of the pendulum decreases causing the frequency
of the beat to increase.
From Table (6.1 – 3), the geometric center of the
composite body’s cross-section in the x direction is
(6.5 – 1) xC 
1
1
xdA    xdA   xdA  ... xdA

( 2)
( n)
A 
A  (1)
The pendulum is a composite body composed of a
stem that can be modeled as a simple line body and a
counterweight that can be modeled as a point body. This section
shows how to calculate the mass integrals of a composite body
given the mass integrals of the simple bodies that make up the
composite body. Similarly, it’s shown how to calculate the
section integrals of a composite body’s cross-section given the
section integrals of the simple shapes that make up the simple
cross-sections.
where
The Geometric Center and Mass Center
and in Eq. (6.5 – 1)
Clearly, the shape of most bodies and their crosssections are far from simple. Whereas, the previous sections
dealt with simple bodies, this section shows how to treat bodies
that are more complicated. The key is to break up the composite
body into simple shapes. Let’s first look at the geometric center
of a composite body’s cross-section.
(6.5 – 3)
(6.5 – 2)
A   dA   dA  
(1)
( 2)
dA  ... 
( n)
dA
In Eqs. (6.5 – 1) and (6.5 – 2) the integrals over the entire
surface of the composite body’s cross section have been divided
into integrals over each of the surfaces of the simple crosssections. Consider the i-th simple cross-section. In Eq. (6.5 – 2),
Ai   dA denotes the area of the i-th simple cross-section
(i )
1

(i) xdA  Ai  A (i) xdA  Ai xCi ,
 i

in which xCi denotes the position of the geometric center of the
i-th simple cross-section in the x direction. Substituting Eq. (6.5
– 3) into (6.5 – 1) yields
(6.5 – 4)
Geometric Center of a Composite Body’s Cross-Section
As shown in Fig. 6.5 – 2, a composite body’s crosssection is made up of n simple cross-sections labeled from 1 to
n. It’s assumed that the locations of the geometric centers rCi =
xCii + yCij (i = 1, 2, …, n) of the simple cross-sections are
known.
Sub-Section
The Geometric Center and the Mass
Center
The Section Integrals and the Mass
Integrals
Computer Analysis
xC 
1
Ai xC1  A2 xC 2  ...  An xCn 
A
By performing similar calculations for the y and z directions, the
geometric center of the composite body’s cross-section is
expressed as
Section Objectives
Objective
To show how to calculate by hand the geometric center of a composite body
cross-section and the mass center of a composite body.
To show how to calculate by hand the section integrals of a composite body.
To show how to calculate by computer the geometric center, section integrals
and mass integrals of a composite body. The calculations make use of a catalog
of the integrals of simple bodies.
(6.5 – 5)
where M i   dm is the mass if the i-th simple body
(i )
1
xdm is the position of its mass center of the
M i (i )
i-th simple body the x direction. Similar equations exist for the y
and z directions. The position of the mass center of the
composite body in the x, y, and z, directions are
1
Ai xC1  A2 xC 2  ...  An xCn ,
A
1
yC  Ai yC1  A2 yC 2  ...  An yCn ,
A
1
zC  Ai zC1  A2 zC 2  ...  An zCn ,
A
and MxCi 
xC 
(6.5 – 8)
where
1
M1xC1  M 2 xC 2  ...  M n xCn ,
M
1
M1 yC1  M 2 yC 2  ...  M n yCn ,
yM 
M
1
M1zC1  M 2 zC 2  ...  M n zCn ,
zM 
M
A  A1  A2 ...  An .
(6.5 – 6)
xM 
Notice that Eq. (6.5 – 5) and the expressions for the geometric
centers of a point body given in Table 6.1 – X are identical. For
the purposes of calculating geometric center, a composite
body can be regarded as a collection of point bodies located
at the geometric centers of the corresponding simple bodies.
Equations (6.5 – 5) and (6.5 – 6) are used to calculate the
geometric center of the composite body’s cross-section given
the geometric centers of the simple cross-sections and their
areas.
Mass Center of a Composite Body
Refer to the illustrative composite body shown in Fig.
(6.5 – 3). In general a composite body is made up of n simple
bodies. The simple bodies are any combination of point bodies,
line bodies, surface bodies, and volume bodies.
where
(6.5 – 9)
M  M1  M 2 ...  M n .
Equations (6.5 – 8) and (6.5 – 9) are used to find the mass
center of a composite body given the mass centers of each of
the simple bodies and their masses. Notice that Eq. (6.5 – 8) and
the expressions for the mass centers of a point body given in
Table 6.1 – X are identical. For the purposes of calculating
mass center, a composite body can be regarded as a
collection of point bodies located at the mass centers of the
corresponding simple bodies.
Figure 6.5 – 3: Composite Body
The Section Integrals and Mass Integrals
Like the geometric center of a composite body’s crosssection, the section integrals of a composite body’s crosssection, namely, its area moments, polar moment, and area
product, are found by first dividing the section integrals into
section integrals over the surfaces of simple cross-sections.
Similarly, the mass integrals of a composite body are found by
first dividing the mass integrals into mass integrals over simple
bodies. In general, the integrals can be written as
It’s assumed that the locations rMi = xMii + yMij + zMik
(i = 1, 2, …, n) of the mass centers of the simple bodies are
known. From Table 6.1 – 2, the mass center of the composite
body in the x direction is located at
(6.5 –7)
xM 
1
M
1 

