The rolling of an aircraft about its fuselage produces centrifugal forces which affect the stiffness of the wings. A number of previous studies explain the effect of centrifugal stiffening in rotating beams and consequently on the frequencies of the beam. Multiple cases of the rotating beam are explored in this thesis to investigate effects of mass distribution and boundary conditions on the frequencies of centrifugally stiffened beams. It is found that for a uniform beam with all degrees of freedom free on both ends, the rigid modes of the beam are affected and are no longer zero when it is stiffened from centrifugal forces. This thesis aims to set up a model to investigate the stiffening effects using the mAEWing2 aircraft. A preliminary analysis is done for the mAEWing2 aircraft and the roll rate, control surface deflection and angle of attack are identified as the parameters to be studied. For a given angle of attack and control surface deflection, the centrifugal forces in the aircraft in steady roll are determined using trim analysis. These are used to pre-stress the model for modal analysis. It is found that in mAEWing2 aircraft in steady roll maneuvers, the centrifugal stiffening effect on the natural frequencies is not significant. It emphasizes the need to conduct a sensitivity analysis to include centrifugal stiffening in the dynamic analysis while designing an aircraft. This, along with some de-stiffening due to gravity loads might be important for the future N+3 aircraft with their high aspect ratio large wingspans. / MS

Active vibration control is a widely implemented method for the helicopter vibration control. Due to the significant progress in microelectronics, this technique outperforms the traditional passive control technique due to weight penalty and lack of adaptability for the changing flight conditions. In this thesis, an optimal controller is designed to attenuate the rotor blade vibration. The mathematical model of the triply coupled vibration of the rotating cantilever beam is used to develop the state-space model of an isolated rotor blade. The required natural frequencies are determined by the modified Galerkin method and only the principal aerodynamic forces acting on the structure are considered to obtain the elements of the input matrix. A linear quadratic regulator is designed to achieve the vibration reduction at the optimum level and the controller is tuned for the hovering and forward flight with different advance ratios.

This thesis deals with the study of two problems in the area of nonlinear dynamics using the method of multiple scales. Accordingly, it consists of two parts. In the first part of the thesis, we explore the asymptotic stability of a planar two-degree- of-freedom robot with two rotary (R) joints following a desired trajectory under feedback control. Although such robots have been extensively studied and there exists stability and other results for position control, there are no analytical results for asymptotic stability when the end of the robot or its joints are made to follow a time dependent trajectory. The nonlinear dynamics of a 2R planar robot, under a proportional plus derivative (PD) and a model based computed torque control, is studied. The method of multiple scales is applied to the two nonlinear second-order ordinary deferential equations which describes the dynamics of the feedback controlled 2R robot. Amplitude modulation equations, as a set of four first order equations, are derived. At a fixed point, the Routh-Hurwitz criterion is used to obtain positive values of proportional and derivative gains at which the controller is asymptotically stable or indeterminate. For the model based control, a parameter representing model mismatch is incorporated and again controller gains are obtained, for a chosen mismatch parameter value, where the controller results in asymptotic stability or is indeterminate. From numerical simulations with gain values in the indeterminate region, it is shown that for some values and ranges of the gains, the non- linear dynamical equations are chaotic and hence the 2R robot cannot follow the desired trajectory and be asymptotically stable. The second part of the thesis deals with the study of the nonlinear dynamics of a rotating flexible link, modeled as a one dimensional beam, undergoing large deformation and with geometric nonlinearities. The partial deferential equation of motion is discretized using a finite element approach to yield four nonlinear, non-autonomous and coupled ordinary deferential equations. The equations are non-dimensional zed using two characteristic velocities – the speed of sound in the material and a speed associated with the trans- verse bending vibration of the beam. The method of multiple scales is used to perform a detailed study of the system. A set of four autonomous equations of the first-order are derived considering primary resonance of the external excitation with one of the natural frequencies of the model and one-to-one internal resonance between two different natural frequencies of the model. Numerical simulations show that for certain ranges of values of these characteristic velocities, the slow flow equations can exhibit chaotic motions. The numerical simulations and the results are related to a rotating wind turbine blade and the approach can be used for the study of the nonlinear dynamics of a single link flexible manipulator. The second part of the thesis also deals with the synchronization of chaos in the equations of motion of the flexible beam. A nonlinear control scheme via active nonlinear control and Lyapunov stability theory is proposed to synchronize the chaotic system. The proposed controller ensures that the error between the controlled and the original system asymptotically go to zero. A numerical example using parameters of a rotating power generating wind turbine blade is used to illustrate the theoretical approach.

