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Humans change joint quasi-stiffness (k joint) and leg stiffness (kleg) when running at different speeds on level ground and during uphill and downhill running. These mechanical properties can inform device designs for running such as footwear, exoskeletons and prostheses. We measured kinetics and kinematics from 17 runners (10 M; 7 F) at three speeds on 0°, ±2°, ±4° and ±6° slopes. We calculated ankle and knee k joint, the quotient of change in joint moment and angular displacement, and theoretical leg stiffness (klegT) based on the joint external moment arms and k joint. Runners increased k ankle at faster speeds (p < 0.01). Runners increased and decreased the ankle and knee contributions to klegT, respectively, by 2.89% per 1° steeper uphill slope (p < 0.01) during the first half of stance. Runners decreased and increased ankle and knee joint contributions to klegT, respectively, by 3.68% during the first half and 0.86% during the second half of stance per 1° steeper downhill slope (p < 0.01). Thus, biomimetic devices require stiffer k ankle for faster speeds, and greater ankle contributions and greater knee contributions to klegT during the first half of stance for steeper uphill and downhill slopes, respectively.

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Older runners (OR) are increasing their participation in races. Aging may impact the adopted running pattern. Hence, the analysis of stiffness and the inter-joint lower limb coordination in the sagittal plane could contribute to investigating this impact. This study aimed to compare the vertical stiffness (Kvert) and the inter-joint lower limb coordination in the sagittal plane between younger runners (YR) and OR. This cross-sectional study recruited 15 YR males and 15 OR males. The pelvis and lower limb motions were assessed while running on a treadmill at self-selected (range OR: 1.94-3.75 m.s-1, YR: 2.08-4.17 m.s-1) and fixed speeds (3.33 m.s-1). Hip-ankle, knee-ankle, and hip-knee coupling angle (CA) and its variability (CAV) were extracted using the vector coding method. Mann-Whitney U tests compared Kvert between groups at each running speed. Watson's U2 tests compared the mean CA between groups in three intervals of the contact phase at each running speed. Statistical Parametric Mapping independent t-test compared the CAV curve between groups at each running speed. OR showed greater Kvert than YR at both speeds. Hip-ankle CA pattern differed between groups during the early stance at both speed conditions. OR showed in-phase, distal dominancy in hip-ankle CA, whereas YR showed anti-phase, proximal dominancy. Knee-ankle CA was distinct only at self-selected speed, in which OR showed in-phase, proximal dominancy, while YR exhibited anti-phase, proximal dominancy. CAV did not differ between groups. The findings showed that OR adopted a stiffer pattern characterized by distinct inter-joint lower limb CA, at early stance, during self-selected and fixed speeds.

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[This corrects the article DOI: 10.3389/fphys.2022.1044363.].

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Duty factor (DF) and step frequency (SF) were previously defined as the key running pattern determinants. Hence, this study aimed to investigate the association of DF and SF on 1) the vertical and fore-aft ground reaction force signals using statistical parametric mapping; 2) the force related variables (peaks, loading rates, impulses); and 3) the spring-mass characteristics of the lower limb, assessed by computing the force-length relationship and leg stiffness, for treadmill runs at several endurance running speeds. One hundred and fifteen runners ran at 9, 11, and 13 km/h. Force data (1000 Hz) and whole-body three-dimensional kinematics (200 Hz) were acquired by an instrumented treadmill and optoelectronic system, respectively. Both lower DF and SF led to larger vertical and fore-aft ground reaction force fluctuations, but to a lower extent for SF than for DF. Besides, the linearity of the force-length relationship during the leg compression decreased with increasing DF or with decreasing SF but did not change during the leg decompression. These findings showed that the lower the DF and the higher the SF, the more the runner relies on the optimization of the spring-mass model, whereas the higher the DF and the lower the SF, the more the runner promotes forward propulsion.

