Location: BondGraph Basic @ 38d8a9aa6515 / BG Tutorial Mechanical Systems / FEM Double Pendulum.cellml
- Author:
- Soroush Safaei <ssaf006@aucklanduni.ac.nz>
- Date:
- 2017-08-17 11:40:48+12:00
- Desc:
- Merge remote-tracking branch 'origin/master'
- Permanent Source URI:
- https://models.fieldml.org/workspace/43b/rawfile/38d8a9aa65151ba0e3b2c342731b9488a3e272c9/BG Tutorial Mechanical Systems/FEM Double Pendulum.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#" xmlns:xlink="http://www.w3.org/1999/xlink">
<!-- Multibody problem: FE solution of driven mass on rail with double pin jointed pendulum-->
<import xlink:href="Units.cellml">
<units name="per_s" units_ref="per_s"/>
<units name="m_per_s" units_ref="m_per_s"/>
<units name="m_per_s2" units_ref="m_per_s2"/>
<units name="rad_per_s" units_ref="rad_per_s"/>
<units name="rad_per_s2" units_ref="rad_per_s2"/>
<units name="J_per_m" units_ref="J_per_m"/>
<units name="J_per_rad" units_ref="J_per_rad"/>
<units name="J_per_m2" units_ref="J_per_m2"/>
<units name="J_s_per_rad" units_ref="J_s_per_rad"/>
<units name="J_s_per_m" units_ref="J_s_per_m"/>
<units name="J_s_per_m2" units_ref="J_s_per_m2"/>
<units name="J_s2_per_m2" units_ref="J_s2_per_m2"/>
</import>
<component name="pendulum">
<variable name="t" units="second"/>
<!-- State variables-->
<variable initial_value="0" name="q_A_x" units="metre"/>
<variable initial_value="0" name="q_A_y" units="metre"/>
<variable initial_value="0" name="q_B_x" units="metre"/>
<variable initial_value="-1" name="q_B_y" units="metre"/>
<variable initial_value="1" name="q_C_x" units="metre"/>
<variable initial_value="-1" name="q_C_y" units="metre"/>
<variable initial_value="0" name="v_A_x" units="m_per_s"/>
<variable initial_value="0" name="v_A_y" units="m_per_s"/>
<variable initial_value="0" name="v_B_x" units="m_per_s"/>
<variable initial_value="0" name="v_B_y" units="m_per_s"/>
<variable initial_value="0" name="v_C_x" units="m_per_s"/>
<variable initial_value="0" name="v_C_y" units="m_per_s"/>
<variable name="u_A_x" units="J_per_m"/>
<variable name="u_A_y" units="J_per_m"/>
<variable name="u_B_x" units="J_per_m"/>
<variable name="u_B_y" units="J_per_m"/>
<variable name="u_C_x" units="J_per_m"/>
<variable name="u_C_y" units="J_per_m"/>
<variable name="u_AB_truss" units="J_per_m"/>
<variable name="u_BC_truss" units="J_per_m"/>
<variable name="q_theta_AB" units="radian"/>
<variable name="q_theta_BC" units="radian"/>
<variable name="q_theta_AB_degree" units="dimensionless"/>
