Sprinting with banked turns

Bank angle effects can attenuate peak running speed on the order of 10%. Experimental and theoretical results are presented here to quantify this phenomenon over a wide range of bank angles theta b and turn radii R. Experimentally, eleven subjects ran on a 34 m long plywood test track with variable radius and bank angle to sample the (R, theta b) space. From another study, ten subjects are borrowed to examine the theta b = 0 degrees case in greater detail. Various gait parameters were measured from high-speed film, and after parallax correction, compared with the theoretical predictions. The theory is a simple two-parameter constant force model requiring only the effective ankle pulley ratio beta and the runner's top speed vm. A closed-form dimensionless solution is presented for the speed ratio (v/vm) as a function of the radius number (Rg/v2) and the bank angle theta b. Agreement between theory and experiment is limited by experimental scatter. For twenty different subjects and twelve different combinations of R and theta b, the apparent ankle pulley ratio is beta = 0.27 +/- 0.22 based on 128 separate trials. Applications are discussed briefly for the design of indoor and outdoor running tracks. The theory allows a calculation of foot force, bone force, and tendon tension for the general case of arbitrary maximum speed, turn radius and bank angle.     

Bipedal locomotion: effects of speed, size and limb posture in birds and humans

Seven species of ground-dwelling birds (body mass range: 0.045-90 kg) were filmed while walking and running on a treadmill. High-speed light films were also taken of humans to compare kinematic patterns of avian with human bipedalism. Consistent patterns of stride frequency, stride length, step length, duty factor and limb excursion were observed in all species, with most of the variation among species being due to differences in body size. In general, smaller bipeds have higher stride frequencies (α M ^−0.18), shorter stride lengths (α M ^0.38) and more limited ranges of speed within each gait than large bipeds. After normalizing for size (based on Froude number, after Alexander, 1977), remaining kinematic variation is largely due to interspecific differences in posture and relative limb segment lengths. For their size, smaller bipeds have greater step lengths, limb excursion angles and duty factors than large bipeds because of their more crouched posture and greater effective limb length. The most notable differences in limb kinematics between birds and humans occur at the walk-run transition and are maintained as running speed increases. Change of gait is smooth and difficult to discern in birds, but distinct in humans, involving abrupt decreases in step length and duty factor (time of contact) and a corresponding increase in limb swing time. These differences appear to reflect a spring-like run that is stiff in humans (favouring elastic energy recovery) but more compliant in birds (increasing time of ground contact). Differences between birds and humans in balance of the body's centre of mass not only affect femoral orientation and motion, but also affect pattern of limb excursion with speed.     

Criteria for dynamic similarity in bouncing gaits

Animals of different sizes tend to move in a dynamically similar manner when travelling at speeds corresponding to equal values of a dimensionless parameter (DP) called the Froude number. Consequently, the Froude number has been widely used for defining equivalent speeds and predicting speeds of locomotion by extinct species and on other planets. However, experiments using simulated reduced gravity have demonstrated that equality of the Froude number does not guarantee dynamic similarity. This has cast doubt upon the usefulness of the Froude number in locomotion research. Here we use dimensional analysis of the planar spring-mass model, combined with Buckingham's Pi-Theorem, to demonstrate that four DPs must be equal for dynamic similarity in bouncing gaits such as trotting, hopping and bipedal running. This can be reduced to three DPs by applying the constraint of maintaining a constant average speed of locomotion. Sensitivity analysis indicates that all of these DPs are important for predicting dynamic similarity. We show that the reason humans do not run in a dynamically similar manner at equal Froude number in different levels of simulated reduced gravity is that dimensionless leg stiffness decreases as gravity increases. The reason that the Froude number can predict dynamic similarity in Earth gravity is that dimensionless leg stiffness and dimensionless vertical landing speed are both independent of size. In conclusion, although equal Froude number is not sufficient for dynamic similarity, it is a necessary condition. Therefore, to detect fundamental differences in locomotion, animals of different sizes should be compared at equal Froude number, so that they can be as close to dynamic similarity as possible. More generally, the concept of dynamic similarity provides a powerful framework within which similarities and differences in locomotion can be interpreted.     

