Equivalent Spring Constants Research Articles

Estimation of local variation in Young's modulus over a gold nanocontact using microscopic nanomechanical measurement method.

Nanoscale materials tend to have a single crystal domain, leading to not only size dependence but also orientation dependence of their mechanical properties. Recently, we developed a microscopic nanomechanical measurement method (MNMM), which enabled us to obtain equivalent spring constants (force gradients) of nanocontacts while observing their atomic structures by transmission electron microscopy (TEM). Therein, we evaluated Young's modulus based on a model that a newly introduced layer at the thinnest section of a nanocontact determined the change in the measured equivalent spring constant, and discussed their size dependence. However, this model is not general for other nanomaterials that do not exhibit the introduction of a new atomic layer while stretching. In this study, using MNMM, we propose a new analytical method to directly retrieve the local Young's modulus of nanomaterials by measuring initial lattice spacing and its displacement of a local region in the TEM image during the stretching of the nanocontact. This reveals the size dependence of local Young's modulus at various positions of the nanocontact at once. As a result, our estimated Young's modulus for a gold [111] nanocontact showed a size dependence similar to the one previously reported. This indicates that this analytical method benefits in revealing the mechanical properties of not only nanomaterials but also structurally heterogeneous materials such as high-entropy alloys.&#xD.

Elastic Wave Propagation in 2D Carbon Nano‐Onion Lattices

AbstractThis study presents a simple analytical model to investigate wave propagation in 2D carbon nano‐onions (CNOs) and nitrogen‐doped carbon nano‐onions (N‐CNOs) lattices. Furthermore, the dispersion relationships of the waves and bandgaps in these lattices are derived based on Bloch's theorem. The CNOs and N‐CNOs lattices are modeled as infinite 2D mass‐in‐mass structures accurately assembled using linear springs. The Lennard–Jones potential energy is employed to obtain equivalent spring constants. A key finding of this research is the identification of bandgaps within all lattice structures, signifying regions where wave propagation is prohibited. The existence of these bandgaps offers potential for the advancement of adjustable nano‐scale metamaterials.

Planets, springs and pendulums

Seeing connections between different areas of physics is a good way to teach physics. In the orbit of a planet, there is a continuous interchange between gravitational potential energy and kinetic energy with the sum being constant. This is essentially the same physics as a mass on the end of a spring, or a pendulum. In this paper, equivalent spring constants are calculated for planetary orbits and the pendulum equation used to derive Kepler’s third law.

Site Specific Non-Linear Ground Response Analysis and Soil Structure Interaction Study of Newly Constructed Dharahara Tower

This study investigates the seismic performance of the recently reconstructed Dharahara tower, a historically significant monument in Nepal that has experienced damage in past earthquakes and was completely destroyed in the recent Gorkha earthquake. Ground Response Analysis (GRA) and Soil-Structure Interaction (SSI) investigations are conducted to assess its structural behavior. GRA includes free field and structure-influenced analyses, with subsequent comparative assessment. Additionally, 3D non-linear finite element analysis is employed to derive equivalent spring constants representing soil and foundation characteristics, which are incorporated into the SSI analysis. These computed spring constants are used to model support conditions, allowing for an evaluation of the super-structural response. The study also employs non-linear dynamic analysis to compare structural responses between fix-based and spring-based models. The results indicate that the presence of the structure significantly influences surface wave motion amplification in GRA, resulting in peak ground accelerations (PGAs) that exhibit de-amplification in the free field and amplification in the presence of the structure. Furthermore, the introduction of an equivalent spring system in the soil-structure model changes the system's vibration period and damping characteristics, leading to enhanced dynamic response compared to the fixed-base model.

