On the response of a harmonically excited two degree-of-freedom system consisting of a linear and a nonlinear quasi-zero stiffness oscillator

There are many systems which consist of a nonlinear oscillator attached to a linear system, examples of which are nonlinear vibration absorbers, or nonlinear systems under test using shakers excited harmonically with a constant force. This paper presents a study of the dynamic behaviour of a specific two degree-of-freedom system representing such a system, in which the nonlinear system does not affect the vibration of the forced linear system. The nonlinearity of the attachment is derived from a geometric configuration consisting of a mass suspended on two springs which are adjusted to achieve a quasi-zero stiffness characteristic with pure cubic nonlinearity. The response of the system at the frequency of excitation is found analytically by applying the method of averaging. The effects of the system parameters on the frequency-amplitude response of the relative motion are examined. It is found that closed detached resonance curves lying outside or inside the continuous path of the main resonance curve can appear as a part of the overall amplitude-frequency response. Two typical situations for the creation of the detached resonance curve inside the main resonance curve, which are dependent on the damping in the nonlinear oscillator, are discussed.     

A study of a nonlinear vibration isolator with a quasi-zero stiffness characteristic

A vibration isolator consisting of a vertical linear spring and two nonlinear pre-stressed oblique springs is considered in this paper. The system has both geometrical and physical nonlinearity. Firstly, a static analysis is carried out. The softening parameter leading to quasi-zero dynamic stiffness at the equilibrium position is obtained as a function of the initial geometry, pre-stress and the stiffness of the springs. The optimal combination of the system parameters is found that maximises the displacement from the equilibrium position when the prescribed stiffness is equal to that of the vertical spring alone. It also satisfies the condition that the dynamic stiffness only changes slightly in the neighbourhood of the static equilibrium position. For these values, a dynamical analysis of the isolator under asymmetric excitation is performed to quantify the undesirable effects of the nonlinearities. It includes considering the possibilities of the appearance of period-doubling bifurcation and its development into chaotic motion. For this purpose, approximate analytical methods and numerical simulations accompanied with qualitative methods including phase plane plots, Poincaré maps and Lyapunov exponents are used. Finally, the frequency at which the first period-doubling bifurcation appears is found and the effect of damping on this frequency determined.     

Approximate analytical solutions to a conservative oscillator using global residue harmonic balance method

In the current research paper, a conservative system comprising of a mass grounded by linear and nonlinear springs in series connection is studied. The equation of motion for the aforementioned system has been derived as a nonlinear ordinary differential equation with inertia and static–type cubic nonlinearities. The global residue harmonic balance method is applied to obtain an approximate analytical frequency and periodic solution of the problem. Using the obtained analytical expressions, the influences of the hardening and softening nonlinear spring on the non–dimensional frequency are investigated. The results show that developing the system nonlinearity leads the displacement of the mass and the deflection of linear spring to approach each other. Moreover, comparison of the results obtained using the proposed procedure with those achieved by other methods such as numerical method, variational iteration method and harmonic balance approach demonstrates the accuracy and advantages of the current approach.     

On the interaction of the responses at the resonance frequencies of a nonlinear two degrees-of-freedom system

This paper describes the dynamic behaviour of a coupled system which includes a nonlinear hardening system driven harmonically by a shaker. The shaker is modelled as a linear single degree-of-freedom system and the nonlinear system under test is modelled as a hardening Duffing oscillator. The mass of the nonlinear system is much less than the moving mass of the shaker and thus the nonlinear system has little effect on the shaker dynamics. The nonlinearity is due to the geometric configuration consisting of a mass suspended on four springs, which incline as they are extended. Following experimental validation, the model is used to explore the dynamic behaviour of the system under a range of different conditions. Of particular interest is the situation when the linear natural frequency of the nonlinear system is less than the natural frequency of the shaker such that the frequency response curve of the nonlinear system bends to higher frequencies and thus interacts with the resonance frequency of the shaker. It is found that for some values of the system parameters a complicated frequency response curve for the nonlinear system can occur; closed detached curves can appear as a part of the overall amplitude-frequency response. These detached curves can lie outside or inside the main resonance curve, and a physical explanation for their occurrence is given.     

