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Evening fellow redpillers. For this Sunday Redpill Theory, we are going to go deeper about the subject of plate spinning in an LTR.
The kinetic equation of the spinning plate is 2M s + J dd2 φ t/dt 2 = 0, where the torque M s is correlated with the structural geometries and elastic deformation of the threads. With the relation between M s and φ t determined, the winding angle φ t and rotational speed ω d of the plate can be solved from the governing equation. The detailed theoretical derivation is given in Supplementary Information. Here we attempt to reveal the influence of loading rate ε˙s=dεs/dtε˙s=dεs/dt on the unwinding dynamics. In the examples, we take E t = 2 GPa, r t0 = 1 mm, φ tm = 3600o, l s0 = 20 cm, m d = 50 g, r d = 5 cm (J d = 6.25 × 10−5 kg · m2), and r h = 1 mm. The strain of the string is assumed to be εs=εmin+(εmax−εmin)tn/tnunwindεs=εmin+(εmax−εmin)tn/tunwindn, where t unwind is the unwinding period and n a dimensionless exponent. The minimal εminεmin and the maximal axial strain εmaxεmax of the string are determined from F s| t=0 s = 0 N and t unwind = 1 s.
The temporal evolutions of the axial tensile force F s and torque M s of the string, the winding angle φ t of the threads, and the angular velocity ω d of the plate are plotted in Fig. 2, where we take n = 1/3, 1/2, 1, 2, and 3 for different loading rates ε˙sε˙s. The maximal axial strain of a string that makes F s = 0 N and M s = 0 N · m is referred to as the critical strain: εcr=1−r¯2t0ϕt(ϕt+2r¯h/r¯t0−2)−−−−−−−−−−−−−−−−−−−−−−√−1εcr=1−r¯t02ϕt(ϕt+2r¯h/r¯t0−2)−1. Only when ε s > ε cr, can the string accelerate the plate. As shown in Fig. 2, the unwinding processes of n = 1/3, 1/2, and 1 are aborted as a result of ε s < ε cr, while that of n = 2 and 3 are completed with ϕt|t=tunwind=0∘ϕt|t=tunwind=0∘. In the early stage of the unwinding process, the strain rates ε˙sε˙s in the case of n ≤ 1 are distinctly greater than that of n > 1, which lead to a rapid decrease in the winding angle φ t (Fig. 2c) and, in turn, increase the critical strain ε cr. If the increasing ε cr exceeds ε s, the string is loosened and featured by F s = 0 N and M s = 0 N · m (Fig. 2a and b). An appropriate strain rate ε˙sε˙s (e.g., n = 2 and 3) enables the plate to be continuously accelerated to the end of the unwinding process (Fig. 2d). It is worthy of mentioning that, less than 200 N is required to twirl the plate to 10000o/s (about 1667 rpm) when n = 1/3, while nearly 400 N is required in the case of n = 3 to make the plate spin at the similar speed (Fig. 2a and d). In the former case, however, the maximal torque M s is much larger than that in the later
The dependence of the unwinding dynamics on the structural geometries, e.g., the length l s0 of the string, the radius r t0 of the threads, and the distance 2r h between the drill holes, is also explored. When ε s = ε cr, the plate will be spun at a constant velocity ω c and the winding angle of the threads φ t = φ tm − ω c t. With ω c and φ t known, the lower limit of the strain rate that keeps the plate accelerating can be determined as ε˙cr=dεcr/dtε˙cr=dεcr/dt. The smaller the ε˙crε˙cr, the easier the unwinding process will be. The ε˙crε˙cr vs. t relations are plotted in Fig. 3a and b, where we take φ tm = 3600o, ω c = 3600o/s, l s0 = 20 cm, and several representative values of r¯h=rh/ls0r¯h=rh/ls0 and r¯t0=rt0/ls0r¯t0=rt0/ls0. The critical strain rate ε˙crε˙cr increases with increasing r¯hr¯h and r¯t0r¯t0. It suggests that threads with a large slenderness (i.e., r¯−1t0=ls0/rt0r¯t0−1=ls0/rt0) and a plate with paracentral holes favor easier unwinding. However, threads with an excessively large length cannot be easily rewound when a lightweight plate is used during the winding process. When the thread radius r t0 is specified, using longer threads, though with a larger slenderness, may not lead to an easier operation. Excellent performance (e.g., faster rotation and easier operation) of the system requires an adequate combination of thread and plate materials.
The angular velocity ω d of the plate is further plotted as a function of the tensile force F s of the string in Fig. 3c and d, where we take E t = 2 GPa, φ tm = 3600o, l s0 = 20 cm, ε s = ε cr| t =0s + t 3/10, J d = 6.25 × 10−5 kg · m2, and several representative r¯hr¯h and r¯t0r¯t0. For a fixed F s, ω d increases with the increasing r¯hr¯h and r¯t0r¯t0. This is because the torque transmitted to the plate will be enlarged if r¯hr¯h and r¯t0r¯t0 increase. When the length l s0 of the string is specified, the plate could be spun at a higher speed by using thicker threads, which is consistent with previous experimental observations1. It suggests that threads with a smaller slenderness and a plate with a larger distance between drill holes are preferable to achieve a high-speed rotation.
