Thomas A. McMahon

The mechanics of running: How does stiffness couple with speed?

A mathematical model for terrestrial running is presented, based on a leg with the properties of a simple spring. Experimental force-platform evidence is reviewed justifying the formulation of the model. The governing differential equations are given in dimensionless form to make the results representative of animals of all body sizes. The dimensionless input parameters are: U, a horizontal Froude number based on forward speed and leg length; V, a vertical Froude number based on vertical landing velocity and leg length, and KLEG, a dimensionless stiffness for the leg-spring. Results show that at high forward speed, KLEG is a nearly linear function of both U and V, while the effective vertical stiffness is a quadratic function of U. For each U, V pair, the simulation shows that the vertical force at mid-step may be minimized by the choice of a particular step length. A particularly useful specification of the theory occurs when both KLEG and V are assumed fixed. When KLEG = 15 and V = 0.18, the model makes predictions of relative stride length S and initial leg angle theta o that are in good agreement with experimental data obtained from the literature.

Scaling stride frequency and gait to animal size: mice to horses.

The stride frequency at which animals of different size change from one gait to another (walk, trot, gallop) changes in a regular manner with body mass. The speed at the transition from trot to gallop can be used as an equivalent speed for comparing animals of different size. This transition point occurs at lower speeds and higher stride frequencies in smaller animals. Plotting stride frequency at the trot-gallop transition point as a function of body mass in logarithmic coordinates yields a straight line.

Tree structures: deducing the principle of mechanical design.

Abstract A statistical description of the branching patterns of trees is proposed in the context of a power-law tapered beam model. Depending on the exponent which describes tapering of the depth of the beam, the model either preserves geometric, elastic, or static stress similarity. A detailed study of the morphometry of three oak, one poplar, one cherry, and one white pine corroborates the stationarity of these branching structures and fits the elastically similar model. A separate study of the natural frequencies of branch segments and whole trees within four species also agrees with the predictions of the elastically similar model.

The influence of track compliance on running

Abstract A model of running is proposed in which the leg is represented as a rack-and-pinion element in series with a damped spring. The rack-and-pinion element emphasizes the role of descending commands, while the damped spring represents the dynamic properties of muscles and the position and the rate sensitivity of reflexes. This model is used to predict separately the effect of track compliance on step length and ground contact time. The predictions are compared with experiments in which athletes ran over tracks of controlled spring stiffness. A sharp spike in foot force up to 5 times body weight was found on hard surfaces, but this spike disappeared as the athletes ran on soft experimental tracks. Both ground contact time and step length increased on very compliant surfaces, leading to moderately reduced running speeds, but a range of track stiffness was discovered which actually enhances speed.

Prediction of Femoral Impact Forces in Falls on the Hip

A major determinant of the risk of hip fracture in a fall from standing height is the force applied to the femur at impact. This force is determined by the impact velocity of the hip and the effective mass, stiffness, and damping of the body at the moment of contact. We have developed a simple experiment (the pelvis release experiment) to measure the effective stiffness and damping of the body when a step change in force is applied to the lateral aspect of the hip. Results from pelvis release experiments with 14 human subjects suggest that both increased soft tissue thickness over the hip and impacting the ground in a relaxed state can decrease the effective stiffness of the body, and subsequently reduce peak impact forces. Comparison between our fall impact force predictions and in-vitro measures of femoral fracture strength suggest that any fall from standing height producing direct, lateral impact on the greater trochanter can fracture the elderly hip.

Ballistic walking: an improved model

Abstract A mathematical model of the swing phase of walking is presented. The lower extremities are represented by links, and the rest of the body by a point mass at the hip joint. It is assumed that no muscular moments are provided to any of the joints of the extremities during the swing phase. The body then moves under the action of gravity alone. The model analyzed here is an improvement of a previous model by the same authors. The range of possible times of swing for each step length is computed for the model, and the results are compared with published experimental data. Typical histograms of forces applied to the ground and angles of the limbs against time are also given. The computed forces and angles have the same general time course as those found experimentally in normal walking, with the exception of the vertical force. The reasons responsible for this are discussed. The comparison of the results of the model with normal walking also suggests the possible effects that the absent determinants have on gait.

