Dynamics of oscillating muscular masses: dependencies of natural frequencies as a function of activity, shock resistance and crushed material

By comparing the frequencies in the tables 2 And 4, it is likely that the frequencies of approximately 60 Hz, 330 Hz, 355 Hz, 450 Hz and 550 Hz reflect the properties of the interaction between the frame itself and the floor material, since these frequencies are always present without the GAS fixed in the frame. In addition, the frequency at \(\approximately\) 60 Hz disappears in the FFT analysis when changing the ground material from polystyrene to aluminum. Thus, the \(\approximately\) 60 Hz in the FFT analysis is attributed to the polystyrene-frame interaction.

In all tests, the frequencies attributed to the SGA are always lower in the passive tests (Table 5). Like the three GAZ frequencies (F2, F3, and \(F4\) \(\Green\) \(F6\)) depend on the properties of the fibrous materials (active and passive) (table 3: through the rigidities of the basic model \(k_{C}\) And \(k_{S}\)Fig. 1c, therefore \(k_{SC}\) And \(k_{SSC}\), see equations. (3) And (4)), the positive correlation between frequency and muscle activity is probably due to the formation of cross-bridges. In active fibrous material, attachment of myosin heads mechanically couples actin and myosin filaments creating cross-bridges, and sliding of the filaments requires their distortion.31,32. Additionally, any added cross-bridge increases the stiffness of the fibrous material.31,33. In a study11 Similar to our current model, we found that the stiffness of fully activated fibrous material only (and not \(k_{S}\) representing the properties of fiber-aponeurosis, but \(k_{C}\) here equal \(k_{CE}\) there11) East \(\approximately\) 370% (\(\frac{15,000\,\text {N m}^{-1}-3200\,\text {N m}^{-1}}{3200\,\text {N \, m}^{ – 1}}\)) higher than in the passive fibrous material11 (see also supplementary table S4). Active muscle contraction also loads the aponeurosis biaxially.27,34and at least one study21 showed in isotonic contraction experiments that the longitudinal rigidity of the probed aponeuroses of wild turkeys M. lateral gastrocnemius increased by \(\approximately\) 64% (\(\frac{180\,\text {N \, m}^{-1}-110\,\text {N \, m}^{-1}}{110\,\text {N \, m }^{-1}}\)(Fig. 5B21)) from passive muscular condition to fully activated muscular condition. We assume that either the aponeuroses17,21 or fibrous material11 dominate F2 due to their estimated stiffness values ​​being at least an order of magnitude lower than those of the longer (proximal) GAS tendon, with tendon stiffness estimates based on published values ​​of Young’s modulus.30,31. Unfortunately, with our current 3DoF model and current experimental setup, we still cannot resolve individual tissue contributions or potential differences in proximal and distal stiffness values.

A frequency (F7) which we found in the passive condition (455 Hz) differed significantly from its active counterpart (470 Hz). These two frequencies are also slightly different from all frequencies of the frame or frame-floor material (polystyrene, board 4), and the 3DoF model does not predict them either. Therefore, either the 3DoF model is too simple (e.g., too few degrees of freedom) to predict these frequencies, or the interaction between the GAS and the two frame extrusions slightly changes the overall stiffness of the muscle preparation. If so, it could also affect F2 in the table 4while the lowest frame-ground-material frequency found remains the same because the mass of the frame and the visco-elasticity of the polystyrene determine it.

For both active and passive tests, changing the shock resistance does not result in any significant change in the frequency spectrum. Our results might differ with higher impact resistance, because for fresh, fully stimulated GAS, the impact resistance in the 4 cm tests is insufficient to induce forced disruption of the crossbridge.17. Here, the impact resistance is not sufficient to significantly interfere with the working stroke: whether by forced rupture of the connections between bridges.35 or influence of the center of oscillation at which the crossbridge generates the maximum GAS contractile force. Changing the impact force can change the oscillation amplitudes, however, issues such as the potential elastic recoil of the oscillating masses, their dissipation energy and wave propagation, all depending on the impact force and therefore their oscillation amplitudes, were not part of this study. although it’s worth investigating next.

