Suppressing the nonlinear vibrations of a compressor blade via a nonlinear saturation controller

2016 ◽  
Vol 24 (8) ◽  
pp. 1488-1504 ◽  
Author(s):  
Ali Kandil ◽  
Hany A El-Gohary
Keyword(s):  
Natural Frequency ◽  
Multiple Scales ◽  
Perturbation Technique ◽  
Time History ◽  
Nonlinear Saturation ◽  
State Equations ◽  
Response Curves ◽  
Rotating Speed ◽  
Main System ◽  
Nonlinear Saturation Controller

A nonlinear saturation controller (NSC) is applied in this work to reduce the oscillations of a rotating blade dynamical system running at unsteady rotating speed. The controller is coupled quadratically to the main system by designing its natural frequency to be one half of the main system natural frequency. This is done to setup an energy bridge between them to make use of saturation phenomenon. That phenomenon is advantageous when the excitation force increases; the whole energy in the main system is channeled to the controller. The two system modes of vibrations are found to be linearly coupled powerfully, so the controller is applied only to the first mode and, consequently, the second mode tracks it. The multiple scales perturbation technique (MSPT) is adopted to derive the steady state equations that describe the modulations of amplitudes and phases of the system before and after control. Then, a stability analysis is achieved via Lyapunov’s indirect method to determine the stable and unstable solutions depending on the real parts of the Jacobian matrix eigenvalues. Time history and different response curves of the controlled system are included for showing the controller effect. Eventually, validation curves and comparison with previously published work are included.

scholarly journals Forced Vibration in Cutting Process considering the Nonlinear Curvature and Inertia of a Rotating Composite Cutter Bar

2020 ◽  
Vol 2020 ◽  
pp. 1-19
Author(s):  
Yongsheng Ren ◽  
Donghui Yao
Keyword(s):  
Cutting Force ◽  
Forced Vibration ◽  
Multiple Scales ◽  
Damping Ratio ◽  
Three Dimensional ◽  
Forced Response ◽  
Cutting Process ◽  
Response Curves ◽  
Rotating Speed ◽  
Cutting System

Forced vibration of the cutting system with a three-dimensional composite cutter bar is investigated. The composite cutter bar is simplified as a rotating cantilever shaft which is subjected to a cutting force including regenerative delay effects and harmonic exciting items. The nonlinear curvature and inertia of the cutter bar are taken into account based on inextensible assumption. The effects of the moment of inertia, gyroscopic effect, and internal and external damping are also considered, but shear deformation is neglected. Equation of motion is derived based on Hamiltonʼs extended principle and discretized by the Galerkin method. The analytical solutions of the steady-state response of the cutting system are constructed by using the method of multiple scales. The response of the cutting system is studied for primary and superharmonic resonances. The effects of length-to-diameter ratio, damping ratio, cutting force coefficients, ply angle, rotating speed, and internal and external damping are investigated. The results show that nonlinear curvature and inertia imposed a significant effect on the dynamic behavior of the cutting process. The equivalent nonlinearity of the cutting system shows hard spring characteristics. Multiple solutions and jumping phenomenon of typical Duffing system are found in forced response curves.

Nonlinear free vibrations analysis of overhung rotors under the influence of gravity

2019 ◽  
Vol 234 (2) ◽  
pp. 575-588
Author(s):  
M Moradi Tiaki ◽  
SAA Hosseini ◽  
H Shaban Ali Nezhad
Keyword(s):  
Natural Frequency ◽  
High Frequency ◽  
Multiple Scales ◽  
Perturbation Technique ◽  
Equations Of Motion ◽  
Beat Frequency ◽  
Dynamical Behavior ◽  
Free Vibrations ◽  
Beam Model ◽  
Rayleigh Beam