Hemiplegic gait: a kinematic analysis using walking speed as a basis.

The kinematics of treadmill ambulation of stroke patients (N = 9) and healthy subjects (N = 4) was studied at a wide range of different velocities (i.e. 0.25-1.5 m s-1), with a focus on the transverse rotations of the trunk. Video recordings revealed, for both stroke patients and healthy subjects, similar relations between walking speed and stride length as well as stride frequency. The phase difference between pelvic and thoracic rotations (i.e. trunk rotation) and the total range of trunk rotation were almost linearly related to the walking speed. Healthy subjects showed a marked increase in pelvic rotation from 1 to 1.5 m s-1. Using dimensional analysis in a comparison between stroke patients and healthy subjects, invariances in the coordination of gait were found for stride length, stride frequency, pelvic rotation, and trunk rotation. Constant relations were obtained between, on the one hand, dimensionless velocity and, on the other, dimensionless stride length as well as stride frequency. Transitions were found between the velocities 0.75 and 1 m s-1 for dimensionless pelvic rotation and trunk rotation, indicating that, from this velocity range onwards, pelvic swing lengthens the stride: rotations of pelvis, thorax and trunk become tightly coordinated. On the basis of the dimensionless stride length, stride frequency, pelvic rotation and trunk rotation, deficits in the gait of stroke patients could be quantified. It is concluded that walking speed is an important control parameter, which should be used as a basic variable in the evaluation of the gait of stroke patients.     

A dynamic similarity hypothesis for the gaits of quadrupedal mammals

The dynamic similarity hypothesis postulates that different mammals move in a dynamically similar fashion whenever they travel at speeds that give them equal values of a dimensionless parameter, the Froude number. Thus, given information about one species, it could be possible to predict for others relationships between size, speed and features of gait such as stride length, duty factor, the phase relationships of the feet and the patterns of force exerted on the ground.
Data for a diverse sample of mammals have been used to test the hypothesis. It is found to be tenable in many cases when comparisons are confined to quadrupedal mammals of the type described by Jenkins (1971) as "cursorial". Most mammals of mass greater than 5 kg are of this type. Although the hypothesis applies less successfully to comparisons between cursorial and non-cursorial mammals it is shown to be a reasonable approximation even for such comparisons and for comparisons between quadrupedal mammals and bipedal mammals and birds.     

Fast locomotion of some African ungulates

Ten species of ungulate were filmed, galloping in their natural habitat. They ranged in size from Thomson's gazelle (about 20 kg) to giraffe (about 1000 kg). They were pursued to make them run as fast as possible. The films have been analysed to determine speed, stride frequency, stride and step lengths, and duty factors. The dependence of these quantities on body size is discussed.  

Summary:

Fast locomotion of zebra, giraffe, warthog and seven species of Bovidae has been studied. The animals were filmed from a pursuing vehicle while galloping in their natural habitat.
Stride frequency was more closely correlated with limb length (represented by hip height) than with body mass. Mean stride frequency was proportional to (hip height)^-0·51 and maximum stride frequency to (hip height) ^-0·63.
Maximum speed was between 10 and 14 m s ^-1 for all species except buffalo (7 m s ^-1). It was not significantly correlated with body mass.
Since the small species ran at least as fast as the large ones they attained higher Froude numbers. Relative stride length was approximately 1·8 (Froude number)^0·39 for all species, irrespective of size. Relative step length was approximately 0·65 (Froude number)^0·2, both for the fore feet and for the hind ones. The vertical forces exerted by the feet are proportional to (body weight)×(Froude number)^0·2 so the forces at maximum speed are larger multiples of body weight for small species than for large ones.     