Chains of oscillators with negative stiffness elements

Negative stiffness is not allowed by thermodynamics and hence materials and systems whose global behaviour exhibits negative stiffness are unstable. However the stability is possible when these materials/systems are elements of a larger system sufficiently stiff to stabilise the negative stiffness elements. In order to investigate the effect of stabilisation we analyse oscillations in a chain of n linear oscillators (masses and springs connected in series) when some of the springs? stiffnesses can assume negative values. The ends of the chain are fixed. We formulated the necessary stability condition: only one spring in the chain can have negative stiffness. Furthermore, the value of negative stiffness cannot exceed a certain critical value that depends upon the (positive) stiffnesses of other springs. At the critical negative stiffness the system develops an eigenmode with vanishing frequency. In systems with viscous damping vanishing of an eigenfrequency does not yet lead to instability. Further increase in the value of negative stiffness leads to the appearance of aperiodic eigenmodes even with light damping. At the critical negative stiffness the low dissipative mode becomes non-dissipative, while for the high dissipative mode the damping coefficient becomes as twice as high as the damping coefficient of the system. A special element with controllable negative stiffness is suggested for designing hybrid materials whose stiffness and hence the dynamic behaviour is controlled by the magnitude of applied compressive force.     

Vibration modelling of helical springs with non-uniform ends

Helical springs constitute an integral part of many mechanical systems. Usually, a helical spring is modelled as a massless, frequency independent stiffness element. For a typical suspension spring, these assumptions are only valid in the quasi-static case or at low frequencies. At higher frequencies, the influence of the internal resonances of the spring grows and thus a detailed model is required. In some cases, such as when the spring is uniform, analytical models can be developed. However, in typical springs, only the central turns are uniform; the ends are often not (for example, having a varying helix angle or cross-section). Thus, obtaining analytical models in this case can be very difficult if at all possible. In this paper, the modelling of such non-uniform springs are considered. The uniform (central) part of helical springs is modelled using the wave and finite element (WFE) method since a helical spring can be regarded as a curved waveguide. The WFE model is obtained by post-processing the finite element (FE) model of a single straight or curved beam element using periodic structure theory. This yields the wave characteristics which can be used to find the dynamic stiffness matrix of the central turns of the spring. As for the non-uniform ends, they are modelled using the standard finite element (FE) method. The dynamic stiffness matrices of the ends and the central turns can be assembled as in standard FE yielding a FE/WFE model whose size is much smaller than a full FE model of the spring. This can be used to predict the stiffness of the spring and the force transmissibility. Numerical examples are presented.     

Modeling and identification of freeplay nonlinearity

A nonlinear analysis is performed for the purpose of identification of the pitch freeplay nonlinearity and its effect on the type of bifurcation of a two degree-of-freedom aeroelastic system. The databases for the identification are generated from experimental investigations of a pitch-plunge rigid airfoil supported by a nonlinear torsional spring. Experimental data and linear analysis are performed to validate the parameters of the linearized equations. Based on the periodic responses of the experimental data which included the flutter frequency and its third harmonics, the freeplay nonlinearity is approximated by a polynomial expansion up to the third order. This representation allows us to use the normal form of the Hopf bifurcation to characterize the type of instability. Based on numerical integrations, the coefficients of the polynomial expansion representing the freeplay nonlinearity are identified.     

