Location: BondGraph Basic @ d8de10edb753 / BG Tutorial Mechanical Systems / FEM Pendulum - flexible.cellml
- Author:
- Soroush <ssaf006@aucklanduni.ac.nz>
- Date:
- 2018-06-25 19:01:14+12:00
- Desc:
- adding new examples
- Permanent Source URI:
- https://models.cellml.org/workspace/43b/rawfile/d8de10edb753792c2741c0619a65288ffcbddecc/BG Tutorial Mechanical Systems/FEM Pendulum - flexible.cellml
<?xml version='1.0'?>
<model name="my_model" xmlns="http://www.cellml.org/cellml/1.1#" xmlns:cellml="http://www.cellml.org/cellml/1.1#">
<!-- Multibody problem: FE solution of driven mass on rail with single pin jointed flexible pendulum-->
<units name="per_s">
<unit exponent="-1" units="second"/>
</units>
<units name="m_per_s">
<unit units="metre"/>
<unit exponent="-1" units="second"/>
</units>
<units name="m_per_s2">
<unit units="metre"/>
<unit exponent="-2" units="second"/>
</units>
<units name="rad_per_s">
<unit units="radian"/>
<unit exponent="-1" units="second"/>
</units>
<units name="rad_per_s2">
<unit units="radian"/>
<unit exponent="-2" units="second"/>
</units>
<units name="Jm">
<unit units="joule"/>
<unit units="metre"/>
</units>
<units name="Js_per_m">
<unit units="joule"/>
<unit units="second"/>
<unit exponent="-1" units="metre"/>
</units>
<units name="Js_per_m2">
<unit units="joule"/>
<unit units="second"/>
<unit exponent="-2" units="metre"/>
</units>
<units name="Js2_per_m3">
<unit units="joule"/>
<unit exponent="2" units="second"/>
<unit exponent="-3" units="metre"/>
</units>
<units name="Js2_per_m2">
<unit units="joule"/>
<unit exponent="2" units="second"/>
<unit exponent="-2" units="metre"/>
</units>
<units name="J_per_m">
<unit units="joule"/>
<unit exponent="-1" units="metre"/>
</units>
<units name="Js2_per_rad">
<unit units="joule"/>
<unit exponent="2" units="second"/>
<unit exponent="-1" units="radian"/>
</units>
<component name="main">
<variable initial_value="0" name="t" units="second"/>
<variable initial_value="3.141592" name="PI" units="dimensionless"/>
<variable name="theta_AB" units="dimensionless"/>
<variable name="theta1_AB" units="dimensionless"/>
<variable name="theta2_AB" units="dimensionless"/>
<variable name="costheta_AB" units="dimensionless"/>
<variable name="sintheta_AB" units="dimensionless"/>
<variable initial_value="9.81" name="g" units="m_per_s2"/>
<variable initial_value="1" name="l_AB_0" units="metre"/>
<variable name="l_AB" units="metre"/>
<variable initial_value="10000" name="E" units="J_per_m"/>
<variable initial_value="10" name="R_truss" units="Js_per_m"/>
<variable initial_value="1" name="R_beam" units="Js_per_m2"/>
<variable name="strain_AB" units="dimensionless"/>
<variable name="strain_rate_AB" units="per_s"/>
<variable initial_value="1" name="m_A" units="Js2_per_m2"/>
<variable initial_value="1" name="m_B" units="Js2_per_m2"/>
<variable initial_value="0" name="F_A" units="J_per_m"/>
<variable name="u_A_1" units="J_per_m"/>
<variable name="u_A_2" units="J_per_m"/>
<variable name="u_truss_AB" units="J_per_m"/>
<variable name="u_beam_AB" units="J_per_m"/>
<variable initial_value="0" name="v_A_1" units="m_per_s"/>
<variable initial_value="0" name="v_A_2" units="m_per_s"/>
<variable initial_value="0" name="v_B_1" units="m_per_s"/>
<variable initial_value="0" name="v_B_2" units="m_per_s"/>
<variable name="x_AB" units="metre"/>
<variable name="y_AB" units="metre"/>
<variable initial_value="0" name="q_A_1" units="metre"/>
<variable initial_value="0" name="q_A_2" units="metre"/>
<variable initial_value="0.707106781" name="q_B_1" units="metre"/>
