scholarly journals Running springs: speed and animal size

1993 ◽  
Vol 185 (1) ◽  
pp. 71-86 ◽  
Author(s):  
C. T. Farley ◽  
J. Glasheen ◽  
T. A. McMahon
Keyword(s):  
Body Mass ◽  
Stance Phase ◽  
Center Of Mass ◽  
Vertical Vibration ◽  
Contact Phase ◽  
Mass System ◽  
Spring System ◽  
Animal Size ◽  
Resonant Period ◽  
Leg Spring

Trotting and hopping animals use muscles, tendons and ligaments to store and return elastic energy as they bounce along the ground. We examine how the musculoskeletal spring system operates at different speeds and in animals of different sizes. We model trotting and hopping as a simple spring-mass system which consists of a leg spring and a mass. We find that the stiffness of the leg spring (k(leg)) is nearly independent of speed in dogs, goats, horses and red kangaroos. As these animals trot or hop faster, the leg spring sweeps a greater angle during the stance phase, and the vertical excursion of the center of mass during the ground contact phase decreases. The combination of these changes to the spring system causes animals to bounce off the ground more quickly at higher speeds. Analysis of a wide size range of animals (0.1-140 kg) at equivalent speeds reveals that larger animals have stiffer leg springs (k(leg) [symbol: see text] M0.67, where M is body mass), but that the angle swept by the leg spring is nearly independent of body mass. As a result, the resonant period of vertical vibration of the spring-mass system is longer in larger animals. The length of time that the feet are in contact with the ground increases with body mass in nearly the same way as the resonant period of vertical vibration.

scholarly journals Estimation of leg stiffness using an approximation to the planar spring–mass system in high-speed running

2020 ◽  
Vol 17 (1) ◽  
pp. 172988141989071
Author(s):  
Wei Guo ◽  
Changrong Cai ◽  
Mantian Li ◽  
Fusheng Zha ◽  
Pengfei Wang ◽  
...  
Keyword(s):  
Analytical Solution ◽  
High Speed ◽  
Stance Phase ◽  
Center Of Mass ◽  
Critical Role ◽  
Line Center ◽  
Motion Model ◽  
Mass System ◽  
Leg Stiffness ◽  
Wide Range

Leg stiffness plays a critical role in legged robots’ speed regulation. However, the analytic solutions to the differential equations of the stance phase do not exist, of course not for the exact analytical solution of stiffness. In view of the challenge in dealing with every circumstance by numerical methods, which have been adopted to tabulate approximate answers, the “harmonic motion model” was used as approximation of the stance phase. However, the wide range leg sweep angles and small fluctuations of the “center of mass” in fast movement were overlooked. In this article, we raise a “triangle motion model” with uniform forward speed, symmetric movement, and straight-line center of mass trajectory. The characters are then shifted to a quadratic equation by Taylor expansion and obtain an approximate analytical solution. Both the numerical simulation and ADAMS-Matlab co-simulation of the control system show the accuracy of the triangle motion model method in predicting leg stiffness even in the ultra-high-speed case, and it is also adaptable to low-speed cases. The study illuminates the relationship between leg stiffness and speed, and the approximation model of the planar spring–mass system may serve as an analytical tool for leg stiffness estimation in high-speed locomotion.

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

1998 ◽  
Vol 201 (21) ◽  
pp. 2935-2944 ◽  
Author(s):  
C R Lee ◽  
C T Farley
Keyword(s):  
Stance Phase ◽  
Center Of Mass ◽  
Human Walking ◽  
Gait Transition ◽  
Mass System ◽  
Pendulum System ◽  
Vertical Movements ◽  
Transition Speed ◽  
Inverted Pendulum System ◽  
Limb Compression

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.

scholarly journals Shape Optimization of Running Specific Prosthesis Based on Force-Displacement Characteristics

Proceedings ◽  
2020 ◽  
Vol 49 (1) ◽  
pp. 7
Author(s):  
Cem Guzelbulut ◽  
Katsuyuki Suzuki ◽  
Satoshi Shimono ◽  
Hiroaki Hobara
Keyword(s):  
Contact Region ◽  
Contact Phase ◽  
Mass System ◽  
Reinforced Plastics ◽  
Non Linear ◽  
Spring System ◽  
Two Phases ◽  
And Performance ◽  
Mass Spring ◽  
Force Displacement

