Simulation of the dynamic interaction of rail vehicle pantograph and catenary through a modal approach

Abstract

Electric trains rely on the pantograph and the overhead catenary system (OCS) to receive energy from the power main lines. The purpose of this article is to elaborate on the simulation of pantograph and catenary dynamic interaction. The main feature of this method is using a fast analytical approach in order to simulate the entire catenary and to avoid using the finite elements method. This method is making use of the system vibration modes. Therefore, vital phenomenon such as the wave propagation and reflections can also be simulated. Additionally, droppers under compression are simulated as buckled columns that can endure certain amount of compression forces. Inclusion of the proper OCS initial conditions is also a unique add on to this procedure. Evaluation of both static and dynamic responses of catenary is by comparing with the results from some other available software programs. Validation of the results is according to EN 50318: 2002 standard document.

Appendix A: Derivation of the stiffness, damping and mass matrices

Variable	Description	Variable	Description
${\mathbf {M}}$	Mass matrix of whole system	$m_{p1i}$	Mass of collector strip of ith pantograph (Figure 5)
${\mathbf {C}}$	Damping matrix of whole system	$m_{p2i}$	Second mass of ith pantograph (Figure 5)
${\mathbf {K}}$	Stiffness matrix of whole system	${\mathbf {slk}}$	Vector shows which dropper is on and which one is off due to slaking (0 or 1)
${\mathbf {f}}$	Excitation factor	${\mathbf {slkm}}$	Slaking vector of droppers which connects to messenger cable (0 or 1)
${\mathbf {x}}$	Space state variable	${\mathbf {slks}}$	Slaking vector of droppers which connects to stitch wires (0 or 1)
${\mathbf {x}}_{{\mathbf {m}}}$	Messenger cable Rayleigh-Ritz coefficient	${\mathbf {slksi}}$	Slaking vector of droppers which connects to ith stitch wires (0 or 1)
${\mathbf {x}}_{{\mathbf {c}}}$	Contact wire Rayleigh–Ritz coefficient	${\mathbf {k}}_{{\mathbf {dr}}}$	Stiffness vector of droppers
${\mathbf {x}}_{{\mathbf {si}}}$	ith Stitch wire Rayleigh–Ritz coefficient	${\mathbf {k}}_{{\mathbf {drm}}}$	Stiffness vector of droppers which connects to messenger cable
${\mathbf {x}}_{{\mathbf {pi}}}$	ith Pantograph Rayleigh–Ritz coefficient	${\mathbf {k}}_{{\mathbf {drs}}}$	Stiffness vector of droppers which connects to stitch wires
${\mathbf {M}}_{{\mathbf {m}}}$	Mass matrix of messenger cable	${\mathbf {k}}_{{\mathbf {drsi}}}$	Stiffness vector of droppers which connects to ith stitch wire
${\mathbf {M}}_{{\mathbf {c}}}$	Mass matrix of contact wire	${\mathbf {k}}_{{\mathbf {su}}}$	Stiffness vector of messenger cable supports
${\mathbf {M}}_{{\mathbf {si}}}$	Mass matrix of ith stitch wire	${\mathbf {k}}_{{\mathbf {stc}}}$	Stiffness vector of stitch wire and messenger cable clamps
${\mathbf {M}}_{{\mathbf {pi}}}$	Mass matrix of ith pantograph	${\mathbf {k}}_{{\mathbf {re}}}$	Stiffness vector of registration arms
${\mathbf {K}}_{{\mathbf {m}}}$	Stiffness matrix of messenger cable	${\mathbf {k}}_{{\mathbf {re}}}$	Stiffness vector of contact point of droppers
${\mathbf {K}}_{{\mathbf {c}}}$	Stiffness matrix of contact wire	$k_{c}$	Stiffness of contact point (pantograph and catenary)
${\mathbf {K}}_{{\mathbf {si}}}$	Stiffness matrix of ith stitch wire	$C_{c}$	Damping of contact point (pantograph and catenary)
${\mathbf {K}}_{{\mathbf {p}}}$	Stiffness matrix of pantograph	$k_{p1i}$	Stiffness of collector strip of ith pantograph (Figure 5)
${\mathbf {K}}_{{\mathbf {mc}}}$	Relative stiffness matrix between messenger cable and contact wire	$k_{p2i}$	Second stiffness of ith pantograph (Figure 5)
${\mathbf {K}}_{{\mathbf {msi}}}$	Relative stiffness matrix between messenger cable and ith stitch wire	${\varvec{c}}_{{\mathbf {dr}}}$	Damping vector of droppers
${\mathbf {K}}_{{\mathbf {csi}}}$	Relative stiffness matrix between contact wire and ith stitch wire	${\varvec{c}}_{{\mathbf {drm}}}$	Damping vector of droppers which connects to messenger cable
${\mathbf {K}}_{{\mathbf {cpi}}}$	Relative stiffness matrix between contact wire and ith pantograph	${\mathbf {c}}_{{\mathbf {drs}}}$	Damping vector of droppers which connects to stitch wires
${{{\varvec{\upalpha }} m}}$	Mass coefficient of messenger cable	${\mathbf {c}}_{{\mathbf {drsi}}}$	Damping vector of droppers which connects to ith stitch wire
${{{\varvec{\upalpha }} c}}$	Mass coefficient of contact wire	${\mathbf {c}}_{{\mathbf {su}}}$	Damping vector of messenger cable supports
${{{\varvec{\upalpha }} s}}$	Mass coefficient of stitch wire	${\mathbf {c}}_{{\mathbf {stc}}}$	Damping vector of stitch wire and messenger cable clamps

