{"id":3130,"date":"2025-09-23T12:32:59","date_gmt":"2025-09-23T10:32:59","guid":{"rendered":"https:\/\/jureknawrocki.com\/?p=3130"},"modified":"2026-02-24T12:07:01","modified_gmt":"2026-02-24T11:07:01","slug":"dynamika-autopipe-czy-caesar-cz-i","status":"publish","type":"post","link":"https:\/\/jureknawrocki.com\/en\/dynamika-autopipe-czy-caesar-cz-i\/","title":{"rendered":"Dynamics: AutoPipe or Caesar Part I"},"content":{"rendered":"<p>I have a plan to write a few posts about dynamics and see what differences in results the AutoPipe and Caesar II provide. The first section will focus on the simplest issue: natural frequency. <\/p>\n\n\n\n<p>The model is identical for both programs: 10\u2032, STD, A106B, air 0.5 MPa, +100C. Length 15m. The natural frequency for a restrained beam is described by the well-known formula given below. For deformation mode I, f = 1.18 Hz<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"244\" src=\"https:\/\/jureknawrocki.com\/wp-content\/uploads\/2-4-1024x244.jpg\" alt=\"\" class=\"wp-image-3134\" srcset=\"https:\/\/jureknawrocki.com\/wp-content\/uploads\/2-4-1024x244.jpg 1024w, https:\/\/jureknawrocki.com\/wp-content\/uploads\/2-4-980x233.jpg 980w, https:\/\/jureknawrocki.com\/wp-content\/uploads\/2-4-480x114.jpg 480w\" sizes=\"(min-width: 0px) and (max-width: 480px) 480px, (min-width: 481px) and (max-width: 980px) 980px, (min-width: 981px) 1024px, 100vw\" \/><\/figure>\n\n\n\n<p>The results in AutoPipe for mode I are 1.16 Hz when dividing the pipe into 10 equal parts<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"903\" height=\"549\" src=\"https:\/\/jureknawrocki.com\/wp-content\/uploads\/3-4.jpg\" alt=\"\" class=\"wp-image-3135\" srcset=\"https:\/\/jureknawrocki.com\/wp-content\/uploads\/3-4.jpg 903w, https:\/\/jureknawrocki.com\/wp-content\/uploads\/3-4-480x292.jpg 480w\" sizes=\"(min-width: 0px) and (max-width: 480px) 480px, (min-width: 481px) 903px, 100vw\" \/><\/figure>\n\n\n\n<p>The results in AutoPipe for mode I are 1.13 Hz without division.<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"635\" src=\"https:\/\/jureknawrocki.com\/wp-content\/uploads\/1-3-1024x635.jpg\" alt=\"\" class=\"wp-image-3133\" srcset=\"https:\/\/jureknawrocki.com\/wp-content\/uploads\/1-3-1024x635.jpg 1024w, https:\/\/jureknawrocki.com\/wp-content\/uploads\/1-3-980x608.jpg 980w, https:\/\/jureknawrocki.com\/wp-content\/uploads\/1-3-480x298.jpg 480w\" sizes=\"(min-width: 0px) and (max-width: 480px) 480px, (min-width: 481px) and (max-width: 980px) 980px, (min-width: 981px) 1024px, 100vw\" \/><\/figure>\n\n\n\n<p>The results in CII for mode I are 1.17 Hz when dividing the tube into 10 equal parts and using a lumped mass model.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"756\" height=\"564\" src=\"https:\/\/jureknawrocki.com\/wp-content\/uploads\/5-4.jpg\" alt=\"\" class=\"wp-image-3137\" style=\"width:823px;height:auto\" srcset=\"https:\/\/jureknawrocki.com\/wp-content\/uploads\/5-4.jpg 756w, https:\/\/jureknawrocki.com\/wp-content\/uploads\/5-4-480x358.jpg 480w\" sizes=\"(min-width: 0px) and (max-width: 480px) 480px, (min-width: 481px) 756px, 100vw\" \/><\/figure>\n\n\n\n<p>The CII results for mode I are 0.82 Hz without splitting and with a lumped mass model.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"735\" height=\"535\" src=\"https:\/\/jureknawrocki.com\/wp-content\/uploads\/4-4.jpg\" alt=\"\" class=\"wp-image-3136\" style=\"width:823px;height:auto\" srcset=\"https:\/\/jureknawrocki.com\/wp-content\/uploads\/4-4.jpg 735w, https:\/\/jureknawrocki.com\/wp-content\/uploads\/4-4-480x349.jpg 480w\" sizes=\"(min-width: 0px) and (max-width: 480px) 480px, (min-width: 481px) 735px, 100vw\" \/><\/figure>\n\n\n\n<p>However, for the consistent mass model, the results are much better. It's true that the analysis takes longer, but for simple models, this doesn't matter.