Distorting limb design for dynamically similar locomotion

Terrestrial mammals of different sizes tend to move in a dynamically similar manner when travelling at speeds corresponding to equal values of the Froude number. This means that certain dimensionless locomotor parameters, including peak vertical ground reaction force relative to body weight, stride length relative to leg length and duty factor, are independent of animal size. The Froude number is consequently used to define equivalent speeds for mammals of different sizes. However, most musculoskeletal-tissue properties, including tendon elastic modulus, do not scale in a dynamically similar manner. Therefore, mammals could not be completely dynamically similar, even if perfectly geometrically similar. We argue that, for mammals to move in a dynamically similar manner, they must exhibit systematic 'distortions' of limb structure with size that compensate for the size independence of the tendon elastic modulus. An implication of this is that comparing mammals at equal Froude numbers cannot remove all size-dependent effects. We show that the previously published allometry of limb moment arms is sufficient to compensate for size-independent tendon properties. This suggests that it is an important factor in allowing mammals of different sizes to move in a dynamically similar manner.     

Modela-r as a Froude and Strouhal dimensionless numbers combination for dynamic similarity in running

The aim of this study was to test the hypothesis that running at fixed fractions of Froude (Nfr) and Strouhal (Str) dimensionless numbers combinations induce dynamic similarity between humans of different sizes. Nineteen subjects ran in three experimental conditions, (i) constant speed, (ii) similar speed (Nfr) and (iii) similar speed and similar step frequency (Nfr and Str combination). In addition to anthropometric data, temporal, kinematic and kinetic parameters were assessed at each stage to measure dynamic similarity informed by dimensional scale factors and by the decrease of dimensionless mechanical parameter variability. Over a total of 54 dynamic parameters, dynamic similarity from scale factors was met for 16 (mean r=0.51), 32 (mean r=0.49) and 52 (mean r=0.60) parameters in the first, the second and the third experimental conditions, respectively. The variability of the dimensionless preceding parameters was lower in the third condition than in the others. This study shows that the combination of Nfr and Str, computed from the dimensionless energy ratio at the center of gravity (Modela-r) ensures dynamic similarity between different-sized subjects. The relevance of using similar experimental conditions to compare mechanical dimensionless parameters is also proved and will highlight the study of running techniques, or equipment, and will allow the identification of abnormal and pathogenic running patterns. Modela-r may be adapted to study other abilities requiring bounces in human or animal locomotion or to conduct investigations in comparative biomechanics.     

Pedestrian locomotion energetics and gait characteristics of a diving bird, the great cormorant, Phalacrocorax carbo

Great cormorants Phalacrocorax carbo are foot propelled diving birds that seem poorly suited to locomotion on land. They have relatively short legs, which are presumably adapted for the generation of high forces during the power stroke of aquatic locomotion, and walk with a pronounced "clumsy waddle". We hypothesise (1) that the speed, independent minimum cost of locomotion (C min, ml O2 m(-1)) will be high for cormorants during treadmill exercise, and (2) that cormorants will have a relatively limited speed range in comparison to more cursorial birds. We measured the rate of oxygen consumption (V02) of cormorants during pedestrian locomotion on a treadmill, and filmed them to determine duty factor (the fraction of stride period that the foot is in contact with the ground), foot contact time (tc), stride frequency (f), swing phase duration and stride length. C min was 2.1-fold higher than that predicted by their body mass and phylogenetic position, but was not significantly different from the C min of runners (Galliformes and Struthioniformes). The extrapolated gamma-intercept of the relationship between V02 and speed was 1.9-fold higher than that predicted by allometry. Again, cormorants were not significantly different from runners. Contrary to our hypothesis, we therefore conclude that cormorants do not have high pedestrian transport costs. Cormorants were observed to use a grounded gait with two double support phases at all speeds measured, and showed an apparent gait transition between 0.17 and 0.25 m s(-1). This transition occurs at a Froude number between 0.016 and 0.037, which is lower than the value of approximately 0.5 observed for many other species. However, despite the use of a limited speed range, and a gait transition at relatively low speed, we conclude that the pedestrian locomotion of these foot propelled diving birds is otherwise generally similar to that of cursorial birds at comparable relative velocities.     

