Multi-scaled analysis of the damped dynamics of an elastic rod with an essentially nonlinear end attachment

We study multi-frequency transitions in the transient dynamics of a viscously damped dispersive finite rod with an essentially nonlinear end attachment. The attachment consists of a small mass connected to the rod by means of an essentially nonlinear stiffness in parallel to a viscous damper. First, the periodic orbits of the underlying hamiltonian system with no damping are computed, and depicted in a frequency–energy plot (FEP). This representation enables one to clearly distinguish between the different types of periodic motions, forming back bone curves and subharmonic tongues. Then the damped dynamics of the system is computed; the rod and attachment responses are initially analyzed by the numerical Morlet wavelet transform (WT), and then by the empirical mode decomposition (EMD) or Hilbert–Huang transform (HTT), whereby, the time series are decomposed in terms of intrinsic mode functions (IMFs) at different characteristic time scales (or, equivalently, frequency scales). Comparisons of the evolutions of the instantaneous frequencies of the IMFs to the WT spectra of the time series enables one to identify the dominant IMFs of the signals, as well as, the time scales at which the dominant dynamics evolve at different time windows of the responses; hence, it is possible to reconstruct complex transient responses as superposition of the dominant IMFs involving different time scales of the dynamical response. Moreover, by superimposing the WT spectra and the instantaneous frequencies of the IMFs to the FEPs of the underlying hamiltonian system, one is able to clearly identify the multi-scaled transitions that occur in the transient damped dynamics, and to interpret them as ‘jumps’ between different branches of periodic orbits of the underlying hamiltonian system. As a result, this work develops a physics-based, multi-scaled framework and provides the necessary computational tools for multi-scaled analysis of complex multi-frequency transitions of essentially nonlinear dynamical systems

Nonlinear MDOF system characterization and identi cation using the Hilbert-Huang transform

The Hilbert transform is one of the most successful approaches to tracking the varying nature of vibration of a large class of nonlinear systems thanks to the extraction of backbone curves from experimental data. Because signals with multiple frequency components do not admit a well-behaved Hilbert transform, it is inherently limited to the analysis of single-degree-of-freedom systems. In this study, the joint application of the complexification-averaging method and the empirical mode decomposition enables us to develop a new technique, the slow-flow model identification method. Through numerical and experimental applications, we demonstrate that the proposed method is adequate for characterizing and identifying multi-degree-offreedom nonlinear systems

The slow-flow method of identification in nonlinear structural dynamics

The Hilbert-Huang transform (HHT) has been shown to be effective for characterizing a wide range of nonstationary signals in terms of elemental components through what has been called the empirical mode decomposition. The HHT has been utilized extensively despite the absence of a serious analytical foundation, as it provides a concise basis for the analysis of strongly nonlinear systems. In this paper, we attempt to provide the missing link, showing the relationship between the EMD and the slow-flow equations of the system. The slow-flow model is established by performing a partition between slow and fast dynamics using the complexification-averaging technique, and a dynamical system described by slowly-varying amplitudes and phases is obtained. These variables can also be extracted directly from the experimental measurements using the Hilbert transform coupled with the EMD. The comparison between the experimental and analytical results forms the basis of a nonlinear system identification method, termed the slow-flowmodel identification method, which is demonstrated using numerical examples

Assessing the Dissipative Capacity of Particle Impact Dampers Based on their Nonlinear Bandwidth Characteristics

