Karamba3D 2.2.0
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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 β€œEigen Modes”-component can be used to detect kinematic modes.
An Eigen-mode
x⃗\vec{x}
is the solution to the matrix-equation
C~⋅x⃗=λ⋅x⃗\utilde{C} \cdot \vec{x} = \lambda \cdot \vec{x}
which is called the special eigen-value problem. Where
C~\utilde{C}
is a matrix,
x⃗\vec{x}
a vector and
Ξ»\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.
x⃗\vec{x}
is an eigen-mode as can be computed with Karamba3D.
Vibration modes
x⃗\vec{x}
of structures result from the solution of a general Eigenvalue problem. This has the form
C~⋅x⃗=ω2⋅M~⋅x⃗\utilde{C} \cdot \vec{x} = \omega^2 \cdot \utilde{M} \cdot \vec{x}
. In a structural context
M~\utilde{M}
is the mass-matrix which represents the effect of inertia. The scalar
Ο‰\omega
can be used to compute the eigenfrequency
ff
of the dynamic system from the equation
f=Ο‰/2Ο€f = \omega / 2\pi
. In the context of structural dynamics eigen-modes are also called normal-modes or vibration-modes.
The β€œBuckling 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~⋅x⃗+λ2⋅CG~⋅x⃗=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.
Last modified 1yr ago
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