Stride length and its determinants in humans, early hominids, primates, and mammals

Primate stride lengths during quadrupedal locomotion are very long when compared to those of nonprimate quadrupedal mammals at the speed of trot/gallop transition. These exceptional lengths are a consequence of the relatively long limbs of primates and the large angular excursions of their limbs during quadrupedalism. When quadrupedal primates employ bipedal gaits they exhibit much lower angular excursions. Consequently their bipedal stride lengths do not appear to be exceptional in length when compared to other mammals. Angular excursions of the lower limbs of modern humans are not exceptionally large. However, when running, humans exhibit relatively long periods of flight (i.e., they have low duty factors) when compared to other mammals including primates. Because of these long periods of flight and their relative long lower limbs, humans have running stride lengths that are at the lower end of the range of stride lengths of quadrupedal primates. The stride length of the Laetoli hominid trails are evaluated in this context.     

Using body size to understand the structural design of animals: quadrupedal locomotion.

Many parameters of gait and performance, including stride frequency, stride length, maximum speed, and rate of O2 uptake are experimentally found to be power-law functions of body weight in running quadrupeds. All of these parameters are reasonably easy to measure except maximum speed, where the question arises whether one means top sprinting speed or top speed for sustained running. Moreover, differences in training and motivation make comparisons of top speed difficult. The problem is circumvented by comparing animals running at the transition between trotting and galloping, a physiologically similar speed. Theoretical models are proposed which preserve either geometric similarity, elastic similarity, or static stress similarity between animals of large and small body weights. The model postulating elastic similarity provides the best correlation with published data on body and bone proportions, body surface area, resting metabolic rate, and basal heart and lung frequencies. It also makes the most successful prediction of stride frequency, stride length, limb excursion angles, and the metabolic power required for running at the trot-gallop transition in quadrupeds ranging in size from mice to horses.     

Force-, EMG-, and elasticity-velocity relationships at submaximal, maximal and supramaximal running speeds in sprinters

The relationships between ground reaction forces, electromyographic activity (EMG), elasticity and running velocity were investigated at five speeds from submaximal to supramaximal levels in 11 male and 8 female sprinters. Supramaximal running was performed by a towing system. Reaction forces were measured on a force platform. EMGs were recorded telemetrically with surface electrodes from the vastus lateralis and gastrocnemius muscles, and elasticity of the contact leg was evaluated with spring constant values measured by film analysis. Data showed increases in most of the parameters studied with increasing running speed. At supramaximal velocity (10.36 +/- 0.31 m X s-1; 108.4 +/- 3.8%) the relative increase in running velocity correlated significantly (P less than 0.01) with the relative increase in stride rate of all subjects. In male subjects the relative change in stride rate correlated with the relative change of IEMG in the eccentric phase (P less than 0.05) between maximal and supramaximal runs. Running with the towing system caused a decrease in elasticity during the impact phase but this was significant (P less than 0.05) only in the female sprinters. The average net resultant force in the eccentric and concentric phases correlated significantly (P less than 0.05-0.001) with running velocity and stride length in the maximal run. It is concluded that increased neural activation in supramaximal effort positively affects stride rate and that average net resultant force as a specific force indicator is primarily related to stride length and that the values in this indicator may explain the difference in running velocity between men and women.     

Humans Running at Stadiums and Beaches and the Accuracy of Speed Estimations from Fossil Trackways

The concept of dynamic similarity between mechanical properties of vertebrates and engineered structures has served in previous work to suggest that there is a power law relationship between vertebrate speeds and stride length. This relationship, with some additional assumptions about hind limb height, has permitted the calculation of speeds from fossil trackways of dinosaurs. However, there are claims that uncertainties are large. In this work we analyze the accuracy of speed calculations for fossil vertebrates based on fossil trackways by using data derived from both athletic competitions and an experiment with humans walking and running on a beach. Our results show that although there are somewhat different running regimes, in general terms human speed can be described in a simple way, and differences between observed and predicted speeds usually are no more than 10–15%. Thus, while recognizing that some uncertainty remains in the estimation of hind limb height, we conclude that reliable speed calculations can be obtained from vertebrate fossil trackways. Our results also show that very reliable speed estimates can be obtained from human fossil trackways directly from stride length measurements.     

