Steady-state Response Amplitude Research Articles

Nonlinear Phase Angle and Amplitude Analysis of Porous Functionally Graded Material Rotating Shaft Under Electromagnetic Loads

In this research, the nonlinear response including amplitude and phase angle of a functionally graded inextensional rotating shaft is studied under the electromagnetic load. Three types of porosity distributions through the radial direction are considered in this work. For the rotating shaft with nonlinear curvature and inertia, the governing equations and corresponding boundary conditions are derived. The Galerkin and the multiple scales methods are used to obtain the modulation equations. The effect of the power law index for a functionally graded shaft fabricated from the mixture of ceramic and metal is presented in the frequency response diagrams for primary resonance under electromagnetic force. Numerical simulations include the effects of the power law index of functionally graded material (FGM), porosity distribution, air-gap length, electromagnetic load, and vibration modes of a rotor on the phase angle and amplitude of steady state responses.

Auditory neuropathy in mice and humans with Friedreich ataxia.

Recent studies have found that human Friedreich ataxia patients have dysfunction of transmission in the auditory neural pathways. Here, we characterize hearing deficits in a mouse model of Friedreich ataxia and compare these to a clinical population. Sixteen mice with a C57BL/6 background were evaluated. Eight were YG8Pook/J animals (Friedreich ataxia phenotype) and eight wild-type mice served as controls. Auditory function was assessed between ages 6 and 12 months using otoacoustic emissions and auditory steady-state responses. At study end, motor deficit was assessed using Rotorod testing and inner ear tissue was examined. Thirty-seven individuals with Friedreich ataxia underwent auditory steady-state evoked potential assessment and response amplitudes were compared with functional hearing ability (speech perception-in-noise) and disease status was measured by the Friedreich Ataxia Rating Scale. The YG8Pook/J mice showed anatomic and functional abnormality. While otoacoustic emission responses from the cochlear hair cells were mildly affected, auditory steady-state responses showed exaggerated amplitude reductions as the animals aged with Friedreich ataxia mice showing a 50-60% decrease compared to controls who showed only a 20-25% reduction (F(2,94) = 17.90, p < 0.00). Furthermore, the YG8Pook/J mice had fewer surviving spiral ganglion neurons, indicating greater degeneration of the auditory nerve. Neuronal density was 20-25% lower depending on cochlear region (F(1, 30) = 45.02, p < 0.001). In human participants, auditory steady-state response amplitudes were correlated with both Consonant-Nucleus-Consonant word scores and Friedreich Ataxia Rating Scale score. This study found degenerative changes in auditory structure and function in YG8Pook/J mice, indicating that auditory measures in these animals may provide a model for testing Friedreich ataxia treatments. In addition, auditory steady-state response findings in a clinical population suggested that these scalp-recorded potentials may serve as an objective biomarker for disease progress in affected individuals.

An increase in the auditory steady-state response amplitudes after a period of listening to binaural beat stimuli in tinnitus patients: a pilot study

BackgroundTinnitus impact on persons’ lifestyle, function, and emotion is of significant importance that has been the leader for conducting an increasing amount of research in the field of tinnitus pathophysiology, assessment, and management. Binaural beats (BB) are one of acoustic neuromodulation approaches used in psychological disorders, such as distress and anxiety. Thus, we hypothesized that binaural beat could be helpful in the relief of tinnitus distress and annoyance.MethodsSeventeen chronic tinnitus subjects participated in this quasi-experimental (quantitative research) study. In this study, the effect of binaural beat stimuli was evaluated subjectively using the tinnitus handicap inventory (THI) scores, the visual analog scale for loudness and annoyance (VAS_L, VAS_A), and objectively by the 40-Hz ASSR after 1 month of listening to binaural beats, and the correlation between these two assessments was evaluated.ResultsAfter 1 month of binaural beat stimuli listening, all of the subjective findings were significantly improved, and the amplitude of 40-Hz ASSR was increased in the right auditory and anterior frontal regions at 2000-Hz carrier frequency. Besides, there was a high correlation between the decreasing of the subjective scores with the rising of the amplitude of 40-Hz ASSR.ConclusionThe use of binaural beat as an acoustic neuromodulation method for tinnitus management may be recommended according to the current study findings. However, more investigations on the effectiveness supported by data from controlled clinical trials and more correlations with ASSR alteration are highly suggested.