The dissipative capacity as quantified by the nonlinear bandwidth measure of impulsively loaded primary structures (PSs) coupled to particle impact dampers (PIDs) is assessed. The considered PIDs are designed by initially placing different numbers of spherical, linearly viscoelastic granules at different 2D initial topologies and clearances. The strongly nonlinear and highly discontinuous dynamics of the PIDs are simulated via the discrete element method taking Hertzian interactions, slipping friction and granular rotations into account. The general definition of nonlinear bandwidth is used to evaluate the energy dissipation capacity of the integrated PS-PID systems. Moreover, the effect of the dynamics of the PIDs on the time-bandwidth product of these systems is studied, as a measure of their capacity to store or dissipate vibration energy. It is found that the initial topologies of the granules in the PID drastically affect the time-bandwidth product, which, depending on shock intensity, may break the classical limit of unity which holds for linear time-invariant dissipative resonators. The optimal PS-PID systems composed of multiple granules produce large nonlinear bandwidths, indicating strong dissipative capacity of broadband input energy by the PIDs. Additionally, in the optimal configurations, the time-bandwidth product, i.e., the measure of the frequency bandwidth of the input shock that is stored in the PS-PID system, in tandem with the amount of time it takes for the system to dissipate (1/e) of the initial energy, can be tuned either above or below unity by varying the applied shock intensity. The implications of these findings on the dissipative capacity of the system considered are discussed, showing that it can be predictively assessed so that PIDs can act as highly effective nonlinear energy sinks capable of rapid and efficient suppression of vibration induced by shocks

PASSIVE SUPPRESSION OF AEROELASTIC INSTABILITIES OF IN-FLOW WINGS BY TARGETED ENERGY TRANSFERS TO LIGHTWEIGHT ESSENTIALLY NONLINEAR ATTACHMENTS

Theoretical and experimental suppression of aeroelastic instabilities by means of broadband passive targeted energy transfers has been recently studied. A single-degree-offreedom (SDOF) nonlinear energy sink (NES) was coupled to a 2-DOF rigid wing modeled in the low-speed, subsonic regime with quasi-steady aerodynamic theory. The nonlinear attachment was designed and optimized to suppress the critical nonlinear modal energy exchanges between the flow and the (pitch and heave) wing modes, thus suppressing the (transient) triggering mechanism of aeroelastic instability. We performed bifurcation analysis to find regions of robust passive aeroelastic suppression in parameter space. Then, we employed multi-degreeof-freedom nonlinear energy sinks (MDOF NESs) to improve robustness of the aeroelastic instability suppression. Bifurcation analysis by a numerical continuation technique demonstrated that controlling the occurrence of a limit point cycle (LPC or saddle-node) bifurcation point above a Hopf bifurcation point is crucial to enhancing suppression robustness. MDOF NESs not only can enhance robustness of suppression against even strong gust-like disturbances, but they require lower NES mass compared to SDOF NES designs. The validity of the theoretical findings was proven by a series of wind tunnel experiments

Earthquake response control of a cable-stayed bridge by using μ-synthesis

Bu çalışmada çeşitli kontrol algoritmalarının etkinliğini birbiriyle karşılaştırmak amacıyla oluşturulmuş bir kablolu köprü kontrol model örneği [1, 2] baz alınmıştır. Köprünün sonlu elemanlar modeli modal önem sırasına göre küçültülerek 1/6 boyutunda bir kontrol tasarım modeli elde edilmiştir. Köprünün deprem hareketi sırasındaki kritik noktaları dikkatle gözlenmiş, ve yol seviyesinde kule kesme kuvvetlerinin oluştuğu noktaların kontrolü en zor noktalar olduğunu ortaya konmuştur. Kontrol amaçlı olarak hidrolik pistonlar kullanılmıştır. Belirsizliği tolere edebilen bir tasarım önerisiyle, tasarım modeli ile değerlendirme modeli arasındaki fark sınanmıştır. Ayrıca μ-sentezi esas alınarak bir kontrol denklemi kurgulanmıştır ve bununla birlikte davranış benzetimi gerçekleştirilmiştir.This study is based on a benchmark control problem for seismic response of cable-stayed bridges that is introduced to compare the effectiveness of various control algorithms [1, 2]. The finite element model of the bridge is arranged according to its modal importance and is reduced to 1/6’th of its size. Special attention is given to the control of critical locations of the bridge response. Among these, the tower shear forces at the deck level were the hardest to reduce by using hydraulic actuators. An uncertainty tolerant design is introduced to account for the design model difference from the evaluation model. Finally, a μ-synthesis based controller is designed and used in the closed loop system simulation of the bridge