Therefore the spring stiffness is characterized through the

stiffness matrix [k] defined as

The constant mass matrix can be obtained using variable separation; however, the expressions of elastic potential energy and the

stiffness matrix become more complex.

A small stiffness coefficient corresponding to the drilling degrees is added to the

stiffness matrix to avoid equation singularity.

In general, [tau] = [K.sub.q][DELTA] and D = J(q)[DELTA] where A is the vector of the joint deformations, and [K.sub.q] = diag([K.sub.1], [K.sub.2], ..., [K.sub.n]) is the joint

stiffness matrix. From (13) and (14), the generalized compliance matrix for the pelvic support mechanism can be derived as

where [{[delta]}.sup.e] is the displacement at the node, {[delta](x, y)} is the displacement at any place in the unit, {[sigma]} is the stress at any place in the unit, {[epsilon]} is the strain at any place in the unit, [[K].sup.e] is the

stiffness matrix of the unit, [B] is the strain matrix of the unit, [D] is the elastic matrix of the unit, and [S] is the stress matrix of the unit, t is the thickness of the unit, the general value is 1.

Substituting (7) into (3), with respect to the displacement q, the expression of the

stiffness matrix of the element is obtained as

[11] proposed a static reanalysis method given the modification of supports by using modified master stiffness matrices, a rank-one decomposition of the corresponding incremental

stiffness matrix, and a sparse Cholesky rank-one update/downdate algorithm.

They found that only the stiffness proportional damping coefficient that is related to the

stiffness matrix is essential to damping out high-frequency oscillations, and the low-frequency oscillation is damped out automatically in an implicit dynamic approach, HHT-[alpha] approach, which they used in their study [24, 25].

where [bar.Q] is a reduced

stiffness matrix obtained by the rotation of the original

stiffness matrix Q:

In general, a cohesive element can be differentiated to a continuum element by a reduced

stiffness matrix based on reduced sectional stresses, Figure 2.

The nondimensional mass matrix is reduced to the unity matrix [I], and the

stiffness matrix [[K.sup.SS.sub.N]], combining the effects of the extensional and spiral springs, obtained by addition of the Winkler soil

stiffness matrix and the spiral spring matrix, for the simply supported case, can be presented as follows [9]:

where [c.sup.E] is the elastic

stiffness matrix, the superscript T denotes transpose, E is the electric field vector, D is the electric displacement vector, [[epsilon].sup.S] is the dielectric constant matrix, and e is the piezo stress/charge constant.