<variable initial_value="0.707106781" name="q_B_2" units="metre"/>
<variable name="y_beam_AB_1" units="metre"/>
<variable initial_value="0.1" name="y_beam_AB_1_ampl" units="metre"/>
<variable name="y_beam_AB_2" units="metre"/>
<variable initial_value="0.1" name="y_beam_AB_2_ampl" units="metre"/>
<variable name="y_beam_AB_3" units="metre"/>
<variable initial_value="0.1" name="y_beam_AB_3_ampl" units="metre"/>
<variable initial_value="0" name="v_beam_AB" units="m_per_s"/>
<variable initial_value="0.03" name="EI" units="Jm"/>
<variable initial_value="1" name="rho" units="Js2_per_m3"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply>
<eq/>
<ci>x_AB</ci>
<apply>
<minus/>
<ci>q_B_1</ci>
<ci>q_A_1</ci>
</apply>
</apply>
<apply>
<eq/>
<ci>y_AB</ci>
<apply>
<minus/>
<ci>q_B_2</ci>
<ci>q_A_2</ci>
</apply>
</apply>
<apply>
<eq/>
<ci>l_AB</ci>
<apply>
<root/>
<apply>
<plus/>
<apply>
<power/>
<ci>x_AB</ci>
<cn cellml:units="dimensionless">2</cn>
</apply>
<apply>
<power/>
<ci>y_AB</ci>
<cn cellml:units="dimensionless">2</cn>
</apply>
</apply>
</apply>
</apply>
<apply>
<eq/>
<ci>costheta_AB</ci>
<apply>
<divide/>
<ci>x_AB</ci>
<ci>l_AB</ci>
</apply>
</apply>
<apply>
<eq/>
<ci>sintheta_AB</ci>
<apply>
<divide/>
<ci>y_AB</ci>
<ci>l_AB</ci>
</apply>
</apply>
<apply>
<eq/>
<ci>strain_AB</ci>
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<divide/>
<apply>
<minus/>
<ci>l_AB</ci>
<ci>l_AB_0</ci>
</apply>
<ci>l_AB_0</ci>
</apply>
</apply>
<apply>
<eq/>
<ci>strain_rate_AB</ci>
<apply>
<divide/>
<apply>
<plus/>
<apply>
<times/>
<ci>x_AB</ci>
<apply>
<minus/>
<ci>v_B_1</ci>
<ci>v_A_1</ci>
</apply>
</apply>
<apply>
<times/>
<ci>y_AB</ci>
<apply>
<minus/>
<ci>v_B_2</ci>
<ci>v_A_2</ci>
</apply>
</apply>
</apply>
<apply>
<times/>
<ci>l_AB</ci>
<ci>l_AB_0</ci>
</apply>
</apply>
</apply>
<apply>
<eq/>
<ci>u_truss_AB</ci>
<apply>
<plus/>
<apply>
<times/>
<ci>E</ci>
<ci>strain_AB</ci>
</apply>
<apply>
<times/>
<ci>R_truss</ci>
<ci>strain_rate_AB</ci>
</apply>
</apply>
</apply>
<apply>
<eq/>
<ci>u_A_1</ci>
<apply>
<plus/>
<apply>
<times/>
<ci>u_truss_AB</ci>
<ci>costheta_AB</ci>
</apply>
<apply>
<times/>
<ci>u_beam_AB</ci>
<ci>sintheta_AB</ci>
</apply>
</apply>
</apply>
<apply>
<eq/>
<ci>u_A_2</ci>
<apply>
<plus/>
<apply>
<times/>
<ci>u_truss_AB</ci>
<ci>sintheta_AB</ci>
</apply>
<apply>
<times/>
<ci>u_beam_AB</ci>
<ci>costheta_AB</ci>
</apply>
</apply>
</apply>
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<eq/>
<ci>y_beam_AB_1</ci>
<apply>
<times/>
<ci>y_beam_AB_1_ampl</ci>
<apply>
<sin/>
<apply>
<times/>
<apply>
<root/>
<apply>
<divide/>
<ci>EI</ci>
<ci>rho</ci>
</apply>
</apply>
<apply>
<power/>
<apply>
<divide/>
<ci>PI</ci>
<ci>l_AB</ci>
</apply>
<cn cellml:units="dimensionless">2</cn>
</apply>
<ci>t</ci>
</apply>
</apply>
</apply>
</apply>
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<eq/>
<ci>y_beam_AB_2</ci>
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<times/>
<ci>y_beam_AB_2_ampl</ci>
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<times/>
<apply>
<root/>
<apply>
<divide/>
<ci>EI</ci>
<ci>rho</ci>
</apply>
</apply>
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<power/>
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<divide/>
<apply>
<times/>
<cn cellml:units="dimensionless">2</cn>
<ci>PI</ci>
</apply>
<ci>l_AB</ci>
</apply>
<cn cellml:units="dimensionless">2</cn>
</apply>
<ci>t</ci>
</apply>
</apply>
</apply>
</apply>
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<eq/>
<ci>y_beam_AB_3</ci>
<apply>
<times/>
<ci>y_beam_AB_3_ampl</ci>
<apply>
<sin/>
