Delayed-Acceleration Feedback for Active-Multimode Vibration Control of Cantilever Beams

2007 ◽  
Author(s):  
Khaled A. Alhazza ◽  
Ali H. Nayfeh ◽  
Mohammed F. Daqaq
Keyword(s):  
Cantilever Beam ◽  
Controller Design ◽  
Equations Of Motion ◽  
Time Integration ◽  
Free Vibrations ◽  
Delayed Feedback Control ◽  
Vibration Modes ◽  
Stability Boundaries ◽  
Single Input Single Output ◽  
The Stability

We present a single-input single-output multimode delayed-feedback control methodology to mitigate the free vibrations of a flexible cantilever beam. For the purpose of controller design and stability analysis, we consider a reduced-order model consisting of the first n vibration modes. The temporal variation of these modes is represented by a set of nonlinearly-coupled ordinary-differential equations that capture the evolving dynamics of the beam. Considering a linearized version of these equations, we derive a set of analytical conditions that are solved numerically to assess the stability of the closed-loop system. To verify these conditions, we characterize the stability boundaries using the first two vibration modes and compare them to damping contours obtained by long-time integration of the full nonlinear equations of motion. Simulations show excellent agreement between both approaches. We analyze the effect of the size and location of the piezoelectric patch and the location of the sensor on the stability of the response. We show that the stability boundaries are highly dependent on these parameters. Finally, we implement the controller on a cantilever beam for different controller gain-delay combinations and assess the performance using time histories of the beam response. Numerical simulations clearly demonstrate the controller ability to mitigate vibrations emanating from multiple modes simultaneously.

scholarly journals Free Vibrations of a Cantilevered SWCNT with Distributed Mass in the Presence of Nonlocal Effect

2015 ◽  
Vol 2015 ◽  
pp. 1-12 ◽  
Author(s):  
M. A. De Rosa ◽  
M. Lippiello ◽  
H. D. Martin
Keyword(s):  
Nonlocal Elasticity ◽  
Equations Of Motion ◽  
Nonlocal Elasticity Theory ◽  
Added Mass ◽  
Free Vibrations ◽  
Point Of View ◽  
Vibration Modes ◽  
Nonlocal Effects ◽  
Single Walled Carbon Nanotube ◽  
Parametric Relationships

The Hamilton principle is applied to deduce the free vibration frequencies of a cantilever single-walled carbon nanotube (SWCNT) in the presence of an added mass, which can be distributed along an arbitrary part of the span. The nonlocal elasticity theory by Eringen has been employed, in order to take into account the nanoscale effects. An exact formulation leads to the equations of motion, which can be solved to give the frequencies and the corresponding vibration modes. Moreover, two approximate semianalytical methods are also illustrated, which can provide quick parametric relationships. From a more practical point of view, the problem of detecting the mass of the attached particle has been solved by calculating the relative frequency shift due to the presence of the added mass: from it, the mass value can be easily deduced. The paper ends with some numerical examples, in which the nonlocal effects are thoroughly investigated.

Nonlinear Planar Dynamics of Fluid-Conveying Cantilevered Extensible Pipes

2013 ◽  
Author(s):  
Mergen H. Ghayesh ◽  
Michael P. Païdoussis ◽  
Marco Amabili
Keyword(s):  
Flow Velocity ◽  
Trivial Solution ◽  
Equations Of Motion ◽  
Continuation Method ◽  
Time Integration ◽  
Axial Direction ◽  
Transverse Displacement ◽  
Lagrange Equations ◽  
Critical Flow Velocity ◽  
The Stability

This paper for the first time investigates the nonlinear planar dynamics of a cantilevered extensible pipe conveying fluid; the centreline of the pipe is considered to be extensible resulting in coupled longitudinal and transverse equations of motion; specifically, the kinetic and potential energies are obtained in terms of longitudinal and transverse displacements and then the extended version of the Lagrange equations for systems containing non-material volumes is employed to derive the equations of motion. Direct time integration along with the pseudo-arclength continuation method are employed to solve the discretized equations of motion. Bifurcation diagrams of the system are constructed as the flow velocity is increased as the bifurcation parameter. As opposed to the case of an inextensible pipe, an extensible pipe elongates in the axial direction as the flow velocity is increased from zero. At the critical flow velocity, the stability of the system is lost via a supercritical Hopf bifurcation, emerging from the trivial solution for the transverse displacement and non-trivial solution for the longitudinal displacement and leading to a flutter.

scholarly journals Dynamic Modelling And Predictive Control For Insect-Like Flapping Wing Aerial Micro Robots

2021 ◽  
Author(s):  
Alborz Sakhaei
Keyword(s):  
Predictive Control ◽  
Controller Design ◽  
Equations Of Motion ◽  
Dynamic Modelling ◽  
Scale Insect ◽  
Flapping Wing ◽  
Confined Space ◽  
Simulation Framework ◽  
The Stability ◽  
Technological Limitations

