We numerically investigate the bandwidth and collimation characteristics of ultrasound beams generated by a simple collimated ultrasound beam source that consists of a piezoelectric disk operated near its radial mode resonances. We simulate the ultrasound beam generated in a fluid medium as a function of the excitation frequency for two cases: 1) free piezoelectric disk that corresponds to zero-traction along the lateral edge, and 2) fixed piezoelectric disk that corresponds to zero-displacement along the lateral edge. We present and discuss the physical mechanism underpinning the frequency-dependent collimation and bandwidth properties of the ultrasound beams. We observe that the collimated beam generated by the free disk repeatedly lengthens/shortens and also extends/retracts sidelobes with increasing frequency. Alternatively, fixing the piezoelectric disk results in a consistent beam profile shape across a broad range of frequencies. This facilitates generating broadband signals such as a Gaussian pulse or chirp, which are common in ultrasound imaging. Thus, the fixed piezoelectric disk finds application as a collimated ultrasound beam source in a wide range of applications including medical ultrasound imaging, scanning acoustic microscopy, sonar detection, and other nondestructive ultrasound inspection techniques.

In this paper, free and forced vibrations of a transversely vibrating Timoshenko beam/frame carrying a discrete two-degree-of-freedom spring-mass system are analyzed using the wave vibration approach, in which vibrations are described as waves that propagate along uniform structural elements and are reflected and transmitted at structural discontinuities. From the wave vibration standpoint, external excitations applied to a structure have the effect of injecting vibration waves to the structure. In the combined beam/frame and two-degree-of-freedom spring-mass system, the vibrating discrete spring-mass system injects waves into the distributed beam/frame through the spring forces at the two spring attached points. Assembling the propagation, reflection, transmission, and external force injected wave relations in the beam/frame provides an analytical solution to vibrations of the combined system. In this study, the effects of rotary inertia and shear deformation on bending vibrations are taken into account, which is important when the combined structure involves short beam element or when higher frequency modes are of interest. Numerical examples are given, with comparisons to available results based on classical vibration theories. The wave vibration approach is seen to provide a systematic and concise solution to both free and forced vibration problems in hybrid distributed and discrete systems.

This paper depicts the application of symbolically computed Lyapunov–Perron (L–P) transformation to solve linear and nonlinear quasi-periodic systems. The L–P transformation converts a linear quasi-periodic system into a time-invariant one. State augmentation and the method of normal forms are used to compute the L–P transformation analytically. The state augmentation approach converts a linear quasi-periodic system into a nonlinear time-invariant system as the quasi-periodic parametric excitation terms are replaced by “fictitious” states. This nonlinear system can be reduced to a linear system via normal forms in the absence of resonances. In this process, one obtains near identity transformation that contains fictitious states. Once the quasi-periodic terms replace the fictitious states they represent, the near identity transformation is converted to the L–P transformation. The L–P transformation can be used to solve linear quasi-periodic systems with external excitation and nonlinear quasi-periodic systems. Two examples are included in this work, a commutative quasi-periodic system and a non-commutative Mathieu–Hill type quasi-periodic system. The results obtained via the L–P transformation approach match very well with the numerical integration and analytical results.

Structures possessing cyclic symmetry such as turbine bladed disks, ultrasonic motors, and toothed gear wheels can experience elevated vibration levels when small deviations from circumferential periodicity exist. Detection of these perturbations via classical system identification approaches is time-consuming, indirect, and exhibits low sensitivity to defects, and is affected by measurement noise. The present work utilizes low-level forces that automatically lock onto a weighted rotating projection of the system modes at resonance frequency to enhance the detectability of small structural imperfections. The spatial localization of defects is exploited to identify multiple, localized, isolated defects’ locations. The defects’ severities are estimated based on the deviation from the circular structure's analytical mode shapes. The fast and enhanced precision of defect identification is obtained by employing the modal-filtered autoresonance technique. To validate the presented method, an experimental system consisting of a ring of coupled Helmholtz acoustic resonators was developed. Experimental results show good agreement with numerical simulations, verifying the method's capabilities to identify the location and severity of multiple defects. Thus, the implementation of the suggested method provides fast and precise structural health monitoring of cyclic-symmetric systems.

