Equivalent Spring Stiffness Research Articles

A Unified Approach of Free Vibration Analysis for Stiffened Cylindrical Shell with General Boundary Conditions

A unified approach of free vibration analysis for stiffened cylindrical shell with general boundary conditions is presented in this paper. The vibration of stiffened cylindrical shell is modeled mathematically involving the first-order shear deformation shell theory. The improved Fourier series is selected as the admissible displacement function while the arbitrary boundary conditions are simulated by adjusting the equivalent spring stiffness. The natural frequencies and modal shapes of the stiffened shell are obtained by solving the dynamic model with the Rayleigh-Ritz procedure. Various numerical results of free vibration analysis for stiffened cylindrical shell are obtained, including natural frequencies and modes under simply supported, free, and clamped boundary conditions. Moreover, the effects of stiffener on natural frequencies are discussed. Compared with several state-of-the-art methods, the feasibility and validity of the proposed method are verified.

Stem and root system response of a Norway spruce tree (Picea abies L.) under static loading

AbstractThe mechanical response of a well-instrumented Norway spruce (Picea abies L.) tree under controlled winch loading was monitored. The main aims of this study were to understand the tree-root-soil response to lateral pull loads, to examine the applicability of simple engineering principles to the tree-root-soil response and to introduce a soil component into the tree stability analysis. The stem response was recorded with tilt sensors at three different heights; two sensors at each height tracked the response in transverse directions. These data were used to derive deflection and bending moment profiles of the stem for different lateral loads. A root on the windward side and one on the leeward side were instrumented using strain gauges. The tree was winched to failure and the data were collected from the strain gauges during winching. Using the data obtained from strain gauges on the roots, strain, bending moment, shear force and deflection profiles the roots with increasing load were calculated. Using the soil reaction force-deflection profiles of the roots, equivalent spring stiffness constants were determined. An estimate of the anchorage strength and moment–rotation relationship was made using a simple characteristic curve equation normalized by the failure moment and rotation that can be modified for different soil conditions.

State of charge-dependent failure prediction model for cylindrical lithium-ion batteries under mechanical abuse

Based on the previous research, the parameters of the equivalent mechanical model are associated with state of charge, thus, the model can simulate the mechanical behavior of cylindrical lithium-ion batteries under continuous state of charge with simulation software. The comparison results show that the average fitting error of the equivalent mechanical model under different state-of-charge is less than 3.91% and the simulation error for failure displacement is less than 4%. Apart from that, the equivalent spring stiffness of the equivalent mechanical model will soften under mechanical abuse. Based on this, the first order derivative of the load force to displacement (df/dx) of the equivalent mechanical model is carried out. The results show that the displacement of df/dx extreme value can better reflect the mechanical failure and soft short circuit of the battery, the average prediction error is 10.91% and 5.46% which always in advance of the actual failure. So, the failure criteria of lithium-ion batteries related to state of charge are proposed. In the future, these criteria can be used as the reference basis for judging the failure of lithium-ion batteries based on mechanical penetration and the simple form of the prediction criteria can be easily programmed into the controller.

Assessment Method for Inelastic Higher Mode Effects in Outrigger Braced Tall Building Under Seismic Loads

Outrigger system is a more popular structural system in tall buildings and higher mode participation in such buildings is an important aspect in design. During severe earthquake impacts, buildings undergo inelastic deformations, and this can lead to lengthening of modal periods of the building. Since higher mode effects have a significant impact on the overall behavior in tall buildings, the impact due to lengthening of modal periods needs to be addressed in the design. In this study, a simple assessment method is presented to quantify the effect of higher mode response on the behavior of outrigger braced buildings under seismic loads. The method comprises of simplifying the building to vertical cantilever with lateral degrees of freedom equal to number of stories and rotational springs with equivalent spring stiffness at corresponding outrigger locations. A pin-based structure was assumed to limit the first mode response to represent the transition from elastic to inelastic status of the building. The proposed method is capable of estimating the inelastic lengthened periods for higher modes and their responses in outrigger braced tall buildings. The developed method was applied to a case study outrigger braced tall building and reasonable results are predicted.

