Maneuver Control and Vibration Suppression of a Smart Flexible Satellite Using Robust Passivity Based Control

2012 ◽  
Vol 488-489 ◽  
pp. 1803-1807
Author(s):  
Mohammad Azadi
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Vibration Suppression ◽  
Equations Of Motion ◽  
Governing Equations ◽  
Lagrange Method ◽  
Piezoelectric Layers ◽  
Simulation Results ◽  
Passivity Based Control ◽  
Flexible Appendages

In this paper a satellite with two flexible appendages and the piezoelectric layers which are attached to them and a central hub is considered. The piezoelectric layers are used as actuators. The governing equations of motion are derived based on Lagrange method. Using Rayleigh-Ritz technique ordinary differential equations of motion are obtained. A robust passivity based control is applied to the system to not only control the three axes maneuver of the satellite but also suppress the vibrations of the flexible appendages. Finally, the system is simulated and simulation results show the good performance of this controller.

Multibody Dynamics on Differentiable Manifolds

2021 ◽  
Vol 16 (4) ◽  
Author(s):  
Edward J. Haug
Keyword(s):  
Euclidean Space ◽  
Differential Equations ◽  
Vector Space ◽  
Ordinary Differential Equations ◽  
Multibody Dynamics ◽  
Equations Of Motion ◽  
Differentiable Manifold ◽  
Kinematics And Dynamics ◽  
Variational Equations ◽  
Differentiable Manifolds

Abstract Topological and vector space attributes of Euclidean space are consolidated from the mathematical literature and employed to create a differentiable manifold structure for holonomic multibody kinematics and dynamics. Using vector space properties of Euclidean space and multivariable calculus, a local kinematic parameterization is presented that establishes the regular configuration space of a multibody system as a differentiable manifold. Topological properties of Euclidean space show that this manifold is naturally partitioned into disjoint, maximal, path connected, singularity free domains of kinematic and dynamic functionality. Using the manifold parameterization, the d'Alembert variational equations of multibody dynamics yield well-posed ordinary differential equations of motion on these domains, without introducing Lagrange multipliers. Solutions of the differential equations satisfy configuration, velocity, and acceleration constraint equations and the variational equations of dynamics, i.e., multibody kinematics and dynamics are embedded in these ordinary differential equations. Two examples, one planar and one spatial, are treated using the formulation presented. Solutions obtained are shown to satisfy all three forms of kinematic constraint to within specified error tolerances, using fourth-order Runge–Kutta numerical integration methods.

scholarly journals Fast Attitude Maneuver of a Flexible Spacecraft with Passive Vibration Control Using Shunted Piezoelectric Transducers

2019 ◽  
Vol 2019 ◽  
pp. 1-18
Author(s):  
Bassam A. Albassam
Keyword(s):  
Differential Equations ◽  
Electromechanical Coupling ◽  
Design Method ◽  
Equations Of Motion ◽  
Control Design ◽  
Piezoelectric Transducers ◽  
Control Input ◽  
Attitude Maneuver ◽  
Shunt Circuit ◽  
Flexible Appendages

This paper is concerned with designing a bang-bang control input to perform a quick rotational maneuver of a rigid spacecraft hub connected with flexible appendages. The control design is based on only the rigid body mode making it very simple to design and at the same time achieve the quickest maneuver possible. The induced vibrations are suppressed using piezoelectric transducers bonded to the appendages and connected to an electric circuit with the objective of converting the vibrational energy to electrical energy and then dissipating it using passive electric elements, such as a resistance and an inductor. The proposed control design method is applied to a spacecraft containing a rigid hub and flexible appendages. The attitude control torque is produced using either the reaction wheels contained inside the rigid hub or jet thrusters mounted outside it. The control design process starts with deriving the nonlinear partial differential equations of motion for the spacecraft using Hamilton’s principle which accounts for the electromechanical coupling and the presence of resistive or resistive-inductive circuits. To simplify the analysis, the nonlinear ordinary differential equations of motion are then obtained using the assumed mode method. The effectiveness of the control design method is numerically tested on a spacecraft that is required to perform a quick attitude maneuver and, simultaneously, suppress the induced vibrations. The simulations show a quick and accurate maneuver has been achieved combined with very low levels of vibrations resulting from the reduced coupling between flexible and rigid motions as well as the damping added as a result of the passive shunt circuit. Furthermore, the resistance-inductance shunt circuit is shown to be more effective in damping the vibrations than the resistance shunt circuit.

