Nonlinear Tuned Vibration Absorber Research Articles

Multi-parametric optimization for controlling bifurcation structures

Bifurcations organize the dynamics of many natural and engineered systems. They induce qualitative and quantitative changes to a system’s dynamics, which can have catastrophic consequences if ignored during design. In this paper, we propose a general computational method to control the local bifurcations of dynamical systems by optimizing design parameters. We define an objective functional that enforces the appearance of local bifurcation points at targeted locations or even encourages their disappearance. The methodology is an efficient alternative to bifurcation tracking techniques capable of handling many design parameters ( > 10 2 ). The method is demonstrated on a Duffing oscillator featuring a hardening cubic nonlinearity and an autonomous van der Pol-Duffing oscillator coupled to a nonlinear tuned vibration absorber. The finite-element model of a clamped-free Euler–Bernoulli beam, coupled with a reduced-order modelling technique, is also used to show the extension to the shape optimization of more complicated structures. Results demonstrate that several local bifurcations of various types can be handled simultaneously by the bifurcation control framework, with both parameter and state target values.

Tracking amplitude extrema of nonlinear frequency responses using the harmonic balance method

AbstractThis work proposes a novel efficient method to track the evolution of amplitude extrema featured by frequency responses of nonlinear systems using the harmonic balance method. Means to compute the amplitude of a Fourier series are first outlined, and a set of equations characterizing a local extremum of a nonlinear frequency response amplitude curve is derived. Efficient numerical procedures are used to evaluate these equations and their derivatives (including second‐order ones) to embed them in a predictor‐corrector continuation framework. The proposed approach is illustrated on three examples of increasing complexity, namely a Helmholtz–Duffing oscillator, a two‐degree‐of‐freedom system with a modal interaction, and a doubly clamped von Kàrmàn beam with a nonlinear tuned vibration absorber.

Towed wheel shimmy suppression through a nonlinear tuned vibration absorber

The implementation of the nonlinear tuned vibration absorber (NLTVA) for the suppression of shimmy vibration in towed wheels is addressed in this study. We adopt a modified straight tangent tyre model of a single-degree-of-freedom towed wheel system with an attached NLTVA. Stability analysis illustrated that the NLTVA can significantly improve the stability of the equilibrium of the wheel. Bifurcation analysis highlighted the existence of large bistable regions, which undermines the system’s safety. However, numerical continuation analysis, coupled with a dynamical integrity investigation, revealed that the addition of an intentional softening nonlinearity in the absorber restoring force characteristic enables the complete suppression of the bistable regions, also reducing the amplitude of shimmy oscillations in the unstable region. Quasiperiodic motions were also identified; however, their practical relevance seems marginal.

Experimental Study on the Optimal-Based Vibration Control of a Wind Turbine Tower Using a Small-Scale Electric Drive with MR Damper Support

The paper presents an experimental implementation of an optimal-based vibration control for a scaled wind turbine tower-nacelle structure. A laboratory model of the approximate power scale of 340 W, equipped with a nonlinear tuned vibration absorber (TVA), is analysed. For control purposes, a combined operation of a small-scale electric servo drive and a magnetorheological (MR) damper is used in the TVA system. Nonlinearities of both the electric drive and the MR damper are intrinsic parts of the adopted nonlinear control concept. The aim of the research is the simple-hardware real-time implementation and the experimental investigation of the simultaneous actuator and damper control, including the analysis of the influence of optimal control law parameters and quality function weights on the vibration attenuation efficiency and actuator energy demand. As a reference, an optimal-based, modified ground-hook control with the single goal of the primary structure deflection minimisation is used along with the passive system with zero MR damper current and idling electric actuator, proving the advantages of the proposed method. The regarded solutions guarantee 57% maximum structure deflection reduction concerning the passive TVA configuration, using an MR damper of 32 N maximum force and an electric drive of 12.5 N nominal force and 0.76 W nominal power. An interesting alternative is the optimal control concept tuned with regard to the actuator power minimisation—it provides 30% maximum structure deflection attenuation (concerning the passive TVA configuration) while using a passive damper of 3.3 N maximum force and an actuator of 0.17 W nominal power only. It makes evident the advantage of the properly tuned optimal control algorithm over the modified ground-hook law—it requires 51% less actuator energy than the latter parametrised to exhibit the same vibration attenuation properties.

