Vertical Harmonic Excitation Research Articles

Numerical investigation on the coupled vibrations of piezoelectric energy harvester with a liquid-filled proof mass

Replacing the solid tip mass of a piezoelectric cantilever beam with a liquid-filled mass can increase its frequency bandwidth due to the effect of nonlinear liquid sloshing. To investigate the coupled vibrations of the piezoelectric beam and the sloshing liquid, as well as their contributions to the output power, a coupled two-dimensional finite element method-smoothed particle hydrodynamics model has been developed in this study. Using this model, the dynamic behavior of a piezoelectric beam with a liquid-filled rectangular container as the tip mass, subjected to vertical harmonic excitation, has been investigated. The effects of parametric sloshing, excitation level, and geometric nonlinearity on the output voltages have been studied in detail. The simulation results indicate that: (a) the parametric sloshing in the liquid container exhibits subharmonic characteristics, which can be triggered by matching the excitation frequency to twice the natural frequency of the sloshing mode; (b) the piezoelectric beam exhibits subharmonic or harmonic oscillations at parametric resonance; (c) due to the effect of coupled vibrations, the energy harvester with a liquid-filled proof mass has a broader bandwidth compared to the traditional harvester; (d) the frequency response diagram of the output voltage shows multiple peaks at high excitation amplitudes, and the bifurcations are caused by parametric sloshing.

SPH modelling of dissipative sloshing flows under violent vertical harmonic excitation

In the present work the damping effect of sloshing flows in tanks subjected to vertical harmonic oscillations is modelled through the Smoothed Particle Hydrodynamics (SPH) numerical method. The prediction of energy dissipation in these problems is of interest, among the others, in the aeronautic field to appropriately address sloshing-induced loads on aircraft wings. To this purpose, an enhanced SPH scheme is applied and extensively validated through a comparison with a recent experimental campaign in which a partially filled tank is subjected to harmonic vertical accelerations ranging from 0.25 g up to 6 g. Conversely to previous works addressing the same phenomenon, in the present work long-time simulations are performed spanning over several periods of oscillations. This approach allows comparing the predicted energy dissipation in terms of averages computed over several tank oscillations, thus providing a robust numerical outcome. The numerical scheme is tested over a large matrix of different frequencies and accelerations covering a wide range of flow regimes and spanning from mildly-deformed free surface to violent shaken flow. Both 2D and 3D numerical frameworks are considered and compared to the experimental reference. It is shown that the 3D solver is able, in most of the cases, to recover the experimental rate of dissipated energy with errors comparable to the intrinsic uncertainties of the problem whereas the 2D solution significantly under-predicts the damping in high-energy sloshing regimes.

Investigation on the influence of the structural characteristics of particle dampers on its damping performance

Under the condition of vertical harmonic excitation, the influence of two structural characteristics of the damping rod and the inner wall on the damping characteristics of the damper is studied. The force signal and acceleration signal of the damper under different excitation conditions are measured through experiments.The damping characteristics such as effective mass, loss power and loss factor of the damper are analyzed by the steady-state energy flow method. The experiment results indicate that, on the basis of ordinary particle dampers, adding damping rods can intensify the friction and extrusion of particles and rods, thereby enhancing the energy dissipation performance. By pasting rubber material on the inner wall of the damper, the soft inner wall can consume the acoustic energy during collision, enhance the energy consumption during collision, and improve the damping effect.

Novel Open Trench Techniques in Mitigating Ground-Borne Vibrations due to Traffic under a Wide Range of Ground Conditions

This study proposes the novel geometrical design of open trenches in contrast to the traditional rectangular type of open trench, in order to improve the efficiency in mitigating the traffic-induced ground vibrations. The novel geometric design involves the variation in cross-sectional shape, cross-sectional area, sidewall inclination, depth of trenches, and the number of trenches. In addition, the efficiency of the multirow trenches with shallower depths, in contrast to the single-row deep trench, is also assessed for further mitigating the low-frequency ground vibrations. It is argued that these novel open trenches would change the wave propagation direction while also weakening the wave propagation capability in the original direction. A two-dimensional (2D) finite-element study is performed to investigate the isolation efficiency of novel open trench techniques with varying cross sections in a linear elastic, isotropic, and homogeneous half-space. A steady-state vertical harmonic excitation of the ground surface was simulated. The numerical results show that compared with the rectangular open trench, the part of the area behind the new cross section trench has better isolation efficiency, Cross section 6 seems to be the most efficient of the six cross sections, and the isolation efficiency is improved by nearly 10.2% compared. Increasing the cross-sectional area could not effectively improve the isolation efficiency of the open trenches. Selection of inclination of the hypotenuse to an appropriate angle, increasing the number and depth of open trenches could improve the isolation efficiency. The vibration isolation performance of the trenches in the soft soil was better than the stiff soil. Besides, an excessively high groundwater table is usually detrimental to the isolation efficiency of the trenches.