In this paper, nonlinear free vibration of a cantilever flexible shaft carrying a rigid disk at its free end (overhung rotor) is investigated. The Rayleigh beam model is used and the rotor has large amplitude vibrations. With the assumption of inextensibility, the effect of nonlinear curvature and inertia is considered. The effect of disk mass on the dynamical behavior of the system is studied in the presence and absence of gravity (horizontal and vertical rotors). By using perturbation technique (method of multiple scales), the main focus is on the influence of gravity on equations of motion and on quantities such as amplitude and damped natural frequency. Here, a different behavior is observed due to the rotor weight. Indeed, the combination effects of gyroscopic term, nonlinearity and gravity are studied on the modal behavior of the system. It is shown that the static deflection creates second order nonlinear terms and changes the nonlinear damped natural frequency. With considering of gravity, both beat and high frequency in beat phenomenon increase. With increasing of the rotor weight, the minimum value of amplitude is extremely amplified in the direction of gravity but in the other transverse direction, amplitude of vibrations decreases. In addition, it is found that the weight has directly influence on beat frequency, while the mass ratio between disk and beam affects the high frequency.

scholarly journals Hybrid rayleigh–van der pol–duffing oscillator: Stability analysis and controller

2021 ◽  
pp. 146134842110264
Author(s):  
Chun-Hui He ◽  
Dan Tian ◽  
Galal M Moatimid ◽  
Hala F Salman ◽  
Marwa H Zekry
Keyword(s):  
Fixed Points ◽  
Multiple Scales ◽  
Duffing Oscillator ◽  
Primary Resonance ◽  
Time History ◽  
Phase Portraits ◽  
Response Curves ◽  
Van Der Pol ◽  
Stability Performance ◽  
Linearized Stability

The current study examines the hybrid Rayleigh–Van der Pol–Duffing oscillator (HRVD) with a cubic–quintic nonlinear term and an external excited force. The Poincaré–Lindstedt technique is adapted to attain an approximate bounded solution. A comparison between the approximate solution with the fourth-order Runge–Kutta method (RK4) shows a good matching. In case of the autonomous system, the linearized stability approach is employed to realize the stability performance near fixed points. The phase portraits are plotted to visualize the behavior of HRVD around their fixed points. The multiple scales method, along with a nonlinear integrated positive position feedback (NIPPF) controller, is employed to minimize the vibrations of the excited force. Optimal conditions of the operation system and frequency response curves (FRCs) are discussed at different values of the controller and the system parameters. The system is scrutinized numerically and graphically before and after providing the controller at the primary resonance case. The MATLAB program is employed to simulate the effectiveness of different parameters and the controller on the system. The calculations showed that NIPPF is the best controller. The validations of time history and FRC of the analysis as well as the numerical results are satisfied by making a comparison among them.

Vibration Suppression of a Four-Degrees-of-Freedom Nonlinear Spring Pendulum via Longitudinal and Transverse Absorbers

2011 ◽  
Vol 79 (1) ◽  
Author(s):  
M. Eissa ◽  
M. Kamel ◽  
A. T. El-Sayed
Keyword(s):  
Degrees Of Freedom ◽  
Vibration Suppression ◽  
Perturbation Technique ◽  
Time History ◽  
Multiple Scale ◽  
Threshold Value ◽  
Response Curves ◽  
Pendulum System ◽  
Nonlinear Spring ◽  
The Stability

An investigation into the passive vibration reduction of the nonlinear spring pendulum system, simulating the ship roll motion is presented. This leads to a four-degree-of-freedom (4-DOF) system subjected to multiparametric excitation forces. The two absorbers in the longitudinal and transverse directions are usually designed to control the vibration near the simultaneous subharmonic and internal resonance where system damage is probable. The theoretical results are obtained by applying the multiple scale perturbation technique (MSPT). The stability of the obtained nonlinear solution is studied and solved numerically. The obtained results from the frequency response curves confirmed the numerical results which were obtained using time history. For validity, the numerical solution is compared with the analytical solution. Effectiveness of the absorbers (Ea) are about 13 000 for the first mode (x) and 10 000 for the second mode (ϕ). A threshold value of linear damping coefficient can be used directly for vibration suppression of both vibration modes. Comparison with the available published work is reported.

