Small Excitation Amplitudes Research Articles

Stabilization of statically unstable columns by axial vibration of arbitrary frequency

The classical Chelomei's problem of stabilization of a statically unstable elastic column by axial harmonic vibration is reconsidered. The excitation frequency is assumed to be arbitrary. Two types of boundary conditions of the column, simply supported and clamped–hinged ends, are considered. Both cases are studied analytically with check by numerical analysis. For undamped columns with the axial force close to the critical stability value, a finite number of triangular stabilization zones appears. This implies that stabilization is attained at medium frequency range (compared to the second eigenfrequency of the unexcited column). The high-frequency stabilization is impossible, as the stabilization regions vanish with the increasing excitation frequency. With addition of internal damping, a continuous stabilization region appears for small excitation amplitudes, which starts at medium frequencies and extends to high-frequency range. The influence of external damping on stabilization regions is shown to be small.

Interaction of discrete breathers with thermal fluctuations

Discrete breathers (DBs) are time-periodic and spatially localized lattice excitations, which can be linearly stable or unstable with respect to either localized or extended perturbations. We analyze the interaction of DBs with a thermalized background of small-amplitude lattice excitations in a one-dimensional lattice of Morse oscillators with nearest-neighbor interaction. We find that stable DBs are barely influenced by the thermal noise. Unstable DBs are starting to propagate through the lattice, without losing their localization character. The instability can be due to localized perturbations as well as to extended perturbations. We discuss these observations in terms of resonances of DBs with localized and delocalized perturbations, and relate them to the issue of DB impact on the statistical properties of nonlinear lattices.

Amplitude reduction of primary resonance of nonlinear oscillator by a dynamic vibration absorber using nonlinear coupling

The paper presents the characteristics of a new type of nonlinear dynamic vibration absorber for a main system subjected to a nonlinear restoring force under primary resonance. The absorber is connected to the main system by a link in order to be excited with twice the frequency of the motion of the main system. The natural frequency of the absorber is tuned to be twice the natural frequency of the main system, in contrast to autoparametric vibration absorber, whose natural frequency is tuned to be one-half the natural frequency of the main system. The presented absorber is not excited through the autoparametric resonance, i.e., no trivial equilibrium state exists. Therefore, the absorber always oscillates because of the motion of the main system and cannot be trapped by Coulomb friction acting on the absorber, in contrast to the autoparametric vibration absorber. Under small excitation amplitude, this absorber does not produce an overhang in the frequency response curve, which occurs because of the use of the conventional autoparametric vibration absorber; the overhang renders the response amplitude larger than that in the case without an absorber. In addition, the absorber removes the hysteresis in the frequency response curve caused by the nonlinearity of the restoring force acting on the main system. Regarding large excitation amplitude, the response amplitude in the main system can be decreased by increasing the damping of the absorber, but that decrease is limited by the nonlinearity in the restoring force acting on the main system. This paper also describes experimental validation of the absorber under small excitation amplitude using a simple apparatus.

Controlling bistability in tapping-mode atomic force microscopy using dual-frequency excitation

This letter discusses an experimental method to suppress spontaneous transitions between low- and high-amplitude oscillatory responses in tapping-mode atomic force microscopy in the absence of feedback control. Here, the cantilever is excited at two frequencies and the dynamic force curves for different excitation amplitudes are recorded. Experimental observations of the dual-frequency excitation strategy are reported for three different cantilevers. These suggest that such transitions may indeed be eliminated from a region of interest of separations between the sample surface and the average position of the cantilever support even with relatively small secondary excitation amplitudes.

An Integrated Floating-Electrode Electric Microgenerator

Microfabricated electric generators, scavenging ambient mechanical energy, are potential power sources for autonomous systems. Described presently are the design, modeling, and implementation of a single-wafer floating-electrode electric microgenerator, integrating a micromechanical resonator and a number of electronic devices. Forming a plate of a variable capacitor, the resonator is responsible for converting mechanical vibration to electricity. A sense transistor and a diode bridge are integrated, respectively, for monitoring the charging of the electrode and for rectification. A lumped electromechanical model of the generator is developed and expressed in terms of a set of nonlinear coupled state equations that are numerically solved. For small-amplitude excitation, a circuit based on a set of linearized equations is developed. The generator is realized using a compatible combination of standard complementary metal-oxide-semiconductor (CMOS) floating gate process and a post-CMOS photoresist molded electroplating process. Adequate agreement between model predictions and measurement results was obtained

Experimental study on identification of stiffness change in a concrete frame experiencing damage and retrofit

This paper describes an experimental study on structural health monitoring of a 1:3-scaled one-story concrete frame subjected to seismic damage and retrofit. The structure is tested on a shaking table by exerting successively enhanced earthquake excitations until severe damage, and then retrofitted using fiber-reinforced polymers (FRP). The modal properties of the tested structure at trifling, moderate, severe damage and strengthening stages are measured by subjecting it to a small-amplitude white-noise excitation after each earthquake attack. Making use of the measured global modal frequencies and a validated finite element model of the tested structure, a neural network method is developed to quantitatively identify the stiffness reduction due to damage and the stiffness enhancement due to strengthening. The identification results are compared with `true` damage severities that are defined and determined based on visual inspection and local impact testing. It is shown that by the use of FRP retrofit, the stiffness of the severely damaged structure can be recovered to the level as in the trifling damage stage.