Escape of a circular cylinder from a potential well via nonlinear vortex-induced vibrations: An experimental investigation

In this study, the basic characteristics of escape from a potential well is investigated experimentally for a circular cylinder undergoing vortex-induced vibrations for Reynolds number ranging between 1000 and 5000. The experimental system consisted of a circular cylinder suspended by two elastic beams and allowed to oscillate in the vicinity of two attracting magnets. The shape of the system’s restoring force and the height of the potential energy barrier are controlled by changing the distance between the magnets and the oscillating cylinder. Escape occurs when the oscillating cylinder overcomes the potential barrier and collapses onto one of the attracting magnets. It is shown that, escape can occur in a range of flow velocities bounded by two speeds, U1<U<U2. At U1, escape occurs due to a sudden jump in the steady-state response amplitude when the flow speed is quasi-statically increased from zero towards higher values. At U2, escape is associated with a gradual increase in the steady-state response amplitude when the flow speed is quasi-statically decreased from high values. The range of flow speed for which escape can occur increases with the system’s effective nonlinearity and decreases with the mass ratio (ratio between the mass of the cylinder and the displaced fluid). At a critical value of the nonlinearity, U1 and U2 merge into a single point causing escape to occur at a single speed regardless of the direction of the flow speed sweep. For values of the nonlinearity below this critical value, escape can only occur at U2. Finally, it is shown that, results of the experiment do not change based on whether the flow speed is quasi-statically or abruptly varied. This is because no overshoot was observed in the transient response of the cylinder when the flow speed was abruptly changed.

The role of medial olivocochlear activity in contralateral suppression of auditory steady-state responses

The auditory steady-state response (ASSR) amplitudes fall in the presence of contralateral noise. However, whether and to what extent medial olivocochlear (MOC) activity involves in contralateral suppression of ASSR remain unclear. Therefore, we assess the role of MOC activity in contralateral suppression of ASSR. Mice were treated with strychnine to completely eliminate MOC activity and then measured ASSR amplitudes in the presence of contralateral noise. The contralateral noise reduces ASSR amplitudes at some stimulus intensity. After treating with the strychnine to eliminate MOC activity, ASSR amplitudes recovered again. MOC activity participated in contralateral suppression of ASSR.

Transverse and longitudinal coupled nonlinear vibration of an axially moving Euler beam under variable tension

When describing the vibration characteristics of the transverse parameters of an axially moving beam, the effects of nonlinear factors need to be introduced. Under normal circumstances, it is difficult to solve the nonlinear vibration equations of horizontal and vertical coupling with mixed partial derivative terms of time and space using approximate analytical methods. The longitudinal displacement in the vibration process is very small and much smaller than the lateral displacement, so the equation is simplified to a nonlinear vibration governing equation only about the lateral displacement. However, in these studies, no specific numerical basis for the reason for the simplification is given. Therefore, this paper considers the coupling model of the Euler beam’s lateral and longitudinal vibrations, chooses the Galerkin truncation method to decouple and numerically solve it, and obtains the numerical solution of the lateral and longitudinal vibration displacements. Based on the calculation results of the direct multi-scale method and the differential quadrature method of the simplified model, and the solution results of the eight-order Galerkin truncation method of the coupled model, the steady-state response amplitudes of the coupled model and the simplified model are compared to verify the accuracy of the simplified model Therefore, it provides a strong theoretical basis for the simplified model that ignores the longitudinal vibration.

