Simple Mass-spring System Research Articles

Damage Localization from the Null Space of Changes in the Transfer Matrix

*A new theorem connecting changes in transfer matrices to the spatial location of stiffness related damage is presented. The theorem states that the span of the null space of the change in the transfer matrix (∆G) contains vectors that are Laplace transforms of excitations for which the dynamic stress field is identically zero over the portion of the domain where the damage is located. An important corollary states that if ∆G proves rank deficient at any s (with the exception of the origin in system with rigid body modes) it is guaranteed to be rank deficient throughout the plane. It is shown that a sufficient condition for rank deficiency in ∆G is that the number of independent measurements be larger than the rank of the change in stiffness resulting from damage. Implementation of the theorem in the time domain involves evaluation of the null space of ∆G along a discretized Bromwich contour, numerical inversion of the resulting s-functions and evaluation of the dynamic response to the computed signals to identify the regions where the stress field is zero. The theorem is most efficiently implemented, however, in the s-domain, where numerical Laplace inversion is avoided and the number of evaluations of the null of ∆G can reduced to one. Application of the theorem to locate damage is illustrated in a simple mass-spring system and in a FEM model of a square plate with free-free boundary conditions.

Matrix Formulation for Minimum Response of Undamped Structure

A useful method for the determination of frequencies that correspond to points of local minima in the frequency response of finite element structure is presented. The method is based on the application of linear matrix algebra methods in conjunction with the fundamental definitions for existence of local minima. The procedure was initially developed for determination of antiresonant frequencies, and is now extended for application to minimum response. The derivations of the procedures for antiresonance and minimum response are both included in this paper. A significant advantage of this method is that it allows for the formulation of sensitivity relationships similar to what is commonly used for resonant frequency. Thus, there is potential for expanded use of minimum response information for structural optimization, damage detection, and other related tasks. In addition, generality in the formulation is retained such that the response may be defined as any linear combination of the coordinate responses, and any arbitrary forcing function may be considered. A numerical example, with a simple one-dimensional spring-mass system, is presented to provide for illustration and validation of the approach.

Patterns of mechanical energy change in tetrapod gait: pendula, springs and work

Kinematic and center of mass (CoM) mechanical variables used to define terrestrial gaits are compared for various tetrapod species. Kinematic variables (limb phase, duty factor) provide important timing information regarding the neural control and limb coordination of various gaits. Whereas, mechanical variables (potential and kinetic energy relative phase, %Recovery, %Congruity) provide insight into the underlying mechanisms that minimize muscle work and the metabolic cost of locomotion, and also influence neural control strategies. Two basic mechanisms identified by Cavagna et al. (1977. Am J Physiol 233:R243-R261) are used broadly by various bipedal and quadrupedal species. During walking, animals exchange CoM potential energy (PE) with kinetic energy (KE) via an inverted pendulum mechanism to reduce muscle work. During the stance period of running (including trotting, hopping and galloping) gaits, animals convert PE and KE into elastic strain energy in spring elements of the limbs and trunk and regain this energy later during limb support. The bouncing motion of the body on the support limb(s) is well represented by a simple mass-spring system. Limb spring compliance allows the storage and return of elastic energy to reduce muscle work. These two distinct patterns of CoM mechanical energy exchange are fairly well correlated with kinematic distinctions of limb movement patterns associated with gait change. However, in some cases such correlations can be misleading. When running (or trotting) at low speeds many animals lack an aerial period and have limb duty factors that exceed 0.5. Rather than interpreting this as a change of gait, the underlying mechanics of the body's CoM motion indicate no fundamental change in limb movement pattern or CoM dynamics has occurred. Nevertheless, the idealized, distinctive patterns of CoM energy fluctuation predicted by an inverted pendulum for walking and a bouncing mass spring for running are often not clear cut, especially for less cursorial species. When the kinematic and mechanical patterns of a broader diversity of quadrupeds and bipeds are compared, more complex patterns emerge, indicating that some animals may combine walking and running mechanics at intermediate speeds or at very large size. These models also ignore energy costs that are likely associated with the opposing action of limbs that have overlapping support times during walking. A recent model of terrestrial gait (Ruina et al., 2005. J Theor Biol, in press) that treats limb contact with the ground in terms of collisional energy loss indicates that considerable CoM energy can be conserved simply by matching the path of CoM motion perpendicular to limb ground force. This model, coupled with the earlier ones of pendular exchange during walking and mass-spring elastic energy savings during running, provides compelling argument for the view that the legged locomotion of quadrupeds and other terrestrial animals has generally evolved to minimize muscle work during steady level movement.

