Multivalued Response Research Articles

Arclength-based response matching of multivalued frequency responses to update models with strong nonlinearities

Model updating of strongly nonlinear systems is a vital tool to establish the precise nonlinear model from the multivalued responses and one major step in the updating process is the matching of the predicted and measured multivalued responses. However, when the multivalued behavior of the predicted and measured responses varies widely, the traditional multivalued response matching methods perform poorly. In this paper, a novel arclength-based multivalued response matching technique is proposed, which can help to update the model from linear to strongly nonlinear and handle both stable and unstable responses. The arclength is obtained from the continuation and the multivalued response is parameterized and transferred from the frequency domain to the arclength domain. The Arclength-Response curve is single-valued and can be easily separated into several parts based on feature points. Transferring the separation results back to the frequency domain, the full multivalued response is divided to single-valued Frequency-Response curves, where single-valued predicted and tested responses may be directly matched. A 2 DOF strong nonlinear model is first updated for verification. Using the proposed method, the linear response is directly updated based on the multivalued response with both softening and hardening regions. The error in the final updated parameters is below 1% and the proposed updating method performs better than two traditional methods. A real 3 DOF nonlinear model with a non-smooth strong nonlinearity with the hardening region is then used for validation. The stable and unstable responses of the system are measured from a shaker voltage-based constant frequency test and model updating is then conducted from linear to strongly nonlinear. During the updating process, the complex nonlinear behavior has six bifurcation points on one predicted response curve, but the updated responses fit the real ones well after 18 iterations. These results show the validity and superiority of the proposed method.

Model updating of dynamic structures with strong nonlinearities using fixed frequency continuation tests

Model updating using multivalued responses is an important approach to construct strongly nonlinear models. During updating, swept frequency tests are usually used to measure the multivalued responses, but the measured responses only contain the stable branches. Accordingly, updating mainly uses these stable responses for the nonlinear model construction. Unstable responses are rarely tested or participate in the updating process, which may lead to errors in the constructed model with strong nonlinearities. Recently, fixed frequency tests using a continuation scheme have been used to measure fully multivalued responses, including both the stable and unstable branches, for structures with the strong nonlinearity. However, the measured responses were then transformed to equivalent swept frequency tests for the existing updating schemes, where this transformation may induce errors in the updating process. In this paper, a novel method for strong nonlinear model updating using the results of the fixed frequency test is proposed. Such updating is based on the directly measured data of the fixed frequency tests and can handle both the stable and unstable responses. Fixed frequency continuation using the Multi-Harmonic Balance Method (MHBM) is developed to predict the multivalued responses and their sensitivities. The multivalued responses of the prediction and measurement are directly matched in sensitivity-based updating. The proposed method is numerically verified on a 2 DOF nonlinear model. Using the proposed method, the nonlinear model is updated from linear to strongly nonlinear and the updated parameters all converge to the simulated values after 7 iterations. The proposed method is then experimentally validated on a 3 DOF nonlinear system with a non-smooth nonlinearity. A nonlinear model with a 7th order polynomial stiffness represents the equivalent system and is directly updated based on the fixed frequency test results. The updated responses are generally a good fit to the tested responses. Compared with updating using only the stable responses, the proposed updating method achieves higher precision and its updated responses are closer to the tested responses. These results show the validity and superiority of the proposed method.

Design of multivalued frequency response filters by using nonlinear feedback

This contribution introduces the design of multivalued multiband frequency response filters exploiting nonlinear feedback design techniques. The interest in this topic is related to the possibility of analysing in real-time sinusoidal signals unveiling whether their main frequency components increase or decrease along different complex paths and in fixed sets of windows. This can be done by using the effects of frequency hysteresis, peculiar characteristic of the proposed devices.

Research on multi-valued response and bursting oscillation of series multi-stable piezoelectric energy harvester

Research on multi-valued response and bursting oscillation of series multi-stable piezoelectric energy harvester

Investigation of the nonlinear phenomena of a Langevin ultrasonic transducer caused by high applied voltage

In the field of power ultrasound, Langevin ultrasonic transducers (LUTs) usually operate at a large displacements output power by applying high voltages. However, empirically, a LUT exhibits nonlinearities such as amplitude jumping and peak hysteresis for high voltages in actual operations. The nonlinearities would reduce the efficiency and output accuracy of an LUT. In this research, the burst-mode method was used to measure the longitudinal vibration velocity of the LUT, which gradually decreased with time after the excitation voltage was turned off. The equivalent mechanical losses and equivalent spring constants were determined using the velocity attenuation rate and resonant frequency and they were found to be the linear functions of velocity, helping to develop a novel nonlinear model. This model contained two quadratic nonlinear terms based on the linear model. Furthermore, the developed nonlinear model was analyzed using the Lagrangian method as well as the multiscale method, which confirmed that the model was effective in describing the nonlinear behavior. It was also found that the frequency-amplitude curve bent when the nonlinear term was taken into account, which resembled the nonlinear phenomenon tested experimentally. From a physical point of view, this bending was meaningful because it led to the formation of multi-valued response regions with jumping phenomena. Additionally, according to the obtained results, the maximum value of the system response was independent of the degree of nonlinearity of the system.