The outstanding potential capability of flapping-wing aerial micro robots to perform gamut [sic] of applications ranging from indoor and confined space missions to perilous environment explorations elevates them from conventional fixed and rotary wing micro aerial vehicles. Despite the remarkable progress in development of manufacturing paradigms to fabricate an at-scale insect-like aerial micro robot, the existing methods are still incompetent to mimic even the most basic maneuvers [sic] of the flying insects. This incompetency comes from technological limitations in terms of size and power density as well as lack of thorough insight into the complex neuromuscular actuation mechanism of the insects' wing. These limitations raise the motivation to develop a simulation framework to be used to analyze the stability and flight dynamics of the insect-like aerial micro robots, and provide a means by which the controller design for these systems could be accomplished. This thesis describes the development of such simulation framework in the context of dynamic modelling and controller design. A consistent set of dynamic and kinematic equations of motion are developed, and the application of the model predictive control strategy for insect-like flapping wing aerial micro robots is investsigated.

Vibrations of an Elastically Mounted Spinning Rotor Partially Filled With Liquid

1982 ◽  
Vol 104 (2) ◽  
pp. 389-396 ◽  
Author(s):  
G. Lichtenberg
Keyword(s):  
Equations Of Motion ◽  
Constant Angular Velocity ◽  
Two Dimensional ◽  
Field Equations ◽  
Stability Boundaries ◽  
Inviscid Incompressible Fluid ◽  
Wide Range ◽  
The Stability ◽  
Elastically Supported ◽  
Dimensional Surface

The stability of a rotor with a cylindrical cavity, spinning with constant angular velocity and partially filled with an inviscid, incompressible fluid is studied. The rotor is elastically supported on a vertically mounted massless shaft in overhung position. A set of coupled linearized spatial equations of motion of the rotor and field equations, as well as boundary conditions of the liquid, is established and solved, leading to a characteristic equation. First numerical results predict a wide range of rotor speeds, where the system performs unstable motions caused by a two-dimensional surface wave of the liquid. The stability boundaries are calculated for a flat rotor in dependence on the mass of the contained liquid and agree extremely well with experimental data.

On the Stability of PD Control for a Two-Link Rigid-Flexible Manipulator

1994 ◽  
Vol 116 (2) ◽  
pp. 208-215 ◽  
Author(s):  
Ahmet S. Yigit
Keyword(s):  
Controller Design ◽  
Equations Of Motion ◽  
Flexible Manipulator ◽  
Simulation Studies ◽  
Parameter Uncertainties ◽  
Pd Control ◽  
System Parameters ◽  
Flexible System ◽  
The Stability

Controller design for a rigid-flexible two-link manipulator is considered. Robustness of independent joint PD control is investigated. It has been shown that the stability of independent joint PD control does not depend explicitly on the system parameters. No discretization or linearization of the equations of motion is required to assure the stability. Simulation studies also show that independent joint PD control gives reasonably good results for the flexible system, and is robust to parameter uncertainties.

Instability Threshold and Stability Boundaries of Rotor-Bearing Systems

1996 ◽  
Vol 118 (1) ◽  
pp. 115-121 ◽  
Author(s):  
W. J. Chen
Keyword(s):  
High Speed ◽  
Equations Of Motion ◽  
Optimization Techniques ◽  
Instability Threshold ◽  
System Stability ◽  
Stability Boundaries ◽  
Practical Applications ◽  
Design Variables ◽  
Rotor Bearing ◽  
The Stability

A direct numerical method for the determination of instability threshold and stability boundaries of flexible rotor-bearing systems is presented. The proposed procedure can also be used to improve the system stability by considering the design variables as operating parameters. The finite element method is utilized in the formulation of system equations of motion. The numerical algorithm is based on nonlinear optimization techniques. Two examples are presented to illustrate the feasibility, desirability, and ability of the proposed algorithm. A simple journal bearing system is used for the parametric study. An industrial high-speed compressor is employed to demonstrate the ability of this algorithm to deal with practical applications. The stability boundaries calculated from this algorithm are in agreement with the experimental results.