Seismic response control of bridges with nonlinear tuned vibration absorbers

The paper investigates seismic response control of bridges with nonlinear tuned vibration absorbers (n-TVAs). A three-span continuous bridge is used as case study. The n-TVAs are designed to act in both longitudinal and transverse directions of the bridge. The n-TVAs are modelled as inelastic springs with bilinear hysteretic behavior. A simple and effective method to design the parameters of the n-TVAs for controlling structural displacement is presented. The performance of the proposed n-TVAs is compared with the optimum linear TVAs (o-TVAs) and elasto-plastic bilinear TVAs (bn-TVAs). The results show that the n-TVAs are at least as effective as o-TVAs in controlling structural response. Because the n-TVAs do not require additional dashpots like the o-TVAs, they provide a more economical solution. It is observed that although effective in controlling structural response, their stroke is of the same order as that of o-TVAs. Addition of a viscous dashpot to the device is found to reduce their stroke, although accompanied by an inherent detrimental effect on the structural control performance.

A nonlinear optimization shooting method for bifurcation tracking of nonlinear systems

A continuation method for bifurcation tracking is presented based on the proposed optimization problem formulation which is designed to locate the bifurcation periodic solution. The bifurcation detection problem is formulated as a constrained optimization problem. The nonlinear constraints of the optimization problem are imposed on the shooting function and bifurcation conditions derived from the Floquet theory whereas the objective function associated with the pseudo-arclength correlation equation is devised to solution continuation. The proposed optimization formulation is integrated with the prediction–correction strategy to achieve bifurcation tracking. Two numerical examples about the Jeffcott rotor and the nonlinear tuned vibration absorber are illustrated to validate the effectiveness of the proposed methodology. Numerical results have demonstrated that the proposed method offers a convenient scheme to follow bifurcation periodic solution.

Impulsive vibration mitigation through a nonlinear tuned vibration absorber

The dynamics of a nonlinear passive vibration absorber conceived to mitigate vibrations of a nonlinear host structure is considered in this paper. The system under study is composed of a primary system, consisting of an undamped nonlinear oscillator of Duffing type, and a nonlinear dynamic vibration absorber, denominated nonlinear tuned vibration absorber (NLTVA). The NLTVA consists of a small mass, attached to the host structure through a linear damper, a linear and a cubic spring. The host structure is subject to free vibrations and the performance of the NLTVA is evaluated with respect to the minimal time required to dissipate a specific amount of the mechanical energy of the system. In order to characterize the dynamics of the system, a combination of numerical and analytical techniques is implemented. In particular, on the basis of the first-order reduced model, slow invariant manifolds of the transient dynamics are identified, which enable to estimate the absorber performance. Results illustrate that two different dynamical paths exist and the system can undergo either of them, depending on the initial conditions and on the value of the absorber nonlinear stiffness coefficient. One path leads to a very fast vibration mitigation, and therefore to a favorable behavior, while the other one causes a very slow energy dissipation.

A multi-parametric recursive continuation method for nonlinear dynamical systems

The aim of this paper is to provide an efficient multi-parametric recursive continuation method of specific solution points of a nonlinear dynamical system such as bifurcation points. The proposed method explores the topology of specific points found on the frequency response curves by tracking extremum points in the successive codimensions of the problem with respect to multiple system parameters. To do so, the characterization of extremum points by a constraint equation and its associated extended system are presented. As a result, a recursive algorithm is generated by successively appending new constraint equations to the extended system at each new level of continuation. Then, the methodology is applied to a nonlinear tuned vibration absorber (NLTVA). The limit of existence of isolated solutions and extremum points optimizing the region without isolated solution are found and used to improve the NLTVA.

Nonlinear magnetic vibration absorber for passive control of a multi–storey structure

A nonlinear magnetic vibration absorber is presented and used to control vibration of a three–storey structure. A distinctive feature of the absorber concerns its versatility for tuning the linear and nonlinear stiffness coefficients, depending on simple geometric design parameters such as the distance between fixed magnets and the moving one. In particular, the absorber can be set either as a nonlinear tuned vibration absorber, a nonlinear energy sink, or a bistable tuned vibration absorber, according to whether the linear stiffness term is positive, vanishing, or negative. The response of the primary structure and the vibration mitigation are investigated in the cases of impulsive shock, free vibration with imposed initial displacement, and single frequency excitation. Significant reductions of the primary structure vibrations are obtained for the three cases investigated, showing the ability of using a vibration absorber only relying on magnetic forces for passive control. The detailed comparisons of the absorbers performance show that, in this case study, no general guidelines can be easily deduced for selecting one of the three tunings for a nonlinear absorber. Depending on the excitation, the vibratory levels, and the frequency content of the excitation, the three configurations show advantages and drawbacks that are discussed.