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Taking as bioinspiration the remarkable acoustic absorption properties of moth wings, we develop a simple analytical model that describes the interaction between acoustic pressure fields, and thin elastic plates incorporating resonant sub-structures. The moth wing is an exemplar of a natural acoustic metamaterial; the wings are deeply subwavelength in thickness at the frequencies of interest, the absorption is broadband and the tiny scales resonate on the moth wing acting in concert. The simplified model incorporates only the essential physics and the scales are idealized to flat rigid rectangular plates coupled via a spring to an elastic plate that forms the wing; all the components are deep-subwavelength at desired frequencies. Based on Fourier analysis, complemented by phenomenological modelling, our theory shows excellent agreement with simulation mimicking the moth-wing structure. Moth wings operate as broadband sound absorbers employing a range of scale sizes. We demonstrate that a random distribution of scale sizes generates a broadband absorption spectrum. To further illustrate the potential of the model, we design a deeply sub-wavelength acoustic counterpart of electromagnetically induced reflectance. This article is part of the theme issue 'Wave generation and transmission in multi-scale complex media and structured metamaterials (part 2)'.

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The volume of active muscle and duration of extensor muscle force well explain the associated metabolic energy expenditure across body mass and velocity during level-ground running and hopping. However, if these parameters fundamentally drive metabolic energy expenditure, then they should pertain to multiple modes of locomotion and provide a simple framework for relating biomechanics to metabolic energy expenditure in bouncing gaits. Therefore, we evaluated the ability of the 'cost of generating force' hypothesis to link biomechanics and metabolic energy expenditure during human running and hopping across step frequencies. We asked participants to run and hop at 85%, 92%, 100%, 108% and 115% of preferred running step frequency. We calculated changes in active muscle volume, duration of force production and metabolic energy expenditure. Overall, as step frequency increased, active muscle volume decreased as a result of postural changes via effective mechanical advantage (EMA) or duty factor. Accounting for changes in EMA and muscle volume better related to metabolic energy expenditure during running and hopping at different step frequencies than assuming a constant EMA and muscle volume. Thus, to ultimately develop muscle mechanics models that can explain metabolic energy expenditure across different modes of locomotion, we suggest more precise measures of muscle force production that include the effects of EMA.

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BACKGROUND: Individuals with lower-limb amputation can use running specific prostheses (RSP) that store and then return elastic energy during stance. However, it is unclear whether varying the stiffness category of the same RSP affects spring-mass behaviour during self-selected, submaximal speed running in individuals with unilateral transtibial amputation. RESEARCH QUESTION: The current study investigates how varying RSP stiffness affects limb stiffness, running performance, and associated joint kinetics in individuals with a unilateral transtibial amputation. METHODS: Kinematic and ground reaction force data were collected from eight males with unilateral transtibial amputation who ran at self-selected submaximal speeds along a 15 m runway in three RSP stiffness conditions; recommended habitual stiffness (HAB) and, following 10-minutes of familiarisation, stiffness categories above (+1) and below (-1) the HAB. Stance-phase centre of mass velocity, contact time, limb stiffness' and joint/RSP work were computed for each limb across RSP stiffness conditions. RESULTS: With increased RSP stiffness, prosthetic limb stiffness increased, whilst intact limb stiffness decreased slightly (p<0.03). Centre of mass forward velocity during stance-phase (p<0.02) and contact time (p<0.04) were higher in the intact limb and lower in the prosthetic limb but were unaffected by RSP stiffness. Intact limb hip joint positive work increased for both the +1 and -1 conditions but remained unchanged across conditions in the prosthetic limb (p<0.02). SIGNIFICANCE: In response to changes in RSP stiffness, there were acute increased mechanical demands on the intact limb, reflecting a reliance on the intact limb during running. However, overall running speed was unaffected, suggesting participants acutely adapted to an RSP of a non-prescribed stiffness.