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

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

Modeling of Nonlinear Dynamics and Temperature Stability of Doped Silicon Microresonators

Silicon is widely used as the device material for many micro resonators applied in timing and frequency referencing. One key disadvantage of silicon resonators compared to quartz resonators is their high thermal sensitivities. Doping silicon is a promising approach for temperature stability. Doped resonators operating at large deformation and finite strain amplitude often go to nonlinear regimes; therefore, nonlinear dynamics must be considered for adequately predicting the system behavior. In this article, the nonlinear vibration analysis is given for rectangular resonators operating in the Lame mode, incorporating both the second- and third-order elastic constant (SOEC and TOEC) components. This article presents an analytic demonstration for the linear and nonlinear lumped mass system equivalent spring constants explicitly in terms of SOEC and TOEC. We show that, for a rectangular resonator in the Lame mode, the first-order nonlinear spring constant would be an explicit expression in terms of TOEC components, which, for a square resonator, will be nullified. We show that there exist optimal doping levels where the anharmonic stiffness coefficient is minimized, implying the most dynamic stable vibrations. Furthermore, this article shows that there exists a tradeoff between dynamical and temperature–frequency stability in terms of the doping level.

Evaluation of bridge support condition using bridge responses

This article presents a method for evaluating the support condition of bridges. This is done by representing the aging and deteriorated supports as rotation springs with equivalent spring constants. Sensitivity analysis was performed to obtain a relationship between the spring constant and the bridge responses (deflections/slopes). From this relationship, measured bridge responses can be used to estimate the equivalent spring constants through interpolation. Numerical analysis was performed to check whether the method can be used to calculate equivalent spring constants. Then, the method was verified by performing laboratory tests on a scale model bridge and field test on an actual bridge. In both tests, spring constants were estimated using the proposed method and then verified by calculating the displacements and frequencies and comparing them to the measured values.

Spatial distribution of Winkler spring stiffness for rectangular mat foundation analysis

Structural design of mat foundations of buildings is often done by performing static analysis of a slab resting on vertical uncoupled Winkler springs. It is already well established that the simplifying assumption of a uniform modulus of subgrade reaction throughout the mat foundation leads to inaccurate results that significantly underestimate the bending moments in the mat. This paper examines the spatial variation of the Winkler spring stiffness constants that is necessary for the mat-on-springs analysis to produce the same slab deflections and bending moment diagrams as finite element analysis that treats the soil as continuum. For this purpose, three-dimensional parametric analyses of slabs resting on elastic soil are performed using the finite element method for various values of soil elastic properties, slab geometrical characteristics and column load configurations. The finite element analysis results were used for back-calculating analytically the equivalent Winkler spring constants at each node of the mat. Based on the numerical results, equations describing the spatial distribution of spring stiffness are proposed. The performance of the proposed equations is compared against existing spring stiffness spatial distribution approaches used in practice.

On the equivalent mass-spring parameters and assumed mode of a cantilevered beam with a tip mass

In the Rayleigh-Ritz approach to the mathematical model of a cantilevered beam with a tip mass, the proper selection of basis functions is critical in representing the original system by an equivalent mass-spring system. Although the fundamental bending mode shape of a beam varies for a different tip mass magnitude, the numerical values of 33/140 and 3 have been conventionally employed as those of the normalized dimensionless equivalent mass and spring constants, respectively, which correspondingly yield errors in its calculated natural frequencies. This work firstly proposes a method to evaluate more accurate values of the equivalent mass and spring for a wide range of the tip mass-to-beam mass ratio by direct use of a fundamental mode, and then proposes a new basis function as a linear combination of two polynomials, which represent static deflection shapes of a beam under a tip force and a uniformly distributed force, respectively, yielding natural frequencies fairly close to those by the continuous beam equation.

Hybrid-model-based stiffness analysis of a three-revolute-prismatic-spherical parallel kinematic machine

