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Using the Rayleigh’s method, the first natural frequency and period corresponding to the first mode of transversal vibrations for cone and wedge beams, having non-uniform cross-section, are calculated in this study. Six different supporting schemes are considered. It has been assumed that the beams are made of a homogeneous and elastic material and the deflection line of the beam axis during vibrations has the same shape as the axis deflected by a static constant uniform continuous load. The values of frequencies, calculated according to the presented procedure, have been compared to the results obtained in the Finite Element Method as well as to the benchmark solutions given in the literature for the natural vibrations of Euler-Bernoulli’s beam. High conformity of results, enough for engineering calculations, is concluded.
In the present study, the mechanisms of interaction between the breathing and stepping movements were investigated. The investigations were carried out in 8 normal subjects and 4 individuals with complete spinal cord lesion. Additionally, experimental studies were performed in 7 decerebrate and 4 spinal cats. Involuntary reflex stepping movements in the air were induced by vibrators attached above a tendon of m. rectus femoris and m. biceps femoris. In the spinal patients, epidural stimulation was carried out with electrodes located in the posterior epidural space at the vertebral level Th11. It was found that voluntary-induced stepping, which are accompanied by visceral reactions, were mediated by the central mechanisms. Changes in breathing pattern in response to vibration-induced involuntary stepping movements point up to the peripheral component being predominant in such interactions. In the spinal patients under epidural spinal cord stimulation delivered below the site of injury, the reaction in the cardio-respiratory system was abolished. Vibrations to the leg muscles also did not evoke respiratory responses. However, in the condition of forced breathing such vibrations evoked the rhythmic bursting EMG activity in m. rectus femoris, which was in harmony with the breathing rhythm. The results confirmed the presence of an interaction between breathing and stepping generators. This interaction is underlain both by central and peripheral components.
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