Location: BondGraph Basic @ 84d1c2103ca3 / BG Tutorial Mechanical Systems / Inverted pendulum wtih FEM.cellml

Author:
Soroush Safaei <ssaf006@aucklanduni.ac.nz>
Date:
2017-06-23 20:14:43+12:00
Desc:
adding new examples
Permanent Source URI:
https://staging.physiomeproject.org/workspace/43b/rawfile/84d1c2103ca337e69ddf1c2b37e9d9fbaf6329e2/BG Tutorial Mechanical Systems/Inverted pendulum wtih FEM.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 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="Js_per_m">
        <unit units="joule"/>
        <unit units="second"/>
        <unit exponent="-1" 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" units="dimensionless"/>
        <variable name="theta1" units="dimensionless"/>
        <variable name="theta2" units="dimensionless"/>
        <variable name="costheta" units="dimensionless"/>
        <variable name="sintheta" units="dimensionless"/>
        <!--var theta_rad: radian;
var omega: rad_per_s;-->
        <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" units="Js_per_m"/>
        <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_B_1" units="J_per_m"/>
        <variable name="u_B_2" units="J_per_m"/>
        <variable name="u_truss_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="1" name="q_A_1" units="metre"/>
        <variable initial_value="0" name="q_A_2" units="metre"/>
        <variable initial_value="0.0" name="q_B_1" units="metre"/>
        <variable initial_value="0.0" name="q_B_2" units="metre"/>
        <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</ci>
                <apply>
                    <divide/>
                    <ci>x_AB</ci>
                    <ci>l_AB</ci>
                </apply>
            </apply>
            <apply>
                <eq/>
                <ci>sintheta</ci>
                <apply>
                    <divide/>
                    <ci>y_AB</ci>
                    <ci>l_AB</ci>
                </apply>
            </apply>
            <apply>
                <eq/>
                <ci>strain_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>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</ci>
                        <ci>strain_rate_AB</ci>
                    </apply>
                </apply>
            </apply>
            <apply>
                <eq/>
                <ci>u_A_1</ci>
                <apply>
                    <times/>
                    <ci>u_truss_AB</ci>
                    <ci>costheta</ci>
                </apply>
            </apply>
            <apply>
                <eq/>
                <ci>u_A_2</ci>
                <apply>
                    <times/>
                    <ci>u_truss_AB</ci>
                    <ci>sintheta</ci>
                </apply>
            </apply>
            <apply>
                <eq/>
                <ci>u_B_1</ci>
                <apply>
                    <times/>
                    <apply>
                        <minus/>
                        <ci>u_truss_AB</ci>
                    </apply>
                    <ci>costheta</ci>
                </apply>
            </apply>
            <apply>
                <eq/>
                <ci>u_B_2</ci>
                <apply>
                    <times/>
                    <apply>
                        <minus/>
                        <ci>u_truss_AB</ci>
                    </apply>
                    <ci>sintheta</ci>
                </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>
                        <apply>
                            <times/>
                            <ci>u_truss_AB</ci>
                            <ci>costheta</ci>
                        </apply>
                    </apply>
                    <ci>m_A</ci>
                </apply>
            </apply>
            <apply>
                <eq/>
                <apply>
                    <diff/>
                    <bvar>
                        <ci>t</ci>
                    </bvar>
                    <ci>v_A_2</ci>
                </apply>
                <apply>
                    <minus/>
                    <apply>
                        <divide/>
                        <apply>
                            <plus/>
                            <ci>F_A</ci>
                            <apply>
                                <times/>
                                <ci>u_truss_AB</ci>
                                <ci>sintheta</ci>
                            </apply>
                        </apply>
                        <ci>m_A</ci>
                    </apply>
                    <ci>g</ci>
                </apply>
            </apply>
            <!--        ode(v_B_1, t) = -u_truss_AB*costheta/m_B;
        ode(v_B_2, t) = -u_truss_AB*sintheta/m_B-g;-->
            <apply>
                <eq/>
                <apply>
                    <diff/>
                    <bvar>
                        <ci>t</ci>
                    </bvar>
                    <ci>q_A_1</ci>
                </apply>
                <ci>v_A_1</ci>
            </apply>
            <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</ci>
                <apply>
                    <divide/>
                    <apply>
                        <times/>
                        <apply>
                            <arccos/>
                            <ci>costheta</ci>
                        </apply>
                        <cn cellml:units="dimensionless">180</cn>
                    </apply>
                    <ci>PI</ci>
                </apply>
            </apply>
            <apply>
                <eq/>
                <ci>theta2</ci>
                <apply>
                    <divide/>
                    <apply>
                        <times/>
                        <apply>
                            <arcsin/>
                            <ci>sintheta</ci>
                        </apply>
                        <cn cellml:units="dimensionless">180</cn>
                    </apply>
                    <ci>PI</ci>
                </apply>
            </apply>
            <apply id="theta_calculation">
                <eq/>
                <ci>theta</ci>
                <piecewise>
                    <piece>
                        <ci>theta1</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</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</ci>
                        </apply>
                        <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</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>
        <!--theta_rad = theta*PI/180{dimensionless};
omega = ode(theta_rad, t);-->
    </component>
</model>