A bandgap switchable elastic metamaterial using shape memory alloys

Adaptive elastic metamaterials are generally tunable but not switchable. Here, the word “switchable” means switching between different bandgap mechanisms, such as from the local resonance bandgaps to the Bragg scattering bandgaps and vice versa. In this work, to achieve switchable bandgaps, we report a new class of elastic metamaterials whose transmission properties can be significantly tuned by curved two-way shape memory alloy (SMA) resonators. The proposed switchable metamaterial possesses bandgaps capable of being switched back and forth between the Bragg scattering and localized resonance ones. Without thermally activating the curved SMA resonators, the metamaterial beam behaves as a phononic crystal beam whose bandgaps are formed by a periodic array of concentrated masses. By heating the SMAs, the SMAs recover their original curved shape and lift the concentrated masses to form the local resonance bandgaps. The reversible dramatic variation in shape and the stiffness of the SMA resonators allows the bandgaps to be switchable and of course tunable. In addition, the equivalent spring stiffness of a curved beam at two possible directions for the first two modes is derived based on Castigliano's second theorem and is experimentally validated. Compared to SMA springs, the curved shape SMAs allow the generation of high-order local resonance bandgaps. If thermally activated at different temperatures, the negative effective mass density can further be tunable. To the author's knowledge, this work is the first theoretical and careful experimental investigation realizing switchable metamaterials using the two-way shape memory alloys.

XFEM Modelling in Multi-bolted Joints Using a Unified Bolt Preload

Multi-bolted joints are adopted and designed to provide efficient load transfer within assembled engineering parts. Bearing failure is favorable during design phase due to more progressive failure mode, however, ability of by-pass stress to be transferred to adjacent bolts in multi-bolted joints prone to catastrophic net-tension failure. Former approach known as equivalent spring stiffness (ESS) was proposed but it requires experimental sliding load value. This has led to semi-empirical approach to require experimental set-up than incorporating a generic bolt preload value. This paper aims to provide a unified bolt preload (UBP) value to be implemented in each bolt independent upon plate properties and bolts arrangements. Strength prediction were taken place by 3-D Extended Finite Element Method (XFEM) framework of various staggered and non-staggered arrangements to include various lay-ups types and plate thickness. The failure loads predictions in each testing series were investigated and then validated against experimental datasets and also compared with previous technique (ESS approach). Crack patterns and failure modes from this approach were consistent with experimental observations, where net-tension failures were observed within all testing series. Less good prediction compared to from ESS technique, partly due to semi-empirical nature in former approach. Nevertheless, reasonable agreement in UBP technique with experimental datasets were obtained (average discrepancy of approximately 20%).

Crack mathematical modeling to study the vibration analysis of cracked micro beams based on the MCST

In this paper, a new approach is presented to capture size effect on the dynamic behavior of a cracked micro beam based on the modified couple stress theory (MCST). Firstly, the stiffness of equivalent torsional spring at the crack location is obtained based on the MCST and also considering the stress intensity factor for different models of the open edge crack. To capture size dependency, the governing equations of motion of the system and also associated boundary conditions are derived based on the MCST and using Hamilton’s principle and stress intensity factor at crack location. Then the natural frequencies of the cracked micro beam are determined. Finally, the variation of natural frequencies of the system are studied with respect to different values of the system parameters such as the material length scale parameter, crack depth, crack location, based on the MCST and considering the newly proposed modeling of the local stiffness at the crack location. The results show that the crack depth, crack location, and also the material length scale parameter are extremely effective on lateral vibrations characteristics of the system.