Significant handle vibrations often occur during mowing operation even with anti-vibration type brush cutters. This is often caused by combined-bending natural mode of the main pipe and driveshaft which is mainly excited by cutting head rotational force. In this study, we focused on the placement span of the rubber bushings that support the driveshaft to suppress this kind of resonance. More specifically, we have designed a new component, called a span-tuning dynamic vibration absorber (ST-DVA), which utilizes the bending mode of the driveshaft that is determined by the placement span of rubber bushings. Analysis results of the finite element method (FEM) showed that the ST-DVA generated anti-resonance at a specific point on the main pipe under the first-order inertial force of the cutting head. We also succeeded in controlling anti-resonance frequency under the excitation. In actual measurements at the target frequency, handle vibration of the first-order component of the cutting head could be reduced by 51% and overall handle vibration could be reduced by 49% compared with those produced via equal-span rubber bushing placement. Hence, our study provides a design method that makes it possible to utilize the driveshaft, which a primary brush cutter component, as a dynamic vibration absorber by altering the placement span of the rubber bushings.

The dynamic interaction between a bridge and a moving train has been widely studied. However, there is a significant gap in our understanding of how the presence of isolation bearings influences the dynamic response, especially when an earthquake occurs. Here, we formulate a coupled model of a train-bridge-bearing system to examine the bearings’ dynamic effects on the system responses. In the analysis, the train is modeled as a moving oscillator, the bridge is a one span simply supported beam and one isolation bearing is installed under each support of the bridge. A mathematical model using fractional derivatives is used to capture the viscoelastic properties of the bearings. The vertical response is the focus of this investigation. Dynamic substructuring is used in modeling to efficiently capture the coupled dynamics of the entire system. Illustrative numerical simulations are carried out to examine the effects of the bearings. The results demonstrate that although the presence of bearings typically decreases the bridge seismic responses, there is a potential to increase the bridge response induced by the moving train.

Due to the great progresses in the fields of smart structures, especially smart soft materials and structures, the parametric control of nonlinear systems attracts extensive attentions in scientific and industrial communities. This paper devotes to the derivation of the optimal parametric control strategy for nonlinear random vibrating systems, in which the excitations are confined to Gaussian white noises. For a prescribed performance index balancing the control performance and control cost, the stochastic dynamic programming equation with respect to the value function is first derived by the principle of dynamic programming. The optimal feedback control law is established according to the extremum condition. The explicit expression of the value function is determined by approximately expressing as a quadratic function of state variables and by solving the final dynamic programming equation. The application and efficacy of the optimal parametric control are illustrated by a random-excited Duffing oscillator and a dielectric elastomer balloon with random pressure. The numerical results show that the optimal parameter control possesses good effectiveness, high efficiency, and high robustness to excitation intensity, and is superior than the associated optimal bounded parametric control.

The Floquet theory has been classically used to study the stability characteristics of linear dynamic systems with periodic coefficients and is commonly applied to Mathieu’s equation, which has parametric stiffness. The focus of this article is to study the response characteristics of a linear oscillator for which the damping coefficient varies periodically in time. The Floquet theory is used to determine the effects of mean plus cyclic damping on the Floquet multipliers. An approximate Floquet solution, which includes an exponential part and a periodic part that is represented by a truncated Fourier series, is then applied to the oscillator. Based on the periodic part, the harmonic balance method is used to obtain the Fourier coefficients and Floquet exponents, which are then used to generate the response to the initial conditions, the boundaries of instability, and the characteristics of the free response solution of the system. The coexistence phenomenon, in which the instability wedges disappear and the transition curves overlap, is recovered by this approach, and its features and robustness are examined.