Nonlinear Oscillation Analysis and Stabilization Control of Magnetic Domain Wall

A chaotic region and stabilization method of domain-wall motion were examined by computer simulation. As a result, the chaotic domain-wall motion was distributed outside of (high frequency)-(large amplitude), and (low frequency)-(small amplitude) excitation region, and the bottom of region supported domain-wall resonance. In addition, the extended delayed feedback control (EDFC) method and Half-period-EDFC method that improved delayed feedback method were shown to be effective as a control method.

Dynamic stability of ring-based angular rate sensors

Dynamic stability of a rotating ring subjected to harmonic perturbations in input angular rate is examined using an asymptotic approach. The governing equations that represent the transverse and tangential in-plane motion of the ring are derived via Hamilton's principle. The equations of motion, after discretization and suitable linearization, represent a two-degree-of-freedom time-varying linear gyroscopic system. Such a system can exhibit instability behaviour characterized by exponential growth in response amplitudes. Employing the method of averaging, conditions for instability are obtained in closed-form. Instability boundaries for the ring in the excitation intensity-frequency space are then established for small excitation amplitudes. In addition, effects of damping, input angular rate variations, and imperfection due to the ring asymmetry are discussed.

Modal reduction of mathematical models of biological molecules

This paper reports a detailed study of modal reduction based on either linear normal mode (LNM) analysis or proper orthogonal decomposition (POD) for modeling a single α- d-glucopyranose monomer as well as a chain of monomers attached to a moving atomic force microscope (AFM) under harmonic excitations. Also a modal reduction method combining POD and component modal synthesis is developed. The accuracy and efficiency of these methods are reported. The focus of this study is to determine to what extent these methods can reduce the time and cost of molecular modeling and simultaneously provide the required accuracy. It has been demonstrated that a linear reduced order model is valid for small amplitude excitation and low frequency excitation. It is found that a nonlinear reduced order model based on POD modes provides a good approximation even for large excitation while the nonlinear reduced order model using linear eigenmodes as the basis vectors is less effective for modeling molecules with a strong nonlinearity. The reduced order model based on component modal synthesis using POD modes for each component also gives a good approximation. With the reduction in the dimension of the system using these methods the computational time and cost can be reduced significantly.

Control and characterization of spatio-temporal disorder in parametrically excited surface waves

The nonlinear interactions of parametrically excited surface waves have been shown to yield a rich family of nonlinear states. When the system is driven by two commensurate frequencies, a variety of interesting superlattice type states are generated via a number of different 3-wave resonant interactions. These states occur either as symmetry-breaking bifurcations of hexagonal patterns composed of a single unstable mode or via nonlinear interactions between the two different unstable modes generated by the two forcing frequencies. Near the system’s bicritical point, a well-defined region of phase space exists in which a highly disordered state, both in space and time, is observed. We first show that this state results from the competition between two distinct nonlinear super-lattice states, each with different characteristic temporal and spatial symmetries. After characterizing the type of spatio-temporal disorder that is embodied in this disordered state, we will demonstrate that it can be controlled. Control to either of its neighboring nonlinear states is achieved by the application of a small-amplitude excitation at a third frequency, where the spatial symmetry of the selected pattern is determined by the temporal symmetry of the third frequency used. This technique can also excite rapid switching between different nonlinear states.

Dynamic characterization of hysteresis elements in mechanical systems. I. Theoretical analysis

