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.     

Pole assignment of friction-induced vibration for stabilisation through state-feedback control

Friction-induced self-excited linear vibration is often governed by a second-order matrix differential equation of motion with an asymmetric stiffness matrix. The asymmetric terms are product of friction coefficient and the normal stiffness at the contact interface. When the friction coefficient becomes high enough, the resultant vibration becomes unstable as frequencies of two conjugate pairs of complex eigenvalues (poles) coalesce (when viscous damping is low).This short paper presents a receptance-based inverse method for assigning complex poles to second-order asymmetric systems through (active) state-feedback control of a combination of active stiffness, active damping and active mass, which is capable of assigning negative real parts to stabilise an unstable system.     

Application of hysteresis functions in vibration problems

In the paper, the notion of the hysteresis function introduced by the author is developed further and applied in some vibration problems for hysteretic materials. The stress–strain relation is built on the basis of well-known extended Masing's rules in which the scaled backbone curve used in the second rule is replaced by a more general function called the hysteresis function. This modification allows us to regulate the dependence of the damping ratio on the strain amplitude in the process of cycle deforming of a material with further extending to arbitrary deformation processes. The three cases of hysteretic systems are considered: the system with limited stress when strain increases without bound, the system with linear backbone curve, and the system with the backbone curve having increasing stiffness for increasing strain. For a number of dynamic examples, a comparison of different hysteresis functions is carried out.     

Vibration isolation via a scissor-like structured platform

More and more attentions are attracted to the analysis and design of nonlinear vibration control/isolation systems for better isolation performance. In this study, an isolation platform with n-layer scissor-like truss structure is investigated to explore novel design of passive/semi-active/active vibration control/isolation systems and to exploit potential nonlinear benefits in vibration suppression. Due to the special scissor-like structure, the dynamic response of the platform has inherent nonlinearities both in equivalent damping and stiffness characteristics (although only linear components are applied), and demonstrates good loading capacity and excellent equilibrium stability. With the mathematical modeling and analysis of the equivalent stiffness and damping of the system, it is shown that: (a) the structural nonlinearity in the system is very helpful in vibration isolation, (b) both equivalent stiffness and damping characteristics are nonlinear and could be designed/adjusted to a desired nonlinearity by tuning structural parameters, and (c) superior vibration isolation performances (e.g., quasi-zero stiffness characteristics etc.) can be achieved with different structural parameters. This scissor-like truss structure can potentially be employed in different engineering practices for much better vibration isolation or control.     

Force distortion in resonance testing of structures with electro-dynamic vibration exciters

Harmonic input force distortion which arises when systems are excited with electrodynamic exciters is shown to be predominantly second harmonic, the major source of the harmonic distortion being due to the vibration exciter characteristics. These are examined by experimentally determining the magnetic field strength properties of a typical exciter and the results show these to be a non-linear even function. This information is used with the equations of motion of the excited which are simulated on an analog computer. The computed force characteristics are shown to compare very closely with experimental results. The amount of second harmonic force distortion generated at a system resonance is analyzed by considering a simple single degree-of-freedom model. It is shown that the amount of force distortion is related to the damping of the system under test and the ratio of the exciter stiffness to the system stiffness. It is also shown that the force input to a system near a system resonance can vary considerably, even though the input current to the exciter is constant. These effects are shown to be due to the forces arising from the mass and stiffness characteristics of the exciter being used. Experimental tests on a simple system confirm the theoretical predictions.     

Passive-active vibration isolation systems to produce zero or infinite dynamic modulus: theoretical and conceptual design strategies

The application of mechanical springs connected in parallel and/or in series with active springs can produce dynamical systems characterised by infinite or zero value stiffness. This mathematical model is extended to more general cases by examining the dynamic modulus associated with damping, stiffness and mass effects. This produces a theoretical basis on which to design an isolation system with infinite or zero dynamic modulus, such that stiffness and damping may have infinite or zero values. Several theoretical designs using a mixture of passive and active systems connected in parallel and/or in series are proposed to overcome limitations of feedback gain experienced in practice to achieve an infinite or zero dynamic modulus. It is shown that such systems can be developed to reduce the weight supported by active actuators as demonstrated, for example, by examining suspension systems of very low natural frequency or with a very large supporting stiffness or with a viscous damper or a self-excited vibration oscillator. A more general system is created by combining these individual systems allowing adjustment of the supporting stiffness and damping using both displacement and velocity feedback controls. Frequency response curves show the effects of active feedback control on the dynamical behaviour of these systems. The theoretical design strategies presented can be applied to design feasible hybrid vibration control systems displaying increased control performance.     

