{"id":48183,"date":"2021-12-01T11:40:25","date_gmt":"2021-12-01T11:40:25","guid":{"rendered":"https:\/\/www.simscale.com\/?page_id=48183"},"modified":"2021-12-06T09:33:34","modified_gmt":"2021-12-06T09:33:34","slug":"remote-point-mass","status":"publish","type":"page","link":"https:\/\/www.simscale.com\/docs\/validation-cases\/remote-point-mass\/","title":{"rendered":"Validation Case: Cantilever Beam With Off-Center Masses &#8211; NAFEMS Test FV4"},"content":{"rendered":"\n\n\n\n\n<p class=\"wp-block-paragraph\">This validation case belongs to solid mechanics, with the case of a circular cantilever beam with two off-center point masses connected at the free end. The test case aims to validate the following parameters:<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>Point Mass boundary condition<\/li><\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">The simulation results of SimScale were compared to the reference results presented in [NAFEMS]\\(^1\\)<\/p>\n\n\n\n<div class=\"hw-block hw-btnWrapper hw-btnWrapper--alignCenter \">\n    <a href=\"https:\/\/www.simscale.com\/workbench\/?pid=3354379970380920362&#038;mi=spec%3A775fe7fa-ae37-4527-8c99-9c392c25ac03%2Cservice%3ASIMULATION%2Cstrategy%3A13\" class=\"hw-btn    \" rel=\"noopener \" target=\"_blank\"    >\n        View Project    <\/a>\n<\/div>\n\n\n\n\n<h2 class=\"wp-block-heading\" id=\"geometry\" >Geometry<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">The geometry used for the case is as follows:<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-large\"><a href=\"https:\/\/frontend-assets.simscale.com\/media\/2021\/11\/image-45.png\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"552\" src=\"https:\/\/frontend-assets.simscale.com\/media\/2021\/11\/image-45-1024x552.png\" alt=\"cantilever beam geometry\" class=\"wp-image-48199\" srcset=\"https:\/\/frontend-assets.simscale.com\/media\/2021\/11\/image-45-1024x552.png 1024w, https:\/\/frontend-assets.simscale.com\/media\/2021\/11\/image-45-300x162.png 300w, https:\/\/frontend-assets.simscale.com\/media\/2021\/11\/image-45-768x414.png 768w, https:\/\/frontend-assets.simscale.com\/media\/2021\/11\/image-45.png 1121w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/a><figcaption>Figure 1: Geometry model of the cantilever beam<\/figcaption><\/figure><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">It consists of a straight beam of length 10 \\(m\\) and a circular, constant cross-section of radius 0.5 \\(m\\).<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"analysis-type-and-mesh\" >Analysis Type and Mesh<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Tool Type: <\/strong>Code_aster<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Analysis Type: <\/strong><a href=\"https:\/\/www.simscale.com\/docs\/analysis-types\/frequency-analysis\/\">Frequency Analysis<\/a><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Mesh and Element Types:<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The tetrahedral meshes were computed using SimScale&#8217;s <a href=\"https:\/\/www.simscale.com\/docs\/simulation-setup\/meshing\/standard\/\">standard mesh algorithm<\/a> and manual sizing. The following table shows an overview of the mesh characteristics:<\/p>\n\n\n\n<figure class=\"wp-block-table is-style-stripes\"><table><thead><tr><th class=\"has-text-align-center\" data-align=\"center\">Case<\/th><th class=\"has-text-align-center\" data-align=\"center\">Mesh Type<\/th><th class=\"has-text-align-center\" data-align=\"center\">Number of Nodes<\/th><th class=\"has-text-align-center\" data-align=\"center\">Element Type<\/th><\/tr><\/thead><tbody><tr><td class=\"has-text-align-center\" data-align=\"center\">A<\/td><td class=\"has-text-align-center\" data-align=\"center\">Standard<\/td><td class=\"has-text-align-center\" data-align=\"center\">32 663<\/td><td class=\"has-text-align-center\" data-align=\"center\">1st order tetrahedral<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">B<\/td><td class=\"has-text-align-center\" data-align=\"center\">Standard<\/td><td class=\"has-text-align-center\" data-align=\"center\">118 696<\/td><td class=\"has-text-align-center\" data-align=\"center\">1st order tetrahedral<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">C<\/td><td class=\"has-text-align-center\" data-align=\"center\">Standard<\/td><td class=\"has-text-align-center\" data-align=\"center\">240 757<\/td><td class=\"has-text-align-center\" data-align=\"center\">2nd order tetrahedral<\/td><\/tr><\/tbody><\/table><figcaption>Table 1: Number of mesh nodes and types of elements for each case.