Parameter Identification of a Strongly Nonlinear Rotor-Bearing System Based on Reconstructed Constant Response Tests

Abstract A strongly nonlinear rotor-bearing system often has multiple solutions under harmonic excitations and jump phenomena. For example, a hardening nonlinearity may include a jump-down in the acceleration process and jump-up in the deceleration process. It is challenging to measure all of these multiple responses and establish an accurate dynamic model from experimental data to predict these phenomena. This paper used a fixed frequency test method to measure all of these multiple responses under harmonic excitations and developed a novel strategy to characterize and identify nonlinearities in a strongly nonlinear rotor-bearing system based on reconstructing constant response tests from fixed frequency test data. The fixed frequency tests are achieved by monotonically increasing the voltage applied to the exciter at a fixed frequency and using the force drop-out phenomenon through the resonance to control the force applied to the structure. This test method could measure multivalued response curves of a strongly nonlinear rotor-bearing system in a nonrotating state. The constant response tests could be reconstructed from these multivalued response curves. The relationship of equivalent stiffness versus displacement can be established, and hence, the nonlinear stiffness is characterized and identified from constant response tests. A rotor-bearing system with a strongly nonlinear support is used to demonstrate the method, and the nonlinear support stiffness parameters are identified and validated in a nonrotating state. The identified nonlinear rotor-bearing model also could predict the jump phenomena in the acceleration or deceleration process. The results demonstrate the feasibility and effectiveness of the approach, and also show the potential for practical applications in engineering.

Measurement of multivalued response curves of a strongly nonlinear system by exploiting exciter dynamics

A strongly nonlinear system often has multiple solutions under harmonic excitation. However, measuring all of these multiple responses in structural dynamics is challenging because often one solution is unstable and difficult to obtain. The standard stepped sine approach is to fix the harmonic excitation force amplitude, and step the excitation frequency up or down. This leads to the well-known jump phenomenon, and captures at most two stable solutions. Alternatively, the excitation frequency can be fixed and the amplitude swept up or down, although this also leads to jumps in the response. Recently, experimental continuation methods have successfully measured all solutions, including the unstable solutions, via active control. This paper takes a different approach and exploits the dynamics of the electromagnetic exciter to both stabilize the unstable solution, and also to track the solutions continuously, without any jumps. This is achieved by monotonically increasing or decreasing the voltage applied to the exciter at a fixed frequency, and using the force drop-out phenomenon through the resonance to control the force applied to the structure. In these tests, the input voltage then defines the continuation parameter, rather than force amplitude or frequency in the standard tests. The obvious advantage of this method is that there is no feedback control of the excitation and it is easy to implement. A strongly nonlinear single degree of freedom system is used to demonstrate this method.

A new class of quasistatic frictional contact problems governed by a variational–hemivariational inequality

A quasistatic nonsmooth frictional contact problem for a viscoelastic material is studied. The contact is modeled by a multivalued normal damped response condition with the Clarke generalized gradient of a locally Lipschitz superpotential and the friction is described by a version of the Coulomb law of dry friction with the friction bound depending on the regularized normal stress. The weak formulation of the contact problem is a history-dependent variational–hemivariational inequality coupled with an operator equation for the stress field. A result on the unique weak solvability to this coupled system is proved through a recent contribution on history-dependent subdifferential inclusions and a fixed point approach.

Variational–hemivariational inequality for a class of dynamic nonsmooth frictional contact problems

In this paper, a dynamic frictional contact problem for viscoelastic materials with long memory is studied. The contact is modeled by a multivalued normal damped response condition with the Clarke generalized gradient of a locally Lipschitz superpotential and the friction is described by a version of the Coulomb law of dry friction with the friction bound depending on the regularized normal stress. The weak formulation of the contact problem is a history-dependent variational–hemivariational inequality for the velocity. A result on the unique weak solvability to this inequality is proved through a recent contribution on evolutionary subdifferential inclusions and a fixed point approach.

