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.