Simple Mass Spring Research Articles

How reflected wave fronts dynamically establish Hooke's law in a spring

A simple benchtop experiment in which a moving cart collides with a fixed spring is described. Force–time and force–distance data recorded during the collision display the transit of compression wave fronts through the spring following impact. These data can be used by students to develop a computational model of the dynamics of this simple mass–spring–sensor system using a simple application of the wave equation and thereby develop an intriguing picture of how a spring realizes Hooke's law approximately in this dynamic physical problem.

Leg stiffness of sprinters using running-specific prostheses

Running-specific prostheses (RSF) are designed to replicate the spring-like nature of biological legs (bioL) during running. However, it is not clear how these devices affect whole leg stiffness characteristics or running dynamics over a range of speeds. We used a simple spring-mass model to examine running mechanics across a range of speeds, in unilateral and bilateral transtibial amputees and performance-matched controls. We found significant differences between the affected leg (AL) of unilateral amputees and both ALs of bilateral amputees compared with the bioL of non-amputees for nearly every variable measured. Leg stiffness remained constant or increased with speed in bioL, but decreased with speed in legs with RSPs. The decrease in leg stiffness in legs with RSPs was mainly owing to a combination of lower peak ground reaction forces and increased leg compression with increasing speeds. Leg stiffness is an important parameter affecting contact time and the force exerted on the ground. It is likely that the fixed stiffness of the prosthesis coupled with differences in the limb posture required to run with the prosthesis limits the ability to modulate whole leg stiffness and the ability to apply high vertical ground reaction forces during sprinting.

Suppression of the primary resonance vibrations of a forced nonlinear system using a dynamic vibration absorber

In a single degree-of-freedom weakly nonlinear oscillator subjected to periodic external excitation, a small-amplitude excitation may produce a relatively large-amplitude response under primary resonance conditions. Jump and hysteresis phenomena that result from saddle-node bifurcations may occur in the steady-state response of the forced nonlinear oscillator. A simple mass–spring–damper vibration absorber is thus employed to suppress the nonlinear vibrations of the forced nonlinear oscillator for the primary resonance conditions. The values of the spring stiffness and mass of the vibration absorber are significantly lower than their counterpart of the forced nonlinear oscillator. Vibrational energy of the forced nonlinear oscillator is transferred to the attached light mass through linked spring and damper. As a result, the nonlinear vibrations of the forced oscillator are greatly reduced and the vibrations of the absorber are significant. The method of multiple scales is used to obtain the averaged equations that determine the amplitude and phases of the first-order approximate solutions to primary resonance vibrations of the forced nonlinear oscillator. Illustrative examples are given to show the effectiveness of the dynamic vibration absorber for suppressing primary resonance vibrations. The effects of the linked spring and damper and the attached mass on the reduction of nonlinear vibrations are studied with the help of frequency response curves, the attenuation ratio of response amplitude and the desensitisation ratio of the critical amplitude of excitation.

Negative mass sound shielding structures: Early results

AbstractElastic meta‐materials or meta‐composites can be designed to have a negative effective mass density at certain vibration frequencies, thus blocking wave propagation through the material within that frequency band. The negative mass behaviour is generated by resonant structures within the material that oscillate 180° out of phase with the acoustic pressure waves applied to the surface. As this research is in its initial stages this paper describes work in progress in both the experimental and numerical domains, and some early results from solids containing geometrically simple spring–mass resonant structures. Behaviour is characterized experimentally by dynamic tests of individual resonators as well as impedance tube measurements of panel‐like structures containing multiple resonant elements. The experimental results exhibit clear evidence of the expected resonances, and partial band gap behaviour. Finite element models of both single resonant elements and impedance tube specimens are being developed, and the current status of these models is described. Their results to date show good agreement with the mass law and qualitative agreement with the experimental results.

Mitigating Vibration Along an Arbitrarily Supported Elastic Structure Using Multiple Two-Degree-of-Freedom Oscillators

Simple spring-mass systems are often deployed as vibration absorbers to quench excess vibration in structural systems. In this paper, multiple two-degree-of-freedom oscillators that translate and rotate are used to mitigate vibration by imposing points of zero displacement, or nodes, along any arbitrarily supported elastic structure during harmonic excitations. Nodes can often be enforced along an elastic structure by attaching suitably chosen two-degree-of-freedom oscillators. In application, however, the actual selection of the oscillator parameters also depends on the tolerable translational and rotational vibration amplitudes of the attached oscillators, because if these vibration amplitudes are large, then theoretically feasible solutions would not be practical to implement. In this paper, an efficient approach is developed that can be used to tune the oscillator parameters that are required to induce nodes, while satisfying the tolerable vibration amplitudes of the oscillators. Instead of solving for the oscillator parameters directly, the restoring forces exerted by the springs are computed instead. The proposed approach is simple to apply, efficient to solve, and more importantly, allows one to easily impose the tolerable translational and rotational vibration amplitudes of the two-degree-of-freedom oscillators. A design guide for choosing the required oscillator parameters is outlined, and numerical experiments are performed to validate the proposed scheme of imposing nodes along a structure at multiple locations during harmonic excitations.

Running on uneven ground: leg adjustment to vertical steps and self-stability

Human running is characterized by comparably simple whole-body dynamics. These dynamics can be modelled with a point mass bouncing on a spring leg. Theoretical studies using such spring-mass models predict that running can be self-stable. In simulations, this self-stability allows for running on uneven ground without paying attention to the ground irregularities. Whether humans actually use this property of the mechanical system in such an irregular environment is, however, unclear. One way to approach this question is to study how the leg stiffness in stance and the leg orientation in flight are changed in response to ground perturbations. Here, for 11 human subjects we studied two consecutive contacts during running on uneven ground with a force plate of adjustable height (step of +5, +10 and +15 cm). We found that runners adjust their leg stiffness to the height of a vertical step. The adjustment is characterized by a 9% increase in leg stiffness in preparation for the perturbation and by a systematic decrease in proportion to the step height. At the highest vertical step (+15 cm), leg stiffness was reduced by about 26%. We also observed that the angle of attack decreased from 68 deg. to 62 deg. with increasing ground height. These leg adjustments are in accordance with the predictions of a stable spring-mass system. Furthermore, we could describe the identified leg forces and leg compressions with a simple spring-mass simulation for a given body mass, leg stiffness, angle of attack and initial conditions. We compared the experimental findings with the self-stabilizing properties of the spring-mass model, and discuss how humans use a combination of strategies that include purely mechanical self-stabilization and active neuromuscular control. Finally, beyond self-stability, we suggest that control may apply to smooth centre of mass kinematics.

Modified SDOF Models for Improved Representation of the Impact Response of Composite Plates

Previous research has shown that simple single degree of freedom (SDOF) models can be used successfully to reproduce the force-time response of composite plates subjected to low-velocity impacts (LVI). In particular, a simple spring-mass (SM) model yields accurate estimates of the peak contact force as a function of impact energy. The major limitation of this model is that it applies only to elastic impact events, and ceases to be valid after the onset of damage. The effort of this investigation is to modify the SM model, while retaining the simplicity of its formulation, and to extend its range of validity to the super-critical regime. Three models are derived, namely the damaged stiffness, the dissipated energy, and the spring-mass-dashpot models, and their validity is examined through comparisons with experimental data. Impact force-time traces for carbon/epoxy laminates exhibiting various combinations of laminate thickness and support span are collected, and it is found that the additional unknowns introduced for each of the models, although determined empirically, are strongly related to the structural parameters for each configuration investigated.

Accurate evaluation of lowest band gaps in ternary locally resonant phononic crystals

Based on a better understanding of the lattice vibration modes, two simple spring–mass models are constructed in order to evaluate the frequencies on both the lower and upper edges of the lowest locally resonant band gaps of the ternary locally resonant phononic crystals. The parameters of the models are given in a reasonable way based on the physical insight into the band gap mechanism. Both the lumped-mass methods and our models are used in the study of the influences of structural and the material parameters on frequencies on both edges of the lowest gaps in the ternary locally resonant phononic crystals. The analytical evaluations with our models and the theoretical predictions with the lumped-mass method are in good agreement with each other. The newly proposed heuristic models are helpful for a better understanding of the locally resonant band gap mechanism, as well as more accurate evaluation of the band edge frequencies.

Ground vibration generated by trains in underground tunnels

A popular method used to reduce vibration transmitted from underground railways into nearby buildings is floating-slab track, whereby a concrete slab supporting the two rails is mounted on rubber bearings or steel springs to isolate it from the tunnel invert. This paper adds a track model to a previously developed three-dimensional tunnel model in order to assess the effectiveness of floating-slab track. A slab beam coupled to the tunnel in the wavenumber domain, with the slab bearings represented by an elastic layer, is examined first. A second beam representing the two rails together is then coupled to the slab, and axle masses representing a train are added to the rail beam. Power-spectral densities and RMS levels of soil vibration due to random roughness-displacement excitation between the masses and the rail beam are calculated. Analytical techniques are used to minimise the computational requirements of the model. The results demonstrate the inadequacy of simple mass–spring and Winkler-beam models with rigid foundations for the assessment of the vibration-isolation performance of railway track. They suggest that the achievable insertion loss is modest and that floating the track slab may in fact cause increased transmission of vibration under certain conditions.

Passive Vibration Control via Electromagnetic Shunt Damping

This work will present a new type of passive vibration control technique based on the concept of electromagnetic shunt damping. The proposed technique is similar to piezoelectric shunt damping, as an appropriately designed impedance is shunted across the terminals of the transducer. Theoretical and experimental results are presented for a simple electromagnetic mass spring damper system.

Simple Spring-mass Model Simulation of Earthquake Cycle along the Nankai Trough in Southwest Japan

Along the Nankai trough in southwest Japan, due to the subduction of the Philippine Sea plate, great earthquakes have occurred repeatedly. The rupture zone is divided into five segments. Historical documents show the characteristic features of the past several earthquake cycles, such as almost simultaneous rupture occurrence in segments in spite of different convergence rates, the recurrence time of 90 to 150 years, the existence of segment pairing in earthquake ruptures and the different coseismic slip behaviors in respective segments.

Wind and remnant tree sway in forest cutblocks. II. Relating measured tree sway to wind statistics

New silvicultural designs use unharvested forest strips to provide wind shelter in harvest cutblocks, and reduce windthrow of remnant trees. In this study we attempted to quantify wind shelter in terms of tree sway. We investigated the relationship between wind statistics and tree sway in two cutblocks in northern Alberta, Canada: a narrow cutblock (1.7 canopy heights in width, X c = 1.7 h), and a wide cutblock ( X c = 6.1 h). We focused on the case of winds oriented across the cutblock width. The instantaneous across- and along-cutblock wind `forces' acting on an understory white spruce ( Picea glauca) were considered as proportional to u| u| and v| v| , where u and v are the across- and along-cutblock wind velocities at height z = 0.4 h. A simple mass–spring–damper displacement model, calibrated to tree sway measurements, was used to diagnose the sway response of a `characteristic' remnant spruce (damping coefficient ζ = 0.11, natural frequency ω n = 0.4 Hz). We determined that in the cutblocks this characteristic tree: (i) had a small average angular displacement 〈 θ〉 relative to its maximum displacement θ max; (ii) exhibited `resonant sway,' where the interaction of tree dynamics and turbulence increased sway by 10–35% over that expected if displacement was proportional to the instantaneous wind force; and (iii) had sway statistics which correlated well with the standard deviation of wind force σ u|u| . Modelled tree sway was used to infer the degree of windthrow protection in the sheltered cutblocks. Our approach was to predict a threshold average wind velocity in the open ( U w, essentially a weather station windspeed) which correlates with the occurrence of windthrow of remnant spruce in the cutblock. Our assumption was that an average velocity of 10 m s −1 causes windthrow of unprotected trees in the open. Larger windspeeds ( U w) are needed to cause windthrow in the cutblocks. In the wide cutblock U w varied from 13 m s −1 near the downwind forest edge to 25 m s −1 near the upwind edge. In the narrow cutblock U w varied from 19 m s −1 at the downwind edge to 30 m s −1 at the upwind edge. The most effective wind shelter, as given by the highest U w, was within three tree heights of the upwind forest. In designing harvest cutblocks to reduce windthrow, we suggest that cutblocks not exceed three tree heights in width.

Eigenvector scaling for mode localization in vibrating systems

A new closed-loop mode localization technique known as eigenvector scaling is presented. Eigenvector scaling is a form of eigenstructure placement that uses feedback to alter portions of the original system eigenvectors. When properly applied, this technique restricts energy propagation in a vibrating system. Active eigenvector scaling also permits the development of an analytic solution for the closed-loop mode-localized discrete system. The ability to produce analytic solutions for both the controlled and the uncontrolled systems permits a direct comparison of the absolute displacements between the two cases. It is demonstrated that closed-loop mode localization can be used to restrict vibrations in certain regions of a discrete structure. Three examples are provided that show how this feedback control technique may be applied to multi-degree-of-freedom simple spring-mass and spring-mass- damper systems, as well as to lumped models of more complicated flexible structures. IBRATION of flexible structures is a common problem in dy- namics and controls. The relative ease with which vibrational energy may propagate through complex systems can lead to exces- sive displacements at critical locations, which in turn can lead to reduced performance or failure. It is not always necessary to control vibrations throughout an entire structure, but rather controlling cer- tain areas within the structure that contain sensitive instruments or machines may be most critical. The motivation for this work, then, is to seek vibration control methods that produce quiet areas within a vibrating system subjected to arbitrary disturbances. One way of controlling flexible structure vibration is through mode localization. Mode localization restricts motion by contain- ing vibrational energy within a small area of the total system. This phenomenon was originally described in solid state physics1 but also exists in flexible structures.2'3 Mode-localized structures exhibit ar- eas of comparatively high displacements, with amplitudes in these parts potentially several magnitudes greater than in the rest of the system. In a large number of studies, mode localization has been considered to be unfavorable, as in mistuned rotor blades.4 How- ever, some studies maintain that mode localization may be useful, since it confines system motion within a specific area, preventing the transmission of potentially harmful vibrational energy.5 Addition- ally, if vibrational energy is contained, then it may be more easily controlled with damping or cancellation. Loosely coupled, slightly mistuned flexible structures are espe- cially susceptible to mode localization.2'6'7 As a result, studies have concentrated on the passive mistuning of this type of structure. It is, however, also possible to create mode localization with active, closed-loop feedback control using a technique known as eigen- structure placement, which is the approach used in this paper.

Dynamics of a state variable frictional law in finite element analysis

This paper discusses a state- and rate-dependent frictional model and its implementation in large deformation finite elements. The model features a state variable to describe fading memory of slip history and a Perzyna viscoplastic regularization to quantify the frictional response to rapid slip changes. Some results in nonlinear dynamics for a simple spring mass frictional slider are very briefly presented as background for some frictional induced instabilities in finite elements that are discussed next. Comparison with available experimental data is performed and an overall good agreement is observed where a simple Coulomb law failed to predict reasonable results.

An explicit time integration technique for dynamic analyses

AbstractA simple explicit solution technique for problems in structural dynamics, based on a Modified Trapezoidal rule Method (MTM) approximation of the governing ordinary differential equations, is developed. The resulting conditionally stable explicit method (MTM) can be easily implemented and is extremely simple to use. Particular attention is focused herein on the concept of numerical stability of the proposed method for a free‐vibrational response of a linear undamped Single‐Degree‐Of‐Freedom system (SDOF). To examine the effectiveness, strengths, and limitations of MTM, error analyses for the natural period, the displacement, the velocity and the associated phase angle for a free undamped simple mass–spring system are derived and compared with Modified Euler Method (MEM) and the well‐known Newmark Beta Method (NBM). Numerical examples for a SDOF system and a Multi‐Degree‐Of‐Freedom (MDOF) system are presented to illustrate the strengths and the limitations of the proposed method.