<\/figcaption><\/figure>\n\n\n\n<figure class=\"wp-block-image size-large\"><a href=\"https:\/\/frontend-assets.simscale.com\/media\/2021\/11\/image-47.png\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"555\" src=\"https:\/\/frontend-assets.simscale.com\/media\/2021\/11\/image-47-1024x555.png\" alt=\"cantilever beam mesh\" class=\"wp-image-48201\" srcset=\"https:\/\/frontend-assets.simscale.com\/media\/2021\/11\/image-47-1024x555.png 1024w, https:\/\/frontend-assets.simscale.com\/media\/2021\/11\/image-47-300x163.png 300w, https:\/\/frontend-assets.simscale.com\/media\/2021\/11\/image-47-768x417.png 768w, https:\/\/frontend-assets.simscale.com\/media\/2021\/11\/image-47.png 1143w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/a><figcaption>Figure 2: Tetrahedral mesh for the beam, in this case corresponding to case C<\/figcaption><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"simulation-setup\" >Simulation Setup<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Material:<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>Linar Elastic Isotropic<ul><li>\\(E\\) = 200 \\(GPa\\)<\/li><li>\\(\\nu\\) = 0.3<\/li><li>\\(\\rho\\) = 8000 \\(kg\/m^3\\)<\/li><\/ul><\/li><\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Boundary Conditions:<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\"><li><em><a href=\"https:\/\/www.simscale.com\/docs\/simulation-setup\/boundary-conditions\/fixed-support-constraint-type\/\">Fixed Support<\/a><\/em>:<ul><li>Face A is fixed <\/li><\/ul><\/li><\/ul>\n\n\n\n<ul class=\"wp-block-list\"><li><a href=\"https:\/\/www.simscale.com\/docs\/simulation-setup\/boundary-conditions\/point-mass\/\"><em>Point Mass<\/em><\/a>:<ul><li>1000 \\(kg\\) connected to face B at a distance of 2 \\(m\\)<\/li><li>10000 \\(kg\\) connected to face B at a disance of 2 \\(m\\)<\/li><\/ul><\/li><\/ul>\n\n\n\n<figure class=\"wp-block-image size-large\"><a href=\"https:\/\/frontend-assets.simscale.com\/media\/2021\/11\/image-46.png\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"546\" src=\"https:\/\/frontend-assets.simscale.com\/media\/2021\/11\/image-46-1024x546.png\" alt=\"simscale validation two off-center point masses boundary conditions\" class=\"wp-image-48200\" srcset=\"https:\/\/frontend-assets.simscale.com\/media\/2021\/11\/image-46-1024x546.png 1024w, https:\/\/frontend-assets.simscale.com\/media\/2021\/11\/image-46-300x160.png 300w, https:\/\/frontend-assets.simscale.com\/media\/2021\/11\/image-46-768x409.png 768w, https:\/\/frontend-assets.simscale.com\/media\/2021\/11\/image-46.png 1208w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/a><figcaption>Figure 3: Boundary conditions and remote point mass locations<\/figcaption><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"reference-solution\" >Reference Solution<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">The reference solution presented in [NAFEMS]\\(^1\\) is of the numerical type. It is presented as the first six natural oscillation frequencies of the cantilever beam with the connected remote point mass. The values are presented in Table 2 alongside the comparison with SimScale results.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"results-comparison\" >Results Comparison<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Comparison of the first six natural frequencies for the run cases with the reference solution:<\/p>\n\n\n\n<figure class=\"wp-block-table is-style-stripes\"><table><thead><tr><th class=\"has-text-align-center\" data-align=\"center\">Mode<\/th><th class=\"has-text-align-center\" data-align=\"center\">NAFEMS<\/th><th class=\"has-text-align-center\" data-align=\"center\">Case A<br>[Deviation]<\/th><th class=\"has-text-align-center\" data-align=\"center\">Case B<br> [Deviation] <\/th><th class=\"has-text-align-center\" data-align=\"center\">Case C<br> [Deviation] <\/th><\/tr><\/thead><tbody><tr><td class=\"has-text-align-center\" data-align=\"center\">1<\/td><td class=\"has-text-align-center\" data-align=\"center\">1.723<\/td><td class=\"has-text-align-center\" data-align=\"center\">1.739<br>[+0.9%]<\/td><td class=\"has-text-align-center\" data-align=\"center\"> 1.731<br>[+0.5%]<\/td><td class=\"has-text-align-center\" data-align=\"center\">1.767<br>[+2.6%]<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">2<\/td><td