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1

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Regulation of flexible structures via nonlinear coupling (1991)

Golnaraghi, M. Farid

Springer

Dynamics and control 1 (1991), S. 405-428

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ISSN: 1573-8450

Source: Springer Online Journal Archives 1860-2000

Topics: Electrical Engineering, Measurement and Control Technology

Notes: Abstract In this article, we propose an active/passive vibration controller for a cantilever beam using a sliding mass-spring-dashpot mechanism. The controller is placed at the free end of the beam, introducing Coriolis, inertia, and centripetal nonlinearities into the system, resulting in nonlinear coupling that may be used to quench the transient vibration of the beam. When the natural frequency of the slider is twice the fundamental beam frequency (2:1 internal resonance), the two systems will be coupled through nonlinearities that cause the oscillatory energy to be transferred back and forth between the beam and the slider. Control is achieved once the vibration of the beam is absorbed by the slider and dissipated through the slider damping. Numerical results show that this technique can improve the effective damping ratio of the structure by a factor of 15. This technique is particularly useful for reducing large-amplitude oscillations to levels that may be managed using conventional methods. Due to the nonlinearities in the system, for small or zero controller damping, chaotic transient oscillations can occur depending on the amplitude of the initial disturbance of the beam.

Type of Medium: Electronic Resource

URL: http://dx.doi.org/10.1007/BF02169768

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2

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Regulation of a lumped parameter cantilever beam via internal resonance using nonlinear coupling enhancement (1994)

Golnaraghi, M. Farid ; Tuer, Kevin ; Wang, David

Springer

Dynamics and control 4 (1994), S. 73-96

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ISSN: 1573-8450

Source: Springer Online Journal Archives 1860-2000

Topics: Electrical Engineering, Measurement and Control Technology

Notes: Abstract In this article, we propose a nonlinear control law to enhance the performance of internal resonance (IR) controllers used for the regulation of vibration in flexible structures. Active IR controllers are composed of sliding or rotational actuators attached to the flexible structure, introducing dynamic nonlinearities into the system. The IR control strategy forces a state of internal resonance at which the transfer of oscillatory energy from the structure to the controller mechanism takes place. Once this energy is transferred to the controller it is dissipated via velocity feedback. These controllers are unconventional since they function at a state of resonance. The resonance condition is established upon tuning the controller natural frequency (using position feedback) to twice the fundamental beam frequency which causes the oscillatory energy to be transferred to the controller via the nonlinear coupling terms where it is dissipated. However, the rate of dissipation is limited since the controllers have a limited “critical” damping coefficient. In this article we study the effect of the IR controller on a simplified lumped mass model representing a cantilever beam. The nonlinear enhancement technique proposed in this article improves the performance of this type of regulatory by increasing the nonlinear coupling effect which is responsible for the energy transfer, and hence, speeding the rate of vibration suppression.

Type of Medium: Electronic Resource

URL: http://dx.doi.org/10.1007/BF02115740

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3

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Development of a generalised active vibration suppression strategy for a cantilever beam using internal resonance (1994)

Tuer, K. L. ; Golnaraghi, M. F. ; Wang, D.

Springer

Nonlinear dynamics 5 (1994), S. 131-151

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ISSN: 1573-269X

Keywords: Internal resonance ; vibration suppression ; generalised control

Source: Springer Online Journal Archives 1860-2000

Topics: Mathematics

Notes: Abstract In this paper, the potential to utilise modal coupling effects in the formulation of a generalised vibration suppression algorithm is investigated. The plant, a flexible cantilever beam undergoing first mode oscillation, is modelled by a second order differential equation with a spring constant and damping coefficient that are representative of the first mode flexibility and material damping of the beam, respectively. In order to establish an internal resonance condition, a second equation, designated the supplementary equation or controller, is appended to the plant to render a two-degree-of-freedom system. The objective is to generate an internally resonant pair. Upon successful completion of this task, a suppression technique is implemented whereby energy is removed from the plant via the supplementary system. The introduction of the supplementary system results in a set of design parameters which are employed to realise a state of internal resonance and to establish the desired dynamic response. The choice of 2:1 internal resonance models results in a unidirectional control torque making this technique particularly attractive for systems using thrusters or tendons as actuators. A similar structural configuration regulated under a PD (Proportional-Derivative) control law is compared to the proposed control scheme via simulation.

