<description/> <author>gioince65@gmail.com</author> <company>Parametric Technology Corporation</company> <keywords/> <revisedBy>UNIBS</revisedBy> </userData> <identityInfo> <revision>77</revision> <documentID>137082F0-3CA8-4452-9B1F-04DA688E9F04</documentID> <versionID>D8BA433F-4B85-414B-91E3-FAD1C1305F20</versionID> <parentVersionID>00000000-0000-0000-0000-000000000000</parentVersionID> <branchID>00000000-0000-0000-0000-000000000000</branchID> </identityInfo> </metadata> <settings> <presentation> <textRendering> <textStyles> <textStyle name="Normal"> <blockAttr margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit"/> <inlineAttr font-family="Times New Roman" font-charset="0" font-size="14" font-weight="normal" font-style="italic" underline="false" line-through="false" vertical-align="baseline" color="#f00"/> </textStyle> <textStyle name="Heading 1"> <blockAttr margin-left="0" margin-right="0" text-indent="inherit" text-align="left" 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text-indent="inherit" text-align="center" list-style-type="inherit" tabs="inherit"/> <inlineAttr font-family="Times New Roman" font-charset="0" font-size="24" font-weight="bold" font-style="normal" underline="false" line-through="false" vertical-align="baseline"/> </textStyle> <textStyle name="Subtitle" base-style="Title"> <blockAttr margin-left="0" margin-right="0" text-indent="inherit" text-align="center" list-style-type="inherit" tabs="inherit"/> <inlineAttr font-family="Times New Roman" font-charset="0" font-size="18" font-weight="normal" font-style="normal" underline="false" line-through="false" vertical-align="baseline"/> </textStyle> </textStyles> </textRendering> <mathRendering equation-color="#000"> <operators multiplication="narrow-dot" derivative="derivative" literal-subscript="large" definition="colon-equal" global-definition="triple-equal" local-definition="left-arrow" equality="bold-equal" symbolic-evaluation="right-arrow"/> <mathStyles> <mathStyle name="Variables" 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local-assignment="MC11"/> </calculationBehavior> <units> <currentUnitSystem name="si" customized="false"/> </units> </calculation> <editor view-annotations="false" view-regions="false"> <ruler is-visible="false" ruler-unit="in"/> <plotTemplate> <xy item-idref="1"/> </plotTemplate> <grid granularity-x="6" granularity-y="6"/> </editor> <fileFormat image-type="image/png" image-quality="75" save-numeric-results="true" exclude-large-results="true" save-text-images="false" screen-dpi="96"/> <miscellaneous> <handbook handbook-region-tag-ub="414" can-delete-original-handbook-regions="true" can-delete-user-regions="true" can-print="true" can-copy="true" can-save="true" file-permission-mask="4294967295"/> </miscellaneous> </settings> <regions> <region region-id="269" left="6" top="15.75" width="166.5" height="18" align-x="20.25" align-y="30" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f size="16"> Trave incastro - appoggio </f> </text> </region> <region region-id="296" left="6" top="48" width="514.5" height="220.5" align-x="6" align-y="48" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <picture> <metafile mapping-mode="8" x-extent="513.326" y-extent="219.3732" item-idref="2"/> </picture> <rendering item-idref="3"/> </region> <region region-id="7" left="6" top="294.75" width="320.25" height="15.75" align-x="39" align-y="306" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math error="La variabile è indefinita." optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:function> <ml:id xml:space="preserve">Y</ml:id> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> 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optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:function> <ml:id xml:space="preserve">Y''</ml:id> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">Y</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:lambda> <ml:degree> <ml:real>2</ml:real> </ml:degree> </ml:apply> </ml:define> </math> <rendering item-idref="7"/> </region> <region region-id="303" left="132" top="373.5" width="404.25" height="21" align-x="222" align-y="390" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:symEval style="default" hide-keywords="false" hide-lhs="false" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:id xml:space="preserve">Y''</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> <ml:command> <ml:sequence> <ml:id xml:space="preserve" font="1">collect</ml:id> <ml:id xml:space="preserve">β</ml:id> </ml:sequence> </ml:command> <ml:symResult> <ml:apply> <ml:mult/> <ml:parens> <ml:apply> <ml:plus/> <ml:apply> <ml:minus/> <ml:apply> <ml:minus/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">C</ml:id> <ml:apply> <ml:id xml:space="preserve">cosh</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">β</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">B</ml:id> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">β</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">A</ml:id> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">β</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">D</ml:id> <ml:apply> <ml:id xml:space="preserve">sinh</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">β</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:parens> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">β</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:symResult> </ml:symEval> </math> <rendering item-idref="8"/> </region> <region region-id="306" left="6" top="432" width="138.75" height="15" align-x="6" align-y="444" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Times New Roman" charset="0" size="14"> Condizioni al contorno: </f> </text> </region> <region region-id="12" left="6" top="462" width="69" height="15" align-x="9.75" align-y="474" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Times New Roman" charset="0" size="14"> in x = 0 </f> </text> </region> <region region-id="18" left="6" top="492.75" width="73.5" height="15.75" align-x="38.25" align-y="504" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:symEval style="default" hide-keywords="false" hide-lhs="false" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:id xml:space="preserve">Y</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:symResult> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">A</ml:id> <ml:id xml:space="preserve">C</ml:id> </ml:apply> </ml:symResult> </ml:symEval> </math> <rendering item-idref="9"/> </region> <region region-id="22" left="210" top="492.75" width="54" height="15.75" 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show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Times New Roman" charset="0" size="14"> in x = l </f> </text> </region> <region region-id="310" left="6" top="618.75" width="305.25" height="15.75" align-x="36" align-y="630" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:symEval style="default" hide-keywords="false" hide-lhs="false" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:id xml:space="preserve">Y</ml:id> <ml:id xml:space="preserve">l</ml:id> </ml:apply> <ml:symResult> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">A</ml:id> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">β</ml:id> <ml:id xml:space="preserve">l</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">B</ml:id> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">β</ml:id> <ml:id xml:space="preserve">l</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">C</ml:id> <ml:apply> <ml:id xml:space="preserve">cosh</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">β</ml:id> <ml:id xml:space="preserve">l</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">D</ml:id> <ml:apply> <ml:id xml:space="preserve">sinh</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">β</ml:id> <ml:id xml:space="preserve">l</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:symResult> </ml:symEval> </math> <rendering