Modelling and Model Reduction of Viscoelastic Composite Rotors - An Operator Based Approach

Abstract

Material damping in rotating structure plays a crucial role in dynamics. It produces a tangential force and acts towards whirl direction. This force is also proportional to spin speed, after certain speed it destabilizes the system, and corresponding speed is known as stability limit of spin speed. Thus, a reliable model is indeed essential for the correct assessment of stability. Modelling was done by many researchers and incorporated material damping through frequency dependent viscous or frequency independent hysteretic behaviour. Both of these damping models are not sufficient to predict the exact dynamic characteristic. The motivation for using general viscoelastic model arises from a need to capture the influence of both types of damping behaviours. The operator based constitutive relationship is used to incorporate such type of damping. The instantaneous stress of the linear viscoelastic material is obtained by operating instantaneous strain. The numerator and dt:;nominator of material modulus are a polynomial of differential time operator, and polynomial coefficients are known as viscoelastic parameters. Those parameters are obtained from frequency-dependent storage modulus and loss factor by using any optimization tool or curve fitting technique. Two types of material (low and high damped) are chosen for a case study. The operator based constitutive relationship is further used to bring down higher order equations of motion of an isotropic viscoelastic rotor by using two distinct approaches, i.e. (a) Classical model (CM) and (b) Finite element model (FEM). Lumped mass approximation is adopted in the CM approach thus it is also known as lumped mass classical model. Whereas, in FEM approach, the distributed effect of the continuum is considered. The Euler-Bernoulli beam theory is used here. The shear deformation effect may easily be included; however, it has been left out for simplicity. The shaft material damping has been tackled in such a way that to consider dissipation effect through all .coordinates. The order of the differential equation exclusively depends on the material behaviour and explained by modulus operator. For better understanding the dynamic characteristics of the rotating system, ·a comparative study is done by using following parameters such as decay rate, campbell diagram, modal damping factor, stability limit of spin-speed, unbalance frequency response, and also time response. This study is affirming compatibility between two approaches with each other up to some extent. According to industrial demand, heavy and bulky rotors are replaced by a light yet strong rotor, where the composite material is only supplementary. This composite is achieved either by reinforcing fibre or multi-layering arrangements. The present idea is also extended to develop multi-layer, multi-disc rotor system with the help of CM and FEM. All layers are isotropic viscoelastic material and assumed to be perfectly bonded. The effect of placement of material and their radius ratio in multi-layering of rotor is being well justified by taking an example of aluminium/steel (as low damped material) and PVC/LORD LD-400 (as high damped material). In general rotor shafts made of composite material are used in aerospace, helicopter rotor, under water vehicle soft toys etc. Since all dynamic behaviours of the rotating system are interlinked with rotation. The present study focuses on finding directivity and some other modal characteristics through complex modal analysis. A complex modal coordinate creates a platform to indicate the directivity of modes and provides better information about the whirl direction. The study of natural modes and directional frequency response function is obtained from free and force vibration analysis. The modal analysis also distinguishes the significance of higher order model over conventional second order model. For achieving a complete study, modal analysis is done under consideration of both rolling and sliding contact bearings. Higher-order finite element model involves a large number of coordinates as well as various types of asymmetry in different orders, resulting a complicated system again. Such problem is overcome by reducing the matrix size. For the sake of finding best methodology to reduce the higher order model, two different reduction techniques are applied here, i.e. a) Iterative Improved Reduction System (IIRS) b) Balanced Realization System (BRS). In the first case, an iterative algorithm is used to improve the transformation matrix by achieving convergence. Whereas, in the second case, Hankel singular value decomposition assures to predict the most controllable and most observable states that help to reduce the system matrices accurately. The dynamic study justifies the effectiveness of the reduced model. The presented work purely shows the mathematical modelling of various viscoelastic rotor bearing system. The mathematical procedure and formulations can _be used for any rotors as per their applications.