Usage of carbon fiber reinforced plastics (CFRPs) in running-specific prostheses increases day by day. The tailorable properties of CFRP blades bring many discussions about design and performance. In this study, the effect of shape on performance is investigated through force-displacement characteristics of the prosthesis. For this purpose, the geometry of prosthesis is defined by using B-splines with an initially given thickness. The prosthesis is exposed to vertical tip load at the mounting point, and contact is defined between the prosthesis and ground without friction. The aim of the simulation is to observe the contact behavior of athletes at different positions during the contact phase of a prosthesis. While the prosthesis is in contact with the ground, two different behaviors are observed: compression occurs at a larger contact zone, whereas release occurs at a smaller contact region (almost only the tip of the prosthesis). Different force-displacement characteristics, such as linear and second order, are obtained and the geometry of the prosthesis is optimized to adjust the behavior in the first region. The releasing phase of a prosthesis is related to the contact angle (angle of attack) and stiffness of the prosthesis. The two phases of contact are combined into a non-linear spring-mass system. Ground reaction forces are estimated through the non-linear mass-spring system. Finally, the importance of contacting area, length of moment arm during contact, and effect of each type of force-displacement characteristics on performance is discussed.

scholarly journals A new method for measuring treadmill belt velocity fluctuations: effects of treadmill type, body mass and locomotion speed

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Steffen Willwacher ◽  
Kai Daniel Oberländer ◽  
Patrick Mai ◽  
Daniela Mählich ◽  
Markus Kurz ◽  
...  
Keyword(s):  
Body Mass ◽  
Stance Phase ◽  
Center Of Mass ◽  
Target Speed ◽  
Physiological Variables ◽  
Locomotion Speed ◽  
Type Interaction ◽  
Novel Method ◽  
Braking Phase ◽  
Treadmill Belt

AbstractTreadmills are essential to the study of human and animal locomotion as well as for applied diagnostics in both sports and medicine. The quantification of relevant biomechanical and physiological variables requires a precise regulation of treadmill belt velocity (TBV). Here, we present a novel method for time-efficient tracking of TBV using standard 3D motion capture technology. Further, we analyzed TBV fluctuations of four different treadmills as seven participants walked and ran at target speeds ranging from 1.0 to 4.5 m/s. Using the novel method, we show that TBV regulation differs between treadmill types, and that certain features of TBV regulation are affected by the subjects’ body mass and their locomotion speed. With higher body mass, the TBV reductions in the braking phase of stance became higher, even though this relationship differed between locomotion speeds and treadmill type (significant body mass × speed × treadmill type interaction). Average belt speeds varied between about 98 and 103% of the target speed. For three of the four treadmills, TBV reduction during the stance phase of running was more intense (> 5% target speed) and occurred earlier (before 50% of stance phase) unlike the typical overground center of mass velocity patterns reported in the literature. Overall, the results of this study emphasize the importance of monitoring TBV during locomotor research and applied diagnostics. We provide a novel method that is freely accessible on Matlab’s file exchange server (“getBeltVelocity.m”) allowing TBV tracking to become standard practice in locomotion research.

A novel tuned mass-conical spring system for passive vibration control of a variable mass structure

2021 ◽  
pp. 107754632110004
Author(s):  
Sanjukta Chakraborty ◽  
Aparna (Dey) Ghosh ◽  
Samit Ray-Chaudhuri
Keyword(s):  
Variable Mass ◽  
Design Procedure ◽  
Time History ◽  
Tuned Mass Damper ◽  
Water Tank ◽  
Mass System ◽  
Mass Damper ◽  
Linear Spring ◽  
Spring System ◽  
Elevated Water