<variable name="q_theta_BC_degree" units="dimensionless"/>
<!-- Constitutive parameters-->
<variable initial_value="1" name="l_AB_0" units="metre"/>
<variable initial_value="1" name="l_BC_0" units="metre"/>
<variable name="l_AB" units="metre"/>
<variable name="l_BC" units="metre"/>
<variable name="epsilon_AB" units="dimensionless"/>
<variable name="epsilon_AB_rate" units="per_s"/>
<variable name="epsilon_BC" units="dimensionless"/>
<variable name="epsilon_BC_rate" units="per_s"/>
<variable initial_value="1" name="m_A" units="J_s2_per_m2"/>
<variable initial_value="1" name="m_B" units="J_s2_per_m2"/>
<variable initial_value="1" name="m_C" units="J_s2_per_m2"/>
<variable initial_value="1E4" name="E_AB" units="J_per_m"/>
<variable initial_value="1E4" name="E_BC" units="J_per_m"/>
<variable initial_value="10" name="R_AB" units="J_s_per_m"/>
<variable initial_value="10" name="R_BC" units="J_s_per_m"/>
<variable initial_value="0" name="F_A_x" units="J_per_m"/>
<variable initial_value="0" name="F_A_y" units="J_per_m"/>
<variable initial_value="0" name="F_B_x" units="J_per_m"/>
<variable initial_value="0" name="F_B_y" units="J_per_m"/>
<variable initial_value="0" name="F_C_x" units="J_per_m"/>
<variable initial_value="0" name="F_C_y" units="J_per_m"/>
<variable initial_value="9.81" name="g" units="m_per_s2"/>
<!-- Beam equations-->
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply>
<eq/>
<ci>l_AB</ci>
<apply>
<root/>
<apply>
<plus/>
<apply>
<power/>
<apply>
<minus/>
<ci>q_B_x</ci>
<ci>q_A_x</ci>
</apply>
<cn cellml:units="dimensionless">2</cn>
</apply>
<apply>
<power/>
<apply>
<minus/>
<ci>q_B_y</ci>
<ci>q_A_y</ci>
</apply>
<cn cellml:units="dimensionless">2</cn>
</apply>
</apply>
</apply>
</apply>
<apply>
<eq/>
<ci>l_BC</ci>
<apply>
<root/>
<apply>
<plus/>
<apply>
<power/>
<apply>
<minus/>
<ci>q_C_x</ci>
<ci>q_B_x</ci>
</apply>
<cn cellml:units="dimensionless">2</cn>
</apply>
<apply>
<power/>
<apply>
<minus/>
<ci>q_C_y</ci>
<ci>q_B_y</ci>
</apply>
<cn cellml:units="dimensionless">2</cn>
</apply>
</apply>
</apply>
</apply>
<apply>
<eq/>
<ci>q_theta_AB</ci>
<apply>
<arccos/>
<apply>
<divide/>
<apply>
<minus/>
<ci>q_A_x</ci>
<ci>q_B_x</ci>
</apply>
<ci>l_AB</ci>
</apply>
</apply>
</apply>
<apply>
<eq/>
<ci>q_theta_BC</ci>
<apply>
<arccos/>
<apply>
<divide/>
<apply>
<minus/>
<ci>q_B_x</ci>
<ci>q_C_x</ci>
</apply>
<ci>l_BC</ci>
</apply>
</apply>
</apply>
<apply>
<eq/>
<ci>q_theta_AB_degree</ci>
<apply>
<divide/>
<apply>
<times/>
<ci>q_theta_AB</ci>
<cn cellml:units="dimensionless">180</cn>
</apply>