$m_\mathrm{ucdr}$	Mass of upper clamp of dropper	${\mathbf {c}}_{{\mathbf {re}}}$	Damping vector of registration arms
$m_\mathrm{lcdr}$	Mass of lower clamp of dropper	${\mathbf {c}}_{{\mathbf {re}}}$	Damping vector of contact point of droppers
$m_\mathrm{cst}$	Mass of connection clamp of stitch wire and messenger cable	$c_{p1i}$	Damping of collector strip of ith pantograph (Figure 5)
${em}_\mathrm{re}$	Equivalent mass of registration arm	$c_{p2i}$	Second Damping of ith pantograph (Figure 5)
$x_\mathrm{drm}$	Location of droppers in messenger cable	${{{\varvec{\Omega }} }}_{{\mathbf {m}}}^{{\mathbf {2}}}$	Diagonal matrix of square of messenger cable natural frequencies.
$x_\mathrm{drc}$	Location of droppers in contact wire	${{{\varvec{\Omega }} }}_{c}^{2}$	Diagonal matrix of square of contact wire natural frequencies
$x_\mathrm{drs}$	Location of droppers in stitch wire	${{{\varvec{\Omega }} }}_{s}^{2}$	Diagonal matrix of square of stitch wire natural frequencies.
$x_\mathrm{drcsi}$	Location of droppers of ith stitch wire in contact wire	${\mathbf {Z}}_{{\mathbf {m}}}^{{\mathbf {2}}}$	Diagonal matrix of square of messenger cable natural frequencies.
$x_\mathrm{rem}$	Location of registration arm in messenger cable	${\mathbf {Z}}_{{\mathbf {c}}}^{{\mathbf {2}}}$	Diagonal matrix of square of contact wire natural frequencies.
$x_\mathrm{rec}$	Location of registration arm in contact wire	${\mathbf {Z}}_{{\mathbf {s}}}^{{\mathbf {2}}}$	Diagonal matrix of square of stitch wire natural frequencies
$x_\mathrm{stcm}$	Location of stitch wires connection on messenger cable	$\mathbf {u}_{{\mathbf {3\times 1}}}$	Input vector of system including weight, dead load of droppers and uplift force
$x_\mathrm{stcmi}$	Location of ith stitch wire connection on messenger cable	$F_\mathrm{uplift}$	Uplift force which is applied to pantograph
$x_\mathrm{stcs}$	Location of stitch wire connection on stitch wire ($l_\mathrm{st}, 0)$.	${\mathbf {f}}_{{\mathbf {gm}}}$	Gravity force on messenger cable
${\mathbf {x}}_{{\mathbf {pac}}}{\mathbf {(}}t{\mathbf {)}}$	Location vector of pantographs on contact wire	${\mathbf {f}}_{{\mathbf {gc}}}$	Gravity force on contact wire
$x_\mathrm{paci}(t)$	Location of ith pantograph on contact wire	${\mathbf {f}}_{{\mathbf {gsi}}}$	Gravity force on ith Stitch wire
${{\varvec{\Phi }} \mathbf {m(x,n)}}$	Mode shape matrix of messenger cable in n mode and x location	${\mathbf {f}}_{{\mathbf {gpi}}}$	Gravity force on ith pantograph
${\varvec{\Phi }} {\varvec{s(x,n)}}$	Mode shape matrix of stitch wire in n mode and x location	${\mathbf {f}}_{{\mathbf {dm}}}$	Dead loads of droppers on messenger cable
${{\varvec{\Phi }} \mathbf {c(x,n)}}$	Mode shape matrix of contact wire in n mode and x location	${\mathbf {f}}_{{\mathbf {dc}}}$	Dead loads of droppers on contact wire
$\varvec{\acute{\Phi }}{\mathbf {c(x,n)}}$	Gradient of mode shape matrix of contact wire in n mode and x location	${\mathbf {f}}_{{\mathbf {dsi}}}$	Dead loads of droppers on stitch wire
nm	Number of considered mode shape for messenger cable	${\rho A}_\mathrm{m}$	Mass per unit length of messenger cable
nc	Number of considered mode shape for contact wire	${\rho A}_\mathrm{c}$	Mass per unit length of contact wire
ns	Number of considered mode shape for stitch wire	${\rho A}_\mathrm{s}$	Mass per unit length of stitch wire
nsp	Number of spans	${F0}_\mathrm{drm}$	Dead load of droppers connected to messenger cable
ndrm	Number of connected dropper to messenger cable	${F0}_\mathrm{drs}$	Dead load of droppers connected to stitch wire
ndrsi	Number of connected dropper to ith stitch wire	$\Delta _\mathrm{m}$	The initial location of dropper clamps on messenger cable
ndr	Number of dropper	$\Delta _\mathrm{cm}$	The initial location of dropper clamps on contact wire (droppers connected to messenger cable)
$l_\mathrm{st}$	Length of stitch wire	$\Delta _\mathrm{cs}$	The initial location of dropper clamps on contact wire (droppers connected to stitch wire)
		$\Delta _\mathrm{s}$	The initial location of dropper clamps on stitch wire