The simulation of the coupling between components modeled by finite elements (FEs) plays an important role for the prediction of the forced response of the assembly in terms of resonant frequencies, vibration amplitudes, and damping. This is particularly critical when the time-varying stress distribution must be limited for vibrating components with thin thickness coupled with large contacts. Typical examples can be found in aeronautical structures (plates, panels, and bladed disk components) assembled with bolted flanges, riveted lap joints, or joints without hole discontinuities like rail-hook joints, lace wire sealings, and strip dampers. In this paper, a new test rig is introduced for the experimental validation of a reduced-order model (ROM) based on the Gram–Schmidt Interface (GSI) modes applied to a friction contact whose dimensions are not negligible with respect to the size of the substructures. In this case, classical approaches like Craig–Bampton technique might be not effective in reducing the size of the problem when many contact nodes subjected to nonlinear contact loads cannot be omitted. The technique is implemented in a solution scheme in the frequency domain using penalty contact elements and the harmonic balance method. The preload on the joint is produced by permanent magnets to enhance the friction contact without introducing uncertainties due to bolting. Measurements are compared with the ROM simulations and with standard time-domain integration of the full FE model. The advantage of using the GSI technique is shown in terms of time computation and accuracy of the simulation.

Excitation force of under-chassis active equipment of railway vehicles has a significant impact on the floor vibration of the car body. In order to improve the accuracy of the excitation force identification of active equipment in engineering practice, a new excitation force identification method was proposed by applying modified Sage-Husa adaptive Kalman filter (MSHAKF). First, the advantages of the MSHAKF over conventional Kalman filter (CKF) are introduced. Simulation shows that the MSHAKF has excellent exactness and robustness for active equipment excitation force identification. Finally, a test device for identifying excitation force was established. The infinite impulse response (IIR) low-pass filter is designed by using the bilinear transformation method to eliminate the identification error caused by the frequency multiplication components in the response signal. The experimental result shows that the proposed method is very effective in engineering practice without mastering the noise characteristics of the system.

In this study, a nonlinear dynamic model of a spur gear transmission system with non-uniform wear is proposed to analyze the interaction between surface wear and nonlinear dynamic characteristics. A quasi-static non-uniform wear model is presented, with consideration of the effects of operating time on mesh stiffness and gear backlash. Furthermore, a nonlinear dynamic model with six degrees-of-freedom is established considering surface friction, time-varying gear backlash, time-varying mesh stiffness, and eccentricity, and the Runge–Kutta method applied to solve this model. The bifurcation and chaos in the proposed dynamic model with the change of the operating time and the excitation frequency are investigated by bifurcation and spectrum waterfall diagrams to analyze the bifurcation characteristics and the dimensionless mesh force. It is found that surface wear is generated with a change in operating time and affects the nonlinear dynamic characteristics of the spur gear system. This study provides a better understanding of nonlinear dynamic characteristics of gear transmission systems operating under actual conditions.

It is well-known that nonlinear dry friction damping has the potential to bound the otherwise unboundedly growing vibrations of self-excited structures. An important technical example are the flutter-induced friction-damped limit cycle oscillations of turbomachinery blade rows. Due to symmetries, natural frequencies are inevitably closely spaced, and they can generally be multiples of each other. Not much is known on the nonlinear dynamics of self-excited friction-damped systems in the presence of such internal resonances. In this work, we analyze this situation numerically by regarding a two degrees-of-freedom system. We demonstrate that in the case of closely-spaced natural frequencies, the self-excitation of the lower-frequency mode gives rise to non-periodic oscillations, and the occurrence of unbounded behavior well before reaching the maximum friction damping value. If the system is close to a 1:3 internal resonance, limit cycles associated with much higher frictional damping appear, however, most of these are unstable. If more than one mode is subjected to self-excitation, the maximum resistance against self-excitation is at least given by the damping capacity of the most weakly friction-damped mode. These results are of high technical relevance, as the prevailing practice is to analyze only periodic limit states and argue the stability solely by the slope of the damping-amplitude curve. Our results demonstrate that this practice leads to considerable mis- and overestimation of the resistance against self-excitation, and a more rigorous stability analysis is required.