Measuring piano hammers and modeling their interaction

Comparison of measured piano string vibration spectra with predictions indicates that modeling the hammer with a simple mass and Hooke's law spring is not very satisfactory [Hall and Askenfelt, J. Acoust. Soc. Am. 83, 1627 (1988)]. Measurements have been made on additional hammer sets to see if values of the exponent p in a nonlinear force law F = Kξp can be connected either with hammer quality or with other aspects of piano design and history. The corresponding theory of hammer/string interaction requires solution of nonlinear equations of motion, and has previously been done only for a few isolated examples. Steps toward a flexible general computer program that can explore the dependence of string spectra upon striking parameters with nonlinear hammers will be discussed.

Strain dependence of the vibrational modes of a diacetylene crystal

AbstractResonance Raman spectroscopy has been used to measure the strain dependence of four high‐frequency vibrational modes in a single crystal of toluene sulfonate diacetylene polymer. The 0.62‐μm thick crystal was stretched elastically to 4% strain before fracture occurred. This was equivalent to an ultimate tensile strength of 2.0 × 109 N m−2. The strain dependence of the vibrational modes has been interpreted using a simple point mass and anharmonic spring constant model. A Fermi resonance was observed for two vibrational modes which are degenerate at 2.1% strain and a 1464 cm−1 frequency.

Pressure and temperature dependence of the electronic and vibrational excitations of a conjugated polymer crystal

The pressure dependence of reflection, absorption and resonant Raman spectra of single crystals of toluene sulphonate dicetylene polymer have been measured under hydrostatic conditions to 5 kbars. The peak of the reflection spectrum near 2 eV shifted to lower energies with increasing pressure at a rate of 7.3+or-0.3 meV per kbar. This result is shown to be in qualitative agreement with the prediction of molecular orbital calculations. The temperature and pressure dependence of the resonant Raman spectra are discussed using a simple point mass and anharmonic spring constant model.

Human tolerance to impact acceleration

Tolerance to brief impact is related to the imposed velocity change, whilst in longer impacts it is the level of acceleration which is critical. This behaviour results from the dynamic characteristics of the body which, for practical purposes, can be represented as a simple mass spring system. For any given acceleration vector, and for any given system of body restraint, whole body (primary) tolerance can be defined by two parameters—a critical velocity change and a critical plateau acceleration level. Appropriate figures are given for 14 different posture and restraint conditions. It is stressed that these are best estimates (or in some cases, best guesses) and that secondary or tertiary mechanisms of injury, in particular head strikes, must not be ignored in any real-life conditions of impact.

Seismic model of the gas cooled fast breeder reactor core support structure

The modelling technique for the seismic analysis of the core support structure of a gas-cooled fast breeder reactor is developed. The core support structure consists of the support cylinder and a perforated grid plate to which 265 fuel and blanket elements are clamped as cantilevered beams. The analysis of the core support structure consists of three steps: (1) analysis of the grid plate, (2) analysis of the core elements, and (3) modal synthesis. The first step in developing a solution to the problem is to assume that the core elements (fuel and blanket) are attached to the grid plate as rigid rods. In this case the influence of the rigid rods can be represented by their masses and rotary inertias. The solution of this problem was developed by applying the dynamic theory of grid plates. This was accomplished by generalizing the Reissner-Mindlin thick-plate theory with orthotropic constants and then modifying the formulations of the rotary inertia expressions to include the rotary inertia effects of the core elements. The numerical results showed that the grid plate's fundamental frequency is in the range of the fundamental frequencies of the core elements so that a dynamic coupling effect exists. Because of this dynamic coupling effect the elastic properties of the grid plate must be included in the seismic analysis of the GCFR' The second step was to develop a mathematical model of the grid-plate core-element system using the method of Rayleigh-Ritz. In this model the elastic coupling effect of the core elements was included. For the final application of the theory, the exact solution of the elastic plate with rigid rods was simulated on the computer by applying the elastic rotary inertias of the core elements to the model of the grid plate. With this technique it is possible to model the grid plate with a reasonable number of fuel and blanket elements and to replace the missing core elements with their equivalent effective rotary inertias. The method includes the capability of modeling the different mass, damping and elastic properties of the fuel and blanket elements. Comparing the results of the present analysis with the preliminary simple spring-mass core model, the amplitudes of vibration obtained, in some cases in the present analysis, are a factor of ten smaller than was previously computed. Applying this more elaborate analysis will lead to a simpler and less expensive design.