$$\begin{aligned}&{\mathbf {M}}\ddot{\mathbf{x}}+{\mathbf {C}}\dot{\mathbf{x}}+{\mathbf {Kx}}={\mathbf {f}}\\&{\mathbf {x}}=\left[ {\begin{array}{*{20}c} {\mathbf {x}}_{{\mathbf {m}}} &{} {\mathbf {x}}_{{\mathbf {c}}} &{} {\mathbf {x}}_{{\mathbf {s1}}} &{} {\mathbf {x}}_{{\mathbf {s2}}} &{} \cdots &{} {\mathbf {x}}_{{\mathbf {p1}}} &{} {\mathbf {x}}_{{\mathbf {p2}}} &{} \cdots \\ \end{array} } \right] ^{T} \\&{\mathbf {M}}=\left[ {\begin{array}{*{20}c} {\mathbf {M}}_{{\mathbf {m}}} &{} {\mathbf {0}} &{} {\mathbf {0}} &{} {\mathbf {0}} &{} {{\cdots }} &{} {\mathbf {0}} &{} {\mathbf {0}} &{} {\mathbf {0}}\\ {\mathbf {0}} &{} {\mathbf {M}}_{{\mathbf {c}}} &{} {\mathbf {0}} &{} {\mathbf {0}} &{} {{\cdots }} &{} {\mathbf {0}} &{} {\mathbf {0}} &{} {\mathbf {0}}\\ {\mathbf {0}} &{} {\mathbf {0}} &{} {\mathbf {M}}_{{\mathbf {s1}}} &{} {\mathbf {0}} &{} {{\cdots }} &{} {\mathbf {0}} &{} {\mathbf {0}} &{} