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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="431" left="492" top="690.75" width="288" height="15.75" align-x="764.25" align-y="702" 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="433" left="6" top="721.5" width="393.75" height="21" align-x="97.5" align-y="738" 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:plus/> <ml:apply> <ml:minus/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">A</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:mult/> <ml:id xml:space="preserve">B</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">C</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:apply> <ml:mult/> <ml:id xml:space="preserve">D</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:parens> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">β</ml:id> <ml:real>3</ml:real> </ml:apply> </ml:apply> </ml:symResult> </ml:symEval> </math> <rendering item-idref="17"/> </region> <region region-id="432" left="492" top="726.75" width="288" height="15.75" align-x="764.25" 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:apply> <ml:plus/> <ml:apply> <ml:minus/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">A</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:mult/> <ml:id xml:space="preserve">B</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">C</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:apply> <ml:mult/> <ml:id xml:space="preserve">D</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:real>0</ml:real> </ml:apply> </math> <rendering item-idref="18"/> </region> <region region-id="316" left="6" top="768" width="392.25" height="30" align-x="29.25" align-y="780" 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"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Riassunto delle condizioni al contorno (che formano un sistema lineare)</p> </text> </region> <region region-id="42" left="6" top="804.75" width="54" height="15.75" align-x="44.25" 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: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="19"/> </region> <region region-id="41" left="6" top="828.75" width="53.25" height="15.75" align-x="43.5" align-y="840" 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="20"/> </region> <region region-id="436" left="6" top="858.75" width="288" height="15.75" align-x="278.25" align-y="870" 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="21"/> </region> <region region-id="438" left="6" top="888.75" width="288" height="15.75" align-x="278.25" align-y="900" 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:minus/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">A</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:mult/> <ml:id xml:space="preserve">B</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">C</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:apply> <ml:mult/> <ml:id xml:space="preserve">D</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:real>0</ml:real> </ml:apply> </math> <rendering item-idref="22"/> </region> <region region-id="320" left="6" top="960" width="243" height="30" align-x="6" align-y="972" 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"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Poniamo per semplicità: α = β l e otteniamo:</p> </text> </region> <region region-id="49" left="6" top="990.75" width="54" height="15.75" align-x="44.25" align-y="1002" 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="23"/> </region> <region region-id="50" left="6" top="1014.75" width="53.25" height="15.75" align-x="43.5" align-y="1026" 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="24"/> </region> <region region-id="441" left="6" top="1038.75" width="252" height="15.75" align-x="242.25" align-y="1050" 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="25"/> </region> <region region-id="442" left="6" top="1062.75" width="252" height="15.75" align-x="242.25" align-y="1074" 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:minus/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">A</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:mult/> <ml:id xml:space="preserve">B</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">C</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:mult/> <ml:id xml:space="preserve">D</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:real>0</ml:real> </ml:apply> </math> <rendering item-idref="26"/> </region> <region region-id="328" left="6" top="1092" width="510" height="15" align-x="6" align-y="1104" 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"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Le quattro condizioni al contorno si possono scrivere in forma matriciale, come segue</p> </text> </region> <region region-id="329" left="6" top="1121.25" width="98.25" height="78.75" align-x="72" align-y="1164" 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="27"/> </region> <region region-id="56" left="6" top="1211.25" width="247.5" height="78.75" align-x="44.25" align-y="1254" 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:neg/> <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:id xml:space="preserve">sin</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:neg/> <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: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>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">sinh</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">cosh</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:matrix> </ml:define> </math> <rendering item-idref="28"/> </region> <region region-id="330" left="6" top="1338" width="267" height="15" align-x="6" align-y="1350" 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"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Infatti, il prodotto matriciale seguente, fornisce:</p> </text> </region> <region region-id="331" left="6" top="1361.25" width="492.75" height="78.75" align-x="243" align-y="1404" 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:neg/> <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:id xml:space="preserve">sin</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:neg/> <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: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>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">sinh</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">cosh</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: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:apply> <ml:plus/> <ml:apply> <ml:plus/> <ml:apply> <ml:minus/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">A</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:mult/> <ml:id xml:space="preserve">B</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">C</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:mult/> <ml:id xml:space="preserve">D</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:matrix> </ml:symResult> </ml:symEval> </math> <rendering item-idref="29"/> </region> <region region-id="333" left="6" top="1524" width="388.5" height="30" align-x="6" align-y="1536" 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"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Si calcola ora il determinante della matrice <b> <inlineAttr font-style="normal">Δ(α)</inlineAttr> </b> e lo si pone uguale a zero, per ricavare l'equazione caratteristica:</p> </text> </region> <region region-id="335" left="6" top="1554" width="92.25" height="16.5" align-x="48" align-y="1566" 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="30"/> </region> <region