Stiffness modelling of particles in the core layer for the manufacturing of wood-reduced particleboard

A great demand for wood material has led to an increase in the price of industrial wood in the past and will raise it even further in future. Industry and research have reacted to this increasing price of wood, for example, by developing weight-reduced particleboard. One approach to achieve this reduction in weight is to use suitable chip geometries in the core layer instead of lightweight, resource-saving filling materials. The geometry of the chips in the core layer must have a high compression resistance here to reach the desired apparent density profile. For this purpose, particle geometries were analysed with regard to this property within a project for the development of weight-reduced particleboards. After determining the curve of the compression resistance, it was possible to establish an equivalent spring stiffness model for different chip geometries to characterise the geometry during compression. Merely due to a low bulk density for chips in the core layer, it is not possible to increase the spring stiffness and hence to reach a required apparent density profile. Only compression of the chips leads to a rapid increase in spring stiffness.

Analysis of Bicycle Cushion System byusing Repulsive Force of Magnetics

One commercial package for magnetic analysis was used to apply repulsive forces of permanent magnetics to bicycle cushion system. Reliabilities of finite element analysis were acquired by comparing with those of experimental measurements. Equivalent spring stiffnesses corresponding to various sizes of magnetics were implemented into the bicycle dynamic model with three degree of freedom. Input force caused at front and rear wheels due to road unevenness was considered in the dynamic model. Dynamic behaviors were observed in terms of vertical displacements of the rider and the front reach as well as pitching displacement of the mass center when the bicycle ran over half-triangular bump. The methodology suggested in this paper by the finite element analysis and numerical model will be an useful tool for more accurate prediction of cushion design for any vehicle system if magnetic forces are utilized.

Vibration analysis of a rectangular-hollow-sectional beam with an external crack

The vibration property of a rectangular-hollow-sectional beam with an external crack were theoretically and experimentally studied in this paper. The external part-through crack and through-thickness crack were considered. Linear elastic fracture mechanics were employed to establish the theoretical formula for computing the equivalent torsion spring stiffness of the two external cracks. Then, the experimental method of obtaining the equivalent torsion spring stiffness was developed. It is seen that the test data match fairly well with the analytical data. On this basis, the transfer matrix method was introduced to obtain the frequency equations of these cracked beams under three classic boundary conditions, and followed by the numerical method for calculating natural frequencies. Experimental tests on clamped–free beams were used to verify the accuracy. It should be noted that a part-through crack may lead to a more distinct natural frequency reduction than the through-thickness crack.

Theoretical and Experimental Investigation on the Use of a Surface Acoustic Wave Sensor for SU-8 Thin-Film Adhesion Characterization

This paper investigates the application of a surface acoustic wave (SAW) sensor for characterizing the adhesion of SU-8 thin films. The proposed sensor consists of a thin aluminum nitride (AlN) film deposited on a silicon substrate with two sets of interdigital (IDT) electrodes patterned on the AlN film in a delay line configuration. The SU-8 layer is patterned between the input and output IDT electrodes. A theoretical model for calculating the wave dispersion profile for the SU-8/AlN/Si configuration is developed. The effect of changing the adhesion of the SU-8 film on the wave velocity is implemented by assuming the SU-8/AlN interface consists of a layer of distributed massless springs. Different levels of adhesion are accounted for by changing the stiffness of the interfacial springs and the corresponding changes in the wave dispersion profiles are plotted. Four designs for the SAW sensor are fabricated, for each design two sensor configurations are developed to investigate the change in adhesion of the SU-8 film. In the first configuration, a thin gold film is patterned above the AlN film prior to patterning the SU-8 layer. In the second configuration, an Omnicoat layer is patterned above the gold film prior to SU-8 patterning. Omnicoat is an adhesion promoter commonly used to increase the adhesion of SU-8 to gold. The shift in the center frequency values from both configurations is used to characterize the adhesion of SU-8. The theoretical model is then used to find the equivalent spring stiffness values that fit the theoretical to the measured wave velocities.