Changing the nonlinear resonant response of an asymmetric mono-stable oscillator by injecting a hard high-frequency harmonic excitation

Nonlinearity is an unvoidable intrinsic property of most dynamical systems. Its mere presence can be detrimental to the performance of many devices since it often complicates the response behavior of the device by introducing multi-valued and aperiodic responses. As such, the ability to passively alter and control the nonlinearity can be of utmost importance for the design of many sensors and actuators. In this paper, we demonstrate that the effective nonlinearity associated with the resonant dynamics of a mono-stable asymmetric oscillator can be adjusted by injecting a hard high-frequency non-resonant excitation. We study the dependence of the effective nonlinearity of the slow dynamics on the hard excitation magnitude and highlight the important design parameters controlling this behavior. We also show that the slow resonant dynamics of the oscillator can be locally linearized by choosing the proper hard excitation parameters.

Resonance responses in a two-degree-of-freedom viscoelastic oscillator under randomly disordered periodic excitations

An investigation is presented for primary resonance and internal resonance of two-degree-of-freedom (TDOF) viscoelastic system with some complex nonlinear coupled terms under randomly disordered periodic excitations. The assumed viscoelastic damping depending on the past history of motion is chosen as the form of convolution integrals over an exponentially decaying kernel function. Then, the steady-state responses of the TDOF system are discussed in detail through the method of multiple scales (MMS), which is employed to derive the determined and stochastic differential equations of amplitude and phase modulation for the internal and external resonance modes. Further, the numerical simulation method is absorbed to test the effectiveness and accuracy of the theoretical analysis solutions for steady-state moments with the changing excitation amplitude. The appearance of (random) jump, (random) saturation, and double-jumping can be explored by the development of steady-state moments with different excitation amplitudes and frequency, which also show the resonance bandwidth increases with the excitation amplitude. Besides, viscoelastic damping parameters have great influence on the steady-state responses under the different detuning conditions, which accelerate or delay the appearance of the above mentioned phenomena. Numerical simulation results of phase portraits tell that the increase of the intensity of the random excitation leads to the steady-state solutions changing from a limit cycle to a diffused limit cycle, thus, such multi-valued steady-state responses produce jump from one stable branch to another also response to the previous analysis conclusion.

Simulation Method for Complex Multivalued Curves in Injection-Locked Oscillators

A new methodology is presented for the efficient harmonic-balance simulation of injection-locked oscillators with complex multivalued and disconnected curves. It is illustrated through its application to high-order subharmonically injection-locked oscillators. A graphical technique is applied to analyze the oscillator-phase sensitivity with respect to the input signal, required for the injection-locked operation. The intricate synchronized-solution curves are obtained with the new method, which enables a global exploration of all the coexistent periodic solutions. These solutions can belong to different curve sections, in a multivalued response, or to disconnected synchronization curves. The method is based on the calculation of a series of phase-dependent outer-tier admittance functions, which provide the oscillator response to the injection signal. Coexistent solutions are simultaneously obtained through a contour-plot intersection, without the need for continuation techniques. The method is illustrated through application to an oscillator synchronized to low-frequency sinusoidal signal by means of a nonlinear-transmission line. The analysis and design techniques have been successfully validated through comparison with independent simulations and measurements.

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.

Sensitivity analysis of primary resonances and bifurcations of a controlled piecewise-smooth system with negative stiffness

In this paper, the combination of the cubic nonlinearity and time delay is proposed to improve the performance of a piecewise-smooth (PWS) system with negative stiffness. Dynamical properties, feedback control performance and symmetry-breaking bifurcation are mainly considered for a PWS system with negative stiffness under nonlinear position and velocity feedback control. For the free vibration system, the homoclinic-like orbits are firstly derived. Then, the amplitude-frequency response of the controlled system is obtained analytically in aspect of the Lindstedt–Poincaré method and the method of multiple scales, which is also verified through the numerical results. In this regard, a softening-type behavior, which directly leads to the multi-valued responses, is illustrated over the negative position feedback. Especially, the five-valued responses in which three branches of them are stable are found. And complex multi-valued characteristics are also observed in the force-amplitude responses. Furthermore, for explaining the effectiveness of feedback control, the equivalent damping and stiffness are also introduced. Sensitivity of the system response to the feedback gain and time delay is comprehensively considered and interesting dynamical properties are found. Relatively, from the perspective of suppressing the maximum amplitude and controlling the resonance stability, the selection of the feedback parameters is discussed. Finally, the symmetry-breaking bifurcation and chaotic motion are considered.

Laplace distribution models for road topography and roughness

Gaussian models are frequently used for road profiles. However, these models are often only valid for short sections of the road. Here we present a comprehensive approach to describe various aspects of road surface/elevation by using extensions of Gaussian models arising from random gamma distributed variances. These random variances result in the Laplace distribution and thus we refer to the so defined models as Laplace models. The approach is shown to perform well in modelling road topography, road roughness and multi-valued responses of forces and bending moments containing transients. The different Laplace models are presented together with numerical examples and Matlab code for simulation.