Experimental amplitude and frequency control of a self-excited microcantilever by linear and nonlinear feedback

It is well known that the micro scale deviations of mechanical properties of a sample can be detected by measurement methods that use microcantilever as resonators. Those methods use the natural frequency shift of a resonator, thus we need to recognize the frequency shift caused by the effects of a sample on a resonator with high sensitivity and accuracy. Experimental approaches based on self-excited oscillation enable the detection of these shifts even when the resonator is immersed in a high-viscosity environment. In the present study, we experimentally and theoretically investigate the nonlinear characteristics of a microcantilever resonator and their control by nonlinear feedback. We show that the steady-state response amplitude and the corresponding response frequency can be controlled by cubic nonlinear velocity feedback and cubic nonlinear displacement feedback, respectively. Furthermore, the amplitude and frequency of the steady-state self-excited oscillation can be controlled separately. These results will expand application of measurement methods that use self-excited resonators.

Effect of tinnitus distress on auditory steady-state response amplitudes in chronic tinnitus sufferers

Tinnitus is a bothersome disorder of primarily unknown etiology that affects a large number of people worldwide. Tinnitus distress is the most common clinical complaint by tinnitus sufferers because it strongly affects their personal and social life. Many studies have been carried out to determine the relation between tinnitus pathophysiology and electrophysiological findings such as the auditory steady-state response (ASSR). The results of such studies have been contradictory. The current study aimed to detect a possible relation between tinnitus distress and ASSR amplitudes. The tinnitus participants were divided into high and low distress subgroups according to their tinnitus handicap inventory (THI) scores. The ASSR stimuli were carrier frequencies with low (500 Hz), mid (2000 Hz), and high (4000 Hz) amplitude-modulated tones. ASSR amplitudes were calculated in anterio-frontal (F3, Fz, F4), centro-frontal (FC3, FCz, FC4), left auditory (T3, C5, C3) and right auditory (C4, T4, C6) regions of interest (ROI). Twenty-four right-handed subjects with non-pulsatile chronic tinnitus and 23 normal matched participants participated in this study. For recording ASSR amplitudes were used from 32-electrode EEG recording. Two-way repeated-measurement ANOVA was used to compare the ASSR amplitudes. The findings showed that the ASSR amplitudes in the tinnitus group with low distress were higher (better) than in the group with high distress (p < 0.001). This finding was seen in anterio-frontal and right auditory regions and at all carrier frequencies. The results indicated that there is a relation between the ASSR amplitude and the degree of tinnitus distress as measured by the THI questionnaire.

Mismatch negativity as an index of target engagement for excitation/inhibition-based treatment development: a double-blind, placebo-controlled, randomized, single-dose cross-over study of the serotonin type-3 receptor antagonist CVN058.

Serotonin type-3 receptor (5-HT3R) antagonists show potential as a treatment for cognitive deficits in schizophrenia. CVN058, a brain-penetrant, potent and selective 5-HT3R antagonist, shows efficacy in rodent models of cognition and was well-tolerated in Phase-1 studies. We evaluated the target engagement of CVN058 using mismatch negativity (MMN) in a randomized, double-blind, placebo-controlled, cross-over study. Subjects were stable outpatients with schizophrenia or schizoaffective disorder treated with antipsychotics. Subjects were not permitted to use other 5-HT3R modulators or serotonin reuptake inhibitors. Each subject received a high (150 mg) and low (15 mg or 75 mg) oral dose of CVN058 and placebo in a randomized order across 3 single-day treatment visits separated by at least 1 week. The primary pre-registered outcome was amplitude of duration MMN. Amplitude of other MMN deviants (frequency, intensity, frequency modulation, and location), P50, P300 and auditory steady-state response (ASSR) were exploratory endpoints. 19 of 22 randomized subjects (86.4%) completed the study. Baseline PANSS scores indicated moderate impairment. CVN058 150 mg led to significant improvement vs. placebo on the primary outcome of duration MMN (p = 0.02, Cohen's d = 0.48). A significant treatment effect was also seen in a combined analysis across all MMN deviants (p < 0.001, d = 0.57). Effects on location MMN were independently significant (p < 0.007, d = 0.46). No other significant effects were seen for other deviants, doses or EEG measures. There were no clinically significant treatment related adverse effects. These results show MMN to be a sensitive target engagement biomarker for 5-HT3R, and support the potential utility of CVN058 in correcting the excitatory/inhibitory imbalance in schizophrenia.

