A.2.5: Natural Vibrations, Eigen Modes and Buckling
The Eigen-modes of a structure describe the shapes to which it can be deformed most easily in ascending order. Due to this the “Eigen Modes”-component can be used to detect kinematic modes.
An Eigen-mode
$\vec{x}$
is the solution to the matrix-equation
$\utilde{C} \cdot \vec{x} = \lambda \cdot \vec{x}$
which is called the special eigen-value problem. Where
$\utilde{C}$
is a matrix,
$\vec{x}$
a vector and
$\lambda$
a-scalar (that is a number) called eigenvalue. The whole thing does not necessarily involve structures. Eigen-modes and eigenvalues are intrinsic properties of a matrix. When applied to structures then
$\utilde{C}$
stands for the stiffness-matrix whose number of rows and columns corresponds to the number of degrees of freedom of the structural system.
$\vec{x}$
is an eigen-mode as can be computed with Karamba3D.
Vibration modes
$\vec{x}$
of structures result from the solution of a general Eigenvalue problem. This has the form
$\utilde{C} \cdot \vec{x} = \omega^2 \cdot \utilde{M} \cdot \vec{x}$
. In a structural context
$\utilde{M}$
is the mass-matrix which represents the effect of inertia. The scalar
$\omega$
can be used to compute the eigenfrequency
$f$
of the dynamic system from the equation
$f = \omega / 2\pi$
. In the context of structural dynamics eigen-modes are also called normal-modes or vibration-modes.
The “Buckling Modes”-component calculates the factor with which the normal forces
$N^{II}$
need to be multiplied in order to cause structural instability. The buckling factors are the eigenvalues of the general Eigenvalue problem
$\utilde{C} \cdot \vec{x} + \lambda^2 \cdot \utilde{C_{G}} \cdot \vec{x} = 0$
. Here
$\utilde{C}$
is the elastic stiffness matrix and
$\utilde{C_{G}}$
the geometric stiffness matrix. The latter captures the influence of normal forces
$N^{II}$
on a structure’s deformation response.