<pi/>
</apply>
</apply>
<apply>
<eq/>
<ci>q_theta_BC_degree</ci>
<apply>
<divide/>
<apply>
<times/>
<ci>q_theta_BC</ci>
<cn cellml:units="dimensionless">180</cn>
</apply>
<pi/>
</apply>
</apply>
<apply>
<eq/>
<ci>epsilon_AB</ci>
<apply>
<divide/>
<apply>
<minus/>
<ci>l_AB</ci>
<ci>l_AB_0</ci>
</apply>
<ci>l_AB_0</ci>
</apply>
</apply>
<apply>
<eq/>
<ci>epsilon_BC</ci>
<apply>
<divide/>
<apply>
<minus/>
<ci>l_BC</ci>
<ci>l_BC_0</ci>
</apply>
<ci>l_BC_0</ci>
</apply>
</apply>
<apply>
<eq/>
<ci>epsilon_AB_rate</ci>
<apply>
<divide/>
<apply>
<plus/>
<apply>
<times/>
<apply>
<minus/>
<ci>q_B_x</ci>
<ci>q_A_x</ci>
</apply>
<apply>
<minus/>
<ci>v_B_x</ci>
<ci>v_A_x</ci>
</apply>
</apply>
<apply>
<times/>
<apply>
<minus/>
<ci>q_B_y</ci>
<ci>q_A_y</ci>
</apply>
<apply>
<minus/>
<ci>v_B_y</ci>
<ci>v_A_y</ci>
</apply>
</apply>
</apply>
<apply>
<times/>
<ci>l_AB</ci>
<ci>l_AB_0</ci>
</apply>
</apply>
</apply>
<apply>
<eq/>
<ci>epsilon_BC_rate</ci>
<apply>
<divide/>
<apply>
<plus/>
<apply>
<times/>
<apply>
<minus/>
<ci>q_C_x</ci>
<ci>q_B_x</ci>
</apply>
<apply>
<minus/>
<ci>v_C_x</ci>
<ci>v_B_x</ci>
</apply>
</apply>
<apply>
<times/>
<apply>
<minus/>
<ci>q_C_y</ci>
<ci>q_B_y</ci>
</apply>
<apply>
<minus/>
<ci>v_C_y</ci>
<ci>v_B_y</ci>
</apply>
</apply>
</apply>
<apply>
<times/>
<ci>l_BC</ci>
<ci>l_BC_0</ci>
</apply>
</apply>
</apply>
<apply>
<eq/>
<ci>u_AB_truss</ci>
<apply>
<plus/>
<apply>
<times/>
<ci>E_AB</ci>
<ci>epsilon_AB</ci>
</apply>
<apply>
<times/>
<ci>R_AB</ci>
<ci>epsilon_AB_rate</ci>
</apply>
</apply>
</apply>
<apply>
<eq/>
<ci>u_BC_truss</ci>
<apply>
<plus/>
<apply>
<times/>
<ci>E_BC</ci>
<ci>epsilon_BC</ci>
</apply>
<apply>
<times/>
<ci>R_BC</ci>
<ci>epsilon_BC_rate</ci>
</apply>
</apply>
</apply>
<apply>
<eq/>
<ci>u_A_x</ci>
<apply>
<times/>
<apply>
<minus/>
<ci>u_AB_truss</ci>
</apply>
<apply>
<cos/>
<ci>q_theta_AB</ci>
</apply>
</apply>
</apply>
<apply>
<eq/>
<ci>u_A_y</ci>
<apply>
<times/>
<apply>
<minus/>
<ci>u_AB_truss</ci>
</apply>
<apply>
<sin/>
<ci>q_theta_AB</ci>
</apply>
</apply>
</apply>
<apply>
<eq/>
<ci>u_B_x</ci>
<apply>
<minus/>
<apply>
<times/>
<ci>u_AB_truss</ci>
<apply>
<cos/>
<ci>q_theta_AB</ci>
</apply>
</apply>
<apply>
<times/>
<ci>u_BC_truss</ci>
<apply>
<cos/>
<ci>q_theta_BC</ci>
</apply>
</apply>
</apply>
</apply>
<apply>
<eq/>
<ci>u_B_y</ci>
<apply>
<minus/>
<apply>
<times/>
<ci>u_AB_truss</ci>
<apply>
<sin/>
<ci>q_theta_AB</ci>
</apply>
</apply>
<apply>
<times/>
<ci>u_BC_truss</ci>
<apply>
<sin/>
<ci>q_theta_BC</ci>
</apply>
</apply>
</apply>