Flutter Control of a Two-Degrees-of-Freedom Airfoil Using a Nonlinear Tuned Vibration Absorber

The influence of a nonlinear tuned vibration absorber (NLTVA) on the airfoil flutter is investigated. In particular, its effect on the instability threshold and the potential subcriticality of the bifurcation is analyzed. For that purpose, the airfoil is modeled using the classical pitch and plunge aeroelastic model together with a linear approach for the aerodynamic loads. Large amplitude motions of the airfoil are taken into account with nonlinear restoring forces for the pitch and plunge degrees-of-freedom. The two cases of a hardening and a softening spring behavior are investigated. The influence of each NLTVA parameter is studied, and an optimum tuning of these parameters is found. The study reveals the ability of the NLTVA to shift the instability, avoid its possible subcriticality, and reduce the limit cycle oscillations (LCOs) amplitude.

Responses of Multiple Nonlinear Tuned Vibration Absorbers under Harmonic Excitation

In this paper, a system consisting of multiple perfectly tuned identical translational vibration absorbers, having both hardening and softening springs, attached to a main mass under harmonic excitation is considered. The existence of absorbers’ synchronous and non-synchronous responses is checked. The method of averaging is employed to reach to the averaged autonomous equations of motion that describe the dynamics of the absorbers. A graphical method is then employed to check the existence of different responses of the absorbers. It is found that for absorbers with hardening springs, only one synchronous response of the absorbers occurs and no other responses take place. However, for the case of absorbers with softening springs, other responses were found to exist. These include multi-valued synchronous responses and a jump instability. These findings are in agreement with those of another study by the author where a similar system was considered using different approach.

Chatter mitigation using the nonlinear tuned vibration absorber

A passive vibration absorber, termed the nonlinear tuned vibration absorber (NLTVA), is designed for the suppression of chatter vibrations. Unlike most passive vibration absorbers proposed in the literature for suppressing machine tool vibrations, the NLTVA comprises both a linear and a nonlinear restoring force. Its linear characteristics are tuned in order to optimize the stability properties of the machining operation, while its nonlinear properties are chosen in order to control the bifurcation behavior of the system and guarantee robustness of stable operation. In this study, the NLTVA is applied to turning machining.

A principle of similarity for nonlinear vibration absorbers

This paper develops a principle of similarity for the design of a nonlinear absorber, the nonlinear tuned vibration absorber (NLTVA), attached to a nonlinear primary system. Specifically, for effective vibration mitigation, we show that the NLTVA should feature a nonlinearity possessing the same mathematical form as that of the primary system. A compact analytical formula for the nonlinear coefficient of the absorber is then derived. The formula, valid for any polynomial nonlinearity in the primary system, is found to depend only on the mass ratio and on the nonlinear coefficient of the primary system. When the primary system comprises several polynomial nonlinearities, we demonstrate that the NLTVA obeys a principle of additivity, i.e., each nonlinear coefficient can be calculated independently of the other nonlinear coefficients using the proposed formula.

Design of a magnetic vibration absorber with tunable stiffnesses

The design and characterisation of a magnetic vibration absorber (MVA), completely relying on magnetic forces, is addressed. A distinctive feature of the absorber is the ability of tuning the linear stiffness together with the nonlinear cubic and quintic stiffnesses by means of repulsive magnets located in the axis of the main vibrating magnetic mass, together with a set of corrective magnets located off the main axis. The tuning methodology is passive and relies only on three geometrical parameters. Consequently the MVA can be adjusted to design either a nonlinear tuned vibration absorber (NLTVA), a nonlinear energy sink (NES), or a bi-stable absorber with negative linear stiffness. The expressions of the stiffnesses are given from a multipole expansion of the magnetic fields of repulsive and corrective magnets. A complete static and dynamic characterisation is performed, showing the robustness of the modelling together with the ability of the MVA to work properly in different vibratory regimes, thus making it a suitable candidate for passive vibration mitigation in a wide variety of contexts.