Fractional-order harmonic resonance in a multi-frequency excited fractional Duffing oscillator with distributed time delay

The general investigation on the vibration mechanism at arbitrary fraction-order harmonic resonance of the fractional Mathieu-Duffing system driven by multiple periodic excitations with distributed delay is reported theoretically and numerically. In terms of numerical scheme, a precise numerical algorithm based on the definition of Caputo derivative is applied. The unified analytical results of the steady-state response amplitude at each resonant frequency are obtained by utilizing the method of direct partition of motions, the self-consistent technique and the Krylov-Bogolubov-Mitropolsky asymptotic method. It relies on two kinds of parametric-forced pattern. It is found that for some parametric setting, both the amplitude of the external high-frequency excitation and the frequency of the external low-frequency excitation can give rise to saddle-node bifurcation of the steady-state amplitude in the analytical results. It is equivalent to amplitude jump phenomenon in the numerical observation. While comparing to the integer-order system, it is more likely to happen in the fractional-order situation. With the variant of the fractional order of the derivative, the steady-state amplitude solutions can be converted from monostability to bistability, then to monostability, and finally to the bistability again. This new multi-saddle-node bifurcation has not yet been reported before. The novel transcritical bifurcation induced by the strength of the distributed delay is discussed and verified in detail for the first time.

Do Room Acoustics Affect the Amplitude of Sound-Field Auditory Steady-State Responses?

The sound-field auditory steady-state response (ASSR) is a promising measure for the objective validation of hearing-aid fitting in patients who are unable to respond to behavioral testing reliably. To record the sound-field ASSR, the stimulus is reproduced through a loudspeaker placed in front of the patient. However, the reverberation and background noise of the measurement room could reduce the stimulus modulation used for eliciting the ASSR. As the ASSR level is heavily dependent on the stimulus modulation, any reduction due to room acoustics could affect the clinical viability of sound-field ASSR testing. This study investigated the effect of room acoustics on the level and detection rate of sound-field ASSR. The study also analyzed whether early decay time and an auditory-inspired relative modulation power model could be used to predict the changes in the recorded ASSR in rooms. A monaural auralization approach was used to measure sound-field ASSR via insert earphones. ASSR was measured for 15 normal-hearing adult subjects using narrow-band CE-Chirps® centered at the octave bands of 500, 1000, 2000, and 4000 Hz. These stimuli were convolved with simulated impulse responses of three rooms inspired by audiological testing rooms. The results showed a significant reduction of the ASSR level for the room conditions compared with the reference anechoic condition. Despite this reduction, the detection rates for the first harmonics of the ASSR were unaffected when sufficiently long recordings (up to 6 min) were made. Furthermore, the early decay time and relative modulation power appear to be useful predictors of the ASSR level in the measurement rooms.

Free and Forced Vibration Analysis of H-type and Hybrid Vertical-Axis Wind Turbines

