Elsevier

Computers & Structures

Volume 75, Issue 4, 1 April 2000, Pages 347-359
Computers & Structures

Energy dissipation through joints: theory and experiments

https://doi.org/10.1016/S0045-7949(99)00096-6Get rights and content

Abstract

A new qualitative non-destructive evaluation (NDE) technique based on a modification of the high-frequency impedance measured with an integrated piezoceramic (PZT) actuator–sensor has been successfully demonstrated at the Center for Intelligent Material Systems and Structures. An important characteristic of this NDE technique is the localization of the PZT’s sensing area due to energy dissipation. For a better understanding of this phenomena, this paper will present an analytical method to determine the energy dissipated through joints at high frequency and its relation to the localized actuation-sensing region. The structure used for this analysis consists of two beams connected with a bolted joint and each having free boundary conditions. Due to the high-frequency range employed, theoretical assessments are made using a wave propagation approach. For the same reason, the Timoshenko beam theory is used to model the structure. Using the equations of motion, the dynamic stiffness matrix is assembled in the frequency domain. Then, the energy dissipated in the joint is modeled linearly using mass-spring-dash pot systems and nonlinearly with the application of nonconservative friction clearance systems. This work is the first effort in the high-frequency modeling of structural joints: other work was only concerned with the modeling of the few first modes of vibration. Case studies and experimental verification will then follow.

Introduction

A unique method for the realization of health monitoring of complex structures using integrated piezoceramic (PZT) actuator–sensors is currently being developed at the Center for Intelligent Material Systems and Structures [1], [2], [3]. This new health monitoring method is based on the modification of the structural high-frequency impedance (15–250 kHz) when incipient damage is present in the structure. By integrating the measured impedance variations over a dynamically active frequency band, a scalar damage metric is obtained, and damage is detected when the damage metric reaches a predetermined threshold value. This new qualitative health monitoring technique has been successfully applied to numerous proof-of-concept demonstrators, such as bridge joints, aircraft structures, high precision gears, and bay trusses. In all of these demonstrators, a high-frequency range is used to enhance the ability to detect damage at a microscopic level, since the wavelength is short enough to be contained within the size of the damage. It has been shown that the sensing area is localized to a region close to the PZT actuator–sensor, therefore enhancing the ability of this technique to detect damage without being affected by far field boundary conditions, external loading, or normal operational vibrations of the system. This localization phenomena has only been observed experimentally and it was attributed to energy dissipation in the system at high frequencies.

Even though the new health monitoring technique has been shown to be very performant, more fundamental research is needed before a broad use of the technique is possible. The fundamental research to be performed is geared towards a better understanding of the energy dissipation in structures at high frequencies. An analytical model is needed for the energy dissipated in joints as the high-frequency wave, generated by the PZT actuator–sensor propagates throughout the structure. The aim of this work is to develop a theoretical technique to model such an energy dissipation mechanism and to assess the energy losses that occur by characterizing the wave propagation in the structure to understand the localized sensitivity of the PZT actuator–sensor. This is believed to be the first effort to model structural joints at high frequency; the previous efforts to analyze the dynamics of bolted connections were done with a frequency range corresponding only to the first few modes of vibration. Thereby, the need to model bolted structures under high frequency excitation led us to a unique and clean method to describe the dynamics of the system. The energy dissipation due to material damping has been modeled by the authors in related work [4], [5].

The structure used for this analysis consists of two beams overlapped and connected together with two bolts (Fig. 1). First, the wave propagation model for this particular case is derived. Then, the dynamics of the bolted section is identified with linear spring-dash pot systems: viscous dampers to provide the necessary flow of energy out of the system, and springs to provide the necessary support in the connection. An extended nonlinear model will then be derived to accommodate for the nonlinear behavior of the bolted joint. Analytical calculations will be carried out with experiments to corroborate the validity of the model.