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Aircraft design must be lightweight and cost-efficient on the condition of aircraft certification. In addition to standard load cases, human-induced loads can occur in the aircraft interior. These are crucial for optimal design but difficult to estimate. In this study, a simple bipedal spring-mass model with roller feet predicted human-induced loads caused by human gait for use within an end-to-end design process. The prediction needed no further experimental data. Gait movement and ground reaction force (GRF) were simulated by means of two parameter constraints with easily estimable input variables (gait speed, body mass, body height). To calibrate and validate the prediction model, experiments were conducted in which 12 test persons walked in an aircraft mock-up under different conditions. Additional statistical regression models helped to compensate for bipedal model limitations. Direct regression models predicted single GRF parameters as a reference without a bipedal model. The parameter constraint with equal gait speed in experiment and simulation yielded good estimates of force maxima (error 5.3%), while equal initial GRF gave a more reliable prediction. Both parameter constraints predicted contact time very well (error 0.9%). Predictions with the bipedal model including full GRF curves were overall as reliable as the reference.

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The spring-mass model is a model of locomotion aimed at giving the essential mathematical laws of the trajectory of the center of mass of an animal during bouncing gaits, such as hopping (one-dimensional) and running (two-dimensional). This reductionist mechanical system has been extensively investigated for locomotion over horizontal surfaces, whereas it has been largely neglected on other ecologically relevant surfaces, including inclines. For example, how the degree of inclination impacts the dynamics of the center of mass of the spring-mass model has not been investigated thoroughly. In this work, we derive a mathematical model which extends the spring-mass model to inclined surfaces. Among our results, we derive an approximate solution of the system, assuming a small angular sweep of the limb and a small spring compression during stance, and show that this approximation is very accurate, especially for small inclinations of the ground. Furthermore, we derive theoretical bounds on the difference between the Lagrangian and Lagrange equations of the true and approximate system, and discuss locomotor stability questions of the approximate solutions. We test our models through a sensitivity analysis using parameters relevant to the locomotion of bipedal animals (quail, pheasant, guinea fowl, turkey, ostrich, and humans) and compare our approximate solution to the numerically derived solution of the exact system. We compare the two-dimensional spring-mass model on inclines with the one-dimensional spring-mass model to which it reduces under the limit of no horizontal velocity; we compare the two-dimensional spring-mass model on inclines with the inverted-pendulum model on inclines towards which it converges in the case of high stiffness-to-mass ratio. We include comparisons with historically prevalent no-gravity approximations of these models, as well. The insights we have gleaned through all these comparisons and the ability of our approximation to replicate some of the kinematic changes observed in animals moving on different inclines (e.g. reduction in vertical oscillation of the center of mass and decreased stride length) underlines the valuable and reasonable contributions that very simple, reductionist models, like the spring-mass model, can provide.

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In this article, a recent formulation for real-time simulation is developed combining the strain energy density of the Spring Mass Model (SMM) with the equivalent representation of the Strain Energy Density Function (SEDF). The resulting Equivalent Energy Spring Model (EESM) is expected to provide information in real-time about the mechanical response of soft tissue when subjected to uniaxial deformations. The proposed model represents a variation of the SMM and can be used to predict the mechanical behavior of biological tissues not only during loading but also during unloading deformation states. To assess the accuracy achieved by the EESM, experimental data was collected from liver porcine samples via uniaxial loading and unloading tensile tests. Validation of the model through numerical predictions achieved a refresh rate of 31 fps (31.49 ms of computation time for each frame), achieving a coefficient of determination R2 from 93.23% to 99.94% when compared to experimental data. The proposed hybrid formulation to characterize soft tissue mechanical behavior is fast enough for real-time simulation and captures the soft material nonlinear virgin and stress-softened effects with high accuracy.