A three-revolute-prismatic-spherical parallel kinematic machine is proposed as an alternative solution for high-speed machining tool due to its high rigidity and high dynamics. Considering the parallel kinematic machine module as a typical compliant parallel mechanism, whose three limb assemblages have bending, extending and torsional deflections, this article proposes a hybrid modeling methodology to establish an analytical stiffness model for the three-revolute-prismatic-spherical device. The developed analytical model is further used to evaluate the stiffness mapping of the three-revolute-prismatic-spherical module over a given work plane which is then validated by experimental tests. The simulations and experiments indicate that the present hybrid methodology can predict the three-revolute-prismatic-spherical parallel kinematic machine’s stiffness in a quick and accurate manner. The solution for eigenvalue problem of the stiffness matrix leads to the stiffness characteristics of the parallel module including eigenstiffnesses and the corresponding eigenscrews as well as the equivalent screw spring constants. Based on the eigenscrew decomposition, the parallel kinematic machine is physically interpreted as a rigid platform suspending by six screw springs. The minimum, maximum and average of the screw spring constants are chosen as indices to assess the three-revolute-prismatic-spherical parallel kinematic machine’s stiffness performance. The distributions of the proposed indices throughout the workspace reveal a strong dependency on the mechanism’s configurations. At the final stage, the effects of some design parameters on system stiffness characteristics are investigated with the purpose of providing useful information for the conceptual design and performance improvement of the parallel kinematic machine.

A modified elasto-dynamic model based static stiffness evaluation for a 3-PRS PKM

This paper proposes a modified elasto-dynamic model for a three-prismatic revolute spherical parallel kinematic machine, in which the flexibility of the prismatic revolute spherical limb structures are accounted in and modeled as a hollowed spatial beam with nonuniform cross section. The governing equations are derived through substructure synthesis and finite element formulation. The stiffness matrix of the platform is then extracted from global stiffness matrix and its characteristics at typical configurations are calculated to reveal complicated coupling effects of diagonal and nondiagonal elements of the stiffness matrix. The concept of principle stiffness and coupled stiffness are proposed and their distributions over the workspace are predicted with numerical simulations in a quick manner. Then the stiffness of the platform is physically interpreted as a kinematically unconstrained rigid body suspended by six screw springs with equivalent spring constants and pitches through eigenscrew decomposition. The distributions of screw spring constants over the workspace are then plotted to demonstrate a duality property. At last, the effects of some design variables such as structural and dimensional parameters on system rigidity performance are investigated with the purpose of providing useful information for the structural design and performance enhancement of the parallel kinematic machine.

Study of an eletrostatically actuated torsional micromirror with compliant planar springs

This paper presents modelling, fabrication and testing of a torsional micromirror suspended by two compliant ortho-planar springs attached at the sides. The proposed cantilevered serpentine spring is a single-chain frame, composed of nine beam segments to form a serpentine together with an initial segment and a final segment. Castigliano theory is used to formulate the spring constants under certain loads. Two equivalent spring constants for the micromirror are obtained according to structure analysis. The 2-DOF model for torsion and bending of the micromirror is presented. Finite element analysis (FEA) of the coupled domain of the structure is performed to verify the analytical results. Further validation of the prediction is carried out through static testing of a fabricated micromirror using a PSD (position sensing detector) sensor based experimental set-up. Due to relative compliancy of the dominant torsion mode, the fabricated micromirror is actuated to some angle at a lower voltage. The comparison between FEA and analytical results shows good agreement, demonstrating the accuracy prediction by the derived model. Small deviation from experiments is expected due to the nonlinear nature of electrostatic field.

Identifying the foundation stiffness of a partially embedded post from vibration measurements

Foundations of posts such as lighting columns can suffer damage or deteriorate over time, rendering the post unsafe. Quick, inexpensive and non-intrusive measurement procedures are required to monitor periodically their structural integrity. This paper investigates the behaviour of beam-like structures with a view to identifying a suitable monitoring technique for post foundations. In this initial study a lighting column is modelled as a uniform rigid beam that is constrained by collocated equivalent translational and rotational springs. Expressions are derived for the equivalent spring constants as functions of foundation profile and depth. Modal and static responses are presented as functions of foundation properties. The inverse problem of identifying the foundation stiffnesses from response measurements is discussed. A simple method is proposed based on quasi-static stiffness measurements obtained from impact tests. The method is validated using measurements of a laboratory scale structure.