item-idref="13"/> </region> <region region-id="327" left="510" top="618.75" width="288" height="15.75" align-x="782.25" align-y="630" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#ff0" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">A</ml:id> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">β</ml:id> <ml:id xml:space="preserve">l</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">B</ml:id> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">β</ml:id> <ml:id xml:space="preserve">l</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">C</ml:id> <ml:apply> <ml:id xml:space="preserve">cosh</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">β</ml:id> <ml:id xml:space="preserve">l</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">D</ml:id> <ml:apply> <ml:id xml:space="preserve">sinh</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">β</ml:id> <ml:id xml:space="preserve">l</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:real>0</ml:real> </ml:apply> </math> <rendering item-idref="14"/> </region> <region region-id="325" left="6" top="643.5" width="391.5" height="21" align-x="95.25" align-y="660" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:symEval style="default" hide-keywords="false" hide-lhs="false" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:id xml:space="preserve">Y''</ml:id> <ml:id xml:space="preserve">l</ml:id> </ml:apply> <ml:command> <ml:sequence> <ml:id xml:space="preserve">collect</ml:id> <ml:id xml:space="preserve">β</ml:id> </ml:sequence> </ml:command> <ml:symResult> <ml:apply> <ml:mult/> <ml:parens> <ml:apply> <ml:plus/> <ml:apply> <ml:minus/> <ml:apply> <ml:minus/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">C</ml:id> <ml:apply> <ml:id xml:space="preserve">cosh</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">β</ml:id> <ml:id xml:space="preserve">l</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">B</ml:id> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">β</ml:id> <ml:id xml:space="preserve">l</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">A</ml:id> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">β</ml:id> <ml:id xml:space="preserve">l</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">D</ml:id> <ml:apply> <ml:id xml:space="preserve">sinh</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">β</ml:id> <ml:id xml:space="preserve">l</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:parens> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">β</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:symResult> </ml:symEval> </math> <rendering item-idref="15"/> </region> <region region-id="326" left="510" top="648.75" width="288" height="15.75" align-x="782.25" align-y="660" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#ff0" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:apply> <ml:plus/> <ml:apply> <ml:minus/> <ml:apply> <ml:minus/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">C</ml:id> <ml:apply> <ml:id xml:space="preserve">cosh</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">β</ml:id> <ml:id xml:space="preserve">l</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">B</ml:id> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">β</ml:id> <ml:id xml:space="preserve">l</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">A</ml:id> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">β</ml:id> <ml:id xml:space="preserve">l</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">D</ml:id> <ml:apply> <ml:id xml:space="preserve">sinh</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">β</ml:id> <ml:id xml:space="preserve">l</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:real>0</ml:real> </ml:apply> </math> <rendering item-idref="16"/> </region> <region region-id="316" left="6" top="690" width="398.25" height="15" align-x="29.25" align-y="702" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Riassunto delle condizioni al contorno (che formano un sistema lineare) </text> </region> <region region-id="42" left="6" top="726.75" width="54" height="15.75" align-x="44.25" align-y="738" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#ff0" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">A</ml:id> <ml:id xml:space="preserve">C</ml:id> </ml:apply> <ml:real>0</ml:real> </ml:apply> </math> <rendering item-idref="17"/> </region> <region region-id="41" left="6" top="750.75" width="53.25" height="15.75" align-x="43.5" align-y="762" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#ff0" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">B</ml:id> <ml:id xml:space="preserve">D</ml:id> </ml:apply> <ml:real>0</ml:real> </ml:apply> </math> <rendering item-idref="18"/> </region> <region region-id="40" left="6" top="774.75" width="288" height="15.75" align-x="278.25" align-y="786" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#ff0" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">A</ml:id> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">β</ml:id> <ml:id xml:space="preserve">l</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">B</ml:id> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">β</ml:id> <ml:id xml:space="preserve">l</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">C</ml:id> <ml:apply> <ml:id xml:space="preserve">cosh</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">β</ml:id> <ml:id xml:space="preserve">l</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">D</ml:id> <ml:apply> <ml:id xml:space="preserve">sinh</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">β</ml:id> <ml:id xml:space="preserve">l</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:real>0</ml:real> </ml:apply> </math> <rendering item-idref="19"/> </region> <region region-id="323" left="6" top="804.75" width="288" height="15.75" align-x="278.25" align-y="816" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#ff0" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:apply> <ml:plus/> <ml:apply> <ml:minus/> <ml:apply> <ml:minus/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">C</ml:id> <ml:apply> <ml:id xml:space="preserve">cosh</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">β</ml:id> <ml:id xml:space="preserve">l</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">B</ml:id> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">β</ml:id> <ml:id xml:space="preserve">l</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">A</ml:id> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">β</ml:id> <ml:id xml:space="preserve">l</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">D</ml:id> <ml:apply> <ml:id xml:space="preserve">sinh</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">β</ml:id> <ml:id xml:space="preserve">l</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:real>0</ml:real> </ml:apply> </math> <rendering item-idref="20"/> </region> <region region-id="320" left="6" top="882" width="250.5" height="15" align-x="6" align-y="894" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Poniamo per semplicità: α = β l e otteniamo: </text> </region> <region region-id="49" left="6" top="912.75" width="54" height="15.75" align-x="44.25" align-y="924" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#ff0" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">A</ml:id> <ml:id xml:space="preserve">C</ml:id> </ml:apply> <ml:real>0</ml:real> </ml:apply> </math> <rendering item-idref="21"/> </region> <region region-id="50" left="6" top="936.75" width="53.25" height="15.75" align-x="43.5" align-y="948" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#ff0" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">B</ml:id> <ml:id xml:space="preserve">D</ml:id> </ml:apply> <ml:real>0</ml:real> </ml:apply> </math> <rendering item-idref="22"/> </region> <region