Shaikh Faruque Ali

Flexible structures are continuum modeled as infinite degrees of freedom. Most of the time finite element models are made with large number of degrees-of-freedom to analyse flexible structures. Continuous monitoring, analysis and control of dynamics of flexible structures need simulation of large degrees of freedom in real time. This is computationally expensive and realtime control fails. This study focuses on the development of a reduced order framework for dynamics of flexible structures and use the reduced order model for controlling the structure. Essentially, in the proposed framework, full order state space obtained from finite element modeling of the flexible structure has been reduced to lower subspace using a reduced order algorithm keeping the dynamical characteristic intact. The transformation matrix for reduction has been calculated using system equivalent reduction expansion process (SEREP). Traditionally, full order dynamical system is being used to find the gain matrix to suppress the uncontrolled dynamics associated with the dynamical system. Here in this methodology, the reduced dimension of state space of the system is used for estimating the gain matrix using optimal linear quadratic regulator (LQR). The gain obtained through the reduced system is subsequently used as a feedback to the attached actuators which produce the required force to control the system. A numerical example of a flexible cantilever beam has been shown to investigate the effectiveness of the algorithm.

This study focusses on the vibration control of linear large scale engineering structures, modelled as continuous system, with varying system parameters. Because of manufacturing limitations or/and measurement errors, system parameters need to be modelled as random variables. Here, system parameters are modelled as non-Gaussian random variables. Mathematically, continuous system is governed by partial differential equations and solved using approximation methods. High fidelity finite element model is the starting point of the analysis. Since numerical approximation involves large number of degrees-of-freedom, solving the system in real time is computationally expensive and application of control algorithm is cumbersome. In this study, a reduced order model is developed to reduce the state space dimension of the problem and further integrated with the control algorithm. Next, controller gain is obtained using linear quadratic regulator in reduced subspace which is used as a feedback to the actuators to produce the required control force for vibration control. A numerical example of flexible cantilever beam is solved to demonstrate the efficacy of the algorithm and probabilistic characterisation is carried out using Monte Carlo simulation.

Bladed disc systems, are designed to be rotationally periodic i.e. each sector repeats itself in an angular manner. This ensures that the response characteristics preserve this rotational periodicity and is defined as a tuned system. However, due to manufacturing limitations, there are invariably random spatial variations in the sectors which constitute the rotor. This in turn destroys the periodicity (angular symmetry) of the rotor. Such a system is defined as a mistuned system, leads to energy localization and makes the rotor susceptible to damage leading to a loss of its service life. This study focusses on developing a methodology for modeling the random spatial inhomogeneities and integrating with finite element analysis for estimating the behavior of a sector of a rotor blade assembly. Approximations for the non-Gaussian random fields for the spatial inhomogeneities are directly developed from the data set of measurements available from coordinate measuring machine (CMM). Subsequently, Rosenblatt’s transformations are applied to each component of the discrete measurement data set for polynomial chaos (PC) representation of the parameter to get the estimated probability density function (pdf). Spearman’s rank correlation coefficient (SRCC) is calculated to obtain the correlation parameter. Kullback – Liebler entropy (KLE) is used as a metric to compare the estimated pdf obtained through the PC representation of the measured data. The in-house PC based developed codes are subsequently integrated with commercially available finite element software which enable quantifying the propagation of the spatial random uncertainties into the response characteristics. Statistical processing of the response quantities is carried out to estimate probabilistic measures of the corresponding response quantities.

This study focusses on probabilistic modelling of the bladed disc system and numerical estimation of the distributions of the response quantities of the system. Stochastic finite element model of the system consisting of all the assemblies and the hub is developed and reported. The spatial inhomogeneity of mistuned structures is modelled as non-Gaussian random field. Experimentally, the system parameters can be measured at the specified locations of the bladed disk structure. In this analysis, a synthetic data is generated which represent this measured data set. Further, Nataf transformation is implemented to each component of the data set to get the polynomial chaos expansion framework of the system parameters. Since, the random field of the system parameter is approximated as correlated random variables, Spearman's rank correlation coefficient is used in this manuscript to obtain that correlation among the random parameters across the domain. The approximated probability density function obtained through the aforementioned methodology is compared with the target probability density function of the parameter using Kullback - Liebler (KL) entropy as a metric. Also, the same KL entropy is used as a metric to check the convergence of polynomial chaos terms in the expansion. Next, the proposed polynomial chaos method is integrated with commercial finite element software to quantify the propagation of randomness associated with system parameters into the response quantities. Subsequently, the statistical processing helps in estimating the probabilistic measure of the required response quantities. The results obtained through the conventional Monte Carlo (MC) simulations have been used as the benchmark to compare the response characteristics obtained through the proposed algorithm.