Section snippets

Wave propagation model

Due to the high-frequency range of the analysis (upto 25 kHz), the research herein presented, uses a wave propagation approach, and the one-dimensional structure (Fig. 1) is treated as waveguides along which a wave can travel without spreading in all directions. Nonuniformities in guides, such as junctions and discontinuities, are characterized by reflected and transmitted waves due to incident waves. A methodology using a formulation similar to that of the finite element method, but in which

Linear joint model

The linear analysis of the joint is made with the use of spring-dash pot systems placed at the matting section and acting in the directions of the degrees of freedom used to describe the motion of the structure. In this system, the dash pots model the viscous damping in the joints, while the springs provide the appropriate rigidity for the connection. The mathematical model shown in Fig. 2(a) is, therefore, assumed for computer simulation.

The matting section is discretized into three elements

Energy dissipation through bolted joints

To interpret the energy associated with the propagating wave, it is necessary to define the transmission and reflection coefficients, which are associated with the amount of energy that has been transmitted and reflected after the incident wave encounters the bolted joint. These coefficients are associated with the power flux of the incident, transmitted and reflected waves, which can be derived from the Timoshenko beam theory [12] and they are defined respectively as:P̂I=ReMIφ̇I+VIẇI≤0P̂T=ReMT

Experimental set-up

The experimental set-up used to correlate the analytical an experimental results is shown in Fig. 5: A single PZT-actuator of (20 × 19 × 0.19) mm and a PZT-sensor of (22 × 19 × 0.19) mm are bonded to the structure and wired to the HP4194A analyzer (gain-phase feature) into the input and output slots, respectively. A sine sweep frequency is applied to the PZT-actuator at a constant voltage to excite the system with a coupled axial force-bending moment loading over various frequency ranges, and

Electro-mechanical coupling

The low-power driven PZT actuator–sensor patch, when bonded to a structure, produces a high frequency excitation level at constant voltage and acquires the electrical impedance, given as the ratio of the voltage source and the current flow modulated by the response of the structure. In this manner, knowledge about the mechanical impedance of the monitored structure can be recorded. Due to the high-frequency content of this technique minor changes of the structure’s configuration are accordingly

Bolted beam results

A preliminary analysis of a single beam element cut from the same specimen as the bolted structure and having the same length was first considered. Analytical and experimental results lead to conclude the validity of the method. Moreover, the resonant frequencies analytically obtained were compared to those obtained experimentally, and the Kelvin–Voight coefficients (ηE and ηG), used to model the material damping in the structure, were modified accordingly, until the analytical and experimental

Conclusions

A methodology to model one dimensional bolted structures’ high-frequency vibrations has been developed. Following a wave propagation approach by assuming a spectral representation of the displacements, an assembly of the equations of motion is made on the basis of continuum mechanics. The linear and nonlinear energy dissipation of the bolted joint was then derived. Material damping is added into the formulation by considering the Kelvin–Voight model. The aim of this work is to investigate the

Acknowledgements

The authors would like to acknowledge the support of the National Science Foundation, Grant No. MSS9157080: Dr. Ken Chong, Program Director.

References (21)

There are more references available in the full text version of this article.

Cited by (63)

  • A thorough comparison between measurements and predictions of the amplitude dependent natural frequencies and damping of a bolted structure

    2023, Journal of Sound and Vibration
    Citation Excerpt :

    While many studies have sought to predict the behavior of joints, and some commercial software packages give the impression that this can be readily done, very few studies have presented a prediction that includes the contact behavior and geometry of an actual joint and have also validated those predictions with nonlinear measurements. This challenge is relevant to many structures including blade-disk assemblies [3,4], automotive brakes [5], bridges [6] and engines [7,8]. Even in high consequence industries such as the aerospace industry, joints are typically approximated as linear and test data is used to obtain the linear damping of each mode, which are used in any subsequent analysis such as flutter analysis [9].

  • Nonlinearity of interfaces and force transmission of bolted flange joints under impact loading

    2017, International Journal of Impact Engineering
    Citation Excerpt :

    The structure introduced different stiffness under tensile and compressive stress, however, testing of the model under high frequency of excitation was not mentioned. Esteban and Rogers [16] established a model with spectral function conducting the energy dissipation of bolted joints. The research showed the response of joints and the energy dissipation were different under low and high frequency of excitation.

View all citing articles on Scopus
View full text