Vertical-axis wind turbines (VAWTs) are compact and efficient and have become increasingly popular for wind energy harvesting. This paper mainly focuses on free and forced vibration analysis of two different types of VAWTs, i.e., an H-type VAWT and a new hybrid VAWT. The H-type VAWT has a lower cost, while the hybrid VAWT has a better self-starting capability at a low wind velocity. Both of them can be used for wind energy harvesting. By using the assumed modes method, the two VAWTs are simplified by a single degree-of-freedom (SDOF) model. By utilizing the method of structural mechanics, a multi-degree-of-freedom (MDOF) model is developed for the two VAWTs and the turbines in them are reasonably simplified. Natural frequency analyses for the SDOF and MDOF models of the two VAWTs are conducted. A beam element model (BEM) of the two VAWTs is created to calculate their natural frequencies and mode shapes and to verify natural frequency results from the SDOF and MDOF models. By using the BEM of the two VAWTs, their amplitude-frequency responses are obtained from harmonic response analysis. To analyze forced vibrations of the two VAWTs, aerodynamic loads on the two VAWTs are obtained from computational fluid dynamics (CFD) simulation. By using solid element models of the two VAWTs, forced transient responses of the two VAWTs are calculated by using the aerodynamic loads from CFD simulation. Steady-state forced response amplitudes of the 1 m-mast hybrid VAWT are 23.8% and 20.5% smaller in X- and Y-directions than those of the 1 m-mast H-type VAWT, respectively. Frequency contents of the aerodynamic loads from CFD simulation are calculated, which confirm that they are periodic, and the power efficiency of the H-type VAWT is about 2.6% higher that of the hybrid VAWT.

Parameter Identification of Structural Nonlinearity by Using Response Surface Plotting Technique

With rigorous dynamic performance of mechanical products, it is important to identify dynamic parameters exactly. In this paper, a response surface plotting method is proposed and it can be applied to identify the dynamic parameters of some nonlinear systems. The method is based on the principle of harmonic balance method (HBM). The nonlinear vibration system behaves linearly under the steady-state response amplitude, which presents the equivalent stiffness and damping coefficient. The response surface plot is over two-dimensional space, which utilizes excitation as the vertical axis and the frequency as the horizontal axis. It can be applied to observe the output vibration response data. The modal parameters are identified by the response surface plot as linearity for different excitation levels, and they are converted into equivalent stiffness and damping coefficient for each resonant response. Finally, the HBM with first-order expansion is utilized for identification of stiffness and damping coefficient of nonlinear systems. The classical nonlinear systems are applied in the numerical simulation as the example, which is used to verify its effectiveness and accuracy. An application of this technique for nonlinearity identification by experimental setup is also illustrated.

Characterization of interstimulus interaction in the multiple auditory steady-state responses at high sound levels.

Multiple auditory steady-state response (MASSR) is recommended to estimate hearing thresholds in difficult-to-test individuals. The multiple stimuli that evoke MASSR may present an interstimulus interaction (ISI) that is able to distort the generation of responses. No consensus exists on the effects of the ISI in MASSR when dealing with high sound level stimuli or cases of sensorineural hearing loss. This study investigated the effects of ISI on the amplitude and detectability of auditory steady-state responses, with a focus at and above 65 dB sound pressure level (SPL). Normal hearing (NH) and sensorineural hearing impaired (SNHI) adults were tested with different stimulus types [amplitude modulation (AM) One octave chirp (OC), and a weighted OC (WOC)], stimulus levels, and modalities (single or multiple stimuli). ISI typically attenuated response amplitude of a control stimulus caused by an interference stimulus one octave above the control stimulus. At and above 80 dB SPL, attenuations of around 50% decreased the number of detectable responses near SNHI thresholds, especially for OC and WOC. AM stimuli obtained a higher detection rate than OC and WOC when presented 10 dB above the behavioral hearing threshold of SNHI participants. Using OC in MASSR when assessing elevated thresholds might diminish accuracy on threshold estimation, and extend test duration.

