Problem 58144. Given the mass and stiffness of an undamped SDOF system, find the natural frequency and the natural period of vibration

Created by cdawg

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{"parts":[{"partUri":"/matlab/document.xml","contentType":"application/vnd.mathworks.matlab.code.document+xml","content":"<?xml version=\"1.0\" encoding=\"UTF-8\"?><w:document xmlns:w=\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\"><w:body><w:p><w:pPr><w:pStyle w:val=\"heading\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>Problem Statement</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>Given the mass </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>m</w:t></w:r></w:customXml><w:r><w:t> and stiffness </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>k</w:t></w:r></w:customXml><w:r><w:t> of an undamped SDOF system, find the system's natural frequency in both Hz and radians per second as well as the natural period of vibration. </w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"heading\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>Background</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>An undamped, linear single degree-of-freedom (SDOF) system can be visualized as a spring with stiffness </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>k</w:t></w:r></w:customXml><w:r><w:t>, fixed at one end and having a mass </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>m</w:t></w:r></w:customXml><w:r><w:t> attached to the other:</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:customXml w:element=\"image\"><w:customXmlPr><w:attr w:name=\"height\" w:val=\"93\"/><w:attr w:name=\"width\" w:val=\"90\"/><w:attr w:name=\"verticalAlign\" w:val=\"baseline\"/><w:attr w:name=\"altText\" w:val=\"\"/><w:attr w:name=\"relationshipId\" w:val=\"rId1\"/></w:customXmlPr></w:customXml></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>Once set into motion, the system will oscillate at its </w:t></w:r><w:r><w:rPr><w:b/></w:rPr><w:t>natural frequency</w:t></w:r><w:r><w:t>, or fundamental frequency: </w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"true\"/></w:customXmlPr><w:r><w:t>$\\omega_n =\\sqrt{\\frac{k}{m}}$</w:t></w:r></w:customXml></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>where the natural frequency </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>$\\omega_n$</w:t></w:r></w:customXml><w:r><w:t> is in radians per second. </w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>The time required for the undamped system to complete one cycle of free vibration is the </w:t></w:r><w:r><w:rPr><w:b/></w:rPr><w:t>natural period</w:t></w:r><w:r><w:t> of vibration of the system, </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>T_n</w:t></w:r></w:customXml><w:r><w:t>, which has units of seconds:</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"true\"/></w:customXmlPr><w:r><w:t>$T_n =\\frac{2\\pi }{\\omega_n }$</w:t></w:r></w:customXml></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>A system executes </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>1/T_n</w:t></w:r></w:customXml><w:r><w:t> cycles in 1 second. This is the </w:t></w:r><w:r><w:rPr><w:b/></w:rPr><w:t>natural cyclic frequency</w:t></w:r><w:r><w:t> of vibration:</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"true\"/></w:customXmlPr><w:r><w:t>$f_n =\\frac{1}{T_n }$</w:t></w:r></w:customXml></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>which has units of Hertz (Hz) or cycles per second. The term natural frequency of vibration applies to both </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>$\\omega_n$</w:t></w:r></w:customXml><w:r><w:t> and </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>f_n</w:t></w:r></w:customXml><w:r><w:t>. They have the relationship:</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"true\"/></w:customXmlPr><w:r><w:t>$f_n =\\frac{\\omega_n }{2\\pi }$</w:t></w:r></w:customXml></w:p><w:p><w:pPr><w:pStyle w:val=\"heading\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>Example</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"code\"/></w:pPr><w:r><w:t><![CDATA[m = 10;\nk = 2;\n[wn,fn,Tn] = udSDOF(m,k)\nwn = 5\nfn = 0.7958\nTn = 1.2566]]></w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t></w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t></w:t></w:r></w:p></w:body></w:document>","relationship":[{"relationshipType":"http://schemas.mathworks.com/matlab/code/2013/relationships/image","target":"/media/image1.jpeg","relationshipId":"rId1"}]},{"partUri":"/media/image1.jpeg","contentType":"image/jpeg","content":"data:image/jpeg;base64,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","relationship":null}],"relationships":[{"relationshipType":"http://schemas.mathworks.com/matlab/code/2013/relationships/document","target":"/matlab/document.xml","relationshipId":"rId1"}]}

