Page 322 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 10.3 Transformation of Coordinates (Global Coordinates) 309

Fig. (10.2-2), we have
y (x ) ^l^’l T ^2^1 ^4^2
= iPiQ] + ^2^2 ^4^4 (10.2-7)

where has been substituted for the end displacements.
To determine the generalized mass, the preceding equation is substituted
into the equation for the kinetic energy.

1 1 r 1 - r

T = 2 j dx = j J L Udidij dx

= 2 ^ (1Ü.2-8)

Thus, the generalized mass which forms the elements of the mass matrix, is
equal to

(10.2-9)
d{\
Substituting the four beam functions into Eq. (10.2-9), the mass matrix for
the uniform beam element is expressed in terms of the end displacements:

156 22/ 1 54 - 13/’
ml 22/ 4/2 i 13/ -3 /2
42Ô ( 10.2- 10)
54 13/ r 156 -2 2 /
- - 13/ -3 /2 ! -2 2 / 4/2 -
The matrix is called consistent mass, because it is based on the same beam
functions used for the stiffness matrix.^

10.3 TRANSFORMATION OF COORDINATES
(GLOBAL COORDINATES)

In determining the stiffness matrix of the entire structure in terms of local
elements, it is necessary first to match the displacements of the adjacent elements
to ensure compatibility. In Chapter 6, this was done by examination of each joint,
taking account of the orientations of the adjoining members at each joint.
In the finite element method, this requirement for displacement compatibility
is simplified by resolving the element displacements and forces into a common
coordinate system known as global coordinates.

j. S. Archer, “Consistent Mass Matrix for Distributed Mass Systems,” J. Struct. Die. ASCE,
Vol. 89, No. STA4 (August 1963), pp. 161-178.