Experimental characterisation of sloshing tank dissipative behaviour in vertical harmonic excitation

The effort to reduce costs and environmental impact of modern commercial aircraft requires an ever better understanding of aircraft physics, including the characterisation of the effective damping introduced on a structure by slosh dynamics. In this paper the experimental characterisation of vertical sloshing dissipative behaviour of a rectangular tank, representative of a scaled wing tank, is addressed. Specifically, a controlled electrodynamic shaker is employed to provide vertical displacement by means of sine-sweep excitation. High vertical acceleration makes the free surface unstable, triggering a nonlinear chaotic regime of the liquid stowed inside that experiences violent impacts with the surfaces of the tank. The obtained time histories of the measured accelerations and dynamic forces are shown to be directly related to the energy dissipated by the impacting fluid.

Analytical and experimental studies on out-of-plane dynamic parametric instability of a circular arch under a vertical harmonic base excitation

The out-of-plane dynamic stability of an arch under a vertical periodical base excitation is investigated by using both analytical and experimental methods. When the excitation frequency is near twice the out-of-plane natural frequency of the arch, an out-of-plane divergent vibration is induced, which will eventually leads to an out-of-plane dynamic parametric instability of the arch. Firstly, Hamiltonian principle is employed to derive the in-plane dynamic equations, and analytical solutions for the dynamic axial compressive force and bending moment in pre-stability state are obtained and verified by numerical simulations. Then the mode shape functions for the out-of-plane dynamic instability are analytically determined and the out-of-plane dynamic instability regions are obtained by Bolotin's method. Factors affecting the critical excitation frequencies of the out-of-plane dynamic instability regions, such as the arch's included angle, the slenderness ratio, and the logarithmic decrement, are thoroughly analyzed and verified via the method of multiple scales. Lastly, laboratory tests are carried out for the out-of-plane parametric resonance instability of fix-ended circular arches with various included angles and slenderness ratios. The critical excitation frequencies of dynamic instability regions of the tested arches are determined by increasing and decreasing the excitation frequency, and the experimental results agree with the theoretical counterparts very well. The main contributions of this work include: (1) obtaining an analytical solution of out-of-plane dynamic instability regions for the arch under a vertical harmonic base excitation; (2) applying a newly-designed experimental system to conduct the out-of-plane dynamic instability tests; (3) proposing a method of time domain analytical solution to determine the critical excitation frequencies of the out-of-plane dynamic instability region, and (4) discussing the dynamic instability mechanism for base-excited arches.

Simplified frequency-independent model for vertical vibrations of surface circular foundations

PurposeThis paper aims to simplify a new frequency-independent model to calculate vertical vibration of rigid circular foundation resting on homogenous half-space and subjected to vertical harmonic excitation is presented in this paper.Design/methodology/approachThe proposed model is an oscillator of single degree of freedom, which comprises a mass, a spring and a dashpot. In addition, a fictitious mass is added to the foundation. All coefficients are frequency-independent. The spring is equal to the static stiffness. Damping coefficient and fictitious mass are first calculated at resonance frequency where the response is maximal. Then, using a curve fitting technique the general formulas of damping and fictitious mass frequency-independent are established.FindingsThe validity of the proposed method is checked by comparing the predicted response with those obtained by the half-space theory. The dynamic responses of the new simplified model are also compared with those obtained by some existing lumped-parameter models.Originality/valueUsing this new method, to calculate the dynamic response of foundations, the engineer only needs the geometrical and mechanical characteristics of the foundation (mass and radius) and the soil (density, shear modulus and the Poisson’s ratio) using just a simple calculator. Impedance functions will no longer be needed in this new simplified method. The methodology used for the development of the new simplified model can be applied for the resolution of other problems in dynamics of soil and foundation (superficial and embedded foundations of arbitrary shape, other modes of vibration and foundations resting on non-homogeneous soil).