This article presents the design of a tuned mass damper with a conical spring to enable tuning to the natural frequency of the system at multiple values, as may be convenient in case of a system with fluctuations in the mass. The principle and design procedure of the conical spring in the context of a varying mass system are presented. A passive feedback control mechanism based on a simple pulley-mass system is devised to cater to the multi-tuning requirements. A design example of an elevated water tank with fluctuating water content, subjected to ground excitation, is considered to numerically illustrate the efficiency of such a tuned mass damper associated with the conical spring. The conical spring is designed based on the tuning requirements at different mass conditions of the elevated water tank by satisfying the allowable load bearing capacity of the spring. Comparisons are made with the conventional passive tuned mass damper with a linear spring tuned to the full tank condition. Results from time history analysis reveal that the conical spring-tuned mass damper can be successfully designed to remain tuned and thereby achieve significant response reductions under stiffening conditions of the primary structure, whereas the linear spring-tuned mass damper suffers performance degradation because of detuning, whenever there is any fluctuation in the system mass.

scholarly journals Vertical Jumping for Legged Robot Based on Quadratic Programming

Sensors ◽  
2021 ◽  
Vol 21 (11) ◽  
pp. 3679
Author(s):  
Dingkui Tian ◽  
Junyao Gao ◽  
Xuanyang Shi ◽  
Yizhou Lu ◽  
Chuzhao Liu
Keyword(s):  
Quadratic Programming ◽  
Stance Phase ◽  
Center Of Mass ◽  
Maximum Error ◽  
Legged Robot ◽  
Flight Phase ◽  
Programming Framework ◽  
Vertical Jumping ◽  
Control Schemes ◽  
And Control

The highly dynamic legged jumping motion is a challenging research topic because of the lack of established control schemes that handle over-constrained control objectives well in the stance phase, which are coupled and affect each other, and control robot’s posture in the flight phase, in which the robot is underactuated owing to the foot leaving the ground. This paper introduces an approach of realizing the cyclic vertical jumping motion of a planar simplified legged robot that formulates the jump problem within a quadratic-programming (QP)-based framework. Unlike prior works, which have added different weights in front of control tasks to express the relative hierarchy of tasks, in our framework, the hierarchical quadratic programming (HQP) control strategy is used to guarantee the strict prioritization of the center of mass (CoM) in the stance phase while split dynamic equations are incorporated into the unified quadratic-programming framework to restrict the robot’s posture to be near a desired constant value in the flight phase. The controller is tested in two simulation environments with and without the flight phase controller, the results validate the flight phase controller, with the HQP controller having a maximum error of the CoM in the x direction and y direction of 0.47 and 0.82 cm and thus enabling the strict prioritization of the CoM.

Impact of Age and Body Mass Index on Anthropometry in Working Adults

2017 ◽  
Vol 61 (1) ◽  
pp. 1341-1345 ◽  
Author(s):  
Zachary Merrill ◽  
April Chambers ◽  
Rakié Cham
Keyword(s):  
Body Mass Index ◽  
Body Mass ◽  
Center Of Mass ◽  
Biomechanical Model ◽  
Whole Body ◽  
Working Adults ◽  
Biomechanical Models ◽  
Scan Data ◽  
Body Segment Parameters ◽  
The Impact

Body segment parameters (BSPs) such as segment mass and center of mass are used as inputs in ergonomic design and biomechanical models to predict the risk of musculoskeletal injuries. These models have been shown to be sensitive to the BSP values used as inputs, demonstrating the necessity of using accurate and representative parameters. This study aims to provide accurate BSPs by quantifying the impact of age and body mass index on torso and thigh mass and center of mass in working adults using whole body dual energy x-ray absorptiometry (DXA) scan data. The results showed significant effects of gender, age, and body mass index (BMI) on torso and thigh mass and center of mass, as well as significant effects of age and BMI within genders, indicating that age, gender, and BMI need to be taken into account when predicting BSPs in order to calculate representative ergonomic and biomechanical model outputs.