The pre-sliding-pre-rolling phase of friction behavior is dominated by rate-independent hysteresis. Many machine elements in common engineering use exhibit, therefore, the characteristic of "hysteresis springs," for small displacements at least. Plain and rolling element bearings that are widely used in motion guidance of machine tools are typical examples. While the presence of a hysteresis element may mark the character of the resulting dynamics, little is to be found about this topic in the literature. The study of the nonlinear dynamics caused by such elements becomes imperative if we wish to achieve accurate control of such machines. In this Part I of the investigation, we examine a single-degree-of-freedom mass-hysteresis-spring system and show that, while the free response case is amenable to an exact solution, the more important case of forced response has no closed form solution and requires other methods of treatment. We consider harmonic-balance analysis methods (which are common analysis tools in engineering) suitable for frequency-domain treatment, in particular the approximate describing function (DF) method, and compare those results with "exact" numerical simulations. The DF method yields basically a linear equation with amplitude-dependent modal parameters. We find that agreement in the frequency response function, between DF and exact solution, is good for small excitation amplitudes and for very large amplitudes. Intermediate values, however, show high sensitivity to amplitude variations and, consequently, no regular solution is obtainable by either approach. This appears to be an inherent property of the system pointing to the need for developing further analysis methods. Experimental verification of the analysis outlined in this Part I is given in Part II of the paper.

Numerical study on the nonlinear oscillations of a torsion absorber filled with a viscoelastic fluid

Results are presented concerning the mechanical behavior of a torsion absorber filled with a viscoelastic fluid. As regards the nonlinear constitutive equation of the viscoelastic filling, a multi-mode fluid model is used, leading to a set of coupled ordinary differential equations, which were solved numerically. The influence of different excitation strengths and frequencies on the energy dissipated per cycle in the periodic stage, as well as the resulting shear in the transient and fully developed stages are given and discussed. The findings for small excitation amplitude were compared to those resulting from linear theory.

Active separation control: an overview of Reynolds and Mach numbers effects

Separation control, by nominally two-dimensional periodic excitation, was studied experimentally by the authors and co-workers at Reynolds numbers ranging from 3 × 10 4 to 4 × 10 7 , including compressibility effects. The tests demonstrated that active control using oscillatory flow excitation can effectively delay flow separation from, and reattach separated flow to, aerodynamic surfaces at various flight conditions. At Reynolds number below 10 5, where transition does not occur naturally and cannot be passively forced, active separation control may be the only effective method for delaying separation and generating useful lift. The essence of active separation control relies on exploiting instabilities that are inherent in the flow, generally requiring relatively small amplitude excitation. Effective excitation frequencies generate one to four vortices over the controlled region at all times, irrespective of Reynolds number, and perturbations should preferably be amplified over the region that is susceptible to separation. Periodic excitation is vastly superior to steady blowing in terms of performance benefits and eliminates abrupt flow responses, which are undesirable from a control point of view. The effects of compressibility in the absence of shocks are weak and undesirable effects accompanying separation, such as vortex-shedding and buffet, can be significantly reduced or completely eliminated. Separation resulting from shock-wave/boundary-layer interaction can be ameliorated, providing that excitation is introduced upstream of separation.

Control of spatiotemporal disorder in parametrically excited surface waves.

Interacting surface waves, parametrically excited by two commensurate frequencies, yield a number of nonlinear states. Near the system's bicritical point, a state, highly disordered in space and time, results from competition between nonlinear states. Experimentally, this disordered state can be rapidly stabilized to a variety of nonlinear states via open-loop control with a small-amplitude third frequency excitation, whose temporal symmetry governs the temporal and the spatial symmetry of the selected nonlinear state. This technique also excites rapid switching between nonlinear states.

Theoretical analysis of the dynamic behavior of hysteresis elements in mechanical systems

Many machine elements in common engineering use exhibit the characteristic of “hysteresis springs”. Plain and rolling element bearings that are widely used in motion guidance of machine tools are typical examples. The study of the non-linear dynamics caused by such elements becomes imperative if we wish to achieve accurate control of such machines. This paper outlines the properties of rate-independent hysteresis and shows that the calculation of the free response of a single-degree-of-freedom (SDOF) mass-hysteresis-spring system is amenable to an exact solution. The more important issue of forced response is not so, requiring other methods of treatment. We consider the approximate describing function method and compare its results with exact numerical simulations. Agreement is good for small excitation amplitudes, where the system approximates to a linear mass-spring-damper system, and for very large amplitudes, where some sort of mass-line is approached. Intermediate values however, show high sensitivity to amplitude variations, and no regular solution is obtained by either approach. This appears thus to be an inherent property of the system pointing to the need for developing further analysis methods.

Dynamics of a dry friction oscillator under two-frequency excitations

In this paper, the dynamic behavior of a dry friction oscillator subjected to simultaneous self and external excitations is studied. The dry friction in the system follows the classical Coulomb’s law, and the external excitation consists of two harmonic forces with different frequencies. The focus of the paper is laid on bifurcation analysis to gain insight into the influence of the two-frequency excitation upon the qualitative features of system dynamics. Numerical simulations are performed and the simulation results are visualized by means of bifurcation diagrams, Poincaré sections and Lyapunov exponents. New and interesting conclusions are reached from the study. It is shown that small excitation frequencies and amplitudes tend to cause chaos in the system, whereas large excitation amplitudes can induce only periodic motions. Furthermore, it is found that the probability of the occurrence of periodic motions increases, as the ratio between the two excitation frequencies increases.