Type of Medium: Electronic Resource

URL: http://dx.doi.org/10.1007/BF00045672

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4

Electronic Resource

On the dynamic behaviour of a flexible beam carrying a moving mass (1994)

Khalily, F. ; Golnaraghi, M. F. ; Heppler, G. R.

Springer

Nonlinear dynamics 5 (1994), S. 493-513

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ISSN: 1573-269X

Keywords: Beam dynamics ; moving load

Source: Springer Online Journal Archives 1860-2000

Topics: Mathematics

Notes: Abstract The behaviour of a system containing a mass traveling on a cantilever beam is considered. The mass is induced to move by an applied force as opposed to the case which has been considered in most literature where the position of the moving mass is assumed to be known and independent of the motion of the beam. Furthermore, the system to be discussed has the unique characteristic that the motions of the mass and the beam are coupled. The mathematical model of the system includes two coupled nonlinear integral/partial differential equations which are impossible to solve analytically and are difficult to solve numerically in their original form. As a remedy, the solution is discretized into space and time functions and the equations of motion are reduced to a set of ordinary differential equations. The shape function is chosen so that it satisfies the boundary conditions of the beam as well as the transient conditions imposed by the traveling mass. This choice of the shape function, which considers the mass-beam interaction, provides an improvement over the conventional method of using a simple cantilever beam mode shapes. The ordinary differential equations of motion using the ‘improved’ shaped functions, are solved numerically to obtain the dynamic behaviour of the system. The results illustrate the validity of the model, and demonstrate the advantages of the ‘improved’ model to the ‘un-improved’ equations.

Type of Medium: Electronic Resource

URL: http://dx.doi.org/10.1007/BF00052456

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5

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Dynamics of a Flexible Cantilever Beam Carrying a Moving Mass (1998)

Siddiqui, Sultan A. Q. ; Golnaraghi, M. Farid ; Heppler, Glenn R.

Springer

Nonlinear dynamics 15 (1998), S. 137-154

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ISSN: 1573-269X

Keywords: Beam carrying a moving mass ; internal resonance ; kinematic nonlinearities

Source: Springer Online Journal Archives 1860-2000

Topics: Mathematics

Notes: Abstract The motion of a flexible cantilever beam carrying a moving spring-mass system is investigated. The beam is assumed to be an Euler–Bernouli beam. The motion of the system is described by a set of two nonlinear coupled partial differential equations where the coupling terms have to be evaluated at the position of the mass. The nonlinearities arise due to the coupling between the mass and the beam. Due to the nonlinearities the system exhibits internal resonance which is investigated in this work. The equations of motion are solved numerically using the Rayleigh–Ritz method and an automatic ODE solver. An approximate solution using the perturbation method of multiple scales is also obtained.

Type of Medium: Electronic Resource

URL: http://dx.doi.org/10.1023/A:1008205904691

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6

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A Theoretical and Experimental Implementation of a Control Method Based on Saturation (1997)

Oueini, Shafic S. ; Nayfeh, Ali H. ; Golnaraghi, M. Farid

Springer

Nonlinear dynamics 13 (1997), S. 189-202

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ISSN: 1573-269X

Keywords: Internal resonance ; saturation ; control ; energy transfer

Source: Springer Online Journal Archives 1860-2000

Topics: Mathematics

Notes: Abstract A novel approach for implementing an active nonlinear vibration absorber is presented. The absorber, which is built in electronic circuitry, takes advantage of the saturation phenomenon that occurs when two natural frequencies of a system with quadratic nonlinearities are in the ratio of two-to-one. When the system is excited at a frequency near the higher natural frequency, there is a small ceiling for the system response at the higher frequency and the rest of the input energy is channeled to the low-frequency mode. A working model of using saturation to suppress the vibrations of a rigid beam connected to a DC motor has been built. An electronic oscillator is built, and its frequency is set at one-half the frequency of the beam. The output from a sensor on the beam is multiplied by the output from the electronic oscillator and a suitable gain, and the result is used as the forcing term for the oscillator. At the same time, the output from the oscillator is squared and multiplied by a suitable gain, and that result is used as the input to the motor. The oscillator/actuator and the beam act as the two modes of a two-degree-of-freedom quadratically coupled system with a 2:1 autoparametric resonance. When the beam is excited by a harmonic force, its motion quickly becomes saturated, and most of the energy imparted to the beam by the harmonic force is transferred to the electronic circuit and from there to the actuator. Thus, the harmonic force is made to work against itself. As a result, the motion of the beam always remains small.

Type of Medium: Electronic Resource

URL: http://dx.doi.org/10.1023/A:1008207124935

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