<div style = "text-align: start; line-height: 20.44px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none solid rgb(0, 0, 0); white-space: normal; "><div style="block-size: 878.625px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 439.312px; transform-origin: 407px 439.312px; vertical-align: baseline; "><div style="block-size: 20px; border-block-end-color: rgb(60, 60, 60); border-block-start-color: rgb(60, 60, 60); border-bottom-color: rgb(60, 60, 60); border-inline-end-color: rgb(60, 60, 60); border-inline-start-color: rgb(60, 60, 60); border-left-color: rgb(60, 60, 60); border-right-color: rgb(60, 60, 60); border-top-color: rgb(60, 60, 60); caret-color: rgb(60, 60, 60); color: rgb(60, 60, 60); column-rule-color: rgb(60, 60, 60); font-family: Helvetica, Arial, sans-serif; font-size: 20px; font-weight: 700; line-height: 20px; margin-block-end: 5px; margin-block-start: 3px; margin-bottom: 5px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 3px; outline-color: rgb(60, 60, 60); perspective-origin: 384px 10px; text-align: left; text-decoration: none; text-decoration-color: rgb(60, 60, 60); text-emphasis-color: rgb(60, 60, 60); transform-origin: 384px 10px; white-space: pre-wrap; margin-left: 4px; margin-top: 3px; margin-bottom: 5px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">Problem Statement</span></span></div><div style="block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">Given the mass </span></span><span style="font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">m</span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style=""> and stiffness </span></span><span style="font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">k</span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style=""> of an undamped SDOF system, find the system's natural frequency in both Hz and radians per second as well as the natural period of vibration. </span></span></div><div style="block-size: 20px; border-block-end-color: rgb(60, 60, 60); border-block-start-color: rgb(60, 60, 60); border-bottom-color: rgb(60, 60, 60); border-inline-end-color: rgb(60, 60, 60); border-inline-start-color: rgb(60, 60, 60); border-left-color: rgb(60, 60, 60); border-right-color: rgb(60, 60, 60); border-top-color: rgb(60, 60, 60); caret-color: rgb(60, 60, 60); color: rgb(60, 60, 60); column-rule-color: rgb(60, 60, 60); font-family: Helvetica, Arial, sans-serif; font-size: 20px; font-weight: 700; line-height: 20px; margin-block-end: 5px; margin-block-start: 20px; margin-bottom: 5px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 20px; outline-color: rgb(60, 60, 60); perspective-origin: 384px 10px; text-align: left; text-decoration: none; text-decoration-color: rgb(60, 60, 60); text-emphasis-color: rgb(60, 60, 60); transform-origin: 384px 10px; white-space: pre-wrap; margin-left: 4px; margin-top: 20px; margin-bottom: 5px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">Background</span></span></div><div style="block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">An undamped, linear single degree-of-freedom (SDOF) system can be visualized as a spring with stiffness </span></span><span style="font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">k</span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">, fixed at one end and having a mass </span></span><span style="font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">m</span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style=""> attached to the other:</span></span></div><div style="block-size: 99px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 49.5px; text-align: left; transform-origin: 384px 49.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><img class="imageNode" width="90" height="93" style="vertical-align: baseline;width: 90px;height: 93px" src="data:image/jpeg;base64,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" data-image-state="image-loaded"></div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">Once set into motion, the system will oscillate at its </span></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="font-weight: 700; ">natural frequency</span></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">, or fundamental frequency: </span></span></div><div style="block-size: 41px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 20.5px; text-align: left; transform-origin: 384px 20.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="vertical-align:-16px"><img src="data:image/png;base64,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" width="62.5" height="41" style="width: 62.5px; height: 41px;"></span></div><div style="block-size: 21.5px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.75px; text-align: left; transform-origin: 384px 10.75px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">where the natural frequency </span></span><span style="vertical-align:-6px"><img src="data:image/png;base64,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" width="18.5" height="20" style="width: 18.5px; height: 20px;"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style=""> is in radians per second. </span></span></div><div style="block-size: 42.5px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21.25px; text-align: left; transform-origin: 384px 21.25px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">The time required for the undamped system to complete one cycle of free vibration is the </span></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="font-weight: 700; ">natural period</span></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style=""> of vibration of the system, </span></span><span style="vertical-align:-6px"><img src="data:image/png;base64,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" width="16.5" height="20" style="width: 16.5px; height: 20px;"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">, which has units of seconds:</span></span></div><div style="block-size: 37px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 18.5px; text-align: left; transform-origin: 384px 18.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="vertical-align:-17px"><img src="data:image/png;base64,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" width="51.5" height="37" style="width: 51.5px; height: 37px;"></span></div><div style="block-size: 21.5px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.75px; text-align: left; transform-origin: 384px 10.75px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">A system executes </span></span><span style="vertical-align:-6px"><img src="data:image/png;base64,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" width="31.5" height="20" style="width: 31.5px; height: 20px;"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style=""> cycles in 1 second. This is the </span></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="font-weight: 700; ">natural cyclic frequency</span></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style=""> of vibration:</span></span></div><div style="block-size: 37px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 18.5px; text-align: left; transform-origin: 384px 18.