Localization of elastic waves in randomly disordered multi-coupled multi-span beams

The wave localization in randomly disordered periodic multi-span continuous beams is studied. The transfer matrix method is used to deduce transfer matrices of two kinds of multi-span beams. To calculate the Lyapunov exponents in discrete dynamical systems, the algorithm for determining all the Lyapunov exponents in continuous dynamical systems presented by Wolf et al is employed. The smallest positive Lyapunov exponent of the corresponding discrete dynamical system is called the localization factor, which characterizes the average exponential rates of growth or decay of wave amplitudes along the randomly mistuned multi-span beams. For two kinds of disordered periodic multi-span beams, numerical results of localization factors are given. The effects of the disorder of span-length, the non-dimensional torsional spring stiffness and the non-dimensional linear spring stiffness on the wave localization are analysed and discussed. It can be observed that the localization factors increase with the increase of the coefficient of variation of random span-length and the degree of localization for wave amplitudes increases as the torsional spring stiffness and the linear spring stiffness increase.     

Localization of elastic waves in randomly disordered multi-coupled multi-span beams

Abstract

The wave localization in randomly disordered periodic multi-span continuous beams is studied. The transfer matrix method is used to deduce transfer matrices of two kinds of multi-span beams. To calculate the Lyapunov exponents in discrete dynamical systems, the algorithm for determining all the Lyapunov exponents in continuous dynamical systems presented by Wolf et al is employed. The smallest positive Lyapunov exponent of the corresponding discrete dynamical system is called the localization factor, which characterizes the average exponential rates of growth or decay of wave amplitudes along the randomly mistuned multi-span beams. For two kinds of disordered periodic multi-span beams, numerical results of localization factors are given. The effects of the disorder of span-length, the non-dimensional torsional spring stiffness and the non-dimensional linear spring stiffness on the wave localization are analysed and discussed. It can be observed that the localization factors increase with the increase of the coefficient of variation of random span-length and the degree of localization for wave amplitudes increases as the torsional spring stiffness and the linear spring stiffness increase.     

The effect of magnetic stress and stiffness modulus on resonant characteristics of Ni-Mn-Ga ferromagnetic shape memory alloy actuators

The prospect of using ferromagnetic shape memory alloys (FSMAs) is promising for a resonant actuator that requires large strain output and a drive frequency below 1 kHz. In this investigation, three FSMA actuators, equipped with tetragonal off-stoichiometric Ni₂MnGa single crystals, were developed to study their frequency response and resonant characteristics. The first actuator, labeled as A1, was constructed with low-k bias springs and one Ni-Mn-Ga single crystal. The second actuator, labeled as A2, was constructed with high-k bias springs and one Ni-Mn-Ga crystal. The third actuator, labeled as A3, was constructed with high-k bias springs and two Ni-Mn-Ga crystals connected in parallel. The three actuators were magnetically driven over the frequency range of 10 Hz-1 kHz under 2 and 3.5 kOe magnetic-field amplitudes. The field amplitude of 2 kOe is insufficient to generate significant strain output from all three actuators; the maximum magnetic-field-induced strain (MFIS) at resonance is 2%. The resonant MFIS output improves to 5% under 3.5-kOe amplitude. The frequency responses of all three actuators show a strong effect of the spring k constant and the Ni-Mn-Ga modulus stiffness on the resonant frequencies. The resonant frequency of the Ni-Mn-Ga actuator was raised from 450 to 650 Hz by increasing bias spring k constant and/or the number of Ni-Mn-Ga crystals. The higher number of the Ni-Mn-Ga crystals not only increases the magnetic force output but also raises the total stiffness of the actuator resulting in a higher resonant frequency. The effective modulus of the Ni-Mn-Ga is calculated from the measured resonant frequencies using the mass-spring equation; the calculated modulus values for the three actuators fall in the range of 50-60 MPa. The calculated effective modulus appears to be close to the average modulus value between the low twinning modulus and high elastic modulus of the untwined Ni-Mn-Ga crystal.     