Etiology and Prevention of Age-Related Hip Fractures

Falls and fall-related injuries are among the most serious and common medical problems experienced by the elderly. Hip fracture, one of the most severe consequences of falling in the elderly, occurs in only about 1% of falls. Despite this, hip fracture accounts for a large share of the disability, death, and medical costs associated with falls. As measured by their frequency, influence on quality of life, and economic cost, hip fractures are a public health problem of crisis proportions. Without successful international initiatives aimed at reducing the incidence of falls and hip fractures, the implications for allocations of health resources in this and the next century are staggering. Identifying those at risk for harmful falls requires an understanding of what kinds of falls result in injury and fracture. In elderly persons who fall, in most of whom hip bone mineral density is already several standard deviations below peak values, fall severity (as reflected in falling to the side and impacting the hip) and body habitus are important risk factors for hip fracture and touch on a domain of risk entirely missed by knowledge of bone mineral density. These findings clearly suggest that factors related to both loading and bone fragility play important roles in the etiology of hip fracture. We provide a strategy, based on engineering approaches to fracture risk prediction, for determining the relative etiologic importance of loading and bone fragility and to summarize some of what is known about both sets of factors. We define a factor of risk, phi, as the ratio of the loads applied to the hip divided by the loads necessary to cause fracture and summarize available data on the numerator and the denominator of phi. We then provide an overview of the complex interplay between the risks associated with the initiation, descent, and impact phases of a fall, thereby suggesting an organized approach for evaluating intervention efforts being used to prevent hip fractures. The findings emphasize the continuing need for combined intervention strategies that focus on fall prevention, reductions in fall severity, and maintaining or increasing femoral bone mass and strength, either through targeted exercise programs, optimal nutrition (Ca, Vitamin D), and/or in the use of osteodynamic agents. By developing and refining the factor of risk, a property that captures both the contributions of bone density and the confounding influences of body habitus and fall severity, we believe these intervention strategies can be targeted more appropriately.

Mechanics of Locomotion

Energetic and mechanical principles of walking and running are reviewed, using information available from force-plate studies. A mathematical model of walking is described that conserves the sum of the kinetic and gravitational potential energies of the body. In running, energy is stored transiently in the elastic deformations of stretched muscles and tendons. Theory and experiments are described using these principles and others to find the range of stiffness values for a running track that both lowers the potential for injuries and increases running speed.

Hip impact velocities and body configurations for voluntary falls from standing height

Fall dynamics have largely been ignored in the study of hip fracture etiology and in the development of hip fracture prevention strategies. In this study, we asked the following questions: (1) What are the ranges of hip impact velocities associated with a sideways fall from standing height? (2) What are the ranges of body configurations at impact? and (3) How do protective reflexes such as muscle activation or using an outstretched hand influence fall kinematics? To answer these questions, we recruited six young healthy athletes who performed voluntary sideways falls on a thick foam mattress. Several categories of falls were investigated: (a) muscle-active vs muscle-relaxed falls; (b) falls from a standing position or from walking; and (c) falls in which an outstretched arm was used to break the fall. Each fall was videotaped at 60 frames s(-1). Fall kinematics parameters were obtained by digitizing markers placed on anatomical points of interest. The mean value for vertical hip impact velocity was 2.75 ms(-1) (+ or - 0.42 ms(-1) [S.D.]). The mean value for trunk angle (the angle between the trunk and the vertical) was 17.3 degrees (+ or - 11.5 degrees [S.D.]). We found a 38 percent reduction in the trunk angle at impact, and a 7 percent reduction in hip impact velocity for relaxed vs muscle-active falls. Finally, regarding the. falls in which an outstretched arm was used, only two out of the six subjects were able to break the fall with their arm or hand. For the remaining subjects hip impact occurred first, followed by contact of the arm or hand.

Biomechanics of the movable pretarsal adhesive organ in ants and bees

Hymenoptera attach to smooth surfaces with a flexible pad, the arolium, between the claws. Here we investigate its movement in Asian weaver ants (Oecophylla smaragdina) and honeybees (Apis mellifera).  When ants run upside down on a smooth surface, the arolium is unfolded and folded back with each step. Its extension is strictly coupled with the retraction of the claws. Experimental pull on the claw-flexor tendon revealed that the claw-flexor muscle not only retracts the claws, but also moves the arolium. The elicited arolium movement comprises (i) about a 90° rotation (extension) mediated by the interaction of the two rigid pretarsal sclerites arcus and manubrium and (ii) a lateral expansion and increase in volume. In severed legs of O. smaragdina ants, an increase in hemolymph pressure of 15 kPa was sufficient to inflate the arolium to its full size. Apart from being actively extended, an arolium in contact also can unfold passively when the leg is subject to a pull toward the body.  We propose a combined mechanical–hydraulic model for arolium movement: (i) the arolium is engaged by the action of the unguitractor, which mechanically extends the arolium; (ii) compression of the arolium gland reservoir pumps liquid into the arolium; (iii) arolia partly in contact with the surface are unfolded passively when the legs are pulled toward the body; and (iv) the arolium deflates and moves back to its default position by elastic recoil of the cuticle.