Nonlinear vibration control of a horizontally supported Jeffcott-rotor system

2017 ◽  
Vol 24 (24) ◽  
pp. 5898-5921 ◽  
Author(s):  
M Eissa ◽  
NA Saeed
Keyword(s):  
Multiple Scales ◽  
Perturbation Technique ◽  
Rotor System ◽  
Response Curves ◽  
Restoring Force ◽  
System A ◽  
Jeffcott Rotor ◽  
Before And After ◽  
System Frequency ◽  
Multiple Scales Perturbation Technique

A positive position feedback (PPF) controller is proposed to control the nonlinear vibrations of a horizontally supported Jeffcott-rotor system. A nonlinear restoring force and the rotor weight are considered in the system model. The controller is coupled to the system with 1:1 internal resonance. A second order approximate solution to the system governing equations is constructed by applying the multiple scales perturbation technique (MSPT). The bifurcation analyses of the Jeffcott-rotor system before and after control are conducted. The effects of the different controller parameters on the system frequency–response curves are investigated. Optimum working conditions of the controlled system are extracted to be used in the design of such systems. Numerical simulations demonstrated a good agreement with the approximate results that obtained by MSPT. A comparison is provided with already published work.

Two-frequency parametric excitation and combination resonance of a nanocomposite laminated piezoelectric trapezoidal plate in subsonic airflow

2020 ◽  
pp. 095440622097543
Author(s):  
Marziye Noroozi ◽  
Firooz Bakhtiari-Nejad ◽  
Morteza Dardel
Keyword(s):  
Parametric Resonance ◽  
Multiple Scales ◽  
Geometrical Nonlinearity ◽  
Time History ◽  
Dynamic Responses ◽  
Linear Potential ◽  
Response Curves ◽  
Aerodynamic Pressure ◽  
Averaged Equations ◽  
The Difference

In this study, an analytical approach is presented to analyze the bifurcations and nonlinear dynamics of a cantilevered piezoelectric nanocomposite trapezoidal actuator subjected to two-frequency parametric excitations in the presence of subsonic airflow. The assumption of uniformly distributed single-walled carbon nanotubes along the thickness is taken into the consideration. The governing equations are built by the von-Karman nonlinear strain-displacement relations to consider the geometrical nonlinearity and the linear potential flow theory. The present study focuses on a specific resonance case deals with the occurrence of simultaneous resonances in the principal parametric resonance of the first mode and combination of the parametric resonance of the difference type involving two modes. The multiple scales method is employed to obtain the four nonlinear averaged equations which are solved by using the Runge-Kutta method. Moreover, the frequency-response curves, bifurcation diagrams, time history responses, and phase portrait are obtained to find the nonlinear dynamic responses of the plate. The effects of the amplitude of piezoelectric excitation, piezoelectric detuning parameter, and aerodynamic pressure are also studied. The results indicate that, the chaotic, quasi-periodic and periodic motions of the plate exist under certain conditions and the variation of controlling parameters can change the form of motions of the nanocomposite piezoelectric trapezoidal thin plate.

scholarly journals Vibration Control of a Compressor Blade Using Position and Velocity Feedback

2019 ◽  
Vol 24 (No 1) ◽  
Author(s):  
Ali Kandil ◽  
Magdy Kamel
Keyword(s):  
Primary Resonance ◽  
Time History ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Nonlinear Behaviour ◽  
Response Curves ◽  
Velocity Feedback ◽  
Rotating Speed ◽  
Coupled Mode ◽  
Different Response

Position and velocity feedback controllers are applied in this work to reduce the oscillations of a rotating blade dynamical system running at an unsteady rotating speed. Both the primary resonance and the principal parametric resonance are controlled as they are the worst cases that were verified numerically. The two modes of vibrations are found to be powerfully linearly coupled, so we have applied the controller to only one mode and the other, coupled mode follows it. The overall nonlinear behaviour of the system with and without control is investigated through the multiple time scales method. Time history and different response curves of the controlled system are included to show the controller effect.

scholarly journals Influence of Time Delay on Controlling the Non-Linear Oscillations of a Rotating Blade

Symmetry ◽  
2021 ◽  
Vol 13 (1) ◽  
pp. 85
Author(s):  
Yasser Salah Hamed ◽  
Ali Kandil
Keyword(s):  
Time Delay ◽  
Natural Frequency ◽  
Multiple Scales ◽  
Control Process ◽  
Control Force ◽  
Specific Level ◽  
Loop Control ◽  
Positive Displacement ◽  
Non Linear ◽  
Linear Vibrations