scholarly journals Flow over a non-uniform sheet with non-uniform stretching (shrinking) and porous velocities

2020 ◽  
Vol 12 (2) ◽  
pp. 168781402090900
Author(s):  
Aftab Alam ◽  
Dil Nawaz Khan Marwat ◽  
Saleem Asghar
Keyword(s):  
Boundary Layer ◽  
Differential Equations ◽  
Viscous Flow ◽  
Ordinary Differential Equations ◽  
Excellent Agreement ◽  
Arbitrary Shape ◽  
Realistic Model ◽  
Governing Equations ◽  
Surface Geometry ◽  
Shrinking Surface

Viscous flow over a porous and stretching (shrinking) surface of an arbitrary shape is investigated in this article. New dimensions of the modeled problem are explored through the existing mathematical analogies in such a way that it generalizes the classical simulations. The latest principles provide a framework for unification, and the consolidated approach modifies the classical formulations. A realistic model is presented with new features in order to explain variety of previous observations on the said problems. As a result, new and upgraded version of the problem is appeared for all such models. A set of new, unusual, and generalized transformations is formed for the velocity components and similarity variables. The modified transformations are equipped with generalized stretching (shrinking), porous velocities, and surface geometry. The boundary layer governing equations are reduced into a set of ordinary differential equations (ODEs) by using the unification procedure and technique. The set of ODEs has two unknown functions f and g. The modeled equations have five different parameters, which help us to reduce the problem into all previous formulations. The problem is solved analytically and numerically. The current simulation and its solutions are also compared with existing models for specific value of the parameters, and excellent agreement is found between the solutions.

scholarly journals Vibration Suppression Control of a Flexible Gantry Crane System with Varying Rope Length

2019 ◽  
Vol 2019 ◽  
pp. 1-8
Author(s):  
Tung Lam Nguyen ◽  
Trong Hieu Do ◽  
Hong Quang Nguyen
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Numerical Simulations ◽  
Vibration Suppression ◽  
Direct Method ◽  
Equations Of Motion ◽  
Gantry Crane ◽  
Partial Differential ◽  
Control Approach ◽  
Lyapunov’S Direct Method

The paper presents a control approach to a flexible gantry crane system. From Hamilton’s extended principle the equations of motion that characterized coupled transverse-transverse motions with varying rope length of the gantry is obtained. The equations of motion consist of a system of ordinary and partial differential equations. Lyapunov’s direct method is used to derive the control located at the trolley end that can precisely position the gantry payload and minimize vibrations. The designed control is verified through extensive numerical simulations.

scholarly journals A New Dynamic Model of a Two-Wheeled Two-Flexible-Beam Inverted Pendulum Robot

2020 ◽  
Author(s):  
Amin Mehrvarz ◽  
Mohammad Javad Khodaei ◽  
William Clark ◽  
Nader Jalili
Keyword(s):  
Differential Equations ◽  
High Performance ◽  
Inverted Pendulum ◽  
Equations Of Motion ◽  
Flexible Beam ◽  
Governing Equations ◽  
Natural Modes ◽  
Wheeled Inverted Pendulum ◽  
Highly Nonlinear ◽  
New Type