This study proposes an input decoupling analysis method based on the pulse test and the traditional operational transfer path analysis (OTPA) method for the OTPA transmission path analysis when the input is strongly coupled, and the algorithm is verified for analysis optimization of results. An experimental model of the dual excitation source is established in the laboratory. Experimental comparison studies show that the OTPA method, which is based on signal decoupling, inhibits the influence of cross-coupling among the source signals during OTPA analysis and effectively improves the accuracy of the transmission path test and analysis. The OTPA method is applied to analyze the transfer path of the vibration and noise of the carriages of the electric multiple unit (EMU) train (a high-speed train in China). The false peak frequency in the traditional OTPA method is eliminated, and the accuracy of the analysis is improved.

Annular tuned liquid dampers (TLDs) may be installed in slender structures with limited floor space, in which people and utilities must pass through the core, such as a wind turbine or observation tower. This study investigates an annular-shaped TLD equipped with damping screens. A linearized equivalent mechanical model capable of capturing the fundamental sloshing mode response of an annular TLD is presented. An experimental shake table testing program is completed to assess the performance of the model. Thirty-six frequency sweep tests consisting of various TLD configurations, excitation amplitudes, and excitation directions are completed. Good agreement is observed between the linearized equivalent mechanical model and experimental wave heights, sloshing forces, and energy dissipated per cycle that have been filtered to include only the fundamental sloshing mode response. The model is also observed to be in good agreement with experimental data for different excitation directions. The model is coupled to a generalized structure to investigate the response of a structure equipped with an annular TLD. The annular TLD is found to reduce the response of a generalized offshore wind turbine structure undergoing harmonic force excitation. The annular TLD provides performance comparable to an optimal linear tuned mass damper (TMD) with the same properties for a range of force excitation amplitudes.

Ladder frame structures are used as models for multistorey buildings. These periodic structures exhibit alternating propagating and attenuating frequency bands. Of the six different wave modes of propagation, two modes strongly attenuate at all frequencies. The other four modes have nonoverlapping stop band characteristics. Thus, it is challenging to isolate such structures when subjected to broadband, multimodal base excitation. In this study, we seek to synthesize a periodic ladder frame structure that has attenuation characteristics over the maximal range of frequencies for all the modes of wave propagation. We synthesize a unit cell of the periodic structure, which comprises two distinct regions having different inertial, stiffness, and geometric properties. The eigenvalues of the transfer matrix of the unit cell determines the attenuating or the nonattenuating characteristics of the structure. A novel pictorial presentation in the form of eigenvalue map is developed. This is used to synthesize the optimal unit cell. Also, design guidelines for suitable selection of the design parameters are presented. It is shown that a large finite periodic structure comprising a unit cell synthesized using the present approach has significantly better isolation characteristics in comparison to the homogeneous or any other arbitrarily chosen periodic structure.

The dynamics of structures with joints commonly show nonlinearity in their responses. This nonlinear behavior can arise from the local dynamics of the contact interfaces. The nonlinear mechanisms at an interface are complicated to study due to the lack of observability within the contact interface itself. In this work, digital image correlation (DIC) is used in combination with a high-speed camera to observe the local motion at the edge of the interface of a bolted lap joint. Results demonstrate that it is possible to use this technique to monitor the localized motion of an interface successfully. It is observed that the two beam parts of the studied lap joint separate when undergoing bending vibrations and that there is a clear asymmetry in the response of the left and the right end of the interface. Profilometry indicates that the asymmetry in the response is due to the mesoscale topography of the contact interface, highlighting the importance of accounting for surface features in order to model the nonlinearities of a contact interface accurately.

We explore the vibration attenuation of a periodic structure when one absorber with nonlinear cubic stiffness is included without increasing the total mass. Metastructures, and specifically periodic structures, present interesting characteristics for vibration attenuation that are not found in classical structures. These characteristics have been explored for automotive and aerospace applications, among others, as structures with low mass are paramount for these industries, and keeping low vibration levels in wide frequency range is also desirable. It has been shown that the addition of vibration absorbers in a periodic arrangement can provide vibration attenuation for shock input without increasing the total mass of a structure. In this work, the dynamical response of a metastructure with one nonlinear vibration absorber, with same mass as original structure, optimized for vibration attenuation under harmonic input is compared with a base metastructure without absorbers and a metastructure with linear absorbers via the evaluation of the H 2 norm of the frequency response. A simplified approach is used to compare linear and nonlinear stiffness based on deformation energy, by considering linear and nonlinear restoring forces to be equal at mean deformation. The dynamical response of the optimal system is obtained numerically, and an optimization procedure based on sequential quadratic programming (SQP) is proposed to find the optimal position and stiffness coefficients of only one nonlinear absorber, showing that it results in lower level of vibrations than original structure and than structure with linear absorbers, while almost the same level as a structure with all nonlinear absorbers.