On the design of a high-static-low-dynamic stiffness isolator using linear mechanical springs and magnets

The frequency range over which a linear passive vibration isolator is effective is often limited by the mount stiffness required to support a static load. This can be improved upon by incorporating a negative stiffness element in the mount such that the dynamic stiffness is much less than the static stiffness. In this case, it can be referred to as a high-static-low-dynamic stiffness (HSLDS) mount. This paper is concerned with a theoretical and experimental study of one such mount. It comprises two vertical mechanical springs between which an isolated mass is mounted. At the outer edge of each spring, there is a permanent magnet. In the experimental work reported here, the isolated mass is also a magnet arranged so that it is attracted by the other magnets. Thus, the combination of magnets acts as a negative stiffness counteracting the positive stiffness provided by the mechanical springs. Although the HSLDS suspension system will inevitably be nonlinear, it is shown that for small oscillations the mount considered here is linear. The measured transmissibility is compared with a comparable linear mass-spring-damper system to show the advantages offered by the HSLDS mount.     

Aeroelastic analysis and nonlinear dynamics of an elastically mounted wing

The response of an elastically mounted wing that is free to plunge and pitch, supported by nonlinear translational and torsional springs, and interacting with an incoming stream is analyzed. A tightly coupled model of the wing flow interaction is developed. A three-dimensional code based on the unsteady vortex lattice method is used for the prediction of the unsteady aerodynamic loads. The response of the wing shows a sequence of static and dynamic bifurcations and chaotic motions when increasing the flow speed. Pairs of stable solutions are observed over the different response regimes. The effects of the gust and structural nonlinearity on the wing's response are also investigated. The results show that gust may lead to jumps between the pairs of solutions for static and dynamic equilibrium responses without impacting the boundaries of the different response regimes. As for the effect of the structural nonlinearity, increasing the nonlinear coefficient of the stiffness of the torsional spring yields lower static deflections and amplitudes of the limit cycle oscillations.     

小位移条件下混沌隔振装置的设计与研究

本文根据混沌隔振原理, 设计出小位移下的混沌隔振装置, 该装置能够在小位移下产生强非线性, 并且其线性部分与非线性部分完全区分开, 实验中易于调节其整体刚度, 以及线性和非线性项比例, 大大增加了该装置在工程中的应用前景. 并利用数值计算方法对特定参数下的隔振装置在简谐激励力作用下的运动进行分析, 证实了该混沌隔振装置的可行性.     

Design and experiment of a compact quasi-zero-stiffness isolator capable of a wide range of loads

This paper proposes the design and experiment of a vibration isolator capable of isolating a wide range of loads. The isolator consists of two oblique springs and one vertical spring to achieve quasi-zero stiffness at the equilibrium position. The quasi-zero-stiffness characteristic makes the isolator attenuate external disturbance more at low frequencies, when compared with linear isolators. Unlike previous studies, this paper focuses on the analysis of the effect of different loads and the implementation of an adjustment mechanism to handle a wide range of loads. To ensure zero stiffness under imperfect stiffness matching, a lateral adjustment mechanism is also proposed. Instead of using coil springs, special planar springs are designed to realize the isolator in a compact space. Static and dynamic models are developed to evaluate the effect of key design parameters so that the isolator can have a wide isolation range without sacrificing its size. A prototype and its associated experiments are presented to validate the transmissibility curves under three different loads. The results clearly show the advantage of quasi-zero-stiffness isolators against linear isolators.     

Vibration based algorithm for crack detection in cantilever beam containing two different types of cracks

In this paper, a simple method for detection of multiple edge cracks in Euler–Bernoulli beams having two different types of cracks is presented based on energy equations. Each crack is modeled as a massless rotational spring using Linear Elastic Fracture Mechanics (LEFM) theory, and a relationship among natural frequencies, crack locations and stiffness of equivalent springs is demonstrated. In the procedure, for detection of m cracks in a beam, 3m equations and natural frequencies of healthy and cracked beam in two different directions are needed as input to the algorithm.     