弹性基础隔振系统的简化性能指标和有源控制力

系统研究了基础弹性对单层隔振系统、双层隔振系统及浮筏隔振系统隔振性能的影响。分析了不同隔振系统与不同弹性基础间的振动耦合特性,讨论了不同隔振系统的振级落差和力传递率特性,给出了振级落差和力传递率的简化计算方法。针对不同隔振系统的有源隔振问题,比较了不同作动器安装方式所需的控制力。研究表明,对于所有隔振系统,增加基础的刚度和阻尼有利于提高振级落差和力传递率;对于浮筏隔振系统,增加筏架的刚度和阻尼有利于提高隔振性能和减少有源隔振所需的控制力。      

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.     

Analysis of a damped pneumatic spring

The complex dynamic stiffness of a damped spring is determined. The damping is produced by transient pressure feedback from an auxiliary tank connected by a capillary to the spring cylinder. From the complex stiffness, the damping and stiffness are determined as functions of excitation frequency. The behavior of a compound spring, consisting of a damped pneumatic spring in parallel with a stiffer linear spring, is also examined. The analysis shows that the damping loss factor depends only on the tank/cylinder volume ratio, and that the capillary dimensions affect only the frequency at which maximum damping occurs. The compound spring is shown to have a maximum loss factor which quickly reaches an asymptotic value as the tank/cylinder volume ratio increases. From this presentation a clearer understanding of the behavior of a damped air spring, and a better sense of how design parameters affect the component's characteristics are obtained.     

A decrement method for quantifying nonlinear and linear damping parameters

A method is presented which can estimate the linear and nonlinear damping parameters in a lightly damped system. Only a single response measurement from a free decay test is required as input. This ensures that the magnitude of the damping parameters is not compromised by phase distortion between measurements. The method uses the instantaneous energy to describe the long-term evolution of the system. Practically this is achieved by using only the peak amplitudes in each period. In this way the stiffness is effectively ignored, and only the damping forces are considered. For this reason, the method is not unlike the familiar decrement method, which can be used to estimate the apparent linear damping. The method is developed in the context of a weakly nonlinear, lightly damped system, with both linear and cubic damping. Simulated response data is used to demonstrate the accuracy of the technique. The nonlinear damping parameter is extracted from the response data to within 5% of the exact value, even though the nonlinear term contributes less than 1% to the total force in the system.     

Physical mechanisms of phonation onset: a linear stability analysis of an aeroelastic continuum model of phonation

In an investigation of phonation onset, a linear stability analysis was performed on a two-dimensional, aeroelastic, continuum model of phonation. The model consisted of a vocal fold-shaped constriction situated in a rigid pipe coupled to a potential flow which separated at the superior edge of the vocal fold. The vocal fold constriction was modeled as a plane-strain linear elastic layer. The dominant eigenvalues and eigenmodes of the fluid-structure-interaction system were investigated as a function of glottal airflow. To investigate specific aerodynamic mechanisms of phonation onset, individual components of the glottal airflow (e.g., flow-induced stiffness, inertia, and damping) were systematically added to the driving force. The investigations suggested that flow-induced stiffness was the primary mechanism of phonation onset, involving the synchronization of two structural eigenmodes. Only under conditions of negligible structural damping and a restricted set of vocal fold geometries did flow-induced damping become the primary mechanism of phonation onset. However, for moderate to high structural damping and a more generalized set of vocal fold geometries, flow-induced stiffness remained the primary mechanism of phonation onset.     