 xdm  M (1) xdm  (2) xdm  ...(n) xdm
 1

 1

1   1
xdm  M 2 
xdm  ...  M n 
M1
(
2
)
M   M 1 (1)
M

 2

 Mn
1

M1x M 1  M 2 x M 2  ...  M n x Mn ,
M




(n) xdm
(6.5 – 10)
I O  I O1  I O 2  ...  I On .
where IO is any one of the integrals of the composite body and
IOi is the corresponding integral of the i-th simple body.
Consider the i-th simple body. The integrals IOi are not
generally found in the Body Integral Tables because they only
necessarily provide integrals of bodies about their geometric
centers. In order to find the integrals IO of the composite body,
the integrals ICi (i = 1, 2, … n) of the simple bodies about their
geometric centers, which are found in the tables, need to be
transformed into integrals IOi (i = 1, 2, … , n) about point O. In
general, the transformation of the i-th body is done in two steps.
Figure 6.5 – 4: Rotations and Translations
Refer to Fig. 6.5 – 4. First, the integral ICi” found in the Body
Integral Table is rotated so that its axes line up with the axes of
the composite body. The rotated section integral of the i-th
simple body is denoted by ICi’. The section integral ICi’ is then
translated so that its origin Ci coincides with the origin O of the
composite body. Symbolically, the order of the calculations is
(6.5 – 11)
I Ci " I Ci '  I Oi , (i  1, 2, ... , n)
The rotations are accomplished using Eqs. (6.4 – 6) through (6.4
– 8) and the translations are accomplished using Eqs. (6.3 – 7)
and (6.3 – 10). These equations are listed below for
convenience.
Computer Analysis
Computer analysis becomes necessary when the number of
simple bodies that make up a composite body is large and in
design settings, in which calculations need to be performed
repeatedly. Two computer programs are discussed in this
section. The first is called sbody. sbody is used to calculate
the section integrals and the geometric center of a composite
body cross-section. The second is called mbody. mbody is
Transformations for the Section Integrals
Rotation
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  ,
I xy  ( I y '  I x' ) cos sin   I x' y ' (cos 2   sin 2  ),
J  J'
used to calculate the mass integrals and the mass center of a
composite body. These programs use the section integrals and
the mass integrals of the simple bodies cataloged in Table X in
the back of the book together with Eqs. (6.5 – 12) and (6.5 – 13)
to rotate and translate the integrals of the simple bodies to the
composite body’s coordinate system. To run either program,
you’ll supply a list of the simple bodies that make up the
composite body, the parameters of the simple bodies, and the
locations of the geometric centers of the simple bodies. Both
computer programs are described in detail in the examples at
the end of the section.
Key Terms
Composite Body, mbody, sbody, Simply Body
Review Questions
1. Assume that a composite body is composed of n simple
bodies whose geometric centers are located at rCi, (i = 1, 2, …,
n). How do the shapes of the simple bodies effect the
calculation of the geometric center of the composite body?
2. Assume that the center of an a × a square is located at the
origin of a coordinate system. Describe the difference between
a) rotating the square 45○ about the origin after which it’s
translated to the right a units, and b) translating the square to the
right a units after which it’s rotated about the origin 45 ○.
Transformations for the Mass Integrals
Rotation
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  ,
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  .
Translation
Translation
I Oxx  I Mxx  rx2 M ,
I Ox  I Cx  yC2 A.
I Oyy  I Myy  ry2 M ,
I Oy  I Cy  xC2 A.
I Ozz  I Mzz  rz2 M ,
J O  J C  ( xC2  yC2 ) A.
I Oxy  I Cxy  xC yC A.
I Oxy  I Mxy  xM y M M ,
I Oyz  I Myz  y M z M M ,
I Ozx  I Mzx  z M xM M .