Dynamical behavior of disordered rotationally periodic structures: A homogenization approach

This paper put forth a new approach, based on the mathematical theory of homogenization, to study the vibration localization phenomenon in disordered rotationally periodic structures. In order to illustrate the method, a case-study structure is considered, composed of pendula equipped with hinge angular springs and connected one to each other by linear springs. The structure is mistuned due to mass and/or stiffness imperfections. Simple continuous models describing the dynamical behavior of the structure are derived and validated by comparison with a well-known discrete model. The proposed models provide analytical closed-form expressions for the eigenfrequencies and the eigenmodes, as well as for the resonance peaks of the forced response. These expressions highlight how the features of the dynamics of the mistuned structure, e.g. frequency split and localization phenomenon, depend on the physical parameters involved.     

Chaotic and self-organized critical behavior of a generalized slider-block model

The dynamical behavior of two-dimensional arrays of slider blocks is considered. The blocks are pulled across a frictional surface by a constant-velocity driver; the blocks are connected to the driver and to each other by springs. Only one block is allowed to slip at a time and its displacement can be obtained analytically; the system is deterministic with no stochastic inputs. Studies of a pair of slider blocks show that they exibit periodic, limit-cycle, or choatic behavior depending upon parameter values and initial conditions. Studies of large, two-dimensional arrays of blocks show self-organized criticality. Positive Lyapunov exponents are found that depend upon the stiffness and size of the array.     

Mechanical model of carbon dioxide vibrational spectrum

Classical dynamics methods have been used to study the nonlinear vibrations of a CO₂ molecule. Consideration includes not only the anharmonicity valence angle, which enables one to explain the Fermi resonance, but also the physical nonlinearity of the force field (stiffness and softness of springs). In the farthest neighbor approximation (with regard to oxygen–oxygen interaction), a set of nonlinear differential equations in the Lagrangian form has been derived. Their analytical solution has been derived using the method of invariant normalization. The occurrence of a strange attractor has been discovered by numerical simulation. Recommendations for the selection of initial conditions are given that take into account the possibility of regular beatings that change into to chaotic beatings.     

Wave localization in randomly disordered multi-coupled multi-span beams on elastic foundations

In this paper, the wave propagation and localization in randomly disordered periodic multi-span beams on elastic foundations are studied. For two kinds of beams, i.e. the multi-span beams on elastic foundations with periodic flexible and simple supports, the transfer matrices between two consecutive sub-spans are obtained by means of the continuity conditions. The algorithm for determining all the Lyapunov exponents in continuous dynamic systems presented by Wolf et al. is employed to calculate those in discrete dynamic systems. The localization factor characterizing the average exponential rates of growth or decay of wave amplitudes along the disordered beams is defined as the smallest positive Lyapunov exponent of the discrete dynamical system. The localization length that represents the distance of elastic waves propagating along the disordered periodic structures is defined as the reciprocal of the smallest positive Lyapunov exponent, i.e. the localization factor. For the two kinds of disordered periodic beams on elastic foundations, the numerical results of the localization factors are presented and analysed by comparing them with the results of the beams without elastic foundations to illustrate the effects of the elastic foundations on the wave propagation and localization. The effects of the disorder of span-length and the dimensionless torsional and linear spring stiffness on the localization factors are discussed. Moreover, the localization lengths are also calculated and discussed for certain structural parameters in disordered periodic structures. It can be observed from the results that ordered periodic multi-span beams have the characteristics of the frequency passbands and stopbands and the localization of elastic waves can occur in disordered periodic systems: the localization degree of elastic waves is strengthened with the increase of the coefficient of variation of the span-length. The influences of the elastic foundations on the wave propagation and localization are more complicated. Generally speaking, in lower-frequency regions the elastic foundations have pronounced effects on the spectral structures, but in higher-frequency regions the effects are negligible. The localization degree increases as the torsional spring stiffness increases. The linear spring has few effects on the spectral structures in higher-frequency regions, but in lower-frequency regions it has prominent effects. The larger the disorder degree, the shorter the non-dimensional localization length.