region-id="51" left="6" top="960.75" width="252" height="15.75" align-x="242.25" align-y="972" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#ff0" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">A</ml:id> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">B</ml:id> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">C</ml:id> <ml:apply> <ml:id xml:space="preserve">cosh</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">D</ml:id> <ml:apply> <ml:id xml:space="preserve">sinh</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:real>0</ml:real> </ml:apply> </math> <rendering item-idref="23"/> </region> <region region-id="52" left="6" top="984.75" width="252" height="15.75" align-x="242.25" align-y="996" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#ff0" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:apply> <ml:plus/> <ml:apply> <ml:minus/> <ml:apply> <ml:minus/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">C</ml:id> <ml:apply> <ml:id xml:space="preserve">cosh</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">B</ml:id> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">A</ml:id> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">D</ml:id> <ml:apply> <ml:id xml:space="preserve">sinh</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:real>0</ml:real> </ml:apply> </math> <rendering item-idref="24"/> </region> <region region-id="328" left="6" top="1020" width="409.5" height="30" align-x="6" align-y="1032" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Le quattro condizioni al contorno si possono scrivere in forma matriciale, come segue </text> </region> <region region-id="329" left="6" top="1049.25" width="98.25" height="78.75" align-x="72" align-y="1092" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#ff0" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:apply> <ml:mult/> <ml:apply> <ml:id xml:space="preserve" font="17">Δ</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> <ml:matrix rows="4" cols="1"> <ml:id xml:space="preserve">A</ml:id> <ml:id xml:space="preserve">B</ml:id> <ml:id xml:space="preserve">C</ml:id> <ml:id xml:space="preserve">D</ml:id> </ml:matrix> </ml:apply> <ml:matrix rows="4" cols="1"> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> </ml:matrix> </ml:apply> </math> <rendering item-idref="25"/> </region> <region region-id="56" left="6" top="1139.25" width="243" height="78.75" align-x="44.25" align-y="1182" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:function> <ml:id xml:space="preserve" font="17">Δ</ml:id> <ml:boundVars> <ml:id xml:space="preserve">α</ml:id> </ml:boundVars> </ml:function> <ml:matrix rows="4" cols="4"> <ml:real>1</ml:real> <ml:real>0</ml:real> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> <ml:apply> <ml:neg/> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> <ml:real>0</ml:real> <ml:real>1</ml:real> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> <ml:apply> <ml:neg/> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> <ml:real>1</ml:real> <ml:real>0</ml:real> <ml:apply> <ml:id xml:space="preserve">cosh</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">cosh</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> <ml:real>0</ml:real> <ml:real>1</ml:real> <ml:apply> <ml:id xml:space="preserve">sinh</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">sinh</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:matrix> </ml:define> </math> <rendering item-idref="26"/> </region> <region region-id="330" left="6" top="1266" width="267" height="15" align-x="6" align-y="1278" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Infatti, il prodotto matriciale seguente, fornisce: </text> </region> <region region-id="331" left="6" top="1289.25" width="488.25" height="78.75" align-x="238.5" align-y="1332" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:symEval style="default" hide-keywords="false" hide-lhs="false" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:mult/> <ml:matrix rows="4" cols="4"> <ml:real>1</ml:real> <ml:real>0</ml:real> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> <ml:apply> <ml:neg/> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> <ml:real>0</ml:real> <ml:real>1</ml:real> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> <ml:apply> <ml:neg/> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> <ml:real>1</ml:real> <ml:real>0</ml:real> <ml:apply> <ml:id xml:space="preserve">cosh</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">cosh</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> <ml:real>0</ml:real> <ml:real>1</ml:real> <ml:apply> <ml:id xml:space="preserve">sinh</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">sinh</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:matrix> <ml:matrix rows="4" cols="1"> <ml:id xml:space="preserve">A</ml:id> <ml:id xml:space="preserve">B</ml:id> <ml:id xml:space="preserve">C</ml:id> <ml:id xml:space="preserve">D</ml:id> </ml:matrix> </ml:apply> <ml:symResult> <ml:matrix rows="4" cols="1"> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">A</ml:id> <ml:id xml:space="preserve">C</ml:id> </ml:apply> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">B</ml:id> <ml:id xml:space="preserve">D</ml:id> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">A</ml:id> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">B</ml:id> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">C</ml:id> <ml:apply> <ml:id xml:space="preserve">cosh</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">D</ml:id> <ml:apply> <ml:id xml:space="preserve">sinh</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:minus/> <ml:apply> <ml:minus/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">C</ml:id> <ml:apply> <ml:id xml:space="preserve">cosh</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">B</ml:id> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">A</ml:id> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">D</ml:id> <ml:apply> <ml:id xml:space="preserve">sinh</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:matrix> </ml:symResult> </ml:symEval> </math> <rendering item-idref="27"/> </region> <region region-id="333" left="6" top="1398" width="388.5" height="30" align-x="6" align-y="1410" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Si calcola ora il determinante della matrice <inlineAttr font-style="normal">Δ(α)</inlineAttr> e lo si pone uguale a zero, per ricavare l'equazione caratteristica: </text> </region> <region region-id="335" left="6" top="1428" width="92.25" height="16.5" align-x="48" align-y="1440" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:function> <ml:id xml:space="preserve">det</ml:id> <ml:boundVars> <ml:id xml:space="preserve">α</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:determinant/> <ml:apply> <ml:id xml:space="preserve" font="17">Δ</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="28"/> </region> <region region-id="228" left="6" top="1452.75" width="272.25" height="15.75" align-x="88.5" align-y="1464" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#ff0" tag=""> <math optimize="false" disable-calc="false"> <ml:symEval style="default" hide-keywords="false" hide-lhs="false" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:id xml:space="preserve">det</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> <ml:command> <ml:id xml:space="preserve">factor</ml:id> </ml:command> <ml:symResult> <ml:apply> <ml:mult/> <ml:real>2</ml:real> <ml:parens> <ml:apply> <ml:minus/> <ml:apply> <ml:mult/> <ml:apply> <ml:id xml:space="preserve">cosh</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:id xml:space="preserve">sinh</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:parens> </ml:apply> </ml:symResult> </ml:symEval> </math> <rendering item-idref="29"/> </region> <region region-id="336" left="6" top="1488" width="201.75" height="15" align-x="6" align-y="1500" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Rielaborando