New interaction model for the annular zone of stepped piles with respect to their vertical dynamic characteristics

Stepped piles with a larger section at the top have been increasingly used as foundations for bridge abutment in China. In this paper, numerical modelling using ABAQUS is firstly carried out to investigate the behavior of a annular zone below the top larger section. Then, a new interaction model with a wedge-shaped soil below the annular zone is proposed for pile–soil system. In the model, the soil wedge and normal pile segment were treated as a generalized pile segment. By combining the Novak plane strain model and Rayleigh-Love rod theory, the dynamic characteristics of a stepped pile subject to a vertical harmonic excitation at the pile head are theoretically investigated. The analytical solutions are deduced using Laplace transform and impedance function transfer method. The proposed model is verified by the sensitivity analysis of the soil wedge dimension and comparison with FEM results. Compared with the conventional Voigt model, the present solution shows a flatter oscillation in complex impedance curves, indicating that the Voigt model may overestimate the dynamic response of stepped piles. Finally, selected results are presented to examine the influence of pile geometry on the vertical dynamic responses.

A novel vibro-penetration test (VPT) for the investigation of cohesionless soils in the field

A novel dynamic penetration testing method for the investigation of cohesionless soils based on the principles of vibratory driving has been developed. During the so-called vibro-penetration test (VPT), a vertical harmonic excitation force drives a rod with a conical tip into the ground. The penetration resistance is defined as the number of excitation cycles Nz10 required for 0.10 m penetration with a reference driving energy. In order to evaluate Nz10, the tip and shaft resistance, the tip acceleration and the depth of the penetrometer tip are recorded during driving and the data are evaluated according to a proposed vibro-penetration model. Experimental and numerical investigations are carried out to investigate the soil response, validate the mechanical model and evaluate the influence of the density, the effective stresses and the machine parameters on the vibro-penetration resistance. The results show that the Nz10 – values are reproducible, correlate unambiguously with the state variables but depend on the machine parameters to some extent. In comparison with the CPT, the VPT shows similar sounding records. However, the required equipment is much lighter, the execution faster and in-situ investigations of medium dense to dense cohesionless soils at large depths are feasible.

Intrinsic Localized Modes of Principal Parametric Resonances in Pendulum Arrays Subjected to Vertical Excitation

Intrinsic localized modes (ILMs) are investigated in an N-pendulum array subjected to vertical harmonic excitation. The pendula behave nonlinearly and are coupled with each other because they are connected by torsional, weak, linear springs. In the theoretical analysis, van der Pol's method is employed to determine the expressions for frequency response curves for the principal parametric resonance, considering the nonlinear restoring moment of the pendula. In the numerical results, frequency response curves for N = 2 and 3 are shown to examine the patterns of ILMs, and demonstrate the influences of the connecting spring constants and the imperfections of the pendula. Bifurcation sets are also calculated to show the excitation frequency range and the conditions for the occurrence of ILMs. Increasing the connecting spring constants results in the appearance of Hopf bifurcations. The numerical simulations reveal the occurrence of ILMs with amplitude modulated motions (AMMs), including chaotic motions. ILMs were observed in experiments, and the experimental data were compared with the theoretical results. The validity of the theoretical analysis was confirmed by the experimental data.

Effect of Embedment Depth on Response of Machine Foundation on Saturated Sand

In this paper, a dynamic analysis of strip machine foundation is carried out. The foundation of multiple thicknesses is placed at different depths above a saturated sand with different states (i.e., loose, medium and dense), and vertical harmonic excitation is applied with build up of the excess pore water pressure being considered. The dynamic analysis is performed numerically by using finite element software, PLAXIS 2D. The soil is assumed as elastic perfectly plastic material obeys Mohr–Coulomb yield criterion. A parametric study is carried out to evaluate the dependency of machine foundation on various parameters including the amplitude of the dynamic load, the frequency of the dynamic load and the embedment of foundation. It was concluded that increasing the embedment ratio causes a reduction in the dynamic response up to a certain embedment depth; when the depth of embedment increases higher than 1 m, the effect become less pronounced and as strength of the soil increases, the effect of embedment depth in reducing dynamic response will decrease also. The vertical displacements decrease obviously by 46, 37 and 40 % for loose, medium and dense sand, respectively, when increasing the embedment of foundation from 0.5 to 1 m, while when the embedment of foundation increases from 1 to 1.5 m, the vertical displacements for loose, medium and dense sand decrease by 45, 38 and 3 %, respectively. Finally, when the embedment of foundation increases from 1.5 to 2 m, the decrements in vertical displacements are also recorded for loose, medium and dense sand by 42, 36 and 18 %, respectively.