Identifying the foundation stiffness of a partially embedded post from vibration measurements

Foundations of posts such as lighting columns can suffer damage or deteriorate over time, rendering the post unsafe. Quick, inexpensive and non-intrusive measurement procedures are required to monitor periodically their structural integrity. This paper investigates the behaviour of beam-like structures with a view to identifying a suitable monitoring technique for post foundations. In this initial study a lighting column is modelled as a uniform rigid beam that is constrained by collocated equivalent translational and rotational springs. Expressions are derived for the equivalent spring constants as functions of foundation profile and depth. Modal and static responses are presented as functions of foundation properties. The inverse problem of identifying the foundation stiffnesses from response measurements is discussed. A simple method is proposed based on quasi-static stiffness measurements obtained from impact tests. The method is validated using measurements of a laboratory scale structure.

Modeling and analysis methods of bridges and their effects on seismic responses: I—Theory

Various methods for modeling and analyzing bridges subjected to earthquakes are first discussed in Part I, followed by parametric studies and the evaluation of their effects on seismic responses discussed in Part II. Also included in Part I are the detailed descriptions on the determination of the equivalent soil spring constants as required by the simplified model, and the proposed formulas for calculating such constants for both spread (shallow) footings and pile-supported (deep) foundations.

A hybrid experimental-finite element method for vibration analysis of structures

A hybrid scheme, which combines the advantages of the finite element method and experimental technique, has been developed for vibration studies on plates and beams. Experimental technique is initially used to determine experimentally the first few frequencies of the structure, leading to the determination of equivalent spring constants, and thereby avoiding the assumption of infinite rigidity at the boundaries. The spring constants are then incorporated into a finite element code to determine the higher mode shapes and natural frequencies. It is found that greater accuracy is achieved in the hybrid method when compared with the theoretical data.

Dynamic behavior of gantry cranes. 2nd report. The study on traversing motion.

This paper analyzes the dynamic behavior and stresses in a gantry crane structure during traversing motion using a simulation model. Mathematical equations for the deflection curve of a crane structure are given, and the equivalent masses and spring constants are obtained through the static condensation method. The theoreticant analysis is in good agreement with the experimental results. Non-dimensional coefficients with regard to vertical and horizontal deflections and a method of describing vertical and horizontal deflections in the static state by using those coefficients will be useful in the design of gantry cranes. In addition, the result shows that the maximum stress in a structure is sensitive to the swing angle of its load, which be a significal point of discussion in the future national and international design standards for gantry-type cranes.

DESIGN FORMULAE OF BURIED PIPES SUBJECT TO LARGE GROUND SETTLEMENT AND ITS APPLICATION

The present paper proposed design formulae and equivalent spring constants of ground used for design propose of buried pipelines subject to large ground settlements. The formulae obtained by a beam theory on an elastic foundation can estimate joint rotation angles and joint expansions as well as the maximum bending strains of the pipes. The equivalent spring constants of ground were so determined that calculated values of pipe responses by the design formulae including spring constants were well matched with many experimental values. The determined equivalent spring constants associated with an amount of ground settlements are rather conservative values from a view point of design of buried pipes. Next, resisting capacities of various kinds of pipes to large ground settlements were evaluated by using the proposed design formulae and the equivalent spring constants.

Design of Machine Foundations on Piles

A practical method for the analysis of pile-supported foundations subjected to dynamic loadings is described. The method uses a single-degree-of-freedom mass-spring-dashpot model of a form similar to that used for shallow foundations to analyze the response of the actual system. The method of analysis uses the concept of an equivalent cantilever, which is a technique to simplify the soil-structure interaction problem, and allow the computation of equivalent spring constants in all modes of vibration. The method is demonstrated with an actual case history where computed and observed motions are compared.

A Matric Formulation of Linearly Coupled Vibration Problems

A LARGE class of aircraft and missile structural vibration problems, and also dynamics problems associated with missile test stands and launch equipment, may be represented ideally by masses inter‐connected by extensional and rotational springs. Generally the spring constants, which represent stiffnesses of structural sections, may be determined analytically; however, experiments are sometimes necessary to determine the magnitude of the spring constants when boundary conditions or end restraints of members are neither pinned nor clamped. In some problems one may measure frequencies, without damaging the structure, and work backwards to determine equivalent spring constants which then may be used in solving static problems.