l'equazione, si ricava: </text> </region> <region region-id="341" left="6" top="1536.75" width="89.25" height="15.75" align-x="46.5" align-y="1548" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#ff0" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:apply> <ml:id xml:space="preserve">tan</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">tanh</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> </math> <rendering item-idref="30"/> </region> <region region-id="340" left="168" top="1536" width="333" height="15" align-x="189" align-y="1548" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Equazione<sp count="2"/>caratteristica (o equazione delle frequenze) </text> </region> <region region-id="343" left="6" top="1596" width="378.75" height="15" align-x="32.25" align-y="1608" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Cerchiamo le soluzioni dell'equazione caratteristica per via grafica: </text> </region> <region region-id="69" left="6" top="1644.75" width="81.75" height="20.25" align-x="44.25" align-y="1656" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:function> <ml:id xml:space="preserve" subscript="1">f</ml:id> <ml:boundVars> <ml:id xml:space="preserve">α</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:id xml:space="preserve">tan</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="31"/> </region> <region region-id="72" left="6" top="1680.75" width="87.75" height="20.25" align-x="44.25" align-y="1692" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:function> <ml:id xml:space="preserve" subscript="2">f</ml:id> <ml:boundVars> <ml:id xml:space="preserve">α</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:id xml:space="preserve">tanh</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="32"/> </region> <region region-id="73" left="6" top="1716.75" width="81.75" height="15.75" align-x="22.5" align-y="1728" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">α</ml:id> <ml:range> <ml:sequence> <ml:real>0</ml:real> <ml:real>0.01</ml:real> </ml:sequence> <ml:real>20</ml:real> </ml:range> </ml:define> </math> <rendering item-idref="33"/> </region> <region region-id="77" left="132" top="1716.75" width="40.5" height="15.75" align-x="156" align-y="1728" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">FS</ml:id> <ml:real>4</ml:real> </ml:define> </math> <rendering item-idref="34"/> </region> <region region-id="263" left="6" top="1752" width="603" height="382.5" align-x="6" align-y="1752" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="35"/> <rendering item-idref="36"/> </region> <region region-id="194" left="6" top="2160.75" width="33" height="15.75" align-x="22.5" align-y="2172" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">α</ml:id> <ml:real>4</ml:real> </ml:define> </math> <rendering item-idref="37"/> </region> <region region-id="195" left="6" top="2190.75" width="219" height="20.25" align-x="28.5" align-y="2202" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="1">α</ml:id> <ml:eval placeholderMultiplicationStyle="default"> <ml:apply> <ml:id xml:space="preserve">root</ml:id> <ml:sequence> <ml:apply> <ml:minus/> <ml:apply> <ml:id xml:space="preserve" subscript="1">f</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve" subscript="2">f</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> <ml:id xml:space="preserve">α</ml:id> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>3.9266023132537806</ml:real> </result> </ml:eval> </ml:define> <resultFormat numeric-only="true"> <general precision="6" show-trailing-zeros="false" radix="dec" complex-threshold="10" zero-threshold="15" imaginary-value="i" exponential-threshold="3"/> <matrix display-style="auto" expand-nested-arrays="false"/> <unit format-units="true" simplify-units="true" fractional-unit-exponent="false"/> </resultFormat> </math> <rendering item-idref="38"/> </region> <region region-id="196" left="6" top="2256.75" width="33" height="15.75" align-x="22.5" align-y="2268" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedUDScalar" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">α</ml:id> <ml:real>7</ml:real> </ml:define> </math> <rendering item-idref="39"/> </region> <region region-id="197" left="6" top="2286.75" width="219" height="20.25" align-x="28.5" align-y="2298" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="2">α</ml:id> <ml:eval placeholderMultiplicationStyle="default"> <ml:apply> <ml:id xml:space="preserve">root</ml:id> <ml:sequence> <ml:apply> <ml:minus/> <ml:apply> <ml:id xml:space="preserve" subscript="1">f</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve" subscript="2">f</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> <ml:id xml:space="preserve">α</ml:id> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>7.0685827458686381</ml:real> </result> </ml:eval> </ml:define> <resultFormat numeric-only="true"> <general precision="6" show-trailing-zeros="false" radix="dec" complex-threshold="10" zero-threshold="15" imaginary-value="i" exponential-threshold="3"/> <matrix display-style="auto" expand-nested-arrays="false"/> <unit format-units="true" simplify-units="true" fractional-unit-exponent="false"/> </resultFormat> </math> <rendering item-idref="40"/> </region> <region region-id="198" left="6" top="2334.75" width="39" height="15.75" align-x="22.5" align-y="2346" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedUDScalar" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">α</ml:id> <ml:real>10</ml:real> </ml:define> </math> <rendering item-idref="41"/> </region> <region region-id="199" left="6" top="2364.75" width="225" height="20.25" align-x="28.5" align-y="2376" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="3">α</ml:id> <ml:eval placeholderMultiplicationStyle="default"> <ml:apply> <ml:id xml:space="preserve">root</ml:id> <ml:sequence> <ml:apply> <ml:minus/> <ml:apply> <ml:id xml:space="preserve" subscript="1">f</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve" subscript="2">f</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> <ml:id xml:space="preserve">α</ml:id> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>10.210176122406375</ml:real> </result> </ml:eval> </ml:define> <resultFormat numeric-only="true"> <general precision="6" show-trailing-zeros="false" radix="dec" complex-threshold="10" zero-threshold="15" imaginary-value="i" exponential-threshold="3"/> <matrix display-style="auto" expand-nested-arrays="false"/> <unit format-units="true" simplify-units="true" fractional-unit-exponent="false"/> </resultFormat> </math> <rendering item-idref="42"/> </region> <region region-id="200" left="6" top="2418.75" width="39" height="15.75" align-x="22.5" align-y="2430" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedUDScalar" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">α</ml:id> <ml:real>13</ml:real> </ml:define> </math> <rendering item-idref="43"/> </region> <region region-id="201" left="6" top="2448.75" width="225" height="20.25" align-x="28.5" align-y="2460" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="4">α</ml:id> <ml:eval placeholderMultiplicationStyle="default"> <ml:apply> <ml:id xml:space="preserve">root</ml:id> <ml:sequence> <ml:apply> <ml:minus/> <ml:apply> <ml:id xml:space="preserve" subscript="1">f</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve" subscript="2">f</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> <ml:id xml:space="preserve">α</ml:id> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>13.351768777743059</ml:real> </result> </ml:eval> </ml:define> <resultFormat numeric-only="true"> <general precision="6" show-trailing-zeros="false" radix="dec" complex-threshold="10" zero-threshold="15" imaginary-value="i" exponential-threshold="3"/> <matrix display-style="auto" expand-nested-arrays="false"/> <unit