Nonlinearity identification of Jesus Christ the King monument on the basis of modal parameters

Jacek Grosel,Jakub Marcinowski,Zbigniew Wójcicki,Wojciech Sawicki,V Andreev,T Matseevich,A Adamtsevich,A Ter-Martirosyan

doi:10.1051/matecconf/201819601054

Abstract

The work presents an attempt to identify the nonlinearity of the structure using a simple autonomous vibration recorder. The vibrations were measured continuously for 4 days in three directions, which easily made it possible to record various load states of the structure. The analysis of the response spectra shows a change in the eigenfrequencies caused by the level of load, which is a measure of the non-linearity of the structure. The presented method is an interesting alternative to expensive multi-channel visit measurements, in which the equipment is compensated by the length of measurements

Highlights

IntroductionEquation of motion for a multi-degree of freedom system could be presented in a matrix form
Equation of motion for a multi-degree of freedom system could be presented in a matrix formB q(2)(t) +C q(1)(t)+ K q(0)(t)= F(t) (1)where: □(i) denotes i-th derivative with respect to time i.e. q(0) – vector of displacements, q(1) – vector of velocity, q(2) – vector of acceleration,B – inertia matrix, C – damping matrix, K – stiffness matrix, and F(t) – vector of excitation.Applying modal transformation method to the equation (1), with assumption that damping matrix is linear combination of inertia and stiffness matrix, one obtains{b} r(2)(t) +{c} r(1)(t) + {k} r(0)(t)= f(t) (2)where: {□} denotes diagonal matrix. r(0)– vector of displacements in the general coordinates base.Equation (2) describe motion of the system as a set of independent Single Degree of Freedom (SDOF) systems
Eigenfrequencies were determined on the measurement performed in 2013 and 2016, Operational Modal Analysis (OMA) was adapted and obtained eigenfrequencies were found using three axes vibration measurements recorded in five points on the structure

Summary

Introduction

Equation of motion for a multi-degree of freedom system could be presented in a matrix form. Applying modal transformation method to the equation (1), with assumption that damping matrix is linear combination of inertia and stiffness matrix, one obtains. Where: □i denotes parameter concerning i-th mode, η = ω / ωi, ωi i-th eigenfrequency, ξ – fraction of critical damping. For system with well separated eigenfrequencies and excitation of white noise form Fourier Transform of measured response equals to. Starting form this relation ωi and ξi could be found by the use of nonlinear fitting, such a way for finding modal parameters are referred later as Method 1 (M1). Results obtained form M1 and M2 are compared in the paragraphs, the influence of load level on determined parameters are shown

Monument of the Jesus the King

Vibration measurement

Results of the vibration analysis

Conclusions