
Dynamics of a cantilever beam with piezoelectric sensor: Parameter identification
Cimrman, Robert ; Kolman, Radek ; Musil, Ladislav ; Kotek, Vojtěch ; Kylar, Jaromír
The piezoelectric materials are electroactive materials often applied for realtime sensing or structural health monitoring, both important research topics in dynamics. Mathematical models of such structures have to allow also for the external electrical circuits and contain several material parameters that need to be identified from experiments. We present a model of a simple experiment involving dynamics of a cantilever beam with an attached piezoelectric sensor excited by a suddenly removed weight. The external circuit can be taken into account as having either a finite or infinite resistance. We also outline the parameter identification procedure based on automatic differentiation and present the experimental and numerical results.


Dynamics of a cantilever beam with piezoelectric sensor: Experimental study
Kolman, Radek ; Kylar, Jaromír ; Kotek, Vojtěch ; Cimrman, Robert ; Musil, Ladislav
Online and realtime sensing and monitoring of the health state of complex structures, such as aircraft and critical parts of power stations, is an essential part of the research in dynamics. Several types of sensors are used for sensing dynamic responses and monitoring response changes during the operation of critical parts of complex systems. The piezoelectric (PZ) materials belong to one group of electroactive materials, which transform mechanical deformation into an electrical response. For example, PZ ceramics or PVDF foils are employed for online sensing of the time history of mechanical deformation. Experimentally obtained response of a cantilever beam structure with a glued PZ sensor is the case of interest in this contribution. During the transient problem of the beam loaded by suddenly interrupted load due to the weight of a mass at the end of the beam, the time history of normal velocity at a point on the beam surface has been measured by a laser vibrometer and parallely, the output voltage on the PZ sensor has been measured by an electric device. The experimental data in the case of the first eigenfrequency is in good agreement with the value given by the formulae from the theoretical modeling of free vibration of a linear beam.

 

Direct construction of reciprocal mass matrix and higher order fininite element method
Cimrman, Robert ; Kolman, Radek ; González, J. A. ; Park, K. C.
When solving dynamical problems of computational mechanics, such as contactimpact problems or cases involving complex structures under fast loading conditions, explicit timestepping algorithms are usually preferred over implicit ones. The explicit schemes are normally combined with the lumped (diagonal) mass matrix so that the calculations are efficient and moreover dispersion errors in wave propagation are partially eliminated. As an alternative to lumping with advantageous properties, the reciprocal mass matrix is an inverse mass matrix that has the same sparsity structure as the original consistent mass matrix, preserves the total mass, captures well the desired frequency spectrum and leads thus to efficient and accurate calculations. In the contribution we comment on the usability of the reciprocal mass matrix in connection with higher order FEM.


Localized formulation of bipenalty method in contactimpact problems
Kolman, Radek ; González, J. A. ; Dvořák, Radim ; Kopačka, Ján ; Park, K.C.
Often, the finite element method together with direct time integration is used for modelling of contactimpact problems of bodies. For direct time integration, the implicit or explicit time stepping are gen\nerally employed. It is well known that the time step size in explicit time integration is limited by the stability limit. Further, the trouble comes with the task of impact of bodies with different critical time step sizes for each body in contact. In this case, this numerical strategy based on explicit time stepping with the same time step size for both bodies is not effective and is not accurate due to the dispersion behaviour and spurious stress oscillations. For that reason, a numerical methodology, which allows independent time stepping for each body with its time step size, is needed to develop. In this paper, we introduce the localized variant of the bipenalty method in contactimpact problems with the governing equations derived based on the Hamilton’s principle. The localized bipenalty method is applied into the impact problems of bars as an onedimensional problem. The definition of localized gaps is presented and applied into the full concept of the localized bipenalty method.


Inverse mass matrix for higherorder finite element method in linear freevibration problems
Kolman, Radek ; González, J.G. ; Cimrman, Robert ; Kopačka, Ján ; Cho, S.S. ; Park, B.G.
In the paper, we present adirect inverse mass matrix in the higherorderfinite element method forsolid mechanics. The direct inverse mass matrix is sparse, has the same structure as the consistent mass matrixand preserves the total mass. The core of derivation of the semidiscrete mixed form is based on the Hamilton’s principle of leastaction. The cardinal issue is finding the relationship between discretized velocities and discretized linear momentum. Finally, the simple formula for the direct inversemass matrix is presented as well as thechoice of densityweighted dual shape functions for linear momentum with respect to the displacement shape functionwith achoice of the lumping mass method for obtaining the correct and positive definitive velocitylinear momentum operator. The application of Dirichlet boundaryconditions into the direct inversemass matrix forafloating system is achieved usingthe projection operator. The suggested methodology is tested on a freevibration problem of heterogeneous bar for different ordersof shape functions.


Dispersion errors for wave propagation in thin plate due to the finite element method
Kruisová, Alena ; Kolman, Radek ; Mračko, Michal
In modelling of wave propagation by finite element method, both the spatial and temporal discretization lead to dispersion errors. For 2D plane strain elements these errors can be stated analytically. These analytical relations derived for harmonical waves in the infinite continuum can be used for error estimation on an example of simulation of Lamb's wave in plate using the implicit time integration method. So the analytical relations for dispersion errors can serve for determination of element size and time step size in wave propagation.

 

Bipenalty stabilized explicit finite element algorithm for onedimensional contactimpact problems
Kolman, Radek ; Kopačka, Ján ; Tkachuk, A. ; Gabriel, Dušan ; Gonzáles, J.A.
In this contribution, a stabilization technique for finite element modelling of contactimpact problems based on the bipenalty method and the explicit predictorcorrector time integration is presented. The penalty method is a standard method for enforced contact constrains in dynamic problems. This method is easily implemented but the solution depends on numerical value of the stiffness penalty parameter and also the stability limit for explicit time integration is effected by a choice of this parameter. The bipenalty method is based on penalized not only stiffness term but also mass term concurrently. By this technique with a special ratio of mass and stiffness penalty parameters, the stability limit of contactfree problem is preserved. In this contribution, we also present a modification of the explicit time scheme based on predictorcorrector form. By meaning of this approach, spurious contact oscillations are eliminated and the results do not depend on numerical parameters.

 