format-units="true" simplify-units="true" fractional-unit-exponent="false"/> </resultFormat> </math> <rendering item-idref="44"/> </region> <region region-id="290" left="6" top="2532" width="57.75" height="15" align-x="6" align-y="2544" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Lunghezza </text> </region> <region region-id="291" left="6" top="2550.75" width="117.75" height="15.75" align-x="18.75" align-y="2562" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIUnit" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">l</ml:id> <ml:eval placeholderMultiplicationStyle="default"> <ml:apply> <ml:mult/> <ml:real>1200</ml:real> <ml:id xml:space="preserve">mm</ml:id> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>1.2</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="meter"/> </unitMonomial> </unitedValue> </result> </ml:eval> </ml:define> </math> <rendering item-idref="45"/> </region> <region region-id="278" left="6" top="2598" width="213" height="15" align-x="21" align-y="2610" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Materiale: acciaio </text> </region> <region region-id="279" left="6" top="2620.5" width="72.75" height="39" align-x="23.25" align-y="2640" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">ρ</ml:id> <ml:apply> <ml:mult/> <ml:real>7800</ml:real> <ml:apply> <ml:div/> <ml:id xml:space="preserve">kg</ml:id> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">m</ml:id> <ml:real>3</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:define> <resultFormat numeric-only="true"> <general precision="6" show-trailing-zeros="false" radix="dec" complex-threshold="10" zero-threshold="15" imaginary-value="i" exponential-threshold="6"/> <matrix display-style="auto" expand-nested-arrays="false"/> <unit format-units="true" simplify-units="true" fractional-unit-exponent="false"/> </resultFormat> </math> <rendering item-idref="46"/> </region> <region region-id="280" left="6" top="2665.5" width="186" height="21" align-x="24" align-y="2682" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">E</ml:id> <ml:eval placeholderMultiplicationStyle="default"> <ml:apply> <ml:mult/> <ml:real>206000</ml:real> <ml:id xml:space="preserve">MPa</ml:id> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>206000000000</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="pascal"/> </unitMonomial> </unitedValue> </result> </ml:eval> </ml:define> </math> <rendering item-idref="47"/> </region> <region region-id="287" 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</unitMonomial> </unitedValue> </result> </ml:eval> </ml:define> </math> <rendering item-idref="48"/> </region> <region region-id="283" left="6" top="2771.25" width="158.25" height="39" align-x="22.5" align-y="2796" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIUnit" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">J</ml:id> <ml:eval placeholderMultiplicationStyle="default"> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">π</ml:id> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">d</ml:id> <ml:real>4</ml:real> </ml:apply> </ml:apply> <ml:real>64</ml:real> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>2.4850488763747385E-09</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="meter" power-numerator="4"/> </unitMonomial> 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<ml:sequence> <ml:id xml:space="preserve">collect</ml:id> <ml:id xml:space="preserve">β</ml:id> </ml:sequence> </ml:command> <ml:symResult> <ml:apply> <ml:mult/> <ml:parens> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">B</ml:id> <ml:id xml:space="preserve">D</ml:id> </ml:apply> </ml:parens> <ml:id xml:space="preserve">β</ml:id> </ml:apply> </ml:symResult> </ml:symEval> </math> <rendering item-idref="11"/> </region> <region region-id="23" left="210" top="528.75" width="53.25" height="15.75" align-x="247.5" align-y="540" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#ff0" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">B</ml:id> <ml:id xml:space="preserve">D</ml:id> </ml:apply> <ml:real>0</ml:real> </ml:apply> </math> <rendering item-idref="12"/> </region> <region region-id="26" left="6" top="582" width="71.25" height="15" align-x="9.75" align-y="594" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f family="Times New Roman" charset="0" size="14"> in x = l </f> </text> </region> <region region-id="310" left="6" top="618.75" width="305.25" height="15.75" align-x="36" align-y="630" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:symEval style="default" hide-keywords="false" hide-lhs="false" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:id xml:space="preserve">Y</ml:id> <ml:id xml:space="preserve">l</ml:id> </ml:apply> <ml:symResult> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">A</ml:id> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">β</ml:id> <ml:id xml:space="preserve">l</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">B</ml:id> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">β</ml:id> <ml:id xml:space="preserve">l</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">C</ml:id> <ml:apply> <ml:id xml:space="preserve">cosh</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">β</ml:id> <ml:id xml:space="preserve">l</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">D</ml:id> <ml:apply> <ml:id xml:space="preserve">sinh</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">β</ml:id> <ml:id xml:space="preserve">l</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:symResult> </ml:symEval> </math> <rendering item-idref="13"/> </region> <region region-id="327" left="510" top="618.75" width="288" height="15.75" align-x="782.25" align-y="630" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#ff0" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">A</ml:id> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">β</ml:id> <ml:id xml:space="preserve">l</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">B</ml:id> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">β</ml:id> <ml:id xml:space="preserve">l</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">C</ml:id> <ml:apply> <ml:id xml:space="preserve">cosh</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">β</ml:id> <ml:id xml:space="preserve">l</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">D</ml:id> <ml:apply> <ml:id xml:space="preserve">sinh</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">β</ml:id> <ml:id xml:space="preserve">l</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:real>0</ml:real> </ml:apply> </math> <rendering item-idref="14"/> </region> <region region-id="325" left="6" top="643.5" width="391.5" height="21" align-x="95.25" align-y="660" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:symEval style="default" hide-keywords="false" hide-lhs="false" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:id xml:space="preserve">Y''</ml:id> <ml:id xml:space="preserve">l</ml:id> </ml:apply> <ml:command> <ml:sequence> <ml:id xml:space="preserve">collect</ml:id> <ml:id xml:space="preserve">β</ml:id> </ml:sequence> </ml:command> <ml:symResult> <ml:apply> <ml:mult/> <ml:parens> <ml:apply> <ml:plus/> <ml:apply> <ml:minus/> <ml:apply> <ml:minus/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">C</ml:id> <ml:apply> <ml:id xml:space="preserve">cosh</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">β</ml:id> <ml:id xml:space="preserve">l</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">B</ml:id> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">β</ml:id> <ml:id xml:space="preserve">l</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">A</ml:id> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">β</ml:id> <ml:id xml:space="preserve">l</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">D</ml:id> <ml:apply> <ml:id xml:space="preserve">sinh</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">β</ml:id> <ml:id