Basic Continuum Kinematics - Continuum Mechanics of Anisotropic Materials - page 29

x (1)

X (1)

x (2)

X (2)

DEFORMATION
OR MOTION

X (3)

x (3)

Fig. 2.4 The experimental measurement of a planar homogeneous motion. The reference frame is
the laboratory reference frame. The three initial positions ( X (1) , X (2) , X (3) ) of the markers are
indicated as well as their positions ( x (1) , x (2) , x (3) ) at time t . In many experiments the markers are
attached to a specimen of soft tissue that is undergoing a planar homogeneous motion in order to
quantify the motion

Example 2.1.2
An experimental technique in widespread use in the measurement of the planar
homogeneous motion of a deformable object is to place three markers (dots or
beads) in triangular pattern (so that the markers are not collinear) on the deformable
object before a motion. The initial locations of the three markers are recorded
relative to a fixed laboratory frame of reference as ð X ð 1 Þ

I

, X ð 1 Þ
II

X ð 2 Þ
I

, X ð 2 Þ
II

X ð 3 Þ
I

Þ , ð Þ , and ð ,

X ð 3 Þ
II

, Fig. 2.4 . If the process is automated a camera is used to follow the motion of
the three markers with time and to digitize the data in real time. The instantaneous
locations of the three markers at a time t is recorded relative to a fixed laboratory
frame of reference as

Þ

x ð 1 Þ
1

, x ð 1 Þ
2

x ð 2 Þ
1

, x ð 2 Þ
2

x ð 3 Þ
1

, x ð 3 Þ
2

ð ð t Þ ð t ÞÞ ð ð t Þ ð t ÞÞ ð ð t Þ ð t ÞÞ

, Fig. 2.4 .
From these data the experimentalist calculates the time-dependent coefficients A ( t ),
B ( t ), C ( t ), D ( t ), E ( t ), and F ( t ) of the homogeneous planar motion ( 2.10 ). Determine
the formulas used in the calculation of the time-dependent coefficients A ( t ), B ( t ),
C ( t ), D ( t ), E ( t ), and F ( t ) from the data

, and

X ð 1 Þ
I

, X ð 1 Þ
II

X ð 2 Þ
I

, X ð 2 Þ
II

X ð 3 Þ
I

, X ð 3 Þ
II

,( x ð 1 Þ
1

ð Þ , ð Þ , ð Þ ð t Þ ,

x ð 1 Þ
2

,( x ð 2 Þ
1

, x ð 2 Þ
2

x ð 3 Þ
1

, x ð 3 Þ
2

ð t ÞÞ ð t Þ ð t ÞÞ , and ð ð t Þ ð t ÞÞ .

Solution : The data on the motion of each marker provide two equations that may be
used for the determination of the time-dependent coefficients. Since there are three
markers, a total of six equations is obtained. Three markers are used because it is
known that six equations will be needed to solve the linear system of equations for
the six unknowns, A ( t ), B ( t ), C ( t ), D ( t ), E ( t ), and F ( t ). Using the notation for the data
and the representation of the homogeneous planar motion ( 2.10 ), these six

equations are as follows:

x ð 1 Þ
1

X ð 1 Þ
I

X ð 1 Þ
II

x ð 1 Þ
2

X ð 1 Þ
I

X ð 1 Þ
II

ð t Þ¼ A ð t Þ þ C ð t Þ þ E ð t Þ; ð t Þ¼ D ð t Þ þ B ð t Þ þ F ð t Þ;

x ð 2 Þ
1

X ð 2 Þ
I

X ð 2 Þ
II

x ð 2 Þ
2

X ð 2 Þ
I

X ð 2 Þ
II

ð t Þ¼ A ð t Þ þ C ð t Þ þ E ð t Þ; ð t Þ¼ D ð t Þ þ B ð t Þ þ F ð t Þ;

x ð 3 Þ
1

X ð 3 Þ
I

X ð 3 Þ x ð 3 Þ

2

X ð 3 Þ
I

X ð 3 Þ ð t Þ¼ A ð t Þ þ C ð t Þ

II þ

E ð t Þ; ð t Þ¼ D ð t Þ þ B ð t Þ

II þ

F ð t Þ: