# 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.


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