{"id":922,"date":"2020-12-14T20:09:21","date_gmt":"2020-12-14T20:09:21","guid":{"rendered":"https:\/\/pickedshares.com\/?p=922"},"modified":"2021-05-08T12:23:26","modified_gmt":"2021-05-08T12:23:26","slug":"engineering-mechanics-1-exercise-32-band-brake-with-double-acting-lever","status":"publish","type":"post","link":"https:\/\/pickedshares.com\/en\/engineering-mechanics-1-exercise-32-band-brake-with-double-acting-lever\/","title":{"rendered":"Band brake with double-acting lever"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">This exercise is about the usage of the belt friction formula. <a href=\"https:\/\/pickedshares.com\/en\/static-and-sliding-friction-values\/\">Common friction values can be found here<\/a>. A tool for the <a href=\"https:\/\/pickedshares.com\/en\/belt-friction-calculator\/\">calculation of belt friction is here<\/a>.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Task<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">A brake with a double-acting operating lever is operated with the force F. The maximum braking torques in clockwise and counter-clockwise rotation must be determined.<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"976\" height=\"834\" src=\"https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/12\/drum_with_band_brake_operating_lever.jpg\" alt=\"Band brake with double-action operating lever\" class=\"wp-image-913\" srcset=\"https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/12\/drum_with_band_brake_operating_lever.jpg 976w, https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/12\/drum_with_band_brake_operating_lever-300x256.jpg 300w, https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/12\/drum_with_band_brake_operating_lever-768x656.jpg 768w, https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/12\/drum_with_band_brake_operating_lever-400x342.jpg 400w, https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/12\/drum_with_band_brake_operating_lever-800x684.jpg 800w\" sizes=\"auto, (max-width: 976px) 100vw, 976px\" \/><figcaption>Band brake with double-action operating lever<\/figcaption><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\">Solution<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">The band brake is divided into two sections: the operating lever and the drum with the band. The equilibrium of moments on the drum are considered separately for right-hand and left-hand rotation.<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"667\" src=\"https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/12\/free_cut_operating_lever-1024x667.jpg\" alt=\"Free-cut double-acting operating lever of the band-brake\" class=\"wp-image-915\" srcset=\"https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/12\/free_cut_operating_lever-1024x667.jpg 1024w, https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/12\/free_cut_operating_lever-300x195.jpg 300w, https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/12\/free_cut_operating_lever-768x500.jpg 768w, https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/12\/free_cut_operating_lever-400x261.jpg 400w, https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/12\/free_cut_operating_lever-800x521.jpg 800w, https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/12\/free_cut_operating_lever.jpg 1122w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><figcaption>Free-cut double-acting operating lever of the band-brake<\/figcaption><\/figure>\n\n\n\n<script src=\"https:\/\/cdnjs.cloudflare.com\/ajax\/libs\/mathjax\/2.7.7\/MathJax.js?config=TeX-AMS_HTML\" async=\"async\">  \/\/ A comment that hinders wxWidgets from optimizing this tag too much.