Abstract Inverted pendulums are traditional dynamic problems. If an inverted pendulum is used in a moving cart, a new type of exciting issues will appear. One of these problems is two-wheeled inverted pendulum systems. Because of their small size, high performance in quick driving, and their stability with controller, researchers and engineers are interested in them. In this paper, a new configuration of one specific robot is modeled, and its dynamic behavior is analyzed. The proposed model can move in two directions, and with a proper controller can keep its stability during the operation. In this robot, two cantilever beams are on the two-wheeled base, and they are excited by voltages to the attached piezoelectric actuators. The mathematical model of this system is obtained using the extended Hamilton’s Principle. The results show that the governing equations of motion are highly nonlinear and contain several coupled partial differential equations (PDEs). In order to extract the natural modes of the beams, the undamped, unforced equations of motion and boundary conditions of the beams are used. If a limited number of modes (N1 and N2) are selected for each beam, the coupled PDEs will be changed to N1 + N2 + 5 ordinary differential equations (ODEs). These complex equations are solved numerically, and the natural frequencies of the system are extracted. The system is then simulated in both lateral and horizontal plane movements. The simulation shows that the governing equations are correct, and the system is ready for designing a proper controller. It should be mentioned that in the future works, the derived equations will be validated experimentally, and a suitable control strategy will be applied to the system to make it automated and more applicable.

scholarly journals A Novel Elastic Metamaterial with Multiple Resonators for Vibration Suppression

2021 ◽  
Vol 2021 ◽  
pp. 1-16
Author(s):  
Saman Ahmadi Nooraldinvand ◽  
Hamid M. Sedighi ◽  
Amin Yaghootian
Keyword(s):  
Vibration Suppression ◽  
Equations Of Motion ◽  
Small Mass ◽  
The Other ◽  
Governing Equations ◽  
System Parameters ◽  
Rectangular Frame ◽  
Adams Software ◽  
Linear Spring ◽  
Base Structure

In this paper, two models of elastic metamaterial containing one and two resonators are proposed to obtain the bandgaps with the aim of providing broadband vibration suppression. The model with one DOF is built by assembling several unite cells in which each unite cell consists of a rectangular frame as the base structure and a rack-and-pinion mechanism that is joined to the frame with a linear spring on both sides. In the second model with two DOF, a small mass is added while its center is attached to the center of the pinion on one side and the other side is connected to the rectangular frame via a linear spring. In the first mechanism, the pinion is considered as the single resonator, and in the 2DOF model, on the other hand, the pinion and small mass acted as multiple resonators. By obtaining the governing equations of motion for a single cell in each model, the dynamic behavior of two metastructures is thoroughly investigated. Therefore, the equations of motion for the two models are written in matrix form, and then, the dispersion relations are presented to analyze the influences of system parameters on the bandgaps’ starting/ending frequencies. Finally, two models are successfully compared and then numerically simulated via MATLAB-SIMULINK and MSC-ADAMS software. With the aid of closed-form expressions for starting/ending frequencies, the correlation between the system parameters and bandgap intervals can be readily recognized.

Vibration Suppression in a Pinned-Pinned Nonlinear Rod Using a Frictionless Slider

2012 ◽  
Author(s):  
Tingli Cai ◽  
Ranjan Mukherjee ◽  
Alejandro R. Diaz
Keyword(s):  
Feedback Loop ◽  
Vibration Suppression ◽  
Equilibrium Configuration ◽  
Equations Of Motion ◽  
Lyapunov Stability Theory ◽  
Governing Equations ◽  
Constraint Force ◽  
Negative Work ◽  
Slider Motion ◽  
Principle Of Virtual Displacements

We propose a new method for vibration suppression in a flexible structure using a frictionless sliding constraint. The constraint force applied by the slider is assumed known from measurements and the slider motion is prescribed to do negative work on the structure. The structure is modeled as a two-dimensional nonlinear rod with pinned-pinned boundary conditions and the slider is assumed to constrain the position of one point on the rod but not its slope. The problem is formulated using variable-length finite elements in the framework of Arbitrary Lagrange-Euler (ALE) description. The governing equations of motion are derived using the principle of virtual displacements and D’Alembert’s principle. Numerical simulation results are presented to demonstrate the effectiveness of the control strategy based on the idea of negative work. To meet the bandwidth requirement of the actuator, a nonlinear filter is placed in the feedback loop and asymptotic stability of the equilibrium configuration is established using Lyapunov stability theory.