region-id="228" left="6" top="1578.75" width="203.25" height="15.75" align-x="93.75" align-y="1590" 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">simplify</ml:id> </ml:command> <ml:symResult> <ml:apply> <ml:plus/> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:real>2</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:apply> <ml:id xml:space="preserve">cos</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:symResult> </ml:symEval> </math> <rendering item-idref="31"/> </region> <region region-id="336" left="6" top="1614" width="201.75" height="15" align-x="6" align-y="1626" 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"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Rielaborando l'equazione, si ricava:</p> </text> </region> <region region-id="443" left="6" top="1662.75" width="118.5" height="15.75" align-x="108.75" align-y="1674" 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:real>1</ml:real> <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">cos</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="32"/> </region> <region region-id="340" left="168" top="1662" width="333" height="15" align-x="189" align-y="1674" 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"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Equazione<sp count="2"/>caratteristica (o equazione delle frequenze)</p> </text> </region> <region region-id="343" left="6" top="1722" width="378.75" height="15" align-x="32.25" align-y="1734" 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"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Cerchiamo le soluzioni dell'equazione caratteristica per via grafica:</p> </text> </region> <region region-id="69" left="6" top="1770.75" width="141" height="15.75" align-x="41.25" align-y="1782" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="#ff0" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIUnit" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:function> <ml:id xml:space="preserve">F</ml:id> <ml:boundVars> <ml:id xml:space="preserve">α</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:plus/> <ml:real>1</ml:real> <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">cos</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="33"/> </region> <region region-id="73" left="6" top="1842.75" width="75.75" height="15.75" align-x="22.5" align-y="1854" 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>5</ml:real> </ml:range> </ml:define> </math> <rendering item-idref="34"/> </region> <region region-id="263" left="6" top="1878" width="520.5" height="334.5" align-x="6" align-y="1878" 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="467" left="654" top="1974.75" width="33" height="15.75" align-x="670.5" align-y="1986" 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>2</ml:real> </ml:define> </math> <rendering item-idref="37"/> </region> <region region-id="466" left="654" top="2004.75" width="171.75" height="20.25" align-x="676.5" align-y="2016" 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:id xml:space="preserve">F</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> <ml:id xml:space="preserve">α</ml:id> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>1.8751040687119824</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="465" left="654" top="2070.75" width="33" height="15.75" align-x="670.5" align-y="2082" 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>4</ml:real> </ml:define> </math> <rendering item-idref="39"/> </region> <region region-id="464" left="654" top="2100.75" width="171.75" height="20.25" align-x="676.5" align-y="2112" 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:id xml:space="preserve">F</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> <ml:id xml:space="preserve">α</ml:id> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>4.694091132974175</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="446" left="6" top="2244.75" width="90.75" height="15.75" align-x="22.5" align-y="2256" 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:range> <ml:sequence> <ml:real>7</ml:real> <ml:real>7.01</ml:real> </ml:sequence> <ml:real>11.2</ml:real> </ml:range> </ml:define> </math> <rendering item-idref="41"/> </region> <region region-id="447" left="6" top="2280" width="549.75" height="337.5" align-x="6" align-y="2280" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="42"/> <rendering item-idref="43"/> </region> <region region-id="463" left="654" top="2364.75" width="33" height="15.75" align-x="670.5" 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">α</ml:id> <ml:real>8</ml:real> </ml:define> </math> <rendering item-idref="44"/> </region> <region region-id="462" left="654" top="2394.75" width="171.75" height="20.25" align-x="676.5" align-y="2406" 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:id xml:space="preserve">F</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> <ml:id xml:space="preserve">α</ml:id> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>7.8547574382376126</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="45"/> </region> <region region-id="461" left="654" top="2448.75" width="39" height="15.75" align-x="670.5" 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 warning="WarnRedefinedUDScalar" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">α</ml:id> <ml:real>11</ml:real> </ml:define> </math> <rendering item-idref="46"/> </region> <region region-id="460" left="654" top="2478.75" width="177.75" height="20.25" align-x="676.5" align-y="2490" 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:id xml:space="preserve">F</ml:id> <ml:id xml:space="preserve">α</ml:id> </ml:apply> <ml:id xml:space="preserve">α</ml:id> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>10.995540734875467</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="47"/> </region> <region region-id="290" left="6" top="2646" width="57.75" height="15" align-x="6" align-y="2658" 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"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Lunghezza</p> </text> </region> <region region-id="291" left="6" top="2664.75" width="117.75" height="15.75" align-x="18.75" align-y="2676" 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="48"/> </region> <region region-id="278" left="6" top="2706" width="213" height="15" align-x="21" align-y="2718" 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"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Materiale: acciaio </p> </text> </region> <region region-id="279" left="6" top="2728.5" width="72.75" height="39" align-x="23.25" align-y="2748" 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="49"/> </region> <region region-id="280" left="6" top="2773.5" width="186" height="21" 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lock-width="true"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Sezione circolare di diametro d:</p> </text> </region> <region region-id="282" left="6" top="2850.75" width="120.75" height="15.75" align-x="21.75" align-y="2862" 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">d</ml:id> <ml:eval placeholderMultiplicationStyle="default"> <ml:apply> <ml:mult/> <ml:real>15</ml:real> <ml:id xml:space="preserve">mm</ml:id> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <unitedValue> <ml:real>0.015</ml:real> <unitMonomial xmlns="http://schemas.mathsoft.com/units10"> <unitReference unit="meter"/> </unitMonomial> </unitedValue> </result> </ml:eval> </ml:define> </math> <rendering 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