Stability of equilibrium states in a simple system with unilateral contact and Coulomb friction

The aim of this paper is to study the stability of equilibrium states in a mechanical system involving unilateral contact with Coulomb friction. Since the assumptions made in classical stability theorems are not satisfied with this class of systems, we return to the basic definitions of stability by studying the time evolution of the distance between a given equilibrium and the solution of a Cauchy problem where the initial conditions are in a neighborhood of the equilibrium. It was recently established that the dynamics is well posed in the case of analytical data. In the present study, we focus in particular on the stability of the equilibrium states under a constant force and deal only with a simple mass-spring system in $$\mathbb{R}^{2}$$ .

Side-Wall Measurement using Tilt-Scanning Method in Atomic Force Microscope

We have developed a novel atomic force microscope (AFM) measurement technique which can examine sidewalls of fine patterns on wafers. This technique uses a sharpen tip tilted at an angle in combination with digital probing mode operation, and is thus referred as “tilt-step-in” mode operation. This method allows one to measure sidewall shape moving along tilted tip axis. We analyzed the slip condition between the tip and the sample using a simple spring-mass system model and finite element method (FEM) with several parameters, such as moving direction, stiffness of tip and cantilever, sidewall angle and frictional coefficient. To verify this method, we then measured several reference samples with perpendicular sidewalls and 105° undercuts. By using this technique three dimensional (3-D) images of low-k etch structure of semiconductor device patterns with 88° sidewall and line edge roughness of ArF resist were clearly observed.

Leg Stiffness and Expertise in Men Jumping

The aim of the present study is to investigate: a) the leg spring behavior in the one-leg vertical jump, b) the contribution of impulse parameters to this behavior, and c) the effect of jumping expertise on leg stiffness. Four categories of experts (handball, basketball, volleyball players, and Fosbury athletes), as well as novice subjects performed a run-and-jump test to touch a ball with the head. Five experimental conditions were tested from 55 to 95% of the maximum jump height. Kinematic and kinetic data were collected using six cameras and a force plate. The mechanical behavior of the musculoskeleton component of the human body can be modeled as a simple mass-spring system, from which leg stiffness values can be extracted to better understand energy transfer during running or jumping. The results indicate that leg stiffness (mean value of 11.5 kN.m) decreased with jumping height. Leg shortening at takeoff also increased with jumping height, whereas contact time decreased (-18%). No difference was found between experts and novices for leg stiffness. However, a principal components analysis (PCA) indicated the contribution of two main factors to the performance. The first factor emerged out of vertical force, stiffness, and duration of impulse. The second factor included leg shortening and jumping height. Differences between experts and novices were observed in terms of the contribution of leg stiffness to jump height, and more importantly, clear differences existed between experts in jumping parameters. The analysis performed on the sport categories indeed revealed different jumping profiles, characterized by specific, sport-related impulse parameters.

Reliability of Uncertain Inelastic Structures under Earthquake Excitation

The assessment of safety for structures subjected to earthquake excitation is a time-variant reliability problem whose practical solution, when the structure behaves inelastically and uncertainty in the mechanical parameters is taken into account, is traditionally sought by means of simulation. In this paper the feasibility of an alternative solution based on the first order reliability method is explored. The problem is expressed in terms of first-excursion event within a conceptual framework, established in recent years, in which random vibration problems are discretized to make them tractable by the well-established methods of time-invariant structural reliability. The necessary extensions to apply the methodology to reinforced concrete frame structures are presented. Applications to a simple spring-mass system and to a complete three-dimensional bridge structure are presented.

Pull-in voltage analysis of electrostatically actuated beam structures with fixed–fixed and fixed–free end conditions

In this paper, a closed-form expression for the pull-in voltage of fixed–fixed beams and fixed–free beams is derived starting from the known expression of a simple lumped spring-mass system. The effects of partial electrode configuration, of axial stress, non-linear stiffening, charge re-distribution and fringing fields are all included in the final expression. Further, the results obtained are summarized and validated with other existing empirical and analytical models as well as with finite element simulation results. The model agrees well with finite element simulation results obtained with COVENTORWARE software.