Dynamic characteristics of a novel damped outrigger system

This paper presents exact analytical solutions for a novel damped outrigger system, in which viscous dampers are vertically installed between perimeter columns and the core of a high-rise building. An improved analytical model is developed by modeling the effect of the damped outrigger as a general rotational spring acting on a Bernoulli-Euler beam. The equivalent rotational spring stiffness incorporating the combined effects of dampers and axial stiffness of perimeter columns is derived. The dynamic stiffness method (DSM) is applied to formulate the governing equation of the damped outrigger system. The accuracy and efficiency are verified in comparison with those obtained from compatibility equations and boundary equations. Parametric analysis of three non-dimensional factors is conducted to evaluate the influences of various factors, such as the stiffness ratio of the core to the beam, position of the damped outrigger, and the installed damping coefficient. Results show that the modal damping ratio is significantly influenced by the stiffness ratio of the core to the column, and is more sensitive to damping than the position of the damped outrigger. The proposed analytical model in combination with DSM can be extended to the study of structures with more outriggers.

Modeling of Axial Spring Stiffness in Active Vibration Controlled Drilling

During drilling process, substantial amount of vibration and shock are induced to the drill string. Active vibration controlled drilling is introduced to reduce the vibration and increase the efficiency of drilling process. In this system, two main components that determine the damping coefficient are magnetorheological (MR) damper and spring assembly. Performance of vibration damping system is depending on the viscosity of MR fluid in the damper and spring constant of spring assembly. One of the key issues that are unclear from the design is the correlation between the axial spring stiffness configuration and the damping force which needs to be tuned actively. There has been lack of studies on how the viscosity of MR fluid on the active vibration damper affects the damping stiffness of the whole system. The objective of the project is to extract the correlations for the viscous damping coefficient, equivalent spring stiffness and power input to the system. Simplified vibration model is thus created using Simulink, together with experimental data fed from APS Technology’s in-house team. Inputs of the simulation such as force exerted, mass of mandrel, spring constant and step time are based on the experimental data and can be adjusted to suit different experiments. By having the model, behavior of the system can be studied and analyzed. From the simulation, it is also observed that the relationship between damping coefficient and power input of the system is linear.

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Impact vibration characteristics of a shrouded blade with asymmetric gaps under wake flow excitations

The vibro-impact mechanism of a rotating shrouded blade with asymmetric gaps is investigated. The Frobenius method is employed to determine the dynamic frequencies and corresponding mode functions of the shrouded blade under the action of dynamic stiffness. The mode functions are used to truncate the governing partial differential equation of the shrouded blade to ordinary differential equations by using the Galerkin method. Taking into account the influence of the rotating speed, the contact stiffness of adjacent blades is assumed to be an equivalent contact spring stiffness which is determined by the bending dynamic stiffness of the blade. Then the average method is employed to obtain the dynamic responses of the primary, the sub-harmonic and the super-harmonic resonances of the shrouded blade with wake flow excitation, respectively. Finally, the dynamical responses of a shrouded blade with a set of typical material constants and geometrical parameters are given to illustrate the influence of the blade shroud gaps on the vibration amplitude.

High-Temperature Structural Analysis of a Small-Scale PHE Prototype under the Test Condition of a Small-Scale Gas Loop

A process heat exchanger (PHE) is a key component for transferring the high-temperature heat generated from a very high-temperature reactor (VHTR) to a chemical reaction for the massive production of hydrogen. The Korea Atomic Energy Research Institute has designed and assembled a small-scale nitrogen gas loop for a performance test on VHTR components and has manufactured a small-scale PHE prototype made of Hastelloy-X alloy. A performance test on the PHE prototype is underway in the gas loop, where different kinds of pipelines connecting to the PHE prototype are tested for reducing the thermal stress under the expansion of the PHE prototype. In this study, to evaluate the high-temperature structural integrity of the PHE prototype under the test condition of the gas loop, a realistic and effective boundary condition imposing the stiffness of the pipelines connected to the PHE prototype was suggested. An equivalent spring stiffness to reduce the thermal stress under the expansion of the PHE prototype was computed from the bending deformation and expansion of the pipelines connected to the PHE. A structural analysis on the PHE prototype was also carried out by imposing the suggested boundary condition. As a result of the analysis, the structural integrity of the PHE prototype seems to be maintained under the test condition of the gas loop.

