http://hdl.handle.net/2005/27 2013-04-30T11:28:28Z http://hdl.handle.net/2005/1832 Title: Closed-form Solutions For Rotating And Non-rotating Beams : An Inverse Problem Approach Authors: Sarkar, Korak Abstract: 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. 2012-12-03T18:30:00Z http://hdl.handle.net/2005/1659 Title: Robust Partial Integrated Guidance And Control Of UAVs For Reactive Obstacle Avoidance Authors: Chawla, Charu Abstract: UAVs employed for low altitude jobs are more liable to collide with the urban structures on their way to the goal point. In this thesis, the problem of reactive obstacle avoidance is addressed by an innovative partial integrated guidance and control (PIGC) approach using the Six-DOF model of real UAV unlike the kinematic models used in the existing literatures. The guidance algorithm is designed which uses the collision cone approach to predict any possible collision with the obstacle and computes an alternate aiming direction for the vehicle. The aiming direction of the vehicle is the line of sight line tangent to the safety ball surrounding the obstacle. The point where the tangent touches the safety ball is the aiming point. Once the aiming point is known, the obstacle is avoided by directing the vehicle (on the principles of pursuit guidance) along the tangent to the safety ball. First, the guidance algorithm is applied successfully to the point mass model of UAV to verify the proposed collision avoidance concept. Next, PIGC approach is proposed for reactive obstacle avoidance of UAVs. The reactive nature of the avoidance problem within the available time window demands simultaneous reaction from the guidance and control loop structures of the system i.e, in the IGC framework (executes in single loop). However, such quick maneuvers cause the faster dynamics of the system to go unstable due to inherent separation between the faster and slower dynamics. On the contrary, in the conventional design (executes in three loops), the settling time of the response of different loops will not be able to match with the stringent time-to-go window for obstacle avoidance. This causes delay in tracking in all the loops which will affect the system performance adversely and hence UAV will fail to avoid the obstacle. However, in the PIGC framework, it overcomes the disadvantage of both the IGC design and the conventional design, by introducing one more loop compared to the IGC approach and reducing a loop compared to the conventional approach, hence named as Partial IGC. Nonlinear dynamic inversion technique based PIGC approach utilizes the faster and slower dynamics of the full nonlinear Six-DOF model of UAV and executes the avoidance maneuver in two loops. In the outer loop, the vehicle guidance strategy attempts to reorient the velocity vector of the vehicle along the aiming point within a fraction of the available time-to-go. The orientation of the velocity vector is achieved by enforcing the angular correction in the horizontal and vertical flight path angles and enforcing turn coordination. The outer loop generates the body angular rates which are tracked as the commanded signal in the inner loop. The enforcement of the desired body rates generates the necessary control surface deflections required to stir the UAV. Control surface deflections are realized by the vehicle through the first order actuator dynamics. A controller for the first order actuator model is also proposed in order to reduce the actuator delay. Every loop of the PIGC technique uses nonlinear dynamic inversion technique which has critical issues like sensitiveness to the modeling inaccuracies of the plant model. To make it robust against the parameter inaccuracies of the system, it is reinforced with the neuro-adaptive design in the inner loop of the PIGC design. In the NA design, weight update rule based on Lyapunov’s theory provides online training of the weights. To enhance fast and stable training of the weights, preflight maneuvers are proposed. Preflight maneuvers provide stabilized pre-trained weights which prevents any misapprehensions in the obstacle avoidance scenario. Simulation studies have been carried out with the point mass model and with the Six-DOF model of the real fixed wing UAV in the PIGC framework to test the performance of the nonlinear reactive guidance scheme. Various simulations have been executed with different number and size of the obstacles. NA augmented PIGC design is validated with different levels of uncertainties in the plant model. A comparative study in NA augmented PIGC design was performed between the pre-trained weights and zero weights as used for weight initialization in online training. Various comparative study shows that the NA augmented PIGC design is quite effective in avoiding collisions in different scenarios. Since the NDI technique involved in the PIGC design gives a closed loop solution and does not operate with iterative steps, therefore the reactive obstacle avoidance is achieved in a computationally efficient manner. 2012-04-23T18:30:00Z http://hdl.handle.net/2005/1250 Title: Development Of A New Finite-Volume Lattice Boltzmann Formulation And Studies On Benchmark Flows Authors: Vilasrao, Patil Dhiraj Abstract: This thesis is concerned with the new formulation of a finite-volume lattice Boltzmann equation method and its implementation on unstructured meshes. The finite-volume discretization with a cell-centered tessellation is employed. The new formulation effectively adopts a total variation diminishing concept. The formulation is analyzed for the modified partial differential equation and the apparent viscosity of the model. Further, the high-order extension of the present formulation is laid out. Parallel simulations of a variety of two-dimensional benchmark flows are carried out to validate the formulation. In Chapter 1, the important notions of the kinetic theory and the most celebrated equation in the kinetic theory, ‘the Boltzmann equation’ are given. The historical developments and the theory of a discrete form of Boltzmann equation are briefly discussed. Various off-lattice schemes are introduced. Various methodologies adopted in the past for the solution of the lattice Boltzmann equation on finite-volume discretization are reviewed. The basic objectives of this thesis are stated. In Chapter2,the basic formulations of lattice Boltzmann equation method with a rational behind different boundary condition implementations are discussed. The benchmark flows are studied for various flow phenomenon with the parallel code developed in-house. In particular, the new benchmark solution is given for the flow induced inside a rectangular, deep cavity. In Chapter 3, the need for off-lattice schemes and a general introduction to the finite-volume approach and unstructured mesh technology are given. A new mathematical formulation of the off-lattice finite-volume lattice Boltzmann equation procedure on a cell-centered, arbitrary triangular tessellation is laid out. This formulation employs the total variation diminishing procedure to treat the advection terms. The implementation of the boundary condition is given with an outline of the numerical implementation. The Chapman-Enskog (CE) expansion is performed to derive the conservation equations and an expression for the apparent viscosity from the finite-volume lattice Boltzmann equation formulation in Chapter 4. Further, the numerical investigations are performed to analyze the apparent viscosity variation with respect to the grid resolution. In Chapter 5, an extensive validation of the newly formulated finite-volume scheme is presented. The benchmark flows considered are of increasing complexity and are namely (1) Posieuille flow, (2) unsteady Couette flow, (3) lid-driven cavity flow, (4) flow past a backward step and (5) steady flow past a circular cylinder. Further, a sensitivity study to the various limiter functions has also been carried out. The main objective of Chapter6is to enhance the order of accuracy of spatio-temporal calculations in the newly presented finite-volume lattice Boltzmann equation formulation. Further, efficient implementation of the formulation for parallel processing is carried out. An appropriate decomposition of the computational domain is performed using a graph partitioning tool. The order-of-accuracy has been verified by simulating a flow past a curved surface. The extended formulation is employed to study more complex unsteady flows past circular cylinders. In Chapter 7, the main conclusions of this thesis are summarized. Possible issues to be examined for further improvements in the formulation are identified. Further, the potential applications of the present formulation are discussed. 2011-06-29T18:30:00Z http://hdl.handle.net/2005/1280 Title: Active Vibration Control Synthesis Using Viscoelastic Damping Phenomena Authors: Vadiraja, G K Abstract: In this thesis, a new method is followed to design an active control system which imparts viscoelastic phenomenological damping in an elastic structure. Properties of a hypothetical viscoelastic system are used to design an active feedback controller for an undamped structural system with distributed sensor, actuator and controller. The variational structure is projected on a solution space of a closed-loop system involving a weakly damped structure with distributed sensor and actuator with controller. These controller components assign the phenomenology based on internal strain rate damping parameter of a viscoelastic system to the undamped elastic structure. An elastic cantilever beam with proportional-derivative controller and displacement feedback is considered in all the design formulations. In the first part of the research, a closed-loop control system is designed using two time domain modern control system design methods, pole placement and optimal pole placement, which use viscoelastic damping parameter. Equation of motion of a viscoelastic system is employed to synthesize the desired closed-loop poles. Desired poles are then assigned to an elastic beam with an active controller. Time domain finite element formulation is used without considering actuator-sensor dynamics and the effect of transducer locations. Characteristics of closed-loop system gains are found as a function of desired damping parameter and realization of damping have been analyzed with closed loop system pole positions. The next part consists of a novel frequency domain active control system design to impart desired viscoelastic characteristics, which uses spectral method and the exact dynamic stiffness matrix of the system. In the first case, a sub-optimal local control system for a cantilever beam, with collocated actuator and sensor is designed. In the second case, a closed-loop local controller for an elastic system with non-collocated transducers is designed. Next, a global controller for non-collocated arrangement of sensor-actuator is designed by considering all the degrees-of freedom in the system, which leads to solving an eigenvalue problem. The reason for the failure of the collocated arrangement in global control is also explained. In this novel control system design method transducer dynamics and locations are considered in the formulation. In frequency domain design, the frequency responses of the system show satisfactory performance of the closed-loop elastic system. The closed-loop system is able to reproduce the desired viscoelastic characteristics as targeted in the design. Optimal and sub-optimal system gains are found as functions of transducer locations, transducer properties, excitation frequency and internal strain rate damping parameter of a hypothetical viscoelastic system. Performance of the closed loop system is established by comparing the specific damping capacity of the hypothetical viscoelastic system with that of the closed-loop elastic system. The novel frequency domain method is simple, accurate, efficient and can be extended to complex structures to achieve desired damping. The method can be a better way of designing structures with variable stiffness which has research potential in designing morphing airplanes/spacecrafts. The ultimate goal of this research is that, if this design method is applied to practical applications such as aircraft wings, where vibration is undesirable, one would be able to achieve strength and desired damping characters simultaneously. 2011-07-11T18:30:00Z