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BACKGROUND: Nike ZoomX Vaporfly (NVF) improves running economy and performance. The biomechanical mechanisms of these shoes are not fully understood, although thicker midsoles and carbon fiber plates are considered to play an important role in the spring-like leg characteristics during running. Leg stiffness (kleg) in the spring-mass model has been commonly used to investigate spring-like running mechanics during running. RESEARCH QUESTION: Does kleg during running differ between NVF and traditional (TRAD) shoes? METHODS: Eighteen male habitual forefoot and/or midfoot strike runners ran on a treadmill at 20 km/h with NVF and TRAD shoes, respectively. kleg, vertical oscillation of the center of mass (∆CoM), spatiotemporal parameters, and mechanical loading were determined. RESULTS: kleg was 4.8% lower in the NVF shoe condition than in the TRAD condition, although no significant difference was observed. ∆CoM was not significantly different between shoe conditions. Spatiotemporal parameters and mechanical loading were also not significantly different between shoe conditions. SIGNIFICANCE: The NVF shoe is well known as improving the running economy and running performance for the cause by characteristics of better spring function. Contrary to expectation, kleg and other parameters were not significantly different during running in the NVF compared to TRAD shoe at 20 km/h. These findings indicate that well-trained runners' spring-like running mechanics would not alter even if wearing the NVF shoes.

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Children often display different whole-body dynamics compared to adults during locomotion such as walking and hopping. However, it is unknown whether these differences result in diverging functional usage of the lower limb joints. This study aimed to compare the mechanical functions of the ankle, knee, and hip joints between children and adults during single-leg hopping in-place at different frequencies. Children aged 5-11 years and adults aged 18-35 years performed hopping at their preferred frequency and slower and faster frequencies. Function of the joint was modeled as a combination of a strut, spring, motor, and damper. At the preferred frequency, children hopped equally with strut and spring functions at the ankle and knee joints while adults primarily used the spring function. When increasing frequency, both children and adults decreased the spring index and increased the strut index at the ankle and knee joints. Across all conditions, both children and adults used the strut function primarily at the hip joint. Results suggest that preadolescent children are still developing the adult-like spring function of their ankle and knee joints during hopping in-place. Quantification of spring function during hopping in-place may present an innovative approach to understand the maturation of the stretch-shortening cycle in children.

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This study aimed to determine the influence of footwear condition, foot-strike pattern and step frequency on running spatiotemporal parameters and lower-body stiffness during treadmill running. Thirty-one amateur endurance runners performed a two-session protocol (shod and barefoot). Each session consisted of two trials at 12 km · h-1 over 5 minutes altering step frequency every minute (150, 160, 170, 180 and 190 spm). First, participants were instructed to land with the heel first; after completion, the same protocol was repeated landing with the forefoot first. Repeated measures ANOVAs showed significant differences for footwear condition, foot-strike pattern and step frequency for each variable: percent contact time, percent flight time, vertical stiffness and leg stiffness (all p < 0.001). The results demonstrate greater estimated vertical and leg stiffness when running barefoot for both foot-strike patterns showing the largest values for barefoot+forefoot condition. Likewise, both vertical and leg stiffness became greater as step frequency increased. The proper manipulation of these variables facilitates our understanding of running performance and assist in training programmes design and injury management.

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This paper proposes, a method for the physical modeling of the perimodiolar electrode, particularly for the process of recovering its preset shape with the guide wire drawn out, based on the composite spring-mass model by employing the virtual-volumetric spring inspired from the traditional spring-mass model. Simulation experiments of modeling and virtual insertion of perimodiolar electrode were carried out. The results indicated that the mean and standard deviation of the difference between the local deformation angles of the simulated and measured sets of mass points, (1, 2, 3), (2, 3, 4), …, (13, 14, 15), were 6.34° and 5.98°, respectively. Additionally, the physical model of the perimodiolar electrode can reflect the overall morphological changes of the real perimodiolar electrode.