\n<\/script>\n\\[ \\require{cancel} \\]\n<p><strong>Moments at the operating lever<\/strong><\/p>\n\n<div style=\"overflow:auto;\">\n<p>\\[ \\tag{1} \\sum M(A) = 0 = F_C \\cdot b + F_B \\cdot a - F \\cdot l \\]<\/p>\n<\/div>\n\n\n\n<h3 class=\"wp-block-heading\">Left-hand rotation<\/h3>\n\n\n\n<figure class=\"wp-block-image size-large custom-flex-image75\"><img loading=\"lazy\" decoding=\"async\" width=\"864\" height=\"764\" src=\"https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/12\/brake_drum_left_turning.jpg\" alt=\"Left-turning brake drum\" class=\"wp-image-909\" srcset=\"https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/12\/brake_drum_left_turning.jpg 864w, https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/12\/brake_drum_left_turning-300x265.jpg 300w, https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/12\/brake_drum_left_turning-768x679.jpg 768w, https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/12\/brake_drum_left_turning-400x354.jpg 400w, https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/12\/brake_drum_left_turning-800x707.jpg 800w\" sizes=\"auto, (max-width: 864px) 100vw, 864px\" \/><figcaption>Left-turning brake drum<\/figcaption><\/figure>\n\n\n\n<script src=\"https:\/\/cdnjs.cloudflare.com\/ajax\/libs\/mathjax\/2.7.7\/MathJax.js?config=TeX-AMS_HTML\" async=\"async\">  \/\/ A comment that hinders wxWidgets from optimizing this tag too much.\n<\/script>\n\\[ \\require{cancel} \\]\n<p>Moments at the brake drum<\/p>\n\n<div style=\"overflow:auto;\">\n<p>\\[ \\tag{2} \\sum M = 0 = F_C \\cdot r - F_B \\cdot r + M_L \\]<\/p>\n<\/div>\n\n<p>The rope friction formula (Euler-Eytelwein) provides the relationship between the forces in the working belt and the loose belt:<\/p>\n\n<p>\\[ \\tag{3} F_B \\leq F_C \\cdot e^{\u00b5 \\cdot \\alpha} \\]<\/p>\n\n<p>F<sub>B<\/sub> used in equation (1) delivers<\/p>\n\n<div style=\"overflow:auto;\">\n<p>\\[ \\tag{4} 0 = F_C \\cdot b + F_C \\cdot e^{\u00b5 \\cdot \\alpha} \\cdot a - F \\cdot l \\]<\/p>\n\n<p>\\[ \\tag{5}  F_C \\cdot b + F_C \\cdot e^{\u00b5 \\cdot \\alpha} \\cdot a = F \\cdot l  \\]<\/p>\n<\/div>\n<p>\\[ \\tag{6}  F_C = \\frac{F \\cdot l}{b + e^{\u00b5 \\cdot \\alpha} \\cdot a}  \\]<\/p>\n\n<p>F<sub>B<\/sub> and F<sub>C<\/sub> can now be used in equation (2)<\/p>\n\n<div style=\"overflow:auto;\">\n<p>\\[ \\tag{7} 0 = \\frac{F \\cdot l}{b + e^{\u00b5 \\cdot \\alpha} \\cdot a} \\cdot r - \\frac{F \\cdot l}{b + e^{\u00b5 \\cdot \\alpha} \\cdot a} \\cdot e^{\u00b5 \\cdot \\alpha} \\cdot r + M_L \\]<\/p>\n\n<p>\\[ \\tag{8} M_L = \\frac{F \\cdot l}{b + e^{\u00b5 \\cdot \\alpha} \\cdot a} \\cdot e^{\u00b5 \\cdot \\alpha} \\cdot r - \\frac{F \\cdot l}{b + e^{\u00b5 \\cdot \\alpha} \\cdot a} \\cdot r \\]<\/p>\n\n<p>\\[ \\tag{9} M_L = \\frac{F \\cdot l \\cdot r}{b + e^{\u00b5 \\cdot \\alpha} \\cdot a} \\cdot \\left(e^{\u00b5 \\cdot \\alpha} - 1 \\right) \\]<\/p>\n<\/div>\n<p>So the maximum breaking torque for the left-hand rotation of the band brake is determined.