2.1 Introduction
In general, the vibration analysis of an engineering
system requires: the idealization of the system into a form that can be
analyzed, the formulation of the governing equilibrium equations of this
idealized system, the solutions of the governing equations, and finally
the interpretation of the results. Physical systems may be broadly
classified into two categories: discrete systems or continuous systems.
Based on laws of physics, an engineering problem is thus represented
either by a discrete system, which is characterized by a set of
algebraic equations involving a finite number of unknowns or degrees of
freedom; or by a continuous system which is very often characterized by
a set of partial differential equations with corresponding boundary
conditions (Bathe, 1982; Hashemi, 2002).
The exact solution of the differential equations
and which satisfies all boundary conditions is only possible for
relatively simple systems, and numerical procedures must in general be
employed to predict the system response. These procedures, in essence,
reduce the continuous system to a discrete idealization that can be
analyzed in the same manner as a discrete physical system. The free
vibration analysis of a discrete or continuous system leads to a so
called Eigenvalue problem.
Critical buckling and undamped free vibration problems are often solved
using finite elements to obtain a discrete model with a finite number of
degrees of freedom. In vibration problems, an alternative discrete model
is often obtained by “lumping” distributed masses at convenient points.
Further, these models usually yield linear eigenvalue problems Hashemi
(2002) as:
(2.1)
which
can be solved by many proven and secure mathematical methods. Here, K
and M represent stiffness and mass matrices of the system,
respectively, and K(w)
is the so-called Dynamic Stiffness Matrix (DSM) of the system. For a
continuous system, the formulation generally leads to (Bathe (1982)):
(2.2)
where L1,
L2, l1, and l2 are linear
differential operators. In the free vibration analysis of structures,
the basic idea is to solve the relevant eigenproblem leading to the
eigenvalues, l,
and eigenvectors, {U}, which represent the natural frequencies,
w,
and the modes of structures, respectively. The characteristics of this
model depend on the analysis to be carried out, in essence, the
actual continuous system is reduced to an appropriate discrete system
where the element equilibrium, constitutive relations and element
interconnectivity requirements are satisfied (this will be discussed in
more details farther in this thesis).
2.3
Analytical Formulation Based on Continuous
Models: Non-Linear Eigenproblems
A practical
structure, assembled from elements possessing distributed mass, will
have an infinite number of degrees of freedom and an infinite number of
natural frequencies. The “Exact” member, or element, equations exist for
structures including plane frames, space frames, grids, and many plate
and shell problems. For plane frames, the member equations often
incorporate the stability functions for buckling problems, and their
dynamic equivalents for vibration problems (Wittrick and Williams
(1983)). In this thesis, the focus is on the free undamped vibration
problems. The exact member equations are then used to assemble the
overall dynamic stiffness matrix, K(w),
of the structure. The natural frequencies, in this case, will be
obtainable from a non-linear eigensystem as in equation 2.1.
The elements of the displacement vector U,
to which K(w)
corresponds, is the finite set of
amplitudes of nodal point displacements, varying sinusoidally with time.
The frequency dependent matrices [K(w)]
resulting from the Dynamic Stiffness Matrix (DSM) method and Dynamic
Finite Element (DFE) approach both lead to non-linear eigenvalue
problems. The Finite Element Method (FEM) based on fixed interpolation
functions leads to a linear eigenvalue problem. In the following
sections, the solution methods for both linear and non-linear
eigenproblems are briefly addressed.
In
the conventional FEM formulation, the basis functions of the
approximation space are generally polynomial expressions. The basis
functions are then used to construct the ‘Fixed’ interpolation functions
(i.e. they only vary with element span-wise position x).
The polynomial shape functions satisfy both completeness and
inter-element continuity conditions. The solution of the natural
frequencies pertaining to this technique is simple considering this is a
linear eigenvalue problem (2.1). For simple systems, setting the
determinant to zero leads to a linear algebraic equation from which the
natural frequencies can be easily extracted. For more complex systems,
with large number of Degrees-Of-Freedom (DOF), one could
solve the resulting
classical linear eigenproblem using an inverse iteration, subspace or
Lanczos method (Bathe ,1982).
It is important to notice that due to the approximate nature of the
conventional FEM, one could only solve for as many natural frequencies
as the total DOF of the system.
As it
was stated in previous sections, the DSM and DFE formulations obtained
from continuous models are different compared to the FEM considering the
stiffness matrix is usually frequency dependent. One of the advantages
of using a dynamic stiffness matrix is that natural frequencies are not
missed. They lead to a non-linear eigenvalue problem as:
(2.3)
There
are two possible sets of solutions pertaining to the above equation.
(2.4)
(2.5)
Then, the method frequently used for determining the natural frequencies
of the system is the Wittrick-Williams algorithm presented in different
occasions by Wittrick and Williams (1971), Wittrick and Williams (1982),
Wittrick and Williams (1983). The method is based on the sturm sequence
properties of the frequency dependent stiffness matrix of the system and
involves the input of a trial frequency. The number of natural
frequencies exceeded by this trial frequency is then calculated as
follows:
(2.6)
where
J represents the total number of natural frequencies of the
system exceeded by the trial frequency,
represents
the total number of clamped-clamped (C-C) natural frequencies of all
elements exceeded by the trial frequency (i.e,
)
and is calculated as
(2.7)
The
term 
is
the sign count of KDSM, and is determined by counting the
number of negative elements along the leading diagonal of the upper
triangularized matrix. This is accomplished after the
is
fully assembled. The upper triangular matrix is sensitive to pivotal
operations, such that, during the gauss elimination procedure, the rows
can be pivoted but not the columns. Then, from equation (2.6) the final
number of natural frequencies exceeded by the trial frequency for the
entire beam is calculated. Using a numerical method any natural
frequency can be converged upon. This research uses the bisection
technique as the convergence method. The bisection method is a no fault
method in determining the solution.
There also exist combined methods to speed up the convergence of the
solution. When the bisection method brings the upper and lower limits on
the eigenvalues sufficiently close, a quicker numerical procedure can be
implemented such as linear interpolation presented by Hoorpah, Henchi
and Dhatt (1994), Newton’s method discussed by Hopper and Williams
(1977), parabolic interpolation discussed by Simpson (1984), or inverse
iteration method (refer to Williams and Kennedy, 1988; Hashemi, 1998).
By implementing the Wittrick-William algorithm the resonant frequencies
are established for the free vibration of a system. Difficulty arises
from solving the equation for the modes of deformation due to the zero
force vector residing on the right hand side:
(2.8)
where, F is the zero force vector corresponding to the free
vibration of the structure.
At
the resonant frequency, the dynamic stiffness matrix cannot be inverted
due to the zero determinant. To obtain a non-trivial solution the
frequency variable is manipulated so that the frequency dependent
stiffness matrix is altered slightly. This perturbation must be small as
to not deviate from the solution significantly.
(2.9)
where
is
the altered frequency and i is any real number sufficiently large
enough such that a small perturbation is created. This new frequency is
then substituted into equation (2.8) leading to:
(2.10)
The
force vector on the right side of equation (2.10) is also altered
slightly.
(2.11)
where
is
the altered force vector.Then the modes can be evaluated by manipulating
equation (2.10) to:
(2.12)
The order of perturbation of the frequency variable
and
the force vector depends
on the numerical precision. Using double precision the 10th
order perturbation is acceptable to accurately describe the modes of
deformation (Hashemi, 1998).
The
Wittrick-William technique plays an important part in determining the
natural frequencies and modes of free vibration. This method is used for
both, but not limited to the Dynamic Finite Element method and Dynamic
Stiffness Matrix method where the use of a frequency dependent stiffness
matrix leads to a non-linear eigenvalue problem. The technique can
equally be used as a solver for the FEM (Roach and Hashemi, 2003). This
method is particularly advantageous with the capability of solving any
range of frequencies.