</apply>
<apply>
<eq/>
<ci>u_C_x</ci>
<apply>
<times/>
<ci>u_BC_truss</ci>
<apply>
<cos/>
<ci>q_theta_BC</ci>
</apply>
</apply>
</apply>
<apply>
<eq/>
<ci>u_C_y</ci>
<apply>
<times/>
<ci>u_BC_truss</ci>
<apply>
<sin/>
<ci>q_theta_BC</ci>
</apply>
</apply>
</apply>
<!-- Constitutive relations-->
<!-- ode(v_A_x, t) = (-F_A_x-u_AB_truss*cos(q_theta_AB))/m_A;
ode(v_A_y, t) = (-F_A_y-u_AB_truss*sin(q_theta_AB))/m_A-g;-->
<apply>
<eq/>
<apply>
<diff/>
<bvar>
<ci>t</ci>
</bvar>
<ci>v_B_x</ci>
</apply>
<apply>
<divide/>
<apply>
<minus/>
<apply>
<plus/>
<ci>F_B_x</ci>
<apply>
<times/>
<ci>u_AB_truss</ci>
<apply>
<cos/>
<ci>q_theta_AB</ci>
</apply>
</apply>
</apply>
<apply>
<times/>
<ci>u_BC_truss</ci>
<apply>
<cos/>
<ci>q_theta_BC</ci>
</apply>
</apply>
</apply>
<apply>
<plus/>
<ci>m_B</ci>
<ci>m_C</ci>
</apply>
</apply>
</apply>
<apply>
<eq/>
<apply>
<diff/>
<bvar>
<ci>t</ci>
</bvar>
<ci>v_B_y</ci>
</apply>
<apply>
<minus/>
<apply>
<divide/>
<apply>
<minus/>
<apply>
<plus/>
<ci>F_B_y</ci>
<apply>
<times/>
<ci>u_AB_truss</ci>
<apply>
<sin/>
<ci>q_theta_AB</ci>
</apply>
</apply>
</apply>
<apply>
<times/>
<ci>u_BC_truss</ci>
<apply>
<sin/>
<ci>q_theta_BC</ci>
</apply>
</apply>
</apply>
<apply>
<plus/>
<ci>m_B</ci>
<ci>m_C</ci>
</apply>
</apply>
<ci>g</ci>
</apply>
</apply>
<apply>
<eq/>
<apply>
<diff/>
<bvar>
<ci>t</ci>
</bvar>
<ci>v_C_x</ci>
</apply>
<apply>
<divide/>
<apply>
<plus/>
<ci>F_C_x</ci>
<apply>
<times/>
<ci>u_BC_truss</ci>
<apply>
<cos/>
<ci>q_theta_BC</ci>
</apply>
</apply>
</apply>
<ci>m_C</ci>
</apply>
</apply>
<apply>
<eq/>
<apply>
<diff/>
<bvar>
<ci>t</ci>
</bvar>
<ci>v_C_y</ci>
</apply>
<apply>
<minus/>
<apply>
<divide/>
<apply>
<plus/>
<ci>F_C_y</ci>
<apply>
<times/>
<ci>u_BC_truss</ci>
<apply>
<sin/>
<ci>q_theta_BC</ci>
</apply>
</apply>
</apply>
<ci>m_C</ci>
</apply>
<ci>g</ci>
</apply>
</apply>
<!-- Conservation laws-->
<apply>
<eq/>
<apply>
<diff/>
<bvar>
<ci>t</ci>
</bvar>
<ci>q_A_x</ci>
</apply>
<ci>v_A_x</ci>
</apply>
<apply>
<eq/>
<apply>
<diff/>
<bvar>
<ci>t</ci>
</bvar>
<ci>q_A_y</ci>
</apply>
<ci>v_A_y</ci>
</apply>
<apply>
<eq/>
<apply>
<diff/>
<bvar>
<ci>t</ci>
</bvar>
<ci>q_B_x</ci>
</apply>
<ci>v_B_x</ci>
</apply>
<apply>
<eq/>
<apply>
<diff/>
<bvar>
<ci>t</ci>
</bvar>
<ci>q_B_y</ci>
</apply>
<ci>v_B_y</ci>
</apply>
<apply>
<eq/>
<apply>
<diff/>
<bvar>
<ci>t</ci>
</bvar>
<ci>q_C_x</ci>
</apply>
<ci>v_C_x</ci>
</apply>
<apply>
<eq/>
<apply>
<diff/>
<bvar>
<ci>t</ci>
</bvar>
<ci>q_C_y</ci>
</apply>
<ci>v_C_y</ci>
</apply>
</math>
</component>
</model>