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This study investigated the influence of heat pre-conditioning on the recovery of muscle torque, microvascular function, movement economy and stride mechanics following exercise-induced muscle damage (EIMD). Twenty male participants were equally assigned to a control (CON) and an experimental group (HEAT), and performed a 30-min downhill run (DHR) to elicit EIMD. HEAT group received three consecutive days of heat exposure (45.1 â€‹± â€‹3.2 â€‹min of hot water immersion at 42 â€‹°C) prior to DHR. Microvascular function (near-infrared spectroscopy), maximal voluntary contraction (MVC) torque of the knee extensors, as well as two treadmill-based steady-state runs performed below (SSR-1) and above (SSR-2) the first ventilatory threshold were assessed prior to DHR and repeated for four consecutive days post-DHR (D1-POST to D4-POST). The decline in MVC torque following EIMD was attenuated in HEAT compared with CON at D1-POST (p â€‹= â€‹0.037), D3-POST (p â€‹= â€‹0.002) and D4-POST (p â€‹= â€‹0.022). Muscle soreness increased in both CON and HEAT, but was significantly attenuated in HEAT compared with CON at D2-POST (p â€‹= â€‹0.024) and D3-POST (p â€‹= â€‹0.013). Microvascular function decreased in CON from D1-POST to D3-POST (p â€‹= â€‹0.009 to 0.018), and was lower compared with HEAT throughout D1-POST to D3-POST (p â€‹= â€‹0.003 to 0.017). Pre-heat treatment decreased the magnitude of strength loss and muscle soreness, as well as attenuated the decline in microvascular function following EIMD. Heat treatment appears a promising pre-conditioning strategy when embarking on intensified training periods or competition.

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This study aimed to determine the influence of arch stiffness on running spatiotemporal parameters at a common speed for a wide range of endurance runners (i.e., 12 km·h-1). In total, 97 runners, 52 men and 45 women, completed a treadmill running protocol at 12 km·h-1. Spatiotemporal parameters were measured using the OptoGait system, and foot structure was assessed by determining arch stiffness. Since between-sex differences were found in anthropometric and foot structure variables, data analysis was conducted separately for men and women, and body mass and height were considered as covariates. For both sexes, a k-means cluster analysis grouped participants according to arch stiffness, by obtaining a group of low-arch stiffness (LAS group) and a group of high-arch stiffness (HAS group), with significant differences in arch stiffness (p < 0.001, for both men and women). No significant differences between LAS and HAS groups were found in running spatiotemporal parameters, regardless of sex (p ≥ 0.05). For both sexes, the partial correlation analysis reported no significant correlations (p ≥ 0.05) between foot structure variables and running spatiotemporal parameters. The results obtained show no differences in spatiotemporal gait characteristics during running at submaximal velocity between runners with low-arch stiffness and those with high-arch stiffness, regardless of sex. These findings may have important implications for clinicians and coaches by adding more evidence to the debate about the use of static foot classification measures when characterizing the foot and its biomechanics during running.

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PURPOSE: To examine the influence of growth and maturation in the trajectory of stretch-shortening cycle capability. METHOD: Using a mixed-longitudinal design, absolute and relative leg stiffness and reactive strength index (RSI) were measured 3 times over a 3-year period in 44 youth team-sport players. Maturation was determined as maturity offset and included within the Bayesian inference analysis as a covariate alongside chronological age. RESULTS: Irrespective of age and maturation, there was no change in absolute leg stiffness, however relative leg stiffness decreased over time. Maturation and age reduced this decline, but the decline remained significant (Bayesian factor [10] = 5097, model averaged R2 = .61). The RSI increased over time and more so in older more mature youth players (Bayesian factor [10] = 9.29e8, model averaged R2 = .657). CONCLUSION: In youth players who are at/post peak height velocity, relative leg stiffness appears to decline, which could have an impact on both performance and injury risk. However, RSI increases during this period, and these data reinforce that leg stiffness and RSI reflect different components of stretch-shortening cycle capability. Practitioners should consider these differences when planning training to maximize stretch-shortening cycle capability during growth and maturation in athletes on the developmental performance pathway.