“Fast excitation” cid in a quadrupole ion trap mass spectrometer

Collision-induced dissociation (CID) in a quadrupole ion trap mass spectrometer is usually performed by applying a small amplitude excitation voltage at the same secular frequency as the ion of interest. Here we disclose studies examining the use of large amplitude voltage excitations (applied for short periods of time) to cause fragmentation of the ions of interest. This process has been examined using leucine enkephalin as the model compound and the motion of the ions within the ion trap simulated using ITSIM. The resulting fragmentation information obtained is identical with that observed by conventional resonance excitation CID. “Fast excitation” CID deposits (as determined by the intensity ratio of the a 4/b 4 ion of leucine enkephalin) approximately the same amount of internal energy into an ion as conventional resonance excitation CID where the excitation signal is applied for much longer periods of time. The major difference between the two excitation techniques is the higher rate of excitation (gain in kinetic energy) between successive collisions with helium atoms with “fast excitation” CID as opposed to the conventional resonance excitation CID. With conventional resonance excitation CID ions fragment while the excitation voltage is still being applied whereas for “fast excitation” CID a higher proportion of the ions fragment in the ion cooling time following the excitation pulse. The fragmentation of the (M + 17H) 17+ of horse heart myoglobin is also shown to illustrate the application of “fast excitation” CID to proteins.

Testing of nonlinear interaction effects of sinusoidal and noise excitation on rubber isolator stiffness

The nonlinear excitation effects on dynamic stiffness and damping of a filled rubber isolator are investigated through measurements. For a single harmonic excitation they are found to exhibit a strong amplitude dependence, following the well-known Payne effect where stiffness is high for small excitation amplitudes and low for large amplitudes while damping displays a maximum at intermediate amplitudes. However, expanding the measurements to a multiple harmonic excitation, the commonly applied superimposition principle of single harmonic responses due to this excitation is shown to be non-valid. On the contrary, it is found that reference stiffness at a small excitation amplitude and high frequency is reduced and damping increased while superimposing a large amplitude low frequency excitation component. Superimposition of low frequency noise signals displays essentially the same influence on the reference characteristics. As a rule of thumb, the largest excitation amplitude over the isolator normally determines the stiffness at the reference frequency while the influence of envelope amplitude is increased as its frequency approaches that of the component of interest.

The ‘Indian rope trick’ for a parametrically excited flexible rod: nonlinear and subharmonic analysis

Recently, Mullin has demonstrated experimentally that an upright column that is longer than its critical length for self–weight buckling can be stabilized by subjecting it to vertical harmonic excitation. This paper extends an earlier linearized analysis of a rod–mechanics model of this set–up to include three dimensionality and geometric nonlinearity. The stability of the upright state is then analysed using weakly nonlinear asymptotic expansions in the limit of small–amplitude excitation. First, the unforced problem is treated, extending the classical result by Greenhill to show that all bifurcations are supercritical. The main results are for the forced problem near the simplest among the potential infinity of dynamic instabilities. These correspond to pure bending modes, and to resonances between a vibration mode of the column and the first harmonic or subharmonic of the drive. The result is an asymptotic description of these instabilities including information on the stability of dynamically bifurcating states in terms of the three dimensionless parameters of the problem (bending stiffness and the amplitude and frequency of excitation). A qualitative explanation is offered of why the earlier linearized analysis fails to quantitatively match the experiments.

Dynamics and Control of Ship-Mounted Cranes

When deep water ports are not accessible, cargo is transferred from/to larger ships to/from smaller ships offshore. A typical arrangement is to place three ships side by side. In the middle is a ship with cranes. Along one side of the crane ship is the large container ship, and along the other side is the small ship or lighter, which shuttles back and forth between the ships and shore. The sea state is a very important factor in this operation. The sea-excited motion of the crane ship can excite large pendulations of the containers while they are suspended by the cables of the cranes. The large motions can be due to either large excitation amplitudes or small excitation amplitudes near resonant conditions. In this study, the cable-suspended load is modeled as an elastic spherical pendulum. The method of multiple scales is used to derive four nonlinear ordinary- differential equations that describe the amplitudes and phases of the excited modes. These equations are used to investigate instabilities and provide information for controlling the pendulations. The results show that a parametric excitation at twice the natural frequency leads to a sudden jump in the response as the cable is unreeled. Introducing a harmonic change in the cable length at the same frequency as the excitation can suppress this dynamic instability and result in a smooth response. The analytical results are verified by numerical simulations of the original full nonlinear equations.