xml:space="preserve">l</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:parens> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">β</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:symResult> </ml:symEval> </math> <rendering item-idref="15"/> </region> <region region-id="326" left="510" top="648.75" width="288" height="15.75" align-x="782.25" align-y="660" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#ff0" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:apply> <ml:plus/> <ml:apply> <ml:minus/> <ml:apply> <ml:minus/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">C</ml:id> <ml:apply> <ml:id xml:space="preserve">cosh</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">β</ml:id> <ml:id xml:space="preserve">l</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">B</ml:id> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">β</ml:id> <ml:id xml:space="preserve">l</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">A</ml:id> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">β</ml:id> <ml:id xml:space="preserve">l</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">D</ml:id> <ml:apply> <ml:id xml:space="preserve">sinh</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">β</ml:id> <ml:id xml:space="preserve">l</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:real>0</ml:real> </ml:apply> </math> <rendering item-idref="16"/> </region> <region region-id="316" left="6" top="690" width="398.25" height="15" align-x="29.25" align-y="702" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Riassunto delle condizioni al contorno (che formano un sistema lineare) </text> </region> <region region-id="42" left="6" top="726.75" width="54" height="15.75" align-x="44.25" align-y="738" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#ff0" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">A</ml:id> <ml:id xml:space="preserve">C</ml:id> </ml:apply> <ml:real>0</ml:real> </ml:apply> </math> <rendering item-idref="17"/> </region> <region region-id="41" left="6" top="750.75" width="53.25" height="15.75" align-x="43.5" align-y="762" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#ff0" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">B</ml:id> <ml:id xml:space="preserve">D</ml:id> </ml:apply> <ml:real>0</ml:real> </ml:apply> </math> <rendering item-idref="18"/> </region> <region region-id="40" left="6" top="774.75" width="288" height="15.75" align-x="278.25" align-y="786" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#ff0" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">A</ml:id> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">β</ml:id> <ml:id xml:space="preserve">l</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">B</ml:id> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">β</ml:id> <ml:id xml:space="preserve">l</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">C</ml:id> <ml:apply> <ml:id xml:space="preserve">cosh</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">β</ml:id> <ml:id xml:space="preserve">l</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">D</ml:id> <ml:apply> <ml:id xml:space="preserve">sinh</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">β</ml:id> <ml:id xml:space="preserve">l</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:real>0</ml:real> </ml:apply> </math> <rendering item-idref="19"/> </region> <region region-id="323" left="6" top="804.75" width="288" height="15.75" align-x="278.25" align-y="816" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#ff0" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:apply> <ml:plus/> <ml:apply> <ml:minus/> <ml:apply> <ml:minus/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">C</ml:id> <ml:apply> <ml:id xml:space="preserve">cosh</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">β</ml:id> <ml:id xml:space="preserve">l</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">B</ml:id> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">β</ml:id> <ml:id xml:space="preserve">l</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">A</ml:id> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">β</ml:id> <ml:id xml:space="preserve">l</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">D</ml:id> <ml:apply> <ml:id xml:space="preserve">sinh</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">β</ml:id> <ml:id xml:space="preserve">l</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> <ml:real>0</ml:real> </ml:apply> </math> <rendering item-idref="20"/> </region> <region region-id="320" left="6" top="882" width="250.5" height="15" align-x="6" align-y="894" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Poniamo per semplicità: α = β l e otteniamo: </text> </region> <region region-id="49" left="6" top="912.75" width="54" height="15.75" align-x="44.25" align-y="924" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#ff0" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">A</ml:id> <ml:id xml:space="preserve">C</ml:id> </ml:apply> <ml:real>0</ml:real> </ml:apply> </math> <rendering item-idref="21"/> </region> <region region-id="50" left="6" top="936.75" width="53.25" height="15.75" align-x="43.5" align-y="948" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#ff0" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">B</ml:id> <ml:id xml:space="preserve">D</ml:id> </ml:apply> <ml:real>0</ml:real> </ml:apply> </math> <rendering item-idref="22"/> </region> <region region-id="51" left="6" top="960.75" width="252" height="15.75" align-x="242.25" align-y="972" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#ff0" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">A</ml:id> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">B</ml:id> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">C</ml:id> <ml:apply> <ml:id xml:space="preserve">cosh</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">D</ml:id> <ml:apply> <ml:id xml:space="preserve">sinh</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:real>0</ml:real> </ml:apply> </math> <rendering item-idref="23"/> </region> <region region-id="52" left="6" top="984.75" width="252" height="15.75" align-x="242.25" align-y="996" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#ff0" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:apply> <ml:plus/> <ml:apply> <ml:minus/> <ml:apply> <ml:minus/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">C</ml:id> <ml:apply> <ml:id xml:space="preserve">cosh</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">B</ml:id> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">A</ml:id> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">D</ml:id> <ml:apply> <ml:id xml:space="preserve">sinh</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:real>0</ml:real> </ml:apply> </math> <rendering item-idref="24"/> </region> <region region-id="328" left="6" top="1020" width="409.5" height="30" align-x="6" align-y="1032" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Le quattro condizioni al contorno si possono scrivere in forma matriciale, come segue </text> </region> <region region-id="329" left="6" top="1049.25" width="98.25" height="78.75" align-x="72" align-y="1092" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#ff0" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:apply> <ml:mult/> <ml:apply> <ml:id xml:space="preserve" font="17">Δ</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> <ml:matrix rows="4" cols="1"> <ml:id xml:space="preserve">A</ml:id> <ml:id xml:space="preserve">B</ml:id> <ml:id xml:space="preserve">C</ml:id> <ml:id xml:space="preserve">D</ml:id> </ml:matrix> </ml:apply> <ml:matrix rows="4" cols="1"> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> <ml:real>0</ml:real> </ml:matrix> </ml:apply> </math> <rendering item-idref="25"/> </region> <region region-id="56" left="6" top="1139.25" width="243" height="78.75" align-x="44.25" align-y="1182" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:function> <ml:id