Any thoughts? I'm considering writing another theory about the duration of frame holding.
The kinetic equation of the spinning plate is 2M s + J dd2 φ t/dt 2 = 0, where the torque M s is correlated with the structural geometries and elastic deformation of the threads. With the relation between M s and φ t determined, the winding angle φ t and rotational speed ω d of the plate can be solved from the governing equation. The detailed theoretical derivation is given in Supplementary Information. Here we attempt to reveal the influence of loading rate ε˙s=dεs/dtε˙s=dεs/dt on the unwinding dynamics. In the examples, we take E t = 2 GPa, r t0 = 1 mm, φ tm = 3600o, l s0 = 20 cm, m d = 50 g, r d = 5 cm (J d = 6.25 × 10−5 kg · m2), and r h = 1 mm. The strain of the string is assumed to be εs=εmin+(εmax−εmin)tn/tnunwindεs=εmin+(εmax−εmin)tn/tunwindn, where t unwind is the unwinding period and n a dimensionless exponent. The minimal εminεmin and the maximal axial strain εmaxεmax of the string are determined from F s| t=0 s = 0 N and t unwind = 1 s.
The temporal evolutions of the axial tensile force F s and torque M s of the string, the winding angle φ t of the threads, and the angular velocity ω d of the plate are plotted in Fig. 2, where we take n = 1/3, 1/2, 1, 2, and 3 for different loading rates ε˙sε˙s. The maximal axial strain of a string that makes F s = 0 N and M s = 0 N · m is referred to as the critical strain: εcr=1−r¯2t0ϕt(ϕt+2r¯h/r¯t0−2)−−−−−−−−−−−−−−−−−−−−−−√−1εcr=1−r¯t02ϕt(ϕt+2r¯h/r¯t0−2)−1. Only when ε s > ε cr, can the string accelerate the plate. As shown in Fig. 2, the unwinding processes of n = 1/3, 1/2, and 1 are aborted as a result of ε s < ε cr, while that of n = 2 and 3 are completed with ϕt|t=tunwind=0∘ϕt|t=tunwind=0∘. In the early stage of the unwinding process, the strain rates ε˙sε˙s in the case of n ≤ 1 are distinctly greater than that of n > 1, which lead to a rapid decrease in the winding angle φ t (Fig. 2c) and, in turn, increase the critical strain ε cr. If the increasing ε cr exceeds ε s, the string is loosened and featured by F s = 0 N and M s = 0 N · m (Fig. 2a and b). An appropriate strain rate ε˙sε˙s (e.g., n = 2 and 3) enables the plate to be continuously accelerated to the end of the unwinding process (Fig. 2d). It is worthy of mentioning that, less than 200 N is required to twirl the plate to 10000o/s (about 1667 rpm) when n = 1/3, while nearly 400 N is required in the case of n = 3 to make the plate spin at the similar speed (Fig. 2a and d). In the former case, however, the maximal torque M s is much larger than that in the later
The dependence of the unwinding dynamics on the structural geometries, e.g., the length l s0 of the string, the radius r t0 of the threads, and the distance 2r h between the drill holes, is also explored. When ε s = ε cr, the plate will be spun at a constant velocity ω c and the winding angle of the threads φ t = φ tm − ω c t. With ω c and φ t known, the lower limit of the strain rate that keeps the plate accelerating can be determined as ε˙cr=dεcr/dtε˙cr=dεcr/dt. The smaller the ε˙crε˙cr, the easier the unwinding process will be. The ε˙crε˙cr vs. t relations are plotted in Fig. 3a and b, where we take φ tm = 3600o, ω c = 3600o/s, l s0 = 20 cm, and several representative values of r¯h=rh/ls0r¯h=rh/ls0 and r¯t0=rt0/ls0r¯t0=rt0/ls0. The critical strain rate ε˙crε˙cr increases with increasing r¯hr¯h and r¯t0r¯t0. It suggests that threads with a large slenderness (i.e., r¯−1t0=ls0/rt0r¯t0−1=ls0/rt0) and a plate with paracentral holes favor easier unwinding. However, threads with an excessively large length cannot be easily rewound when a lightweight plate is used during the winding process. When the thread radius r t0 is specified, using longer threads, though with a larger slenderness, may not lead to an easier operation. Excellent performance (e.g., faster rotation and easier operation) of the system requires an adequate combination of thread and plate materials.
The angular velocity ω d of the plate is further plotted as a function of the tensile force F s of the string in Fig. 3c and d, where we take E t = 2 GPa, φ tm = 3600o, l s0 = 20 cm, ε s = ε cr| t =0s + t 3/10, J d = 6.25 × 10−5 kg · m2, and several representative r¯hr¯h and r¯t0r¯t0. For a fixed F s, ω d increases with the increasing r¯hr¯h and r¯t0r¯t0. This is because the torque transmitted to the plate will be enlarged if r¯hr¯h and r¯t0r¯t0 increase. When the length l s0 of the string is specified, the plate could be spun at a higher speed by using thicker threads, which is consistent with previous experimental observations1. It suggests that threads with a smaller slenderness and a plate with a larger distance between drill holes are preferable to achieve a high-speed rotation.
Any thoughts? I'm considering writing another theory about the duration of frame holding.
