Animesh Chatterjee

Non-linear parameter estimation in multi-degree-of-freedom systems using multi-input Volterra series

Abstract Functional form input–output representation through Volterra series has been widely used for non-linear system analysis and non-parametric system identification. Recent research work shows that the series representation can be suitably employed for parametric identification also. However, the classical Volterra series is based on a single-input and its application is limited to analysis and identification of single-degree-of-freedom system only. The concept of single-input Volterra series has been extended to multi-input Volterra series by Worden et al. through definition of direct and cross-kernels. The present study employs the multi-input Volterra series and develops a structured response representation of various harmonics under multi-input harmonic excitations. Kernel synthesis formulations are developed for a polynomial form non-linearity with general square and cubic terms. It is shown that higher-order direct and cross-kernel transforms are functions of the first-order kernel transforms and the non-linear parameter vectors. A parameter estimation procedure based on recursive iteration is suggested and illustrated for a two-degree-of-freedom system with square and cubic stiffness non-linearity. Numerical simulations and error analysis are presented for typical rotor-bearing system parameters.

Non-linear parameter estimation with Volterra series using the method of recursive iteration through harmonic probing

Volterra series provides a platform for non-linear response representation and definition of higher order frequency response functions (FRFs). It has been extensively used in non-parametric system identification through measurement of first and higher order FRFs. A parametric system identification approach has been adopted in the present study. The series response structure is explored for parameter estimation of polynomial form non-linearity. First and higher order frequency response functions are extracted from the measured response harmonic amplitudes through recursive iteration. Relationships between higher order FRFs and first order FRF are then employed to estimate the non-linear parameters. Excitation levels are selected for minimum series approximation error and the number of terms in the series is controlled according to convergence requirement. The problem of low signal strength of higher harmonics is investigated and a measurability criterion is proposed for selection of excitation level and range of excitation frequency. The procedure is illustrated through numerical simulation for a Duffing oscillator. Robustness of the estimation procedure in the presence of measurement noise is also investigated.

Vibration analysis of single lap adhesive joint: Experimental and analytical investigation

An analytical model to study the coupled transverse and longitudinal vibrations of a single lap adhesive joint is described in this paper, which includes partial differential form of the motion equations. A joint consists of two identical adherents of mild steel that are lap jointed over a certain length by a viscoelastic material, epoxy resin (araldite). Adherents are modeled as Euler-Bernoulli free-free beam. Both transverse and axial deformation of adherents and longitudinal shear and transverse peel stresses at the adhesive joint interface are considered in deriving the equations of motion. The numerical solutions of the governing equations for free vibrations yield the system’s natural frequency and mode shapes. Experimentation was carried out on both monolithic and adhesively jointed beams to observe the effect of the joint; natural frequencies of the system were measured experimentally and compared with those obtained theoretically. The fundamental frequency of a free-free jointed beam was more sensitive to joint overlap ratio. However natural frequency depended on the accelerometer location in the system, which was attributed to its mass contribution to the overall system mass. Theoretical frequency response function is generated for a beam with and without accelerometer mass to show the mass loading effect of the transducer (accelerometer).

Sensitivity and Error Analysis of a Coupled Micro-Resonator Array for Ultra-Sensitive Mass Detection Using Matrix Perturbation Theory

A coupled micro-resonator array of cantilevers presents a cyclic symmetrical eigen system, which exhibits a characteristic modal spectrum. However, addition of even a small mass can perturb the modal characteristics significantly and this can be employed to design an ultrasensitive mass sensor. Researchers have investigated a varied range of micro-resonator arrays and have shown that shift in eigenvalues and eigenvectors in the modified spectrum depends on the target mass and its location. However, exact formulation of sensitivity as a calibration constant for a multi-resonator array in general has not been presented. In addition, error analysis, which is an important aspect of any sensor design, has not been studied so far. In this paper, matrix perturbation theory is employed to derive the sensor sensitivity parameters for different target locations and different eigenshift measurements. An optimum sensor array for maximum sensitivity is selected, and error analysis is done for the selected resonator to decide a threshold of target mass for a given error bound. Numerical simulation study shows that interconnection stiffness does not affect the sensitivity as derived, but it affects error of mass detection.

Non-Parametric Identification of Rotor-Bearing System through Volterra-Wiener Theories

The structure of the Volterra and Wiener series, which model the relationship between system response and input in terms of series of first and higher order convolution integrals, provide analytical platforms which can be utilized for parameter estimation. These are non-parametric forms of response representation. Non-parametric identification concerns modeling in a function space by input-output mapping, for systems where sufficient information on the mathematical structure or class is not available. Parametric identification, on the other hand, refers to systems where sufficient a-prioriinformation about the mathematical structure of the class to which the system belongs, is available. In the present study, structured Volterra and Wiener response representations are employed to develop identification and parameter estimation procedures for nonlinear rotor systems. Experimental investigations and validation of algorithms have been carried out on a laboratory test rig. Linear and nonlinear stiffness parameters are estimated and compared with approximate theoretical formulations and some previous experimental results.

Analytical and Numerical Investigation of Lacing Wire Damage Induced Mistuning in Turbine Blade Packet

Investigations of modal parameters for a mistuned packet of turbine blades due to lacing wire damage are reported using analytical and numerical studies with a simplified model. The turbine blade is assumed to be an Euler-Bernoulli beam connected with a lacing wire which is modeled as a mass less linear elastic spring. Thus, the blade is considered as a continuous system and lacing wire as a discrete system. The analytical results using Eigen value analysis are compared with numerical results obtained using commercial finite element package. In real life situation, though not reported in the literature, it is the failure of lacing wire that occurs quite often compared to the turbine blade and acts as precursor to the subsequent blade damage if it goes undetected. Therefore, studying the modal parameters of the grouped turbine blades in the context of lacing wire failure becomes important. The effect of variation of lacing wire location and stiffness indicative of damage resulting in the loss of stiffness on modal parameters is investigated. The study reveals a lot of fundamental understandings pertaining to dynamic behavior of grouped blades compared to the stand-alone blade under the influence of damaged lacing wire.

A Method for Joint Stiffness Identification

The most troublesome problem encountered in the dynamic simulation of a real system is the difficulty of knowing the accurate system parameters. Complex structures are usually composed of several substructures with joints to connect them together. These joints have considerable effect on the behavior of the assembled structure and must be accurately modeled.In this paper an efficient method is proposed for estimating joint stiffness parameter based on regression analysis; only natural frequencies are required in extracting the joint stiffness parameters. Numerical simulation results are presented involving identification of the boundary conditions of an elastically restrained beam.