Dynamics of a Basketball Rolling Around the Rim

2005 ◽  
Vol 128 (2) ◽  
pp. 359-364
Author(s):  
C. Q. Liu ◽  
Fang Li ◽  
R. L. Huston
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Initial Conditions ◽  
Equations Of Motion ◽  
Dynamical Equations ◽  
First Order ◽  
Coaching Strategies ◽  
Ball Motion

Governing dynamical equations of motion for a basketball rolling on the rim of a basket are developed and presented. These equations form a system of five first-order, ordinary differential equations. Given suitable initial conditions, these equations are readily integrated numerically. The results of these integrations predict the success (into the basket) or failure (off the outside of the rim) of the basketball shot. A series of examples are presented. The examples show that minor changes in the initial conditions can produce major changes in the subsequent ball motion. Shooting and coaching strategies are recommended.

Aeroelastic Flutter Analysis of Thick Porous Plates in Supersonic Flow

2019 ◽  
Vol 11 (10) ◽  
pp. 1950096 ◽  
Author(s):  
Reza Bahaadini ◽  
Ali Reza Saidi ◽  
Kazem Majidi-Mozafari
Keyword(s):  
Supersonic Flow ◽  
Plate Theory ◽  
Equations Of Motion ◽  
Functionally Graded ◽  
Governing Equations ◽  
Aerodynamic Pressure ◽  
Aeroelastic Flutter ◽  
Piezoelectric Layers ◽  
Flutter Analysis ◽  
Porosity Coefficient

The aeroelastic flutter analysis of thick porous plates surrounded with piezoelectric layers in supersonic flow is studied. In order to aeroelastic analysis of the thick porous-cellular plate, Reddy’s third-order shear deformation plate theory and first-order piston theory are used. Furthermore, the plate is composed of two face piezoelectric layers and three functionally graded porous distributions core. Applying the extended Hamilton’s principle and Maxwell’s equation, the governing equations of motion are obtained. The partial differential governing equations are transformed into a set of ordinary differential equations by applying Galerkin’s approach. The effects of porosity coefficient, porosity distributions, piezoelectric layers, geometric dimensions, electrical and mechanical boundary conditions on the flutter aerodynamic pressure and natural frequencies of porous-cellular plates are investigated.

scholarly journals Nonlinear Response of Cantilever Beams to Combination and Subcombination Resonances

1998 ◽  
Vol 5 (5-6) ◽  
pp. 277-288 ◽  
Author(s):  
Ali H. Nayfeh ◽  
Haider N. Arafat
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Virtual Work ◽  
Equations Of Motion ◽  
Nonlinear Ordinary Differential Equations ◽  
Response Curves ◽  
First Order ◽  
Aspect Ratios

The nonlinear planar response of cantilever metallic beams to combination parametric and external subcombination resonances is investigated, taking into account the effects of cubic geometric and inertia nonlinearities. The beams considered here are assumed to have large length-to-width aspect ratios and thin rectangular cross sections. Hence, the effects of shear deformations and rotatory inertia are neglected. For the case of combination parametric resonance, a two-mode Galerkin discretization along with Hamilton’s extended principle is used to obtain two second-order nonlinear ordinary-differential equations of motion and associated boundary conditions. Then, the method of multiple scales is applied to obtain a set of four first-order nonlinear ordinary-differential equations governing the modulation of the amplitudes and phases of the two excited modes. For the case of subcombination resonance, the method of multiple scales is applied directly to the Lagrangian and virtual-work term. Then using Hamilton’s extended principle, we obtain a set of four first-order nonlinear ordinary-differential equations governing the amplitudes and phases of the two excited modes. In both cases, the modulation equations are used to generate frequency- and force-response curves. We found that the trivial solution exhibits a jump as it undergoes a subcritical pitchfork bifurcation. Similarly, the nontrivial solutions also exhibit jumps as they undergo saddle-node bifurcations.

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