Time delay is an obstacle in the way of actively controlling non-linear vibrations. In this paper, a rotating blade’s non-linear oscillations are reduced via a time-delayed non-linear saturation controller (NSC). This controller is excited by a positive displacement signal measured from the sensors on the blade, and its output is the suitable control force applied onto the actuators on the blade driving it to the desired minimum vibratory level. Based on the saturation phenomenon, the blade vibrations can be saturated at a specific level while the rest of the energy is transferred to the controller. This can be done by adjusting the controller natural frequency to be one half of the blade natural frequency. The whole behavior is governed by a system of first-order differential equations gained by the method of multiple scales. Different responses are included to show the influences of time delay on the closed-loop control process. Also, a good agreement can be noticed between the analytical curves and the numerically simulated ones.

Study of plate vibrating in fluids having part-through crack at random angles and locations

2021 ◽  
pp. 095440622110127
Author(s):  
Ruqia Ikram ◽  
Asif Israr
Keyword(s):  
Frequency Response ◽  
Multiple Scales ◽  
Peak Amplitude ◽  
Nonlinear Frequency ◽  
Response Curves ◽  
Crack Location ◽  
Cracked Plate ◽  
Nonlinear Frequency Response ◽  
Angle Crack ◽  
Crack Angle

This study presents the vibration characteristics of plate with part-through crack at random angles and locations in fluid. An experimental setup was designed and a series of tests were performed for plates submerged in fluid having cracks at selected angles and locations. However, it was not possible to study these characteristics for all possible crack angles and crack locations throughout the plate dimensions at any fluid level. Therefore, an analytical study is also carried out for plate having horizontal cracks submerged in fluid by adding the influence of crack angle and crack location. The effect of crack angle is incorporated into plate equation by adding bending and twisting moments, and in-plane forces that are applied due to antisymmetric loading, while the influence of crack location is also added in terms of compliance coefficients. Galerkin’s method is applied to get time dependent modal coordinate system. The method of multiple scales is used to find the frequency response and peak amplitude of submerged cracked plate. The analytical model is validated from literature for the horizontally cracked plate submerged in fluid as according to the best of the authors’ knowledge, literature lacks in results for plate with crack at random angle and location in the presence of fluid following validation with experimental results. The combined effect of crack angle, crack location and fluid on the natural frequencies and peak amplitude are investigated in detail. Phenomenon of bending hardening or softening is also observed for different boundary conditions using nonlinear frequency response curves.

The Static and Dynamic Behavior of MEMS Arches Under Electrostatic Actuation

2009 ◽  
Author(s):  
Hassen M. Ouakad ◽  
Mohammad I. Younis ◽  
Fadi M. Alsaleem ◽  
Ronald Miles ◽  
Weili Cui
Keyword(s):  
Multiple Scales ◽  
Perturbation Technique ◽  
Dynamical Behavior ◽  
Primary Resonance ◽  
Reduced Order Model ◽  
Harmonic Load ◽  
Order Model ◽  
Reduced Order ◽  
Electrostatically Actuated ◽  
Model Equations

In this paper, we investigate theoretically and experimentally the static and dynamic behaviors of electrostatically actuated clamped-clamped micromachined arches when excited by a DC load superimposed to an AC harmonic load. A Galerkin based reduced-order model is used to discretize the distributed-parameter model of the considered shallow arch. The natural frequencies of the arch are calculated for various values of DC voltages and initial rises of the arch. The forced vibration response of the arch to a combined DC and AC harmonic load is determined when excited near its fundamental natural frequency. For small DC and AC loads, a perturbation technique (the method of multiple scales) is also used. For large DC and AC, the reduced-order model equations are integrated numerically with time to get the arch dynamic response. The results show various nonlinear scenarios of transitions to snap-through and dynamic pull-in. The effect of rise is shown to have significant effect on the dynamical behavior of the MEMS arch. Experimental work is conducted to test polysilicon curved microbeam when excited by DC and AC loads. Experimental results on primary resonance and dynamic pull-in are shown and compared with the theoretical results.

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