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[几何] 受力的弹簧所在圆柱面展开成平面

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realnumber posted 2020-5-11 10:46 |Read mode
受力的弹簧所在圆柱面展开成平面,此时弹簧在这个平面的曲线方程f(x)是?
起因:被初一的小朋友难倒了,同样的两根钢丝卷成中经比为1:2(都一定范围内)的两条弹簧,问弹性系数比为?
查了下百度,不太理解k与$D_m^3$成反比,有没数学方面一个合理的解释,(其它G,d,$N_c$不管它,标题方向也不见得对,如果真能展开的话,似乎是k与$D_m$成反比 )
弹性系数(劲度系数,见百度百科)\[  k=\frac{Gd^4}{8N_cD_m^3}  \]
G=线材的刚性模数,单位N/mm^2(即切变模量):碳素弹簧钢丝(如65Mn)以及常用弹簧钢丝79000 ;不锈钢丝71000 ,硅青铜线G=41000 【其他详见机械设计手册(第五版)第三卷P11-10】
d=线径(mm),制作弹簧钢丝的直径
$D_0$=外径(mm)
$D_m$=中径=$D_0-d$(mm)
N=总圈数
有效圈数$N_c$=N-2
业余的业余 posted 2020-5-15 03:59
不会 如果是从圆柱面上平行于中心轴的一条线剪开,并摊平,直觉应该是一系列相连的线段。
a.png

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original poster realnumber posted 2020-5-19 15:59
是我没表达好,
弹簧有几圈,就沿着弹簧所在的圆柱展开几圈,估计有人研究过吧

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hbghlyj posted 2023-1-28 09:29
Young's Modulus as a Spring Constant
The number of springs in parallel is proportional to the cross-sectional area $ S$ of the bar. Therefore, the force applied to each spring is proportional to the total applied force $ F$ divided by the cross-sectional area $ S$ . Thus, Hooke's law for each spring in the bundle can be written
$$\frac{F}{S} = Y \frac{\Delta L}{L} $$
where $ Y$ is Young's modulus.
Finding the diameter of a material using tension stiffness and Young's Modulus

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hbghlyj posted 2023-1-28 09:42
en.wikipedia.org/wiki/Helix#Examples最后的一张图片是A helical coil spring
Helix $(a\cos t,a\sin t,bt)$ 沿圆柱面展开是直线 $(at,bt)$.

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hbghlyj posted 2024-12-7 10:10

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