Ride behaviour of trucks transporting liquids

Liquid motion in partially filled containers generates slosh forces and moments which affect the ride behaviour and stability of fluid cargo trucks. The main objective of the paper is to investigate the ride behaviour of trucks carrying two spherical fluid containers. A mathematical model of a medium weight truck has been formulated which takes into account the physical and dynamic characteristics of the fluid cargo. In this model, the fluid cargoes in both tanks are simulated as nonlinear pendulums with specific damping coefficients. The Runge–Kutta method has been used to solve the nonlinear equations of motion of the vehicle system in the time domain. The vehicle is excited by two sources at the same time. The first is a vertical harmonic excitation expressing road irregularities. The second is a relatively small longitudinal deceleration in the direction of motion. The partial filling ratio and liquid cargo viscosity are shown to play major roles in the vehicle ride behaviour. Finally, result analysis and discussions are made.

Prediction of boundary zone parameters and soil–pile separation under horizontal and rocking vibrations

This paper attempts to study the dynamic soil–pile interaction on the non-linear response under coupled vibration by both experimental and analytical study. Dynamic response characteristics of reinforced concrete piles under varying levels of vertical and horizontal harmonic excitation are investigated in the present study. Single and group (2×2) piles are used in the investigation. The horizontal exciting force applied on the pile cap through the oscillator causes coupled motion to the foundation. The measured response is compared by the two different analytical approaches: linear analysis – Novak’s plane strain theory with static interaction factor approach and non-linear analysis – Novak’s continuum model with dynamic interaction factor approach. For non-linear analysis, allowance is made for the separation between the pile and soil in boundary zone model. Reasonable match between the measured and predicted response by non-linear analysis is found. The test data are used to establish the empirical relationship between the extent of soil separation around the pile and the maximum vibration amplitudes under coupled vibration.

이중저 형상 구조물의 음향방사효율과 수중방사소음 해석

Recently, reducing underwater radiated noise (URN) of ships has become an environmental issue to protect marine wildlife. URN of ships can be predicted by various methods considering its generating mechanism and frequency ranges. For URN prediction due to ship structural vibration in low frequency range, the fluid-structure interaction analysis technique based on finite element and boundary element methods (FE/BEM) is regarded as an useful technique. In this paper, URN due to a double bottom-shaped structure vibration has been numerically investigated based on a coupled method of FE/BEM to enhance the prediction accuracy of URN due to the vibration of real ship engine room structure. Acoustic radiation efficiency and URN transfer function in case of vertical harmonic excitation on the top plate of double bottom structure have been evaluated. Using the results, the validity of an existing empirical formula for acoustic radiation efficiency estimation and a simple URN transfer function, which are usually adopted for URN assessment in initial design stage, is discussed.

Vertical Vibration of Founations on Homogeneous Elastic Half-Space

The stiffness and damping coefficients for a rigid massless circular foundation resting on homogeneous elastic half-space under vertical harmonic excitation are evaluated using one-dimensional wave propagation in cones (cone model), based on strength of material approach. Using these coefficients, the frequency dependant displacement amplitude as well as the magnification factor of a massive foundation is computed. The resonant frequencies and peak amplitudes are also studied varying the influencing parameters such as the type of soil and the mass of the foundation. To verify the validity of the model developed, the computed response is compared with reported analytical results with wide variation of influencing parameters, which shows a good agreement. Thus, the simplified cone model can be used in day-to-day analysis and design of foundations under vertical dynamic loading.

230 鉛直励振を受ける柔軟構造物と正方形容器内液面の非線形連成振動(非線形振動,OS-1 振動基礎,総合テーマ「伝統を,未来へ!」)

Nonlinear vibrations of an elastic structure, with a nearly square liquid tank, under vertical harmonic excitation are investigated. When the tuning condition, a ratio 2:1:1, is satisfied among the natural frequencies of the structure and two sloshing modes (1, 0) and (0, 1), the equation of motion for the structure and the modal equations for sloshing are derived considering the nonlinearity of sloshing. Then, van der Pol's method is employed to determine the frequency response curves. The effectiveness of a square liquid tank as a tuned liquid damper is shown by investigating the influences of the system parameters on the frequency response curves.