Spatio-temporal gait characteristics of the hind-limb cycles during voluntary bipedal and quadrupedal walking in bonobos (Pan paniscus)

Spatio-temporal gait characteristics (step and stride length, stride frequency, duty factor) were determined for the hind-limb cycles of nine bonobos (Pan paniscus) walking quadrupedally and bipedally at a range of speeds. The data were recalculated to dimensionless quantities according to the principle of dynamic similarity. Lower leg length was used as the reference length. Interindividual variability in speed modulation strategy of bonobos appears to be low. Compared to quadrupedal walking, bipedal bonobos use smaller steps to attain a given speed (differences increase with speed), resulting in shorter strides at a higher frequency. In the context of the ("hybrid") dynamic pattern approach to locomotion (Latach, 1998) we argue that, despite these absolute differences, intended walking speed is the basic control variable which elicits both quadrupedal and bipedal walking kinematics in a similar way. Differences in the initial status of the dynamic system may be responsible for the differences in step length between both gaits. Comparison with data deduced from the literature shows that the effects of walking speed on stride length and frequency are similar in bonobos, common chimpanzees, and humans. This suggests that (at least) within extant homininae, spatio-temporal gait characteristics are highly comparable, and this in spite of obvious differences in mass distribution and bipedal posture.     

Similarity in multilegged locomotion: Bouncing like a monopode

Despite impressive variation in leg number, length, position and type of skeleton, similarities of legged, pedestrian locomotion exist in energetics, gait, stride frequency and ground-reaction force. Analysis of data available in the literature showed that a bouncing, spring-mass, monopode model can approximate the energetics and dynamics of trotting, running, and hopping in animals as diverse as cockroaches, quail and kangaroos. From an animal's mechanical-energy fluctuation and ground-reaction force, we calculated the compression of a virtual monopode's leg and its stiffness. Comparison of dimensionless parameters revealed that locomotor dynamics depend on gait and leg number and not on body mass. Relative stiffness per leg was similar for all animals and appears to be a very conservative quantity in the design of legged locomotor systems. Differences in the general dynamics of gait are based largely on the number of legs acting simultaneously to determine the total stiffness of the system. Four- and six-legged trotters had a greater whole body stiffness than two-legged runners operating their systems at about the same relative speed. The greater whole body stiffness in trotters resulted in a smaller compression of the virtual leg and a higher natural frequency and stride frequency.     

Groucho running

An important determinant of the mechanics of running is the effective vertical stiffness of the body. This stiffness increases with running speed. At any one speed, the stiffness may be reduced in a controlled fashion by running with the knees bent more than usual. In a series of experiments, subjects ran in both normal and flexed postures on a treadmill. In other experiments, they ran down a runway and over a force platform. Results show that running with the knees bent reduces the effective vertical stiffness and diminishes the transmission of mechanical shock from the foot to the skull but requires an increase of as much as 50% in the rate of O2 consumption. A new dimensionless parameter (u omega 0/g) is introduced to distinguish between hard and soft running modes. Here, omega 0 is the natural frequency of a mass-spring system representing the body, g is gravity, and u is the vertical landing velocity. In normal running, this parameter is near unity, but in deep-flexed running, where the aerial phase of the stride cycle almost disappears, u omega 0/g approaches zero.     

Oxygen delivery does not limit peak running speed during incremental downhill running to exhaustion

Oxygen consumption (VO2), ventilation (VI), respiratory exchange ratio (R), stride frequency and blood lactate concentrations were measured continuously in nine trained athletes during two continuous incremental treadmill runs to exhaustion on gradients of either 0 degree or -3 degrees. Compared to the run at 0 degree gradient, the athletes reached significantly higher maximal treadmill velocities but significantly lower VO2, VI, R and peak blood lactate concentrations (P less than 0.001) during downhill running. These lower VO2 and blood lactate concentrations at exhaustion indicated that factors other than oxygen delivery limited maximal performance during the downhill run. In contrast, stride frequencies were similar at each treadmill velocity; the higher maximal speed during the downhill run was achieved with a significantly longer stride length (P less than 0.001); maximal stride frequency was the same between tests. Equivalent maximal stride frequencies suggested that factors determining the rate of lower limb stride recovery may have limited maximal running speed during downhill running and, possibly, also during horizontal running.     