Rotating Euler-Bernoulli beams and non-homogeneous Timoshenko beams are widely used to model important engineering structures. Hence the vibration analyses of these beams are an important problem from a structural dynamics point of view. The governing differential equations of both these type of beams do not yield any simple closed form solutions, hence we look for the inverse problem approach in determining the beam property variations given certain solutions. Firstly, we look for a rotating beam, with pinned-free boundary conditions, whose eigenpair (frequency and mode-shape) is same as that of a uniform non-rotating beam for a particular mode. It is seen that for any given mode, there exists a flexural stiffness function (FSF) for which the ith mode eigenpair of a rotating beam with uniform mass distribution, is identical to that of a corresponding non-rotating beam with same length and mass distribution. Inserting these derived FSF's in a finite element code for a rotating pinned-free beam, the frequencies and mode shapes of a non-rotating pinned-free beam are obtained. For the first mode, a physically realistic equivalent rotating beam is possible, but for higher modes, the FSF has internal singularities. Strategies for addressing these singularities in the FSF for finite element analysis are provided. The proposed functions can be used as test functions for rotating beam codes and also for targeted destiffening of rotating beams. Secondly, we study the free vibration of rotating Euler-Bernoulli beams, under cantilever boundary condition. For certain polynomial variations of the mass per unit length and the flexural stiffness, there exists a fundamental closed form solution to the fourth order governing differential equation. It is found that there are an infinite number of rotating beams, with various mass per unit length variations and flexural stiffness distributions, which share the same fundamental frequency and mode shape. The derived flexural stiffness polynomial functions are used as test functions for rotating beam numerical codes. They are also used to design rotating cantilever beams which may be required to vibrate with a particular frequency. Thirdly, we study the free vibration of non-homogeneous Timoshenko beams, under fixed-fixed and fixed-hinged boundary conditions. For certain polynomial variations of the material mass density, elastic modulus and shear modulus, there exists a fundamental closed form solution to the coupled second order governing differential equations. It is found that there are an infinite number of non-homogeneous Timoshenko beams, with various material mass density, elastic modulus and shear modulus distributions, which share the same fundamental frequency and mode shape. They can be used to design non-homogeneous Timoshenko beams which may be required for certain engineering applications.

A partial differential equation in space and time represents the physics of rotating beams. Mostly, the numerical solution of such an equation is an available option as analytical solutions are not feasible even for a uniform rotating beam. Although the numerical solutions can be obtained with a number of combinations (in space and time), one tries to seek for a better alternative. In this work, various numerical techniques are applied to the rotating beam problems: finite element method, meshless methods, and B-spline finite element methods. These methods are applied to the governing differential equations of a rotating Euler-Bernoulli beam, rotating Timoshenko beam, rotating Rayleigh beam, and cracked Euler-Bernoulli beam. This work provides some elegant alternatives to the solutions available in the literature, which are more efficient than the existing methods: the p-version of finite element in time for obtaining the time response of periodic ordinary differential equations governing helicopter rotor blade dynamics, the symmetric matrix formulation for a rotating Euler-Bernoulli beam free vibration problem using the Galerkin method, and solution for the Timoshenko beam governing differential equation for free vibration using the meshless methods. Also, the cracked Euler-Bernoulli beam free vibration problem is solved where the importance of higher order polynomial approximation is shown. Finally, the overall response of rotating blades subjected to aerodynamic forcing is obtained in uncoupled trim where the response is independent of the overall helicopter configuration. Stability analysis for the rotor blade in hover and forward flight is also performed using Floquet theory for periodic differential equations.

In this work, rotating beams which are isospectral to non-rotating beams are studied. A rotating beam is isospectral to a non-rotating beam if both the beams have the same spectral properties i.e; both the beams have the same set of natural frequencies under a given boundary condition. The Barcilon-Gottlieb transformation is extended, so that it converts the fourth order governing equation of a rotating beam (uniform or non-uniform), to a canonical fourth order eigenvalue equation. If the coefficients in this canonical equation match with the coefficients of the non-rotating beam (non-uniform or uniform) equation, then the rotating and non-rotating beams are isospectral to each other. The conditions on matching the coefficients lead to a pair of coupled differential equations. We solve these coupled differential equations for a particular case, and thereby obtain a class of isospectral rotating and non-rotating beams. However, to obtain isospectral beams, the transformation must leave the boundary conditions invariant. We show that the clamped end boundary condition is always invariant, and for the free end boundary condition to be invariant, we impose certain conditions on the beam characteristics. The mass and stiffness functions for the isospectral rotating and non-rotating beams are obtained. We use these mass and stiffness functions in a finite element analysis to verify numerically the isospectral property of the rotating and non-rotating beams. Finally, the example of beams having a rectangular cross section is presented to show the application of our analysis. Since experimental determination of rotating beam frequencies is a difficult task, experiments can be easily conducted on these rectangular non-rotating beams, to calculate the frequencies of the isospectral rotating beams.