Regression trees for multivalued numerical response variables

In the framework of regression trees, this paper provides a recursive partitioning methodology to deal with a non-standard response variable. Specifically, either multivalued numerical or modal response of the type histogram will be considered. These data are known as symbolic data, which special cases are classical data, imprecise data, conjunctive data as well as fuzzy data. In spite of pre-processing data in order to deal with standard regression tree methodology, this paper provides, as main contribution, a definition of the impurity measure and of the splitting criterion allowing for building the regression tree for multivalued numerical response variable. We analyze and evaluate the performance of our proposal, using simulated data as well as a real-world case studies.

On the analysis of a high-static-low-dynamic stiffness vibration isolator with time-delayed cubic displacement feedback

The cubic displacement feedback with time delay is proposed in this paper to improve the isolation performance of the high-static-low-dynamic stiffness (HSLDS) vibration isolator. First of all, the frequency response of the controlled HSLDS vibration isolator is obtained by using the multiple scales method and further verified by numerical simulation. Multi-valued responses, jump phenomenon and frequency island could coexist and their stabilities are analysed. Then, the effects of feedback gain and time delay on the frequency response are studied. The jump avoidance condition is obtained with the purpose of eliminating the adverse effects of excitation frequency fluctuation. Finally, the force transmissibility is defined to evaluate the isolation performance of the controlled HSLDS vibration isolator. The results show that with feedback parameters properly designed, such feedback control can play the role of damping force and the controlled HSLDS vibration isolator outperforms the passive counterpart.

Multi-valued responses and dynamic stability of a nonlinear vibro-impact system with a unilateral non-zero offset barrier**Project supported by the National Natural Science Foundation of China (Grant Nos. 11472212, 11532011, 11302171, and 11302172).

In this paper, multi-valued responses and dynamic properties of a nonlinear vibro-impact system with a unilateral nonzero offset barrier are studied. Based on the Krylov–Bogoliubov averaging method and Zhuravlev non-smooth transformation, the frequency response, stability conditions, and the equation of backbone curve are derived. Results show that in some conditions impact system may have two or four steady-state solutions, which are interesting and not mentioned for a vibro-impact system with the existence of frequency island phenomena. Then, the classification of the steady-state solutions is discussed, and it is shown that the nontrivial steady-state solutions may lose stability by saddle node bifurcation and Hopf bifurcation. Furthermore, a criterion for avoiding the jump phenomenon is derived and verified. Lastly, it is found that the distance between the system’s static equilibrium position and the barrier can lead to jump phenomenon under hardening type of nonlinearity stiffness.

Responses and energy transmissibility of a viscoelastic isolation system with a power-form restoring force under delayed feedback control

In this paper, combination of cubic nonlinearity and time delay is designed to improve the performance of a viscoelastic isolation system with a power-form restoring force. By the method of multiple scales, the amplitude-frequency response, stability, backbone curve and energy transmissibility are considered. More specifically, three nonlinear cubic delayed feedback control methodologies are examined: position, velocity and acceleration delayed feedback. It is found that the viscoelastic damping coefficient can induce multi-valued response, especially frequency island phenomenon. In this regard, the isolation system indicates the softening behavior for under-linear restoring force and hardening behavior for over-linear restoring force. And equivalent damping and jump avoidance condition are first proposed to interpret the effect of feedback control loop on dynamical behaviors. Furthermore, with the purpose of improving the stability and reducing the vibration, suitable feedback parameter pairs are determined by the frequency response together with stability conditions. Finally, the vibration isolation property is predicted based on energy transmissibility in different cases. Results show that the strategy proposed in this paper is practicable and feedback control parameters are significant factors to alter dynamical behaviors, and more importantly, to improve the isolation effectiveness for the viscoelastic isolation system.

Principal resonance response of a stochastic elastic impact oscillator under nonlinear delayed state feedback**Project supported by the National Natural Science Foundation of China (Grant Nos. 11172233, 11302170, and 11302172).

In this paper, the principal resonance response of a stochastically driven elastic impact (EI) system with time-delayed cubic velocity feedback is investigated. Firstly, based on the method of multiple scales, the steady-state response and its dynamic stability are analyzed in deterministic and stochastic cases, respectively. It is shown that for the case of the multi-valued response with the frequency island phenomenon, only the smallest amplitude of the steady-state response is stable under a certain time delay, which is different from the case of the traditional frequency response. Then, a design criterion is proposed to suppress the jump phenomenon, which is induced by the saddle-node bifurcation. The effects of the feedback parameters on the steady-state responses, as well as the size, shape, and location of stability regions are studied. Results show that the system responses and the stability boundaries are highly dependent on these parameters. Furthermore, with the purpose of suppressing the amplitude peak and governing the resonance stability, appropriate feedback gain and time delay are derived.