Damping characteristics of a spray-deposited shape memory alloy beam

The damping characteristics of an Ni–Ti shape memory alloy (SMA) beam are theoretically and experimentally studied with interest in identifying an appropriate damping model for the material. The SMA beam is manufactured by a spray deposition method followed by heat treatment and found to have nanocrystalline structure in which damping capacity is high. The beam is then tested to obtain an impulse response and the frequency response function (FRF). By using the Hilbert transform technique it is shown that damping of the beam is almost amplitude independent in the tested range of displacement. It is also shown from the FRF that the damping of the spray-deposited shape memory alloy beam is well represented by a model including both linear viscous and hysteretic dampings.     

A general mass law for broadband energy harvesting

It is well known that the power absorbed by a linear oscillator when excited by white noise base acceleration depends only on the mass of the oscillator and the spectral density of the base motion. This places an upper bound on the energy that can be harvested from a linear oscillator under broadband excitation, regardless of the stiffness of the system or the damping factor. It is shown here that the same result applies to any multi-degree-of-freedom nonlinear system that is subjected to white noise base acceleration: for a given spectral density of base motion the total power absorbed is proportional to the total mass of the system. The only restriction to this result is that the internal forces are assumed to be a function of the instantaneous value of the state vector. The result is derived analytically by several different approaches, and numerical results are presented for an example two-degree-of-freedom-system with various combinations of linear and nonlinear damping and stiffness.     

Nonlinear dynamics of complex hysteretic systems: Oscillator in a magnetic field

Complex hysteresis is a well-known phenomenon in many branches of science. The most prominent examples come from materials with a complex microscopic structure such as magnetic materials, shape-memory alloys, or, porous materials. Their hysteretic behavior is characterized by the existence of multiple internal system states for a given external parameter and by a non-local memory. The input-output behavior of such systems is well studied and in a standard phenomenological approach described by the so-called Preisach operator. What is not well understood, are situations, where such a hysteretic system is dynamically coupled to its environment. Since the hysteretic sub-system provides a complicated form of nonlinearity, one expects non-trivial, possibly chaotic behavior of the combined dynamical system. We study such a combined dynamical system with hysteretic nonlinearity. In this original contribution a simple differential-operator equation with hysteretic damping, which describes a magnetic pendulum is considered. We find, for instance, a fractal dependence of the asymptotic behavior as function of the starting values. The sensitivity of the system to perturbations is investigated by several methods, such as the 0–1 test for chaos and sub-Lyapunov exponents. The power spectral density is also calculated and compared with analytical results for simple input-output scenarios.     

Numerical investigation of coexisting high and low amplitude responses and safe basin erosion for a coupled linear oscillator and nonlinear absorber system

Over the last half century, numerous nonlinear variants of the tuned mass damper have been developed in order to improve attenuation characteristics. In the present study, the performance of a linear oscillator and an absorber with a strongly nonlinear cubic stiffness is evaluated by using numerical methods. This configuration has been of recent interest due to its capability of wide-band energy absorption. However, high amplitude solutions, which would amplify the response of the system, have been shown to often coexist with the low amplitude solutions. The present research is focused on numerically determining the relative strength of the coexisting solutions. Erosion profiles are presented, quantifying the integrity of the system, i.e. the likelihood of converging to a safe, low amplitude response, and providing an indication of the structural safety of a practical absorber system. The results indicate that the high amplitude solutions not only exist but significantly influence the response of the system within the range of expected operating conditions, particularly at excitation frequencies lower than the natural frequency of the linear oscillator. The erosion profiles indicate a 20–40% increase in system integrity for the case of zero damping compared to a small amount of damping, no significant integrity change when adding a small linear stiffness component to the nonlinear absorber, and no significant change in integrity between the midpoint and extreme of the bi-stable range. Additional higher-period solutions are also discovered and evidence of a chaotic response is presented.     

