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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.
is the solution to the matrix-equation
which is called the special eigen-value problem. Where
is a matrix,
a vector and
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
stands for the stiffness-matrix whose number of rows and columns corresponds to the number of degrees of freedom of the structural system.
is an eigen-mode as can be computed with Karamba3D.
of structures result from the solution of a general Eigenvalue problem. This has the form
. In a structural context
is the mass-matrix which represents the effect of inertia. The scalar
can be used to compute the eigenfrequency
of the dynamic system from the equation
. 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
need to be multiplied in order to cause structural instability. The buckling factors are the eigenvalues of the general Eigenvalue problem
is the elastic stiffness matrix and
the geometric stiffness matrix. The latter captures the influence of normal forces
on a structure’s deformation response.