Phase-specific manipulation of rhythmic brain activity by transcranial alternating current stimulation

BackgroundOscillatory phase has been proposed as a key parameter defining the spatiotemporal structure of neural activity. To enhance our understanding of brain rhythms and improve clinical outcomes in pathological conditions, modulation of neural activity by transcranial alternating current stimulation (tACS) emerged as a promising approach. However, the phase-specificity of tACS effects in humans is still critically debated. ObjectiveHere, we investigated the phase-specificity of tACS on visually evoked steady state responses (SSRs) in 24 healthy human participants. MethodsWe used an intermittent electrical stimulation protocol and assessed the influence of tACS on SSR amplitude in the interval immediately following tACS. A neural network model served to validate the plausibility of experimental findings. ResultsWe observed a modulation of SSR amplitudes dependent on the phase shift between flicker and tACS. The tACS effect size was negatively correlated with the strength of flicker-evoked activity. Supported by simulations, data suggest that strong network synchronization limits further neuromodulation by tACS. Neural sources of phase-specific effects were localized in the parieto-occipital cortex within flicker-entrained regions. Importantly, the optimal phase shift between flicker and tACS associated with strongest SSRs was correlated with SSR phase delays in the tACS target region. ConclusionsOverall, our data provide electrophysiological evidence for phase-specific modulations of rhythmic brain activity by tACS in humans. As the optimal timing of tACS application was dependent on cortical SSR phase delays, our data suggest that tACS effects were not mediated by retinal co-stimulation. These findings highlight the potential of tACS for controlled, phase-specific modulations of neural activity.

Study on Steady-State Responses of High-Speed Vehicle Using Infinite Long Track Model

The coupled vehicle/track dynamic model is formulated through integrating a high-speed rail vehicle model with a slab track model via the wheel/rail contact model. The sliding window method is improved using the least square criterion to simulate the vehicle travelling along the infinite long track. The steady-state responses of a high-speed vehicle induced by the discrete sleepers and slab segments are investigated through numerical simulation and analysis of the experimental results. Also the validity of the coupled vehicle/track model is examined through comparing the simulation results with those acquired from field test measurements. The experimental and numerical results show that the wheel/rail contact forces fluctuate considerably as long as the sleeper passing frequency approaches the frequency of P2 resonance (wheelset and rail bouncing in phase on the slab). Increasing the damping of rail pads and primary suspension can lower the steady-state response amplitudes at the resonance region. The oscillations in the wheel/rail normal forces arising from the discrete slab segment excitation can be reduced by increasing the support stiffness of the CAM (cement asphalt mortar) layer under the slab.

Simultaneous vortex- and wake-induced vibration suppression of tandem-arranged circular cylinders using active feedback control system

This paper deals with the active flow-induced vibration (FIV) suppression of two identical tandem-arranged circular cylinders at a low Reynolds number of Re=100. The adopted control system employs fuzzy PID controller in which the desired control force is computed based on the cylinder displacement amplitude, and applied transversely for effective attenuation of flow-induced instabilities in both transverse and streamwise directions. Moreover, the effect of FIV suppression of each cylinder on the response amplitude of other one, and the simultaneous vibration reduction of both cylinders are investigated based on fully-coupled fluid-structure interaction (FSI) simulations. The rigid body motion equations in streamwise and transverse directions are incorporated into the computational fluid dynamics solver to treat the coupling which exists between the fluid flow and cylinders movement. Next, in the utilized co-simulations, the active fuzzy PID controller implemented in Matlab/Simulink is coupled with the FSI oscillator model constructed in computational fluid dynamics solver package. It was shown that, the sudden suppression of the upstream cylinder vibration causes quick distortion in the flow field around the rear cylinder, and consequently, its steady-state response amplitude is destabilized and experiences transient fluctuations once again. On the contrary, the decrease in the transverse oscillation of the downstream cylinder leaves no sensible effect on the vibration of upstream cylinder. In particular, the active control system effectively reduces the transverse displacement amplitudes of upstream and downstream cylinders by as much as 99.9% and 98.7%, respectively, for Vr=5.5, while the corresponding values are 99.8% and 99.7% for Vr=7.5. This superb performance of active FIV control systems is mainly due to de-synchronization-type action in which the vortex shedding frequency is deviated from the oscillator natural frequency. Lastly, it is demonstrated that the vortex shedding patterns for uncontrolled cylinders, which primarily depend on reduced velocity, are turned very similar to that of the stationary cylinders after adopting the fully active control systems.