The paper presents a statistical method for determining a specific variation of random excitations that leads to large transient enhancements (peaks) of a particular dynamical response in a stochastic mechanical system. Such a variation is found by calculating the weighted mean of the excitation variations close to a small number of largest peaks of the response obtained for a single long realization of the system motion. This statistical formula is derived by using the conditional expectation with respect to the rare event of unusually large response values and the ergodic theorem; optionally, a minimal interpeak distance is introduced. A similar formula gives the specific variations of other system variables around the peaks, and it can also be generalized to investigate any multivariable stochastic dynamical system or any set of correlated random signals. This method is applied to transient enhancements of quantities related to running safety and ride comfort of a railway vehicle: the derailment coefficient and the vertical acceleration of the vehicle body, respectively, obtained in simulations of the vehicle motion along a track with random irregularities. The averaged variations of the lateral irregularities and track superelevation close to the track locations of largest peaks of the derailment coefficient show characteristic oscillations leading to enhanced wheelset hunting in a short track section before the peak occurrence. A different pattern is found for the average variation of vertical track irregularities in the vicinity of the track points where largest maxima (or minima) of the vertical body acceleration occur.

Transient vibro-thermography for nondestructive evaluation and super-resolution imaging of material defects invariably employs nonlinear contact dynamics involving the ultrasonic actuator (horn) and the surface of the target structure. It produces nonlinear resonant modes of vibration in the target structural component. Vibration-induced heat generation is one phenomenon involved here. However, the contribution of nonlinear vibration on the thermal signature is poorly understood. In this study, we consider a metallic component with a thin-walled cavity as a representative sharp feature tuned to the main excitation frequency of the ultrasonic actuator. We have developed a mathematical model to simulate transient thermal signature of structural discontinuity/cavity/defect. The model incorporates a coupled thermo-viscoelastic heat generation process in the bulk material based on the Helmholtz free energy formulation. To capture the source of nonlinear resonant modes, we incorporate the stick-separation contact dynamics due to the ultrasonic horn and the target structural component. Commercial finite element simulation ( comsol multiphysics ) is used to quantitatively understand the nonlinear vibration response and the thermal transport behavior of the target structure with the cavity. The proposed model accounts for the effects of both the normal and the shear components of deformation contributing on heat generation and captures the nonlinear modal contribution on the heat flux map. The study shows how the geometric feature and material parameters produce an evolution of the nonlinear subsuper harmonics along with the primary harmonics tuned to the excitation frequency. Results obtained from numerical simulations are compared with the experimental results.

The reactivity control system is a vital safety system for a nuclear reactor. One of the most challenging aspects in the design of these systems is the operation during critical situations, in particular during earthquakes to safely shut-down the reactor. To study these situations, the toolbox python Implementation for Reliability Assessment Tools (PIRAT) is used to model two types of excitation: single frequency and realistic. The main focus of this work is the comparison of the implementation of the contact models used to describe the interaction between the subsystems. For the dynamic tool in PIRAT (dynamic Euler–Bernoulli for seismic event (DEBSE)), this is done with a two-stage linear spring or Lankarani and Nikravesh-based models. For the sine excitation, the results show four distinct response types with the maximum displacement varying between the models. Low-frequency excitation showed little variance while higher frequency excitation showed large variations. The realistic excitation, however, did not show these variations and showed nearly identical results for the contact models tested. This gives confidence in the simulations since the user selected contact model did not greatly affect the simulation results for a realistic excitation.