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Landing is a critical phase of movement for injury occurrence, in which lower limbs should be used equally to better absorb the shock. However, it has been suggested that fatigue can lead to the appearance of asymmetries. The aim of this study was to verify the acute and delayed effects of fatigue on the lower limb asymmetry indexes of peak ground reaction force, leg stiffness and intra-limb coordination during a landing task. Fifteen physically active men performed a fatigue protocol composed of 14 sets of 10 continuous vertical jumps, with a one-minute rest interval between the sets. A step-off landing task was performed before, immediately after, 24 h and 48 h after the fatigue protocol. Two force plates and a video analysis system were used. The symmetry index equation provided the asymmetry indexes. For statistical analysis, ANOVA and effect size analysis were utilized. Inferential statistics did not show the effect of fatigue in the asymmetry indexes for any variable or condition (p > .05). Moderate effect sizes were observed for peak ground reaction force (0.61) and leg stiffness (0.61) immediately after the application of the protocol. In conclusion, fatigue does not seem to significantly change the asymmetries of lower limbs, especially regarding intra-limb coordination. The moderate effects observed for peak ground reaction force and leg stiffness asymmetries suggest that these variables may be acutely affected by fatigue.

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The role of trunk orientation during uneven running is not well understood. This study compared the running mechanics during the approach step to and the step down for a 10 cm expected drop, positioned halfway through a 15 m runway, with that of the level step in 12 participants at a speed of 3.5 m s-1 while maintaining self-selected (17.7±4.2 deg; mean±s.d.), posterior (1.8±7.4 deg) and anterior (26.6±5.6 deg) trunk leans from the vertical. Our findings reveal that the global (i.e. the spring-mass model dynamics and centre-of-mass height) and local (i.e. knee and ankle kinematics and kinetics) biomechanical adjustments during uneven running are specific to the step nature and trunk posture. Unlike the anterior-leaning posture, running with a posterior trunk lean is characterized by increases in leg angle, leg compression, knee flexion angle and moment, resulting in a stiffer knee and a more compliant spring-leg compared with the self-selected condition. In the approach step versus the level step, reductions in leg length and stiffness through the ankle stiffness yield lower leg force and centre-of-mass position. Contrariwise, significant increases in leg length, angle and force, and ankle moment, reflect in a higher centre-of-mass position during the step down. Plus, ankle stiffness significantly decreases, owing to a substantially increased leg compression. Overall, the step down appears to be dominated by centre-of-mass height changes, regardless of having a trunk lean. Observed adjustments during uneven running can be attributed to anticipation of changes to running posture and height. These findings highlight the role of trunk posture in human perturbed locomotion relevant for the design and development of exoskeleton or humanoid bipedal robots.

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Stiffness describes the resistance of a body to deformation. In regard to athletic performance, a stiffer leg-spring would be expected to augment performance by increasing utilisation of elastic energy. Two-dimensional spring-mass and torsional spring models can be applied to model whole-body (vertical and/or leg stiffness) and joint stiffness. Various tasks have been used to characterise stiffness, including hopping, gait, jumping, sledge ergometry and change of direction tasks. Appropriate levels of reliability have been reported in most tasks, although they vary between investigations. Vertical stiffness has demonstrated the strongest reliability across tasks and may be more sensitive to changes in high-velocity running performance than leg stiffness. Joint stiffness demonstrates the weakest reliability, with ankle stiffness more reliable than knee stiffness. Determination of stiffness has typically necessitated force plate analyses; however, validated field-based equations permit determination of whole-body stiffness without force plates. Vertical, leg and joint stiffness measures have all demonstrated relationships with performance measures. Greater stiffness is typically demonstrated with increasing intensity (i.e., running velocity or hopping frequency). Greater stiffness is observed in athletes regularly subjecting the limb to high ground reaction forces (i.e., sprinters). Careful consideration should be given to the most appropriate assessment of stiffness on a team/individual basis.

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