Analytical approach to coupled bending-torsional vibrations of cracked Timoshenko beam

The explicit expressions of steady-state responses of a coupled bending-torsional Timoshenko beam with cracks and damping effect subjected to external harmonic loadings are presented in this paper. The Green's functions method is employed to obtain the analytical solutions. The beam is split to several segments due to the existence of cracks. General Green's functions of each segment with unknown boundary constants are given by using Laplace transform method. A mixture line-spring is introduced to obtain the compatibility conditions of the cracked cross-section between the segments. Based on the transfer matrix method, those constants in the Green's functions for each segment are determined with those matching conditions, as well as boundary relationships and end boundary conditions. Then, the solutions of the whole cracked beam with one or multiple cracks can be expressed in terms of the piecewise functions. The solutions of the multi-cracked beam cover those of the beam with only one crack and the uncracked beam. The theory developed can be used for the beam with classical boundary conditions, but for illustration, results for a cantilever cracked beam with T shape cross-section are illustrated, emphasizing the effect of cracks on the solutions. Comparisons between results in the published literature and those obtained from the developed method show a good agreement, which confirms the validity and accuracy of the proposed approach. The influences of crack depth, location and number on the natural frequencies are discussed. The changes of the steady-state responses of beam are investigated due to the existence of crack. Moreover, the symmetric property of the Green's functions and damping effect on the amplitude of steady-state responses of the cracked beam are studied particularly.

Nonlinear Parametric Vibration of a Fluid-Conveying Pipe Flexibly Restrained at the Ends

In this paper, the nonlinear parametric vibration of fluid-conveying pipes flexibly restrained at both ends and subjected to the pulsation flow excitation is investigated. The nonlinear equation of motion is derived using Hamilton’s principle by considering the Kevin–Voigt viscoelastic damping, the geometric nonlinearity and the translational and rotational springs supported at the ends. The mode functions and eigen-frequencies are determined by the assumed mode method according to the elastic boundary conditions. The Galerkin method is implemented to obtain the natural frequencies and mode shapes of the pipe conveying fluid with different flow velocities. The effects of flexibly restrained conditions on stability of the pipe are analyzed. The nonlinear responses of the pipe under pulsating flow excitation are solved by the direct numerical method. The vibration behaviors are discussed in details, such as time history, frequency spectrum, phase-plane portrait, Poincare map and motion trajectory. The results show that the responses of sub-harmonic resonance and combination resonance can also be reflected in the rigidly supported pipes. The 1/5, 1/8 and 1/13 sub-harmonic resonances can occur at certain excitation frequencies of the nonlinear parametric vibration system. The steady-state response amplitudes increase by a large margin and significantly affect the stability of the pipe. The effects of different spring stiffness coefficients on the parametric resonance responses are presented. For larger translational springs and rotational stiffness coefficients, the resonance frequencies shift to higher regions and the resonance amplitudes may reduce by a certain extent in accordance with the rigid-body motion. This study can provide helpful guidance on the analysis and design of piping systems subject to vibrations.

Dynamic Modeling and Characteristic Analysis of Cantilever Piezoelectric Bimorph

The analytical model of an axially precompressed cantilever bimorph is established using the Hamilton’s principle in this study, and the static characteristics are obtained. The dynamic equations of the cantilever bimorph in generalized coordinates are established using a numerical method, and the dynamic characteristics are analyzed. Finally, simulations are performed and experiments are conducted to verify the validity of the theory. The results show that increase of axial force has significant amplification effects on the steady-state response amplitude of the displacement, and it reduces the resonance frequency. The response time is still in the millisecond range under a large axial force, which indicates that the bimorph has excellent dynamic characteristics as an actuator.