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 x\vec{x} is the solution to the matrix-equation C~x=λx\utilde{C} \cdot \vec{x} = \lambda \cdot \vec{x} which is called the special eigen-value problem. Where C~\utilde{C} is a matrix, x\vec{x} a vector and λ\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 C~\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. x\vec{x} is an eigen-mode as can be computed with Karamba3D.

Vibration modes x\vec{x} of structures result from the solution of a general Eigenvalue problem. This has the form C~x=ω2M~x\utilde{C} \cdot \vec{x} = \omega^2 \cdot \utilde{M} \cdot \vec{x}. In a structural contextM~\utilde{M}is the mass-matrix which represents the effect of inertia. The scalarω\omegacan be used to compute the eigenfrequencyffof the dynamic system from the equationf=ω/2πf = \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 NIIN^{II} need to be multiplied in order to cause structural instability. The buckling factors are the eigenvalues of the general Eigenvalue problem C~x+λ2CG~x=0\utilde{C} \cdot \vec{x} + \lambda^2 \cdot \utilde{C_{G}} \cdot \vec{x} = 0. Here C~\utilde{C} is the elastic stiffness matrix and CG~\utilde{C_{G}} the geometric stiffness matrix. The latter captures the influence of normal forces NIIN^{II} on a structure’s deformation response.

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