xml:space="preserve" font="17">Δ</ml:id> <ml:boundVars> <ml:id xml:space="preserve">α</ml:id> </ml:boundVars> </ml:function> <ml:matrix rows="4" cols="4"> <ml:real>1</ml:real> <ml:real>0</ml:real> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> <ml:apply> <ml:neg/> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> <ml:real>0</ml:real> <ml:real>1</ml:real> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> <ml:apply> <ml:neg/> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> <ml:real>1</ml:real> <ml:real>0</ml:real> <ml:apply> <ml:id xml:space="preserve">cosh</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">cosh</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> <ml:real>0</ml:real> <ml:real>1</ml:real> <ml:apply> <ml:id xml:space="preserve">sinh</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">sinh</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:matrix> </ml:define> </math> <rendering item-idref="26"/> </region> <region region-id="330" left="6" top="1266" width="267" height="15" align-x="6" align-y="1278" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Infatti, il prodotto matriciale seguente, fornisce: </text> </region> <region region-id="331" left="6" top="1289.25" width="488.25" height="78.75" align-x="238.5" align-y="1332" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:symEval style="default" hide-keywords="false" hide-lhs="false" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:mult/> <ml:matrix rows="4" cols="4"> <ml:real>1</ml:real> <ml:real>0</ml:real> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> <ml:apply> <ml:neg/> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> <ml:real>0</ml:real> <ml:real>1</ml:real> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> <ml:apply> <ml:neg/> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> <ml:real>1</ml:real> <ml:real>0</ml:real> <ml:apply> <ml:id xml:space="preserve">cosh</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">cosh</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> <ml:real>0</ml:real> <ml:real>1</ml:real> <ml:apply> <ml:id xml:space="preserve">sinh</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">sinh</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:matrix> <ml:matrix rows="4" cols="1"> <ml:id xml:space="preserve">A</ml:id> <ml:id xml:space="preserve">B</ml:id> <ml:id xml:space="preserve">C</ml:id> <ml:id xml:space="preserve">D</ml:id> </ml:matrix> </ml:apply> <ml:symResult> <ml:matrix rows="4" cols="1"> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">A</ml:id> <ml:id xml:space="preserve">C</ml:id> </ml:apply> <ml:apply> <ml:plus/> <ml:id xml:space="preserve">B</ml:id> <ml:id xml:space="preserve">D</ml:id> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">A</ml:id> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">B</ml:id> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">C</ml:id> <ml:apply> <ml:id xml:space="preserve">cosh</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">D</ml:id> <ml:apply> <ml:id xml:space="preserve">sinh</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:minus/> <ml:apply> <ml:minus/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">C</ml:id> <ml:apply> <ml:id xml:space="preserve">cosh</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">B</ml:id> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">A</ml:id> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">D</ml:id> <ml:apply> <ml:id xml:space="preserve">sinh</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:matrix> </ml:symResult> </ml:symEval> </math> <rendering item-idref="27"/> </region> <region region-id="333" left="6" top="1398" width="388.5" height="30" align-x="6" align-y="1410" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Si calcola ora il determinante della matrice <inlineAttr font-style="normal">Δ(α)</inlineAttr> e lo si pone uguale a zero, per ricavare l'equazione caratteristica: </text> </region> <region region-id="335" left="6" top="1428" width="92.25" height="16.5" align-x="48" align-y="1440" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:function> <ml:id xml:space="preserve">det</ml:id> <ml:boundVars> <ml:id xml:space="preserve">α</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:determinant/> <ml:apply> <ml:id xml:space="preserve" font="17">Δ</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="28"/> </region> <region region-id="228" left="6" top="1452.75" width="272.25" height="15.75" align-x="88.5" align-y="1464" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#ff0" tag=""> <math optimize="false" disable-calc="false"> <ml:symEval style="default" hide-keywords="false" hide-lhs="false" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:id xml:space="preserve">det</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> <ml:command> <ml:id xml:space="preserve">factor</ml:id> </ml:command> <ml:symResult> <ml:apply> <ml:mult/> <ml:real>2</ml:real> <ml:parens> <ml:apply> <ml:minus/> <ml:apply> <ml:mult/> <ml:apply> <ml:id xml:space="preserve">cosh</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:id xml:space="preserve">sinh</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:parens> </ml:apply> </ml:symResult> </ml:symEval> </math> <rendering item-idref="29"/> </region> <region region-id="336" left="6" top="1488" width="201.75" height="15" align-x="6" align-y="1500" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Rielaborando l'equazione, si ricava: </text> </region> <region region-id="341" left="6" top="1536.75" width="89.25" height="15.75" align-x="46.5" align-y="1548" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="#ff0" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:apply> <ml:id xml:space="preserve">tan</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">tanh</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> </math> <rendering item-idref="30"/> </region> <region region-id="340" left="168" top="1536" width="333" height="15" align-x="189" align-y="1548" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Equazione<sp count="2"/>caratteristica (o equazione delle frequenze) </text> </region> <region region-id="343" left="6" top="1596" width="378.75" height="15" align-x="32.25" align-y="1608" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Cerchiamo le soluzioni dell'equazione caratteristica per via grafica: </text> </region> <region region-id="69" left="6" top="1644.75" width="81.75" height="20.25" align-x="44.25" align-y="1656" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:function> <ml:id xml:space="preserve" subscript="1">f</ml:id> <ml:boundVars> <ml:id xml:space="preserve">α</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:id xml:space="preserve">tan</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="31"/> </region> <region region-id="72" left="6" top="1680.75" width="87.75" height="20.25" align-x="44.25" align-y="1692" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:function> <ml:id xml:space="preserve" subscript="2">f</ml:id> <ml:boundVars> <ml:id xml:space="preserve">α</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:id xml:space="preserve">tanh</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="32"/> </region> <region region-id="73" left="6" top="1716.75" width="81.75" height="15.75" align-x="22.5" align-y="1728" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">α</ml:id> <ml:range> <ml:sequence> <ml:real>0</ml:real> <ml:real>0.01</ml:real> </ml:sequence> <ml:real>20</ml:real> </ml:range> </ml:define> </math> <rendering item-idref="33"/> </region> <region region-id="77" left="132" top="1716.75" width="40.5" height="15.75" align-x="156" align-y="1728" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">FS</ml:id> <ml:real>4</ml:real> </ml:define> </math> <rendering