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VIBRATION BASED DAMAGE DETECTION OF 3D BEAMS
Last modified: 2017-12-07
Abstract
The modeling of the dynamic behavior of beams considering their deformation in three dimensional space is very important for many engineering applications as wind power generators, bridges, airplanes, etc. The early detection and localization of damages of these structures is essential for their operation and maintenance. Vibration based methods for damage detection are among the most popular methods for health monitoring of structures. In the current work, numerical methods for damage detection of beams that vibrate in three dimensional space are developed and compared.
The beam can vibrate in space, hence it can experience longitudinal and non-planar bending displacements, torsion and warping. The equation of motion is derived by the principle of virtual work. The Timoshenko’s beam theory is assumed and it is considered that the beam cross section rotates as rigid body due to torsion but may deform in longitudinal direction due to warping. Geometrical type of nonlinearity is included in the model and the equation of motion is discretized by the finite element method. As a result, the following system of nonlinear ordinary differential equations is obtained:
Mq ̈+Cq ̇+K(q)q=F (1)
where M represents the mass matrix, C represents the damping matrix, K represents the nonlinear stiffness matrix that depends on the displacement components, q is the vector of generalized coordinates and F is the vector of generalized external forces.
It is well known that shear locking appears in the cases of thin beams. The shear locking problem is avoided by using selective and reduced integration for the terms related with shear strain energy. This approach is equivalent to the mixed formulation of the finite element method. The damage of the beam is modelled by reducing the elastic properties of the material on the damaged area.
The most popular modal based methods are tested and compared with time domain methods. The modal based methods which are tested are the modal displacement method, the modal curvatures method and the strain energy method. They are compared with the Poincaré map based methods developed previously by the authors of this study and with a new developed method.
The equation of motion is solved in time domain by the Newmark’s time integration method.
Discussion about the applicability of the methods to three-dimensional beam deformations is conducted.
The beam can vibrate in space, hence it can experience longitudinal and non-planar bending displacements, torsion and warping. The equation of motion is derived by the principle of virtual work. The Timoshenko’s beam theory is assumed and it is considered that the beam cross section rotates as rigid body due to torsion but may deform in longitudinal direction due to warping. Geometrical type of nonlinearity is included in the model and the equation of motion is discretized by the finite element method. As a result, the following system of nonlinear ordinary differential equations is obtained:
Mq ̈+Cq ̇+K(q)q=F (1)
where M represents the mass matrix, C represents the damping matrix, K represents the nonlinear stiffness matrix that depends on the displacement components, q is the vector of generalized coordinates and F is the vector of generalized external forces.
It is well known that shear locking appears in the cases of thin beams. The shear locking problem is avoided by using selective and reduced integration for the terms related with shear strain energy. This approach is equivalent to the mixed formulation of the finite element method. The damage of the beam is modelled by reducing the elastic properties of the material on the damaged area.
The most popular modal based methods are tested and compared with time domain methods. The modal based methods which are tested are the modal displacement method, the modal curvatures method and the strain energy method. They are compared with the Poincaré map based methods developed previously by the authors of this study and with a new developed method.
The equation of motion is solved in time domain by the Newmark’s time integration method.
Discussion about the applicability of the methods to three-dimensional beam deformations is conducted.