Analytical solutions for free and forced vibrations of a multiple cracked Timoshenko beam subject to a concentrated moving load

An analytical approach for evaluating the forced vibration response of uniform beams with an arbitrary number of open edge cracks excited by a concentrated moving load is developed in this research. For this purpose, the cracked beam is modeled using beam segments connected by rotational massless linear elastic springs with sectional flexibility, and each segment of the continuous beam is assumed to satisfy Timoshenko beam theory. In this method, the equivalent spring stiffness does not depend on the frequency of vibration and is obtained from fracture mechanics. Considering suitable compatibility requirements at cracked sections and corresponding boundary conditions, characteristic equations of free vibration response are derived. Then, forced vibration response is treated under a moving load with a constant velocity. Using the determined eigenfunctions, the forced vibration response may be obtained by the modal superposition method. Finally, some parametric studies are presented to show the effects of crack parameters and moving load velocity.

Constraint Analysis on the Steel Mid Tower of a Tri Tower Suspension Bridge

A certain project of tri-tower suspension bridge of which the main span was 1080m was taken as a case study of the mid tower stability. A 3D beam FEM model has been carried out to investigate the global mechanical behavior of the bridge in which the internal force response to several different typical load cases were specifically concerned. Moreover, the cable spring stiffness to the mid tower was defined as the ratio of the initial horizontal forces acting on the top end of mid tower to the corresponding horizontal displacement. And the effect of this equivalent spring stiffness on the stability performance of the mid tower was compared to other tow constraint conditions to the tower including free constraint and pin constraint. The comparison showed that the cable equivalent stiffness was 4342kN/m when living load was acting along the whole main girder of the bridge while this value varied with different load cases. The difference between the effect caused by the cable spring stiffness and the free constraint was small indicating that the stability of the mid tower was not sensitive to the cable constraint. In addition, with the increase of the equivalent spring stiffness, the longitudinal buckling eigenvalue also became larger and gradually approached the condition of pin constraint.

Modelling impact sound reduction in floating wood floors

The transmission of footfall-excited impact sound is commonly reduced using floating wood floors. These constructions contain three layers: a load-bearing wood floor, a resilient material and a floating wood slab. For wood construction excited by transient load (e.g. footsteps), sound reduction estimations may contain uncertainties due to inadequate modelling of footsteps and floor components. This paper presents a multiple degree-of-freedom system model using Working Model 2D® (WM2D). The model includes orthotropic behavior of wood and equivalent spring stiffness determination of the resilient layer. Sound reduction is determined and it is shown that larger reductions are possible through the inclusion of floating wood slabs in the load-bearing floor.

Closed-form solutions for crack detection problem of Timoshenko beams with various boundary conditions

An analytical approach for crack identification procedure in uniform beams with an open edge crack, based on bending vibration measurements, is developed in this research. The cracked beam is modeled as two segments connected by a rotational mass-less linear elastic spring with sectional flexibility, and each segment of the continuous beam is assumed to obey Timoshenko beam theory. The method is based on the assumption that the equivalent spring stiffness does not depend on the frequency of vibration, and may be obtained from fracture mechanics. Six various boundary conditions (i.e., simply supported, simple–clamped, clamped–clamped, simple–free shear, clamped–free shear, and cantilever beam) are considered in this research. Considering appropriate compatibility requirements at the cracked section and the corresponding boundary conditions, closed-form expressions for the characteristic equation of each of the six cracked beams are reached. The results provide simple expressions for the characteristic equations, which are functions of circular natural frequencies, crack location, and crack depth. Methods for solving forward solutions (i.e., determination of natural frequencies of beams knowing the crack parameters) are discussed and verified through a large number of finite-element analyses. By knowing the natural frequencies in bending vibrations, it is possible to study the inverse problem in which the crack location and the sectional flexibility may be determined using the characteristic equation. The crack depth is then computed using the relationship between the sectional flexibility and the crack depth. The proposed analytical method is also validated using numerical studies on cracked beam examples with different boundary conditions. There is quite encouraging agreement between the results of the present study and those numerically obtained by the finite-element method.