item-idref="34"/> </region> <region region-id="263" left="6" top="1752" width="603" height="382.5" align-x="6" align-y="1752" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="35"/> <rendering item-idref="36"/> </region> <region region-id="194" left="6" top="2160.75" width="33" height="15.75" align-x="22.5" align-y="2172" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">α</ml:id> <ml:real>4</ml:real> </ml:define> </math> <rendering item-idref="37"/> </region> <region region-id="195" left="6" top="2190.75" width="219" height="20.25" align-x="28.5" align-y="2202" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="1">α</ml:id> <ml:eval placeholderMultiplicationStyle="default"> <ml:apply> <ml:id xml:space="preserve">root</ml:id> <ml:sequence> <ml:apply> <ml:minus/> <ml:apply> <ml:id xml:space="preserve" subscript="1">f</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve" subscript="2">f</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> <ml:id xml:space="preserve">α</ml:id> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>3.9266023132537806</ml:real> </result> </ml:eval> </ml:define> <resultFormat numeric-only="true"> <general precision="6" show-trailing-zeros="false" radix="dec" complex-threshold="10" zero-threshold="15" imaginary-value="i" exponential-threshold="3"/> <matrix display-style="auto" expand-nested-arrays="false"/> <unit format-units="true" simplify-units="true" fractional-unit-exponent="false"/> </resultFormat> </math> <rendering item-idref="38"/> </region> <region region-id="196" left="6" top="2256.75" width="33" height="15.75" align-x="22.5" align-y="2268" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedUDScalar" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">α</ml:id> <ml:real>7</ml:real> </ml:define> </math> <rendering item-idref="39"/> </region> <region region-id="197" left="6" top="2286.75" width="219" height="20.25" align-x="28.5" align-y="2298" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="2">α</ml:id> <ml:eval placeholderMultiplicationStyle="default"> <ml:apply> <ml:id xml:space="preserve">root</ml:id> <ml:sequence> <ml:apply> <ml:minus/> <ml:apply> <ml:id xml:space="preserve" subscript="1">f</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve" subscript="2">f</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> <ml:id xml:space="preserve">α</ml:id> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>7.0685827458686381</ml:real> </result> </ml:eval> </ml:define> <resultFormat numeric-only="true"> <general precision="6" show-trailing-zeros="false" radix="dec" complex-threshold="10" zero-threshold="15" imaginary-value="i" exponential-threshold="3"/> <matrix display-style="auto" expand-nested-arrays="false"/> <unit format-units="true" simplify-units="true" fractional-unit-exponent="false"/> </resultFormat> </math> <rendering item-idref="40"/> </region> <region region-id="198" left="6" top="2334.75" width="39" height="15.75" align-x="22.5" align-y="2346" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedUDScalar" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">α</ml:id> <ml:real>10</ml:real> </ml:define> </math> <rendering item-idref="41"/> </region> <region region-id="199" left="6" top="2364.75" width="225" height="20.25" align-x="28.5" align-y="2376" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="3">α</ml:id> <ml:eval placeholderMultiplicationStyle="default"> <ml:apply> <ml:id xml:space="preserve">root</ml:id> <ml:sequence> <ml:apply> <ml:minus/> <ml:apply> <ml:id xml:space="preserve" subscript="1">f</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve" subscript="2">f</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> <ml:id xml:space="preserve">α</ml:id> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>10.210176122406375</ml:real> </result> </ml:eval> </ml:define> <resultFormat numeric-only="true"> <general precision="6" show-trailing-zeros="false" radix="dec" complex-threshold="10" zero-threshold="15" imaginary-value="i" exponential-threshold="3"/> <matrix display-style="auto" expand-nested-arrays="false"/> <unit format-units="true" simplify-units="true" fractional-unit-exponent="false"/> </resultFormat> </math> <rendering item-idref="42"/> </region> <region region-id="200" left="6" top="2418.75" width="39" height="15.75" align-x="22.5" align-y="2430" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedUDScalar" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">α</ml:id> <ml:real>13</ml:real> </ml:define> </math> <rendering item-idref="43"/> </region> <region region-id="201" left="6" top="2448.75" width="225" height="20.25" align-x="28.5" align-y="2460" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="4">α</ml:id> <ml:eval placeholderMultiplicationStyle="default"> <ml:apply> <ml:id xml:space="preserve">root</ml:id> <ml:sequence> <ml:apply> <ml:minus/> <ml:apply> <ml:id xml:space="preserve" subscript="1">f</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve" subscript="2">f</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> <ml:id xml:space="preserve">α</ml:id> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>13.351768777743059</ml:real> </result> </ml:eval> </ml:define> <resultFormat numeric-only="true"> <general precision="6" show-trailing-zeros="false" radix="dec" complex-threshold="10" zero-threshold="15" imaginary-value="i" exponential-threshold="3"/> <matrix display-style="auto" expand-nested-arrays="false"/> <unit format-units="true" simplify-units="true" fractional-unit-exponent="false"/> </resultFormat> </math> <rendering item-idref="44"/> </region> <region region-id="290" left="6" top="2532" width="57.75" height="15" align-x="6" align-y="2544" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Lunghezza </text> </region> <region region-id="291" left="6" top="2550.75" width="117.75" height="15.75" align-x="18.75" align-y="2562" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIUnit" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">l</ml:id> <ml:eval placeholderMultiplicationStyle="default"> <ml:apply> <ml:mult/> <ml:real>1200</ml:real> <ml:id xml:space="preserve">mm</ml:id> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>1.2</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="meter"/> </unitMonomial> </unitedValue> </result> </ml:eval> </ml:define> </math> <rendering item-idref="45"/> </region> <region region-id="278" left="6" top="2598" width="213" height="15" align-x="21" align-y="2610" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Materiale: acciaio </text> </region> <region region-id="279" left="6" top="2620.5" width="72.75" height="39" align-x="23.25" align-y="2640" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">ρ</ml:id> <ml:apply> <ml:mult/> <ml:real>7800</ml:real> <ml:apply> <ml:div/> <ml:id xml:space="preserve">kg</ml:id> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">m</ml:id> <ml:real>3</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:define> <resultFormat numeric-only="true"> <general precision="6" show-trailing-zeros="false" radix="dec" complex-threshold="10" zero-threshold="15" imaginary-value="i" exponential-threshold="6"/> <matrix display-style="auto" expand-nested-arrays="false"/> <unit format-units="true" simplify-units="true" fractional-unit-exponent="false"/> </resultFormat> </math> <rendering item-idref="46"/> </region> <region region-id="280" left="6" top="2665.5" width="186" height="21" align-x="24" align-y="2682" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">E</ml:id> <ml:eval placeholderMultiplicationStyle="default"> <ml:apply> <ml:mult/> <ml:real>206000</ml:real> <ml:id xml:space="preserve">MPa</ml:id> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>206000000000</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="pascal"/> </unitMonomial> </unitedValue> </result> </ml:eval> 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