Multimode velocity-delayed feedback vibration control of plates using a single sensor and a single actuator

2020 ◽  
pp. 107754632096234
Author(s):  
Majed A Majeed ◽  
Khaled Alhazza ◽  
Emad Khorshid
Keyword(s):  
Feedback Control ◽  
Equations Of Motion ◽  
Linear Equations ◽  
Optimization Method ◽  
Delayed Feedback ◽  
Delayed Feedback Control ◽  
Structural Vibrations ◽  
Single Input Single Output ◽  
Coupled Equations ◽  
Single Sensor

Controlling multimode vibrations using a single actuator and a single sensor is challenging. Most researchers use multiactuators and multisensors to control multimode structural vibrations. In the present work, a multimode delayed feedback control using a single actuator and a single sensor, both attached at the top surface of a simply supported thin plate, is developed. The linear equations of motion of the plate are derived and then discretized using Galerkin’s method. The resulting coupled equations are controlled with a velocity-delay feedback control to mitigate multimode structural vibrations. A sensed accelerometer signal is integrated and then filtered to include only the effect of the targeted vibration frequency. A global optimization method is then used by minimizing the root mean square of the total controlled response of the system. Many parameters, such as size, location, and orientation of the sensor/actuator as well as time delay and controller gain, play an essential role in the controller performance. The results showed that the proposed velocity delay feedback controller was efficiently used to control multimode vibrations using a single sensor and a single actuator. The proposed single-input single-output controller is also capable of focusing on a given vibration mode rather than treating them all equally.

scholarly journals Aircraft Engine Structural Rotor-Stator Modal Interaction

2005 ◽  
Author(s):  
Mathias Legrand ◽  
Se´bastien Roques ◽  
Bernard Peseux ◽  
Christophe Pierre
Keyword(s):  
Equations Of Motion ◽  
Time Integration ◽  
Aircraft Engine ◽  
Travelling Wave ◽  
Integration Scheme ◽  
Frequency Domain Method ◽  
Vibration Modes ◽  
Jet Engines ◽  
Bladed Disk ◽  
Time Integration Scheme

In modern turbo machines such as aircraft jet engines, contact between the casing and bladed disk may occur through a variety of mechanisms: coincidence of vibration modes, thermal deformation of the casing, rotor imbalance, etc. These nonlinear interactions may result in severe damage to both structures and it is important to understand the physical mechanisms that cause them and the circumstances under which they occur. In this study, we focus on the phenomenon of interaction caused by modal coincidence. A simple two-dimensional model of the casing and bladed disk structures is introduced in order to predict the occurrence of the interaction phenomenon versus the rotation speed of the rotor. Each structure is represented in terms of its two k-nodal diameter vibration modes, which are characteristic of axi-symmetric structures and allow for travelling wave motions that may interact through direct contact. The equations of motion are solved using an explicit time integration scheme in conjunction with the Lagrange multiplier method where friction is considered. Results of the numerical tool and theory show good agreement in the prediction of rotational speed to be avoided. To conclude, the mathematical statements of a multi-frequency domain-method are proposed. This method is to be used to circumvent numerical issues inherent to time-marching procedures.

Stability of an Axially Accelerating String Subjected to Frictional Guiding-Forces

2005 ◽  
Author(s):  
Giampaolo Zen ◽  
Sinan Mu¨ftu¨
Keyword(s):  
Floquet Theory ◽  
Equations Of Motion ◽  
Time Integration ◽  
Time History ◽  
The Finite Element Method ◽  
Initial Tension ◽  
Axial Acceleration ◽  
The Stability ◽  
Rigid Supports ◽  
Adaptive Step

The dynamic response of an axially translating continuum subjected to the combined effects of a pair of spring supported frictional guides and axial acceleration is investigated; such systems are both non-conservative and gyroscopic. The continuum is modeled as a tensioned string translating between two rigid supports with a time dependent velocity profile. The equations of motion are derived with the extended Hamilton’s principle and discretized in the space domain with the finite element method. The stability of the system is analyzed with the Floquet theory for cases where the transport velocity is a periodic function of time. Direct time integration using an adaptive step Runge-Kutta algorithm is used to verify the results of the Floquet theory. Results are given in the form of time history diagrams and instability point grids for different sets of parameters such as the location of the stationary load, the stiffness of the elastic support, and the values of initial tension. This work showed that presence of friction adversely affects stability, but using non-zero spring stiffness on the guiding force has a stabilizing effect.

Modeling and Stability Analysis of an Axially Moving Beam With Frictional Contact

2008 ◽  
Vol 75 (3) ◽  
Author(s):  
Gottfried Spelsberg-Korspeter ◽  
Oleg N. Kirillov ◽  
Peter Hagedorn
Keyword(s):  
Stability Analysis ◽  
Frictional Contact ◽  
Equations Of Motion ◽  
Axially Moving Beam ◽  
Perturbation Techniques ◽  
Stability Boundaries ◽  
Analytical Approximations ◽  
Moving Beam ◽  
Axially Moving ◽  
The Stability

This paper considers a moving beam in frictional contact with pads, making the system susceptible for self-excited vibrations. The equations of motion are derived and a stability analysis is performed using perturbation techniques yielding analytical approximations to the stability boundaries. Special attention is given to the interaction of the beam and the rod equations. The mechanism yielding self-excited vibrations does not only occur in moving beams, but also in other moving continua such as rotating plates, for example.

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