class=\"has-text-align-center\" data-align=\"center\">1.727<\/td><td class=\"has-text-align-center\" data-align=\"center\">1.750<br>[+1.3%]<\/td><td class=\"has-text-align-center\" data-align=\"center\">1.734<br>[+0.4%]<\/td><td class=\"has-text-align-center\" data-align=\"center\">1.771<br>[+2.5%]<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">3<\/td><td class=\"has-text-align-center\" data-align=\"center\">7.413<\/td><td class=\"has-text-align-center\" data-align=\"center\">7.573<br>[+2.1%]<\/td><td class=\"has-text-align-center\" data-align=\"center\">7.448<br>[+0.5%]<\/td><td class=\"has-text-align-center\" data-align=\"center\">7.593<br>[+2.4%]<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">4<\/td><td class=\"has-text-align-center\" data-align=\"center\">9.972<\/td><td class=\"has-text-align-center\" data-align=\"center\">10.040<br>[+0.7%]<\/td><td class=\"has-text-align-center\" data-align=\"center\">9.982<br>[+0.1%]<\/td><td class=\"has-text-align-center\" data-align=\"center\">10.186<br>[+2.1%]<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">5<\/td><td class=\"has-text-align-center\" data-align=\"center\">18.155<\/td><td class=\"has-text-align-center\" data-align=\"center\">18.344<br>[+1.0%]<\/td><td class=\"has-text-align-center\" data-align=\"center\">18.141<br>[-0.1%]<\/td><td class=\"has-text-align-center\" data-align=\"center\">18.524<br>[+2.0%]<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">6<\/td><td class=\"has-text-align-center\" data-align=\"center\">26.957<\/td><td class=\"has-text-align-center\" data-align=\"center\">26.941<br>[-0.1%]<\/td><td class=\"has-text-align-center\" data-align=\"center\">26.768<br>[-0.7%]<\/td><td class=\"has-text-align-center\" data-align=\"center\">27.302<br>[+1.3%]<\/td><\/tr><\/tbody><\/table><figcaption>Table 2. Natural frequencies ( \\(Hz\\) ) results comparison<\/figcaption><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">Following are the deformed shape plots for each mode, as taken from case C:<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><a href=\"https:\/\/frontend-assets.simscale.com\/media\/2021\/11\/image-48.png\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"346\" src=\"https:\/\/frontend-assets.simscale.com\/media\/2021\/11\/image-48-1024x346.png\" alt=\"simscale validation remote mass mode shape 1\" class=\"wp-image-48203\" srcset=\"https:\/\/frontend-assets.simscale.com\/media\/2021\/11\/image-48-1024x346.png 1024w, https:\/\/frontend-assets.simscale.com\/media\/2021\/11\/image-48-300x101.png 300w, https:\/\/frontend-assets.simscale.com\/media\/2021\/11\/image-48-768x259.png 768w, https:\/\/frontend-assets.simscale.com\/media\/2021\/11\/image-48.png 1238w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/a><figcaption>Figure 4: Mode shape 1, 1.767  \\(Hz\\) , simple bending in the XY plane<\/figcaption><\/figure>\n\n\n\n<figure class=\"wp-block-image size-large\"><a href=\"https:\/\/frontend-assets.simscale.com\/media\/2021\/11\/image-49.png\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"395\" src=\"https:\/\/frontend-assets.simscale.com\/media\/2021\/11\/image-49-1024x395.png\" alt=\"simscale validation off-center point mass mode shape 2\" class=\"wp-image-48204\" srcset=\"https:\/\/frontend-assets.simscale.com\/media\/2021\/11\/image-49-1024x395.png 1024w, https:\/\/frontend-assets.simscale.com\/media\/2021\/11\/image-49-300x116.png 300w, https:\/\/frontend-assets.simscale.com\/media\/2021\/11\/image-49-768x297.png 768w, https:\/\/frontend-assets.simscale.com\/media\/2021\/11\/image-49.png 1191w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/a><figcaption> Figure 5: Mode shape 2, 1.771  \\(Hz\\) , simple bending in the XZ plane<\/figcaption><\/figure>\n\n\n\n<figure class=\"wp-block-image size-large\"><a href=\"https:\/\/frontend-assets.simscale.com\/media\/2021\/11\/image-50.png\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"403\" src=\"https:\/\/frontend-assets.simscale.com\/media\/2021\/11\/image-50-1024x403.png\" alt=\"simscale validation remote mass mode shape 3\" class=\"wp-image-48205\" srcset=\"https:\/\/frontend-assets.simscale.com\/media\/2021\/11\/image-50-1024x403.png 1024w, https:\/\/frontend-assets.simscale.com\/media\/2021\/11\/image-50-300x118.png 300w, https:\/\/frontend-assets.simscale.com\/media\/2021\/11\/image-50-768x302.png 768w, https:\/\/frontend-assets.simscale.com\/media\/2021\/11\/image-50.png 