There were no significant differences in the frequencies of active GAS when comparing the aluminum (2 mm) trials and the 1.5 cm polystyrene trials (Supplementary Table S2). A likely explanation and advantage of our setup is the controlled environment in which we can design GAS impact situations that are complicated or impossible to specifically target in situ. For example, the muscles of the lower limbs are not pre-activated appropriately.1,12,13,14 in our experiments, and the lower limb is not pre-inclined36,37, respectively, which allows us to reduce the expected complexity of the soft tissue vibration response (here, for example, focusing only on longitudinal oscillation modes). In our results, we then see that in the space of low-dimensional solutions of only a few frequencies, only one of them, namely: F2 in the 1.5 cm passive tests (135 Hz) differs significantly from the also passive aluminum tests (141 Hz), with approximately the same impact resistance (Table 2). We suspect that this (slight but significant: p = 0.043, supplementary table S1) Difference of F2 can be explained either by the time between the last active trial and the passive one performed, or by the time between dissection and passive experiments, or by the total number of experiments for a specific muscle. Such a suspected mechanism of memory or history could be inherent to the structure: an appropriate hypothesis would be that it comes from the third filament, the giant titin molecule. Titin probably contributes to this \(\ge\) 75% of the passive stiffness of the fibrous material at \(L_{opt}\)11and titin is a viscoelastic material38,39. The mechanical effects of titin, for example through its interaction with actin40are determined by contractile history (both mechanical and activity) and, therefore and accordingly, vary depending on the time between each stretching/shortening contraction38,41. Unfortunately, again, due to formal restrictions (so far only a few animals are available in each experimental session) and the overall methodical complexity of the setup, our sample sizes are currently simply too small and our There is too little data to explore any of these potential correlations so far.

In general, the natural frequencies of GAS muscles predicted by our model match well in the two cases probed experimentally, the passive (P.1) and the fully active (A1). We show that treating the muscle as a system clamped by an alternating sequence of pre-stressed linear springs and portions of muscle mass (i.e. a system of harmonic oscillators coupled in series) estimates the natural frequencies of the GAS quite accurately. in response to impact at TD, when at least two criteria are met: First, the muscle must be initially (at TD) in quasi-isometric conditions, which is in agreement with the literature, because the flexion angle of the knee and ankle within 20 ms following TD is correct \(\approximately\) 3° for Tupaia glis trot at 1 ms−142; that is, the GAS is stiff at TD and both angular velocities of the joints are close to zero. The second criterion likely allows for a greater extent of active muscle length before TD, as contraction of the fibrous material can compensate for any initial relaxation. We also show that the elasticities of the aponeuroses and fibrous material dominate the natural frequencies of the high amplitude GAS and that any elasticity of the tendon material on the natural frequencies of the GAS is likely non-existent due to differences in stiffness and mass compared to others. GAS tissues. As the length of the rat GAS (distal) tendon is not extraordinarily short, this should also apply to any other vertebrate of rat size or smaller.18,43 because Young’s modulus18,19 then completely determines the stiffness. For purely mechanical reasons (inertia and compliance), in larger animals with lower natural frequencies17tendons may well play an important role in muscle oscillation mass natural frequencies where \(k_{t,d}\) and even \(k_{t,p}\)with \(k_{t,p} \ne k_{t,d}\)must be taken into account in the first instance when analyzing the natural frequencies of muscles larger than the rat GAS.

While the first (\(frequency _{1}\)) and third (\(frequency _{3}\)) natural frequencies estimated by the model (table 3) are in almost perfect agreement with the two comparable natural frequencies found experimentally (F2 and \(F4\) \(\Green\) \(F6\)Painting 2), the second estimated natural frequency (\(frequency _{2}\)) deviates further from the measured value F3 than its counterparts. As we assume proximal and distal symmetry in terms of stiffness and mass, the absolute displacements of \(m_{S,p}\) And \(m_{S,d}\) are always the same (see Fig. 1c and the eigenvectors of our 3DoF model in the supplementary equations. S20, S22, S23, p. 5, respectively). Therefore, our current model cannot cover a possible non-homogeneous distribution of fiber stresses (wave propagation) across the central part (the fiber belly: CE, Fig. 1c) bar for the case of the eigenvector of \(frequency _{2}\) where the displacement of \(m_{S,p}\) And \(m_{S,d}\) is in opposite directions (supplementary equation. S20, p. 5). Not allowing fiber deformation in an in-phase eigenvector contradicts the literature which found \(\approximately\) 0.3% fiber deformation at \(F_{max}\) in wobbling mass experiments similar to our group 3 trials17. To address certain fiber constraints, proximal-distal mass asymmetry or proximal-distal stiffness asymmetry must be considered in the model. Any break in the proximal-distal symmetry of the model may include the need to improve the explanations of our model with respect to the measured natural frequencies and modes; however, this requires increasing the complexity of the model by adding degrees of freedom. Increasing the complexity of the model by adding degrees of freedom would, in turn, allow us to better distinguish anatomically defined tissue regions within the GAS (i.e., distinguish and accurately represent tendons, as well as regions mixed aponeurosis fibers and pure fibers, respectively) and to examine in a more transparent manner, and thus understand, their contribution to the dynamics of the wobbling masses.


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