Suppression of limit cycle oscillations using the nonlinear tuned vibration absorber.

The objective of this study is to mitigate, or even completely eliminate, the limit cycle oscillations in mechanical systems using a passive nonlinear absorber, termed the nonlinear tuned vibration absorber (NLTVA). An unconventional aspect of the NLTVA is that the mathematical form of its restoring force is not imposed a priori, as it is the case for most existing nonlinear absorbers. The NLTVA parameters are determined analytically using stability and bifurcation analyses, and the resulting design is validated using numerical continuation. The proposed developments are illustrated using a Van der Pol-Duffing primary system.

Performance, robustness and sensitivity analysis of the nonlinear tuned vibration absorber

The nonlinear tuned vibration absorber (NLTVA) is a recently developed nonlinear absorber which generalizes Den Hartog׳s equal peak method to nonlinear systems. If the purposeful introduction of nonlinearity can enhance system performance, it can also give rise to adverse dynamical phenomena, including detached resonance curves and quasiperiodic regimes of motion. Through the combination of numerical continuation of periodic solutions, bifurcation detection and tracking, and global analysis, the present study identifies boundaries in the NLTVA parameter space delimiting safe, unsafe and unacceptable operations. The sensitivity of these boundaries to uncertainty in the NLTVA parameters is also investigated.

Nonlinear generalization of Den Hartog׳s equal-peak method

This study addresses the mitigation of a nonlinear resonance of a mechanical system. In view of the narrow bandwidth of the classical linear tuned vibration absorber, a nonlinear absorber, termed the nonlinear tuned vibration absorber (NLTVA), is introduced in this paper. An unconventional aspect of the NLTVA is that the mathematical form of its restoring force is tailored according to the nonlinear restoring force of the primary system. The NLTVA parameters are then determined using a nonlinear generalization of Den Hartog׳s equal-peak method. The mitigation of the resonant vibrations of a Duffing oscillator is considered to illustrate the proposed developments.

Generalization of Den Hartog’s Equal-Peak Method for nonlinear primary systems

This study addresses the mitigation of one problem nonlinear resonance of a mechanical system. In view of the narrow bandwidth of the classical linear tuned vibration absorber, a new nonlinear absorber, termed the nonlinear tuned vibration absorber (NLTVA), is introduced in this paper. One unconventional aspect of the NLTVA is that the mathematical form of its restoring force is tailored according to the nonlinear restoring force of the primary system. The NLTVA parameters are then determined using a nonlinear generalization of Den Hartog’s equal-peak method. The mitigation of the resonant vibrations of a Duffing oscillator is considered to illustrate the proposed developments.

Design and Analysis of a Simple Nonlinear Vibration Absorber

In this work, a system consisting of a main structure with a nonlinear tuned vibration absorber having a hardening spring attached to it is considered. The nonlinearity in the absorber's spring is a cubic nonlinearity and is obtained by attaching two additional, relatively soft, similar linear springs to the absorber's mass. These added springs are attached to the absorber's mass in such a way that they are initially normal to the absorber's displacement axis. The dynamics of main structure and that of the nonlinear absorber are analyzed using the first order harmonic balancing method (HB1) and the results are compared against those of a system composing of a linear vibration absorber attached to a similar main structure. A numerical example is considered. The results obtained are in agreement with the known features of nonlinear absorbers. They showed that the absorber's effective frequency band width is increased when a nonlinear absorber is employed instead of a linear one. The results also showed that the displacement amplitude of the absorber during operation is decreased when using a nonlinear absorber. It is also found that the absorber's effective band width depends on the original length of the added linear springs, that are used to obtain the nonlinearity in the absorber, in such a manner that it increases as the original length of the two springs is decreased.

The Nonlinear Tuned Vibration Absorber

The objective of this paper is to introduce a new nonlinear dynamical absorber, the nonlinear tuned vibration absorber, through a rigorous nonlinear extension of the tuning rule for the linear tuned vibration absorber. This nonlinear tuning methodology combined with the increased suppression bandwidth brought by the intentional use of nonlinearity leads to the development of an absorber that is effective in wide ranges of frequencies and motion amplitudes. The results are illustrated using a one-degree-of-freedom primary system.