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Analysis and Solution of Dynamic Problems Considering Friction of Fixed Pulley Bearings in Theoretical Mechanics
DOI: 10.12677/IJM.2021.103018, PDF, HTML, XML, 下载: 253  浏览: 620

Abstract: Examples and exercises of dynamics problems of fixed pulley-rope weight systems are common in many theoretical mechanics textbooks. However, in the setting of these examples and exercises, the friction factors of bearing are not considered, so the calculation results can not reflect the movement law of the system truly and objectively. The dynamics problem of a specific fixed pulley-rope weight system being taken as an example, it is shown how to incorporate the friction factors into the analysis and solution, so that the solution results can more objectively reflect the motion law of the system. The analysis and solution ideas given in this paper can also be applied to the analysis and solution of other fixed pulley-rope weight system dynamics in which the influence of bearing friction factors needs to be included.

1. 引言

2. 推导和计算

$\frac{\text{d}{L}_{z}}{\text{d}t}={m}_{A}gR-{m}_{B}gR-Fr$ (1)

${L}_{z}={J}_{O}\omega +{m}_{A}{v}_{A}R+{m}_{B}{v}_{B}R=\left({J}_{O}+{m}_{A}{R}^{2}+{m}_{B}{R}^{2}\right)\omega$ (2)

$\left({J}_{O}+{m}_{A}{R}^{2}+{m}_{B}{R}^{2}\right)\alpha =\left({m}_{A}-{m}_{B}\right)gR-Fr$ (3)

Figure 1. Fixed pulley-rope weight system

Figure 2. Free-body diagram of fixed pulley-rope weight system

$\frac{\text{d}{P}_{x}}{\text{d}t}=-N\mathrm{sin}\theta +F\mathrm{cos}\theta$ (4)

$\frac{\text{d}{P}_{y}}{\text{d}t}=N\mathrm{cos}\theta +F\mathrm{sin}\theta -\left(m+{m}_{A}+{m}_{B}\right)g$ (5)

${P}_{x}=0$ (6)

${P}_{y}=-{m}_{A}{v}_{A}+{m}_{B}{v}_{B}=\left({m}_{B}-{m}_{A}\right)R\omega$ (7)

$N\mathrm{sin}\theta -F\mathrm{cos}\theta =0$ (8)

$N\mathrm{cos}\theta +F\mathrm{sin}\theta =\left(m+{m}_{A}+{m}_{B}\right)g+\left({m}_{B}-{m}_{A}\right)R\alpha$ (9)

${N}^{2}+{F}^{2}={\left[\left(m+{m}_{A}+{m}_{B}\right)g+\left({m}_{B}-{m}_{A}\right)R\alpha \right]}^{2}$ (10)

$F=N\text{ }f$ (11)

$F=\frac{f\left[\left(m+{m}_{A}+{m}_{B}\right)g+\left({m}_{B}-{m}_{A}\right)R\alpha \right]}{\sqrt{1+{f}^{2}}}$ (12)

$\left({J}_{O}+{m}_{A}{R}^{2}+{m}_{B}{R}^{2}\right)\alpha =\left({m}_{A}-{m}_{B}\right)gR-\frac{f\text{​}r\left[\left(m+{m}_{A}+{m}_{B}\right)g+\left({m}_{B}-{m}_{A}\right)R\alpha \right]}{\sqrt{1+{f}^{2}}}$ (13)

$\alpha =\frac{g\left[R\left({m}_{A}-{m}_{B}\right)\sqrt{1+{f}^{2}}-f\text{​}r\left(m+{m}_{A}+{m}_{B}\right)\right]}{\left({J}_{O}+{m}_{A}{R}^{2}+{m}_{B}{R}^{2}\right)\sqrt{1+{f}^{2}}-f\text{​}rR\left({m}_{A}-{m}_{B}\right)}$ (14)

$\stackrel{¨}{\phi }=\frac{g\left[R\left({m}_{A}-{m}_{B}\right)\sqrt{1+{f}^{2}}-f\text{​}r\left(m+{m}_{A}+{m}_{B}\right)\right]}{\left({J}_{O}+{m}_{A}{R}^{2}+{m}_{B}{R}^{2}\right)\sqrt{1+{f}^{2}}-f\text{​}rR\left({m}_{A}-{m}_{B}\right)}$ (15)

$\phi ={\phi }_{0}+{\omega }_{0}t+\frac{g{t}^{2}\left[R\left({m}_{A}-{m}_{B}\right)\sqrt{1+{f}^{2}}-f\text{​}r\left(m+{m}_{A}+{m}_{B}\right)\right]}{2\left[\left({J}_{O}+{m}_{A}{R}^{2}+{m}_{B}{R}^{2}\right)\sqrt{1+{f}^{2}}-f\text{​}rR\left({m}_{A}-{m}_{B}\right)\right]}$ (16)

${T}_{A}={m}_{A}g\frac{\left({J}_{O}+2{m}_{B}{R}^{2}\right)\sqrt{1+{f}^{2}}+f\text{​}rR\left(m+2{m}_{B}\right)}{\left({J}_{O}+{m}_{A}{R}^{2}+{m}_{B}{R}^{2}\right)\sqrt{1+{f}^{2}}-f\text{​}rR\left({m}_{A}-{m}_{B}\right)}$ (17)

${T}_{B}={m}_{B}g\frac{\left({J}_{O}+2{m}_{A}{R}^{2}\right)\sqrt{1+{f}^{2}}-f\text{​}rR\left(m+2{m}_{A}\right)}{\left({J}_{O}+{m}_{A}{R}^{2}+{m}_{B}{R}^{2}\right)\sqrt{1+{f}^{2}}-f\text{​}rR\left({m}_{A}-{m}_{B}\right)}$ (18)

$\alpha =\frac{gR\left({m}_{A}-{m}_{B}\right)}{{J}_{O}+{m}_{A}{R}^{2}+{m}_{B}{R}^{2}}$ (19)

${T}_{A}={m}_{A}g\frac{{J}_{O}+2{m}_{B}{R}^{2}}{{J}_{O}+{m}_{A}{R}^{2}+{m}_{B}{R}^{2}}$ (20)

${T}_{B}={m}_{B}g\frac{{J}_{O}+2{m}_{A}{R}^{2}}{{J}_{O}+{m}_{A}{R}^{2}+{m}_{B}{R}^{2}}$ (21)

3. 结束语

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