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Right-hand rotation<\/h3>\n\n\n\n<figure class=\"wp-block-image size-large custom-flex-image75\"><img loading=\"lazy\" decoding=\"async\" width=\"860\" height=\"869\" src=\"https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/12\/brake_drum_right_turning.jpg\" alt=\"Right-turning brake drum\" class=\"wp-image-911\" srcset=\"https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/12\/brake_drum_right_turning.jpg 860w, https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/12\/brake_drum_right_turning-297x300.jpg 297w, https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/12\/brake_drum_right_turning-768x776.jpg 768w, https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/12\/brake_drum_right_turning-400x404.jpg 400w, https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/12\/brake_drum_right_turning-800x808.jpg 800w\" sizes=\"auto, (max-width: 860px) 100vw, 860px\" \/><figcaption>Right-turning brake drum<\/figcaption><\/figure>\n\n\n\n<script src=\"https:\/\/cdnjs.cloudflare.com\/ajax\/libs\/mathjax\/2.7.7\/MathJax.js?config=TeX-AMS_HTML\" async=\"async\">  \/\/ A comment that hinders wxWidgets from optimizing this tag too much.\n<\/script>\n\\[ \\require{cancel} \\]\n<p>Moments at the brake drum<\/p>\n\n<div style=\"overflow:auto;\">\n<p>\\[ \\tag{10} \\sum M = 0 = F_C \\cdot r - F_B \\cdot r - M_R \\]<\/p>\n<\/div>\n<p>The rope friction formula is in this case:<\/p>\n\n<p>\\[ \\tag{11} F_C \\leq F_B \\cdot e^{\u00b5 \\cdot \\alpha} \\]<\/p>\n<p>\\[ \\tag{12} F_B = F_C \\cdot e^{-\u00b5 \\cdot \\alpha} \\]<\/p>\n\n<p>F<sub>B<\/sub> used in equation (1) delivers<\/p>\n\n<div style=\"overflow:auto;\">\n<p>\\[ \\tag{13} 0 = F_C \\cdot b + F_C \\cdot e^{-\u00b5 \\cdot \\alpha} \\cdot a - F \\cdot l \\]<\/p>\n\n<p>\\[ \\tag{14} F_C  = \\frac{F \\cdot l}{b + e^{-\u00b5 \\cdot \\alpha} \\cdot a}  \\]<\/p>\n<\/div>\n<p>F<sub>B<\/sub> and F<sub>C<\/sub> can now be used in equation (10) <\/p>\n\n<div style=\"overflow:auto;\">\n<p>\\[ \\tag{15} 0 = \\frac{F \\cdot l}{b + e^{-\u00b5 \\cdot \\alpha} \\cdot a} \\cdot r - \\frac{F \\cdot l}{b + e^{-\u00b5 \\cdot \\alpha} \\cdot a} \\cdot e^{-\u00b5 \\cdot \\alpha} \\cdot r - M_R \\]<\/p>\n<p>\\[ \\tag{16} M_R = \\frac{F \\cdot l}{b + e^{-\u00b5 \\cdot \\alpha} \\cdot a} \\cdot r - \\frac{F \\cdot l}{b + e^{-\u00b5 \\cdot \\alpha} \\cdot a} \\cdot e^{-\u00b5 \\cdot \\alpha} \\cdot r  \\]<\/p>\n\n<p>\\[ \\tag{17} M_R =  \\frac{F \\cdot l \\cdot r}{b + e^{-\u00b5 \\cdot \\alpha}} \\cdot \\left( 1 - e^{-\u00b5 \\cdot \\alpha} \\right) \\]<\/p>\n<\/div>\n\n\n\n<p class=\"wp-block-paragraph\">So the maximum breaking torque for the right-hand rotation of the band brake is determined.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Here is a collection of further interesting <a href=\"https:\/\/pickedshares.com\/en\/tag\/friction\/\">exercises and solutions regarding friction<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p> ... <a title=\"Band brake with double-acting lever\" class=\"read-more\" href=\"https:\/\/pickedshares.com\/en\/engineering-mechanics-1-exercise-32-band-brake-with-double-acting-lever\/\" aria-label=\"Read more about Band brake with double-acting lever\">Read more<\/a><\/p>\n","protected":false},"author":1,"featured_media":913,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[21,60],"tags":[27,48],"class_list":["post-922","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-engineering-mechanics-i","category-exercises","tag-friction","tag-resulting-moments","infinite-scroll-item","masonry-post","generate-columns","tablet-grid-50","mobile-grid-100","grid-parent","grid-33"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.8 - 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