<apply>
<times/>
<apply>
<root/>
<apply>
<divide/>
<ci>EI</ci>
<ci>rho</ci>
</apply>
</apply>
<apply>
<power/>
<apply>
<divide/>
<apply>
<times/>
<cn cellml:units="dimensionless">3</cn>
<ci>PI</ci>
</apply>
<ci>l_AB</ci>
</apply>
<cn cellml:units="dimensionless">2</cn>
</apply>
<ci>t</ci>
</apply>
</apply>
</apply>
</apply>
<apply>
<eq/>
<ci>u_beam_AB</ci>
<apply>
<plus/>
<apply>
<times/>
<ci>EI</ci>
<apply>
<power/>
<apply>
<divide/>
<ci>PI</ci>
<ci>l_AB</ci>
</apply>
<cn cellml:units="dimensionless">3</cn>
</apply>
<ci>y_beam_AB_1</ci>
</apply>
<apply>
<times/>
<ci>EI</ci>
<apply>
<power/>
<apply>
<divide/>
<apply>
<times/>
<cn cellml:units="dimensionless">2</cn>
<ci>PI</ci>
</apply>
<ci>l_AB</ci>
</apply>
<cn cellml:units="dimensionless">3</cn>
</apply>
<ci>y_beam_AB_2</ci>
</apply>
<apply>
<times/>
<ci>EI</ci>
<apply>
<power/>
<apply>
<divide/>
<apply>
<times/>
<cn cellml:units="dimensionless">3</cn>
<ci>PI</ci>
</apply>
<ci>l_AB</ci>
</apply>
<cn cellml:units="dimensionless">3</cn>
</apply>
<ci>y_beam_AB_3</ci>
</apply>
</apply>
</apply>
<apply>
<eq/>
<apply>
<diff/>
<bvar>
<ci>t</ci>
</bvar>
<ci>v_A_1</ci>
</apply>
<apply>
<divide/>
<apply>
<plus/>
<ci>F_A</ci>
<ci>u_A_1</ci>
</apply>
<ci>m_A</ci>
</apply>
</apply>
<apply>
<eq/>
<apply>
<diff/>
<bvar>
<ci>t</ci>
</bvar>
<ci>v_B_1</ci>
</apply>
<apply>
<divide/>
<apply>
<minus/>
<ci>u_A_1</ci>
</apply>
<ci>m_B</ci>
</apply>
</apply>
<apply>
<eq/>
<apply>
<diff/>
<bvar>
<ci>t</ci>
</bvar>
<ci>v_B_2</ci>
</apply>
<apply>
<minus/>
<apply>
<divide/>
<apply>
<minus/>
<ci>u_A_2</ci>
</apply>
<ci>m_B</ci>
</apply>
<ci>g</ci>
</apply>
</apply>
<apply>
<eq/>
<apply>
<diff/>
<bvar>
<ci>t</ci>
</bvar>
<ci>q_A_1</ci>
</apply>
<ci>v_A_1</ci>
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<apply>
<eq/>
<apply>
<diff/>
<bvar>
<ci>t</ci>
</bvar>
<ci>q_A_2</ci>
</apply>
<ci>v_A_2</ci>
</apply>
<apply>
<eq/>
<apply>
<diff/>
<bvar>
<ci>t</ci>
</bvar>
<ci>q_B_1</ci>
</apply>
<ci>v_B_1</ci>
</apply>
<apply>
<eq/>
<apply>
<diff/>
<bvar>
<ci>t</ci>
</bvar>
<ci>q_B_2</ci>
</apply>
<ci>v_B_2</ci>
</apply>
<apply>
<eq/>
<ci>theta1_AB</ci>
<apply>
<divide/>
<apply>
<times/>
<apply>
<arccos/>
<ci>costheta_AB</ci>
</apply>
<cn cellml:units="dimensionless">180</cn>
</apply>
<ci>PI</ci>
</apply>
</apply>
<apply>
<eq/>
<ci>theta2_AB</ci>
<apply>
<divide/>
<apply>
<times/>
<apply>
<arcsin/>
<ci>sintheta_AB</ci>
</apply>
<cn cellml:units="dimensionless">180</cn>
</apply>
<ci>PI</ci>
</apply>
</apply>
<apply id="theta_calculation">
<eq/>
<ci>theta_AB</ci>
<piecewise>
<piece>
<ci>theta1_AB</ci>
<apply>
<and/>
<apply>
<gt/>
<ci>x_AB</ci>
<cn cellml:units="metre">0</cn>
</apply>
<apply>
<gt/>
<ci>y_AB</ci>
<cn cellml:units="metre">0</cn>
</apply>
</apply>
</piece>
<piece>
<apply>
<minus/>
<ci>theta1_AB</ci>
</apply>
<apply>
<and/>
<apply>
<gt/>
<ci>x_AB</ci>
<cn cellml:units="metre">0</cn>
</apply>
<apply>
<lt/>
<ci>y_AB</ci>
<cn cellml:units="metre">0</cn>
</apply>
</apply>
</piece>
<piece>
<apply>
<minus/>
<ci>theta1_AB</ci>
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<apply>
<and/>
<apply>
<lt/>
<ci>x_AB</ci>
<cn cellml:units="metre">0</cn>
</apply>
<apply>
<lt/>
<ci>y_AB</ci>
<cn cellml:units="metre">0</cn>
</apply>
</apply>
</piece>
<piece>
<apply>
<minus/>
<ci>theta1_AB</ci>
<cn cellml:units="dimensionless">360</cn>
</apply>
<apply>
<and/>
<apply>
<lt/>
<ci>x_AB</ci>
<cn cellml:units="metre">0</cn>
</apply>
<apply>
<gt/>
<ci>y_AB</ci>
<cn cellml:units="metre">0</cn>
</apply>
</apply>
</piece>
</piecewise>
</apply>
</math>
</component>
</model>