1031w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/a><figcaption> Figure 6: Mode shape 3, 7.539  \\(Hz\\) , bending in the XZ plane, plus torsion<\/figcaption><\/figure>\n\n\n\n<figure class=\"wp-block-image size-large\"><a href=\"https:\/\/frontend-assets.simscale.com\/media\/2021\/11\/image-51.png\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"436\" src=\"https:\/\/frontend-assets.simscale.com\/media\/2021\/11\/image-51-1024x436.png\" alt=\"simscale validation off-center point mass mode shape 4\" class=\"wp-image-48206\" srcset=\"https:\/\/frontend-assets.simscale.com\/media\/2021\/11\/image-51-1024x436.png 1024w, https:\/\/frontend-assets.simscale.com\/media\/2021\/11\/image-51-300x128.png 300w, https:\/\/frontend-assets.simscale.com\/media\/2021\/11\/image-51-768x327.png 768w, https:\/\/frontend-assets.simscale.com\/media\/2021\/11\/image-51.png 1098w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/a><figcaption>  Figure 7: Mode shape 4, 10.186  \\(Hz\\) , double bending in the XY plane<\/figcaption><\/figure>\n\n\n\n<figure class=\"wp-block-image size-large\"><a href=\"https:\/\/frontend-assets.simscale.com\/media\/2021\/11\/image-52.png\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"399\" src=\"https:\/\/frontend-assets.simscale.com\/media\/2021\/11\/image-52-1024x399.png\" alt=\"simscale validation off-center point mass mode shape 5\" class=\"wp-image-48207\" srcset=\"https:\/\/frontend-assets.simscale.com\/media\/2021\/11\/image-52-1024x399.png 1024w, https:\/\/frontend-assets.simscale.com\/media\/2021\/11\/image-52-300x117.png 300w, https:\/\/frontend-assets.simscale.com\/media\/2021\/11\/image-52-768x299.png 768w, https:\/\/frontend-assets.simscale.com\/media\/2021\/11\/image-52.png 1067w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/a><figcaption>  Figure 8: Mode shape 5, 18.524  \\(Hz\\) , double bending in the XZ plane and torsion<\/figcaption><\/figure>\n\n\n\n<figure class=\"wp-block-image size-full\"><a href=\"https:\/\/frontend-assets.simscale.com\/media\/2021\/11\/image-53.png\"><img loading=\"lazy\" decoding=\"async\" width=\"1009\" height=\"446\" src=\"https:\/\/frontend-assets.simscale.com\/media\/2021\/11\/image-53.png\" alt=\"simscale validation remote mass mode shape 6\" class=\"wp-image-48208\" srcset=\"https:\/\/frontend-assets.simscale.com\/media\/2021\/11\/image-53.png 1009w, https:\/\/frontend-assets.simscale.com\/media\/2021\/11\/image-53-300x133.png 300w, https:\/\/frontend-assets.simscale.com\/media\/2021\/11\/image-53-768x339.png 768w\" sizes=\"auto, (max-width: 1009px) 100vw, 1009px\" \/><\/a><figcaption>  Figure 9: Mode shape 6, 27.302 \\(Hz\\), double bending in the XY plane, plus torsion<\/figcaption><\/figure>\n\n\n\n\n<div class='hw-block hw-references hw-references'>\n    <p class='hw-references__title'>References<\/p>\n    <ul class='hw-references__list'>\n\n        <li><cite>NAFEMS publication TNSB, Rev. 3, \u201cThe Standard NAFEMS Benchmarks\u201d, October 1990.<\/cite><\/li>\n    <\/ul>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>This validation case belongs to solid mechanics, with the case of a circular cantilever beam with two off-center point...","protected":false},"author":115,"featured_media":48209,"parent":17191,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"templates\/template-documentation.php","meta":{"_acf_changed":false,"_crdt_document":"","inline_featured_image":false,"footnotes":""},"class_list":["post-48183","page","type-page","status-publish","has-post-thumbnail","hentry"],"acf":[],"_links":{"self":[{"href":"https:\/\/www.simscale.com\/wp-json\/wp\/v2\/pages\/48183","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.simscale.com\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/www.simscale.com\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/www.simscale.com\/wp-json\/wp\/v2\/users\/115"}],"replies":[{"embeddable":true,"href":"https:\/\/www.simscale.com\/wp-json\/wp\/v2\/comments?post=48183"}],"version-history":[{"count":0,"href":"https:\/\/www.simscale.com\/wp-json\/wp\/v2\/pages\/48183\/revisions"}],"up":[{"embeddable":true,"href":"https:\/\/www.simscale.com\/wp-json\/wp\/v2\/pages\/17191"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.simscale.com\/wp-json\/wp\/v2\/media\/48209"}],"wp:attachment":[{"href":"https:\/\/www.simscale.com\/wp-json\/wp\/v2\/media?parent=48183"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}