Angular Circulation Speed of Tablets in a Vibratory Tablet Coating Pan

In this work, a single tablet model and a discrete element method (DEM) computer simulation are developed to obtain the angular circulation speed of tablets in a vibratory tablet coating pan for range of vibration frequencies and amplitudes. The models identify three important dimensionless parameters that influence the speed of the tablets: the dimensionless amplitude ratio (a/R), the Froude number (aω ²/g), and the tablet–wall friction coefficient, where a is the peak vibration amplitude at the drum center, ω is the vibration angular frequency, R is the drum radius, and g is the acceleration due to gravity. The models predict that the angular circulation speed of tablets increases with an increase in each of these parameters. The rate of increase in the angular circulation speed is observed to decrease for larger values of a/R. The angular circulation speed reaches an asymptote beyond a tablet–wall friction coefficient value of about 0.4. Furthermore, it is found that the Froude number should be greater than one for the tablets to start circulating. The angular circulation speed increases as Froude number increases but then does not change significantly at larger values of the Froude number. Period doubling, where the motion of the bed is repeated every two cycles, occurs at a Froude number larger than five. The single tablet model, although much simpler than the DEM model, is able to predict the maximum circulation speed (the limiting case for a large value of tablet–wall friction coefficient) as well as the transition to period doubling.     

The effect of stride length on the dynamics of barefoot and shod running

A number of interventions and technique changes have been proposed to attempt to improve performance and reduce the number of running related injuries. Running shoes, barefoot running and alterations in spatio-temporal parameters (stride frequency and stride length) have been associated with significant kinematic and kinetic changes, which may have implications for performance and injury prevention. However, because footwear interventions have been shown to also affect spatio-temporal parameters, there is uncertainty regarding the origin of the kinematic and kinetic alterations. Therefore, the purpose of this study was to independently evaluate the effects of shoes and changes in stride length on lower extremity kinetics. Eleven individuals ran over-ground at stride lengths ±5 and 10% of their preferred stride length, in both the barefoot and shod condition. Three-dimensional motion capture and force plate data were captured synchronously and used to compute lower extremity joint moments. We found a significant main effect of stride length on anterior–posterior and vertical GRFs, and sagittal plane knee and ankle moments in both barefoot and shod running. When subjects ran at identical stride lengths in the barefoot and shod conditions we did not observe differences for any of the kinetic variables that were measured. These findings suggest that barefoot running triggers a decrease in stride length, which could lead to a decrease in GRFs and sagittal plane joint moments. When evaluating barefoot running as a potential option to reduce injury, it is important to consider the associated change in stride length.     

Validated biomechanical model for efficiency and speed of rowing

The speed of a competitive rowing crew depends on the number of crew members, their body mass, sex and the type of rowing—sweep rowing or sculling. The time-averaged speed is proportional to the rower?s body mass to the 1/36^th power, to the number of crew members to the1/9^th power and to the physiological efficiency (accounted for by the rower?s sex) to the 1/3^rd power. The quality of the rowing shell and propulsion system is captured by one dimensionless parameter that takes the mechanical efficiency, the shape and drag coefficient of the shell and the Froude propulsion efficiency into account. We derive the biomechanical equation for the speed of rowing by two independent methods and further validate it by successfully predicting race times. We derive the theoretical upper limit of the Froude propulsion efficiency for low viscous flows. This upper limit is shown to be a function solely of the velocity ratio of blade to boat speed (i.e., it is completely independent of the blade shape), a result that may also be of interest for other repetitive propulsion systems.