Transverse hysteretic damping characteristics of a serpentine belt: Modeling and experimental investigation

In this paper, the transverse dynamic hysteretic damping characteristics (HDC) of a serpentine belt are investigated. The variable stiffness and variable damping model (VSDM) constituted of a variable-stiffness spring and a variable-damping damper is developed to estimate the HDC of the belt. A test rig is designed to test the force–displacement hysteresis damping curve and resonance frequencies of serpentine belts with different lengths under diverse loading conditions. The force–displacement hysteresis damping curve getting from the experiment is then used to determine the transverse stiffness and damping coefficients needed for the VSDM. The experiment particularly shows that the orientation of the hysteresis curve swings left and right around each natural frequency as it is a symmetrical point. This interesting phenomenon is explicated in detail with the loss angle which is calculated by two methods. Moreover, two sub-analytical models included in the VSDM are proposed to model the dependence of transverse dynamic stiffness and damping coefficient of a belt on belt length, pretension and excitation frequency. A comparison of the hysteresis curves obtained from the VSDM and experiment indicates that they are in good agreement.     

An experimental switchable stiffness device for shock isolation

The development of an experimental switching stiffness device for shock isolation is presented. The system uses magnetic forces to exert a restoring force, which results in an effective stiffness that is used to isolate a payload. When the magnetic force is turned on and off, a switchable stiffness is obtained. Characterization of the physical properties of the device is presented. They are estimated in terms of the percentage stiffness change and effective damping ratio when switched between two constant stiffness states. Additionally, the setup is used to implement a control strategy to reduce the shock response and minimize residual vibration. The system was found to be very effective for shock isolation. The response is reduced by around 50 percent compared with passive isolation showing good correlation with theoretical predictions, and the effective damping ratio in the system following the shock was increased from about 4.5 percent to 13 percent.     

Simplified performance indices and active control force of vibration isolation systems with elastic base

Influence of the elasticity of the base on vibration isolation performances of single layer, double layer and floating raft vibration isolation systems is investigated systematically.Characteristics of vibration coupling between different vibration isolation systems and different elastic bases are analyzed. Moreover the characteristics of vibration acceleration level difference and force transmissibility of different vibration isolation systems are discussed and their simplified expressions are given. In addition the required control forces of active vibration isolation under different installations of actuators for different vibration isolation systems are compared.The results show that for all vibration isolation systems, the addition of the stiffness and damping of the base can enhance their vibration acceleration level difference and force transmissibility.Moreover for floating raft vibration isolation system, the addition of the stiffness and damping of the raft can enhance its vibration isolation performance and reduce the control force required by active vibration isolation.     

Improving the accuracy of the n-dB method for determining damping of non-lightly damped systems

Damping measurements using the spectral magnitude remain popular and are studied here for non-lightly damped systems using the variable bandwidth n-dB method, which is advantageous for non-lightly damped systems. The most commonly used estimator (based on normalised bandwidth) provides significant errors for non-lightly damped systems. An existing more accurate method (using the squares of the frequencies used in the former method) is exact for hysteretic damping, but still provides significant error for viscous damping. Improved estimators are developed in order to correct either exactly, or to insignificant errors, measurements taken with existing estimators. Neither further data nor the individual frequencies are required; the previously calculated damping values are corrected. The application of the improved estimators is dependent upon the existing estimator used and the damping type; however a strategy is suggested to reduce errors if the latter is unknown.     

A comparison of two nonlinear damping mechanisms in a vibration isolator

The vibration transmissibility characteristics of a single-degree-of-freedom (SDOF) passive vibration isolation system with different nonlinear dampers are investigated in this paper. In one configuration, the damper is assumed to be linear and viscous, and is connected to the mass so that it is perpendicular to the spring (horizontal damper). The vibration is in the direction of the spring. The second configuration is one in which the damper is in parallel with the spring but the damping force is proportional to the cube of the relative velocity across the damper (cubic damping). Both configurations are studied for small amplitudes of excitation, when some analysis can be conducted based on analytical expressions, and for large amplitudes of excitation, where the analysis is based on numerical simulations. It is found that the two nonlinear systems can outperform the linear system when force transmissibility is considered. However, for displacement transmissibility, the system with the horizontal damper exhibits some desirable properties, but the system with cubic damping does not.