Optimization in Bifurcation Problems using a Continuation Method

The performance of the cantilever beam autoparametric vibration absorber with a lumped mass attached at an arbitrary point on the beam span is investigated. The absorber would have a distinct feature that in addition to the two-to-one internal resonance, the one-to-three and one-to-five internal resonances would also occur between flexural modes of the beam by tuning the mass and position of the lumped mass. Special attention is paid on studying the effect of these resonances on increasing the effectiveness and extending the range of excitation amplitudes at which the autoparametric vibration absorber remains effective. The problem is formulated based on the third-order nonlinear Euler–Bernoulli beam theory, where the assumed-mode method is used for deriving the discretized equations of motion. The numerical continuation method is then applied to obtain the frequency response curves and detect the bifurcation points. The harmonic balance method is also employed for detecting the type of internal resonances between flexural modes by inspecting the frequency response curves corresponding to different harmonics of the response. Parametric studies on the performance of the absorber are conducted by varying the position and mass of the lumped mass, while the frequency ratio of the primary system to the first mode of the beam is kept equal to two. Results indicated that the one-to-five internal resonance is especially responsible for the considerable enhancement of the performance.

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Asymmetric and chaotic responses of dry friction oscillators with different static and kinetic coefficients of friction

Meccanica ◽

10.1007/s11012-021-01382-8 ◽

2021 ◽

Author(s):

Gábor Csernák ◽

Gábor Licskó

Keyword(s):

Dry Friction ◽

Continuation Method ◽

Friction Model ◽

Lugre Friction Model ◽

Kinetic Coefficients ◽

Friction Oscillator ◽

Lugre Friction ◽

Friction Parameters ◽

Two Parameter ◽

Parameter Bifurcation

AbstractThe responses of a simple harmonically excited dry friction oscillator are analysed in the case when the coefficients of static and kinetic coefficients of friction are different. One- and two-parameter bifurcation curves are determined at suitable parameters by continuation method and the largest Lyapunov exponents of the obtained solutions are estimated. It is shown that chaotic solutions can occur in broad parameter domains—even at realistic friction parameters—that are tightly enclosed by well-defined two-parameter bifurcation curves. The performed analysis also reveals that chaotic trajectories are bifurcating from special asymmetric solutions. To check the robustness of the qualitative results, characteristic bifurcation branches of two slightly modified oscillators are also determined: one with a higher harmonic in the excitation, and another one where Coulomb friction is exchanged by a corresponding LuGre friction model. The qualitative agreement of the diagrams supports the validity of the results.

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Determination of catastrophic sets of a tubular chemical reactor by two-parameter continuation method

International Journal of Chemical Reactor Engineering ◽

10.1515/ijcre-2020-0135 ◽

2020 ◽

Vol 0 (0) ◽

Author(s):

Marek Berezowski

Keyword(s):

Continuation Method ◽

Extreme Points ◽

Chemical Reactor ◽

Stationary States ◽

Parameter Continuation Method ◽

It Systems ◽

Parametric Continuation ◽

Parameter Continuation ◽

Two Parameter

AbstractThe work relates to development and presentation a two-parameter continuation method for determining catastrophic sets of stationary states of a tubular chemical reactor with mass recycle. The catastrophic set is a set of extreme points occurring in the bifurcation diagrams of the reactor. There are many large IT systems that use the parametric continuation method. The most popular is AUTO’97. However, its use is sometimes not convenient. The method developed in this work allows to eliminate the necessity to use huge IT systems from the calculations. Unlike these systems, it can be inserted into the program as a short subroutine. In addition, this method eliminates time-consuming iterations from the calculations.

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The effects of nonlinear energy sink and piezoelectric energy harvester on aeroelastic instability of an airfoil

Journal of Vibration and Control ◽

10.1177/1077546321993585 ◽

2021 ◽

pp. 107754632199358

Author(s):

Ali Fasihi ◽

Majid Shahgholi ◽

Saeed Ghahremani

Keyword(s):

Energy Harvester ◽

Equations Of Motion ◽

Continuation Method ◽

Dynamical Behavior ◽

Harmonic Balance Method ◽

Nonlinear Energy Sink ◽

Piezoelectric Energy ◽

Energy Sink ◽

Two Degree Of Freedom ◽

Nonlinear Energy

The potential of absorbing and harvesting energy from a two-degree-of-freedom airfoil using an attachment of a nonlinear energy sink and a piezoelectric energy harvester is investigated. The equations of motion of the airfoil coupled with the attachment are solved using the harmonic balance method. Solutions obtained by this method are compared to the numerical ones of the pseudo-arclength continuation method. The effects of parameters of the integrated nonlinear energy sink-piezoelectric attachment, namely, the attachment location, nonlinear energy sink mass, nonlinear energy sink damping, and nonlinear energy sink stiffness on the dynamical behavior of the airfoil system are studied for both subcritical and supercritical Hopf bifurcation cases. Analyses demonstrate that absorbing vibration and harvesting energy are profoundly affected by the nonlinear energy sink parameters and the location of the attachment.

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