Simple Leg Placement Strategy for a One Legged Running Model

2012 ◽  
Author(s):  
Timothy Sullivan ◽  
Justin Seipel
Keyword(s):  
Stance Phase ◽  
Energy State ◽  
Center Of Mass ◽  
Mass Movement ◽  
Legged Locomotion ◽  
Energy Conserving ◽  
Lower Torque ◽  
The Stability ◽  
Time Required ◽  
Torque Values

The Spring Loaded Inverted Pendulum (SLIP) model was developed to describe center of mass movement patterns observed in animals, using only a springy leg and a point mass. However, SLIP is energy conserving and does not accurately represent any biological or robotic system. Still, this model is often used as a foundation for the investigation of improved legged locomotion models. One such model called Torque Damped SLIP (TD-SLIP) utilizes two additional parameters, a time dependent torque and dampening to drastically increase the stability. Forced Damped SLIP (FD-SLIP), a predecessor of TD-SLIP, has shown that this model can be further simplified by using a constant torque, instead of a time varying torque, while still maintaining stability. Using FD-SLIP as a base, this paper explores a leg placement strategy using a simple PI controller. The controller takes advantage of the fact that the energy state of FD-SLIP is symmetric entering and leaving the stance phase during steady state conditions. During the flight phase, the touch down leg angle is adjusted so that the energy dissipation due to dampening, during the stance phase, compensates for any imbalance of energy. This controller approximately doubles the region of stability when subjected to velocity perturbations at touchdown, enables the model to operate at considerably lower torque values, and drastically reduces the time required to recover from a perturbation, while using less energy. Finally, the leg placement strategy used effectively imitates the natural human response to velocity perturbations while running.

scholarly journals Numerical Method of Fabric Dynamics Using Front Tracking and Spring Model

2013 ◽  
Vol 14 (5) ◽  
pp. 1228-1251 ◽  
Author(s):  
Yan Li ◽  
I-Liang Chern ◽  
Joung-Dong Kim ◽  
Xiaolin Li
Keyword(s):  
Upper Bound ◽  
Mesh Size ◽  
Front Tracking ◽  
Time Step ◽  
Mass System ◽  
Ode System ◽  
Spring System ◽  
Fabric Surface ◽  
Mass Spring ◽  
Eigen Frequency

AbstractWe use front tracking data structures and functions to model the dynamic evolution of fabric surface. We represent the fabric surface by a triangulated mesh with preset equilibrium side length. The stretching and wrinkling of the surface are modeled by the mass-spring system. The external driving force is added to the fabric motion through the “Impulse method” which computes the velocity of the point mass by superposition of momentum. The mass-spring system is a nonlinear ODE system. Added by the numerical and computational analysis, we show that the spring system has an upper bound of the eigen frequency. We analyzed the system by considering two spring models and we proved in one case that all eigenvalues are imaginary and there exists an upper bound for the eigen-frequency This upper bound plays an important role in determining the numerical stability and accuracy of the ODE system. Based on this analysis, we analyzed the numerical accuracy and stability of the nonlinear spring mass system for fabric surface and its tangential and normal motion. We used the fourth order Runge-Kutta method to solve the ODE system and showed that the time step is linearly dependent on the mesh size for the system.

HELICITY AMPLITUDES FOR COLLIDING BEAM SCATTERING OF TWO SPIN 1/2 PARTICLES

1993 ◽  
Vol 08 (30) ◽  
pp. 5383-5407
Author(s):  
T.B. ANDERS ◽  
A.O. BARUT ◽  
W. JACHMANN
Keyword(s):  
Center Of Mass ◽  
Mass System ◽  
Intermediate Transformation ◽  
Beam System ◽  
Pair Annihilation ◽  
Helicity Amplitudes ◽  
Reaction Channels ◽  
Invariant Coupling ◽  
Exchange Scattering ◽  
The One

As a generalization and extension of the extensive tables of polarization asymmetries given in a previous work,1 we present here tables of helicity amplitudes for the scattering of two spin 1/2 particles in the colliding beam system (i.e. two incoming particles with opposite directions but not necessarily of equal momenta). The particles belonging to the same current may have different masses in order to describe particle excitations. The amplitudes are given for six different basic couplings connecting two vector vertices, a vector vertex at the one current and a derivative vector vertex at the other current, two derivative vector vertices, two tensor vertices, and two scalar vertices. The vertices include axial couplings by factors of type 1+cγ5. The amplitudes are written as expressions with 16 components in the six different reaction channels, namely the scattering of two fermions, of two antifermions, and of a fermion and an antifermion, the pair creation by pair annihilation, as well as the exchange scattering for two identical fermions or antifermions. The formulas may be used for an analysis which extracts the invariant coupling functions from the experimental data obtained in the colliding beam system directly without an intermediate transformation to the center of mass system.

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