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="vertical-align:-17px"><img src="data:image/png;base64,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" width="44" height="37" style="width: 44px; height: 37px;"></span></div><div style="block-size: 42.5px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21.25px; text-align: left; transform-origin: 384px 21.25px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">which has units of Hertz (Hz) or cycles per second. The term natural frequency of vibration applies to both </span></span><span style="vertical-align:-6px"><img src="data:image/png;base64,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" width="18.5" height="20" style="width: 18.5px; height: 20px;"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style=""> and </span></span><span style="vertical-align:-6px"><img src="data:image/png;base64,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" width="12" height="20" style="width: 12px; height: 20px;"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">. They have the relationship:</span></span></div><div style="block-size: 38px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 19px; text-align: left; transform-origin: 384px 19px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="vertical-align:-15px"><img src="data:image/png;base64,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" width="47.5" height="38" style="width: 47.5px; height: 38px;"></span></div><div style="block-size: 20px; border-block-end-color: rgb(60, 60, 60); border-block-start-color: rgb(60, 60, 60); border-bottom-color: rgb(60, 60, 60); border-inline-end-color: rgb(60, 60, 60); border-inline-start-color: rgb(60, 60, 60); border-left-color: rgb(60, 60, 60); border-right-color: rgb(60, 60, 60); border-top-color: rgb(60, 60, 60); caret-color: rgb(60, 60, 60); color: rgb(60, 60, 60); column-rule-color: rgb(60, 60, 60); font-family: Helvetica, Arial, sans-serif; font-size: 20px; font-weight: 700; line-height: 20px; margin-block-end: 5px; margin-block-start: 20px; margin-bottom: 5px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 20px; outline-color: rgb(60, 60, 60); perspective-origin: 384px 10px; text-align: left; text-decoration: none; text-decoration-color: rgb(60, 60, 60); text-emphasis-color: rgb(60, 60, 60); transform-origin: 384px 10px; white-space: pre-wrap; margin-left: 4px; margin-top: 20px; margin-bottom: 5px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">Example</span></span></div><div style="background-color: rgb(247, 247, 247); block-size: 122.625px; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-end-end-radius: 4px; border-end-start-radius: 4px; border-start-end-radius: 4px; border-start-start-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; margin-block-end: 10px; margin-block-start: 10px; margin-bottom: 10px; margin-inline-end: 3px; margin-inline-start: 3px; margin-left: 3px; margin-right: 3px; margin-top: 10px; perspective-origin: 404px 61.3125px; transform-origin: 404px 61.3125px; margin-left: 3px; margin-top: 10px; margin-bottom: 10px; margin-right: 3px; "><div style="background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2188px; transform-origin: 404px 10.2188px; white-space: nowrap; "><span style="block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; "><span style="margin-inline-end: 0px; margin-right: 0px; ">m = 10;</span></span></div><div style="background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2188px; transform-origin: 404px 10.2188px; white-space: nowrap; "><span style="block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; "><span style="margin-inline-end: 0px; margin-right: 0px; ">k = 2;</span></span></div><div style="background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2188px; transform-origin: 404px 10.2188px; white-space: nowrap; "><span style="block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; "><span style="margin-inline-end: 0px; margin-right: 0px; ">[wn,fn,Tn] = udSDOF(m,k)</span></span></div><div style="background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2188px; transform-origin: 404px 10.2188px; white-space: nowrap; "><span style="block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; "><span style="margin-inline-end: 0px; margin-right: 0px; ">wn = 5</span></span></div><div style="background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2188px; transform-origin: 404px 10.2188px; white-space: nowrap; "><span style="block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; "><span style="margin-inline-end: 0px; margin-right: 0px; ">fn = 0.7958</span></span></div><div style="background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2188px; transform-origin: 404px 10.2188px; white-space: nowrap; "><span style="block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; "><span style="margin-inline-end: 0px; margin-right: 0px; ">Tn = 1.2566</span></span></div></div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 10px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 10px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 10px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style=""></span></span></div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style=""></span></span></div></div></div>

Problem Statement
Given the mass  and stiffness  of an undamped SDOF system, find the system's natural frequency in both Hz and radians per second as well as the natural period of vibration. 
Background
An undamped, linear single degree-of-freedom (SDOF) system can be visualized as a spring with stiffness , fixed at one end and having a mass  attached to the other:

Once set into motion, the system will oscillate at its natural frequency, or fundamental frequency:

where the natural frequency  is in radians per second. 
The time required for the undamped system to complete one cycle of free vibration is the natural period of vibration of the system, , which has units of seconds:

A system executes  cycles in 1 second. This is the natural cyclic frequency of vibration:

which has units of Hertz (Hz) or cycles per second. The term natural frequency of vibration applies to both  and . They have the relationship:

Example
m = 10;
k = 2;
[wn,fn,Tn] = udSDOF(m,k)
wn = 5
fn = 0.7958
Tn = 1.2566

Solve

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Last Solution submitted on Aug 24, 2023

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