Dynamic vertical response of model piles — experimental and analytical investigations

Dynamic response characteristics of reinforced concrete model piles were investigated in the field under varying levels of vertical harmonic excitation. In the investigations single piles and 2 × 2 group piles with length to diameter ratios 10, 15 and 20. For group piles, spacing to diameter ratios of 2, 3 and 4 for each length to diameter ratio were used. In both the cases, two different conditions of pile cap — embedded into soil and above the ground surface, were considered in this investigation. The measured response was compared with the response obtained by different analytical methods. The stiffness and damping of piles under vertical vibration were computed by two different analytical approaches — (i) Novak's frequency independent solutions with static interaction factor for parabolic soil profile and (ii) Novak's complex frequency dependent analytical solutions with dynamic interaction factor approach for layered media. From the comparison of these theories with the experimental data it was found that the Novak's frequency dependent solutions with dynamic interaction factor approach produced reasonable estimates of the experimental results. In this study the influence of excitation intensity, static load on pile and different contact condition of pile cap with soil on the dynamic behaviour of pile foundation are reported.

Nonlinear Parametric Vibrations of Elastic Structures Containing Two Cylindrical Liquid-Filled Tanks

Nonlinear vibrations of elastic structures containing two cylindrical liquid-filled tanks subjected to vertical harmonic excitation are investigated. When a 2:1:1 ratio of internal resonance is satisfied between the natural frequencies of the structure and the sloshing in the two liquid tanks, nonlinear parametric vibrations can be observed. The equation of motion for the structure is derived considering the nonlinearity of the hydrodynamic force, and the modal equations for sloshing are derived by using Galerkin's method. Then, frequency response curves are determined by using van der Pol's method. The respective influences of the liquid levels and the deviation of the internal resonance ratio 2:1:1 on the frequency response curves are discussed. The results of the response curves differed from those obtained in the case where a structure had only one liquid-filled tank installed. It is found that two types of vibrations may occur depending on the excitation frequency and the liquid levels: a single-mode solution (sloshing occurs in either tank) and a double-mode solution (sloshing occurs in both tanks). Amplitude and phase modulated motion appears in the single-mode solution when some deviation of the internal resonance ratio exists. The validity of the theoretical results was verified by the experiments.

Influence of Horizontal Excitations on Dynamic Stability of a Slender Beam Under Vertical Excitation

Experimental studies have been conducted to clarify the influence of horizontal harmonic excitations on the dynamic stability of a slender cantilever beam under vertical harmonic excitation. Three kinds of aluminum test beams with rectangular cross section have been used. The test beam being clamped at one end and free at the other end, was vertically stood, and was harmonically excited to both vertical and horizontal directions simultaneously. The direction of the horizontal excitation was taken parallel to one of the beam side faces, i.e. two directions were considered as X and Y directions which have the largest and smallest flexural rigidity, respectively. By varying the horizontal excitation amplitude, keeping the amplitude of excitation in the vertical direction, the influence of the horizontal excitation has been investigated on the principal instability regions in which unstable vibration of the fundamental vibration mode occurs. The excitation frequency in the vertical excitation was taken around twice the fundamental natural frequency 2f Y 1 in smallest rigidity direction, while that in the horizontal direction was taken around both the fundamental natural frequency f Y 1 and twice of it 2f Y 1. Obtained experimental results present useful fundamental data for aseismatic design of structures under earthquake containing both vertical and horizontal excitation components.

Dynamic Response of Machine Foundation on Layered Soil: Cone Model Versus Experiments

This paper presents the experimental validation of analytical solution based on cone model for machine foundation vibration analysis on layered soil. Impedance functions for a rigid massless circular foundation resting on a two layered soil system subjected to vertical harmonic excitation are found using cone model. Linear hysteretic material damping is introduced using correspondence principle. The frequency-amplitude response of a massive foundation is then computed using impedance functions. To verify the solution field experiments are conducted in two different layered soil systems such as gravel layer over in situ soil and gravel layer over concrete slab (rigid base). A total 72 numbers of vertical vibration tests on square model footing were conducted using Lazan type mechanical oscillator, varying the influencing parameters such as depth of top layer, static weight of foundation and dynamic force level. The frequency-amplitude response in general and in particular the resonant frequencies and resonant amplitudes predicted by cone model is compared with the results of experimental investigation, which shows a close agreement. Thus the cone model is reliable in its application to machine foundation vibration on layered soil.