Muscle pre- and coactivity during downward stepping are associated with leg stiffness in aging

We have previously reported that elderly compared to young women executed downward stepping with substantially greater leg stiffness. Because antagonist muscle coactivity increases joint stiffness we hypothesized that increased leg stiffness in aging is associated with increased muscle coactivity. We also explored the possibility that the magnitude of the preparatory muscle activity preceding impact also differed between young and old subjects. Young ( n=11, 20.8 yr) and old ( n=12, 69 yr) women performed downward stepping from a platform set at 20% body height. The leg was modeled as a simple mass-spring system. From video and ground reaction force data leg stiffness was computed as the ratio of force under the foot and the linear shortening of the limb. EMG activity of the vastus lateralis, biceps femoris, gastrocnemius lateralis, and tibialis anterior were recorded with a telemetric system. Elders compared to young subjects had 64% greater leg stiffness during downward stepping. Muscle activity over a 200-ms period preceding touch down was 136% greater in elderly than in young subjects. Biceps femoris and tibialis anterior coactivity during ground contact was 120% greater in the elders. Muscle pre- and coactivity, respectively, accounted for about 50% of the variance in leg stiffness. In conclusion, elderly people elevate muscle pre- and coactivity during downward stepping to stiffen the leg in compensation for impaired neuromotor functions.

On Decoupling Control for a Mechanical System with Parameter Perturbations

Parameter perturbations on decoupling control problem of linear multivariable systems are discussed in this paper. Fabian & Wonham had considered this problem. But the class of parameter perturbations that they studied preserved the decoupling property restrictive. The new class of parameter perturbation which preserves only the decoupling property is derived here. The proposed procedure for calculating the class is useful to mechanical systems when parameter perturbations are confined to specific elements. Experimental results show that this method is effective for a simple mass-spring system. Robust stability is also discussed in this experiment.

Determinants of the center of mass trajectory in human walking and running.

Walking is often modeled as an inverted pendulum system in which the center of mass vaults over the rigid stance limb. Running is modeled as a simple spring-mass system in which the center of mass bounces along on the compliant stance limb. In these models, differences in stance-limb behavior lead to nearly opposite patterns of vertical movements of the center of mass in the two gaits. Our goal was to quantify the importance of stance-limb behavior and other factors in determining the trajectory of the center of mass during walking and running. We collected kinematic and force platform data during human walking and running. Virtual stance-limb compression (i.e. reduction in the distance between the point of foot-ground contact and the center of mass during the first half of the stance phase) was only 26% lower for walking (0.091 m) than for running (0.123 m) at speeds near the gait transition speed. In spite of this relatively small difference, the center of mass moved upwards by 0.031 m during the first half of the stance phase during walking and moved downwards by 0.073 m during the first half of the stance phase during running. The most important reason for this difference was that the stance limb swept through a larger angle during walking (30.4 degrees) than during running (19.2 degrees). We conclude that stance-limb touchdown angle and virtual stance-limb compression both play important roles in determining the trajectory of the center of mass and whether a gait is a walk or a run.

Structural Design Techniques for Reducing Mecanical Vibration of Automotive Transmission Cases.

The passenger car transmission today must be more compact in size, lighter in weight and operate more quietly than ever befor. The structure of the transmission case influences gear noise, rumbling noise and booming noise, as these noise phenomena are caused by various powerplant resonance modes. These resonance modes are characterized by the fundamental structure of the powerplant, and their natural frequencies can be estimated using a simple spring-mass system. This paper presents different approrches to the structural design of the transmission case which achieve weight reduction while at the same time satisfying the requirements for noise and vibration perfor mance.

Integration of VLSI circuits and mechanics for vibration control of flexible structures

The approach to modeling and control of smart flexible structures presented in this paper is based on the concept that an intelligent structure requires an internal knowledge of self to act intelligently. This knowledge can be acquired from local analog models of substructure dynamics and can be used in model-based controller designs. The key to this approach is the synergistic integration of analog VLSI circuit models and control with the sensing and actuation which are then to be embedded into the mechanical structure. This paper presents the motivation, development, and test results of analog VLSI circuit models for use in model-based control of smart flexible structures. Furthermore, control applications for these VLSI circuits are developed and simulation results are presented in which the VLSI circuits are used in adaptive vibration control of a simple mass-spring system.

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.

Dynamic analyses of multiple-degree-of-freedom structures using a modified heun method

A simple explicit solution technique for problems in structural dynamics, based on Heun's numerical method for ordinary differential equations, is developed. The resulting conditionally stable modified Heun method (MHM) is easily implemented and provides a more accurate dynamic response than the modified Euler's method. To examine the effectiveness, strengths, and limitations of MHM, error analyses for the natural period, the displacement, the velocity and associated phase angle for a free and undamped simple mass-spring system are derived. Numerical examples for a single-degree-of-freedom system and a multi-degree-of-freedom (MDOF) system are presented. In direct integration of MDOF systems, the amplitude decay characteristic exhibited by MHM provides a total system solution in which the low mode response is accurately calculated.

Hopping frequency in humans: a test of how springs set stride frequency in bouncing gaits

The storage and recovery of elastic energy in muscle-tendon springs is important in running, hopping, trotting, and galloping. We hypothesized that animals select the stride frequency at which they behave most like simple spring-mass systems. If higher or lower frequencies are used, they will not behave like simple spring-mass systems, and the storage and recovery of elastic energy will be reduced. We tested the hypothesis by having humans hop forward on a treadmill over a range of speeds and hop in place over a range of frequencies. The body was modeled as a simple spring-mass system, and the properties of the spring were measured by use of a force platform. Our subjects used nearly the same frequency (the "preferred frequency," 2.2 hops/s) when they hopped forward on a treadmill and when they hopped in place. At this frequency, the body behaved like a simple spring-mass system. Contrary to our predictions, it also behaved like a simple spring-mass system when the subjects hopped at higher frequencies, up to the maximum they could achieve. However, at the higher frequencies, the time available to apply force to the ground (the ground contact time) was shorter, perhaps resulting in a higher cost of generating muscular force. At frequencies below the preferred frequency, as predicted by the hypothesis, the body did not behave in a springlike manner, and it appeared likely that the storage and recovery of elastic energy was reduced. The combination of springlike behavior and a long ground contact time at the preferred frequency should minimize the cost of generating muscular force.

Understanding natural frequency and damping and how they relate to the measurement of blood pressure.

The model that describes the physical behavior of a fluid-filled catheter-transducer blood pressure monitoring system is a simple mass-spring system. When the mass is displaced and then released, there results a characteristic motion called simple harmonic motion. The full description of this motion requires defining the concepts of undamped and damped natural frequency, as well as of damping itself. Once these concepts are defined and the mass-spring system clearly understood, their relevance to recording blood pressure measurement by fluid-filled catheters is explained. The apparent paradox of how damping can affect undamped natural frequency is clarified. Finally, impedance matching is explained in the context of how some damping devices work. Detailed mathematical proofs are relegated to an appendix.

The reconstruction of a tridiagonal system from its frequency response at an interior point

It is known that a simple spring-mass system may be reconstructed uniquely (apart from a single scaling factor) from the poles and zeros of the frequency response function corresponding to sinusoidal forcing at an end. The (squares of the) poles yield the eigenvalues of a tridiagonal matrix, A, while the zeros yield the eigenvalues of the matrix, A*, with the last row and column deleted. There are proven numerical methods for reconstructing A. The authors show that, if forcing is applied at an interior point, then the reconstruction is unique if that point is not a node of any eigenmode; if it is, there is a family of systems with the required properties. In either case the system may be constructed using modifications of proven techniques.

Response of a replicated human heart system to dynamic loading

The response of the human heart and attached major vasculature to rapid acceleration loading was studied by means of a non-pulsatile replica consisting of leotard fabric components. The system was designed to prevent leakage of water intended to simulate the blood present. This model was emplaced inside a rigid thorax and mounted on a moving cart, with sudden arrest produced by barrier impact. Large displacements of the ventricles and other attached vessels and pressures in the heart compartments were measured. Twelve impacts to the front, side, rear and at oblique incidence for a condition corresponding to the end of diastole were executed. The tests indicated that the strains in the right ventricle and the interchamber pressure were significant. Strains in the inferior vena cava were relatively high with lower values for the superior vena cava. Significant extensions were also observed in the brachiocephalic, left common carotid and left subclavian arteries. These findings qualitatively substantiate clinical experience for humans subjected to rapid non-invasive loading. A simple spring-mass system was used to model the dynamic response of the replicated heart unit. Predicted values were found to be in fair agreement with test data.

The Adaptation of Kron’s Method for Use With Large Finite-Element Models

Kron’s method of dynamic substructure coupling is modified and extended to a form which is well suited for use with large finite-element substructure models. It is shown how both mass condensation and modal truncation can be applied at the substructure level in a manner compatible with the Kron coupling procedure. Either master or slave freedoms may be used as coupling coordinates in the system model, thereby allowing complete flexibility at the substructure analysis stage and in particular, allowing the use of automatic master selection procedures. Substructures may be coupled either directly or through a flexible interlayer. System damping may thus be represented in a fairly general way with each substructure having its own (uniform) damping level but with the further provision of additional damping at the joining surfaces between substructures. The theory is illustrated by examples with simple mass-spring systems.