<\/p>\n\n\n\n<figure class=\"wp-block-image aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"414\" height=\"266\" src=\"https:\/\/jureknawrocki.com\/wp-content\/uploads\/6-4.jpg\" alt=\"\" class=\"wp-image-3144\" srcset=\"https:\/\/jureknawrocki.com\/wp-content\/uploads\/6-4.jpg 414w, https:\/\/jureknawrocki.com\/wp-content\/uploads\/6-4-300x193.jpg 300w, https:\/\/jureknawrocki.com\/wp-content\/uploads\/6-4-18x12.jpg 18w\" sizes=\"(max-width: 414px) 100vw, 414px\" \/><\/figure>\n\n\n\n<p>This simple comparison shows that without dividing long sections into relatively short ones, Caesar II, and using a simplified mass model (lumped), produces results that differ significantly from those described by the formula, but also from those of AutoPipe. Furthermore, AutoPipe provides a very nice figure for higher strain modes (see below), which CII does not.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"928\" height=\"760\" src=\"https:\/\/jureknawrocki.com\/wp-content\/uploads\/0.jpg\" alt=\"\" class=\"wp-image-3132\" srcset=\"https:\/\/jureknawrocki.com\/wp-content\/uploads\/0.jpg 928w, https:\/\/jureknawrocki.com\/wp-content\/uploads\/0-480x393.jpg 480w\" sizes=\"(min-width: 0px) and (max-width: 480px) 480px, (min-width: 481px) 928px, 100vw\" \/><\/figure>","protected":false},"excerpt":{"rendered":"<p>Mam zamiar napisa\u0107 kilka post\u00f3w odno\u015bnie dynamiki i zobaczy\u0107 jakie s\u0105 r\u00f3\u017cnice w wynikach daj\u0105 AutoPipe i Caesar II. W pierwszym odcinku rzecz najprostrza, czyli cz\u0119stotliwo\u015b\u0107 drga\u0144 w\u0142asnych. Model jest identyczny dla obu program\u00f3w: 10&#8242;, STD, A106B, powietrze 0,5 MPa, +100C. D\u0142ugo\u015b\u0107 15m. Cz\u0119stotliwo\u015b\u0107 drga\u0144 w\u0142asnych dla belki utwierdzonej opisuje dobrze znany wz\u00f3r podany poni\u017cej. [&hellip;]<\/p>\n","protected":false},"author":3,"featured_media":3131,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_et_pb_use_builder":"off","_et_pb_old_content":"","_et_gb_content_width":"","footnotes":""},"categories":[123,15],"tags":[105,122,16],"class_list":["post-3130","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-obliczenia-wytrzymalosciowe-rur-i-aparatow","category-projekty","tag-autopipe","tag-caesar-ii","tag-naprezenia"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.6 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Dynamika: AutoPipe czy Caesar cz.I - Jerzy Nawrocki<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/jureknawrocki.com\/en\/dynamika-autopipe-czy-caesar-cz-i\/\" \/>\n<meta property=\"og:locale\" content=\"en_GB\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Dynamika: AutoPipe czy Caesar cz.I - Jerzy Nawrocki\" \/>\n<meta property=\"og:description\" content=\"Mam zamiar napisa\u0107 kilka post\u00f3w odno\u015bnie dynamiki i zobaczy\u0107 jakie s\u0105 r\u00f3\u017cnice w wynikach daj\u0105 AutoPipe i Caesar II. W pierwszym odcinku rzecz najprostrza, czyli cz\u0119stotliwo\u015b\u0107 drga\u0144 w\u0142asnych. Model jest identyczny dla obu program\u00f3w: 10&#8242;, STD, A106B, powietrze 0,5 MPa, +100C. D\u0142ugo\u015b\u0107 15m. Cz\u0119stotliwo\u015b\u0107 drga\u0144 w\u0142asnych dla belki utwierdzonej opisuje dobrze znany wz\u00f3r podany poni\u017cej. 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Jerzy Nawrocki","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/jureknawrocki.com\/en\/dynamika-autopipe-czy-caesar-cz-i\/","og_locale":"en_GB","og_type":"article","og_title":"Dynamika: AutoPipe czy Caesar cz.I - Jerzy Nawrocki","og_description":"Mam zamiar napisa\u0107 kilka post\u00f3w odno\u015bnie dynamiki i zobaczy\u0107 jakie s\u0105 r\u00f3\u017cnice w wynikach daj\u0105 AutoPipe i Caesar II. W pierwszym odcinku rzecz najprostrza, czyli cz\u0119stotliwo\u015b\u0107 drga\u0144 w\u0142asnych. Model jest identyczny dla obu program\u00f3w: 10&#8242;, STD, A106B, powietrze 0,5 MPa, +100C. D\u0142ugo\u015b\u0107 15m. Cz\u0119stotliwo\u015b\u0107 drga\u0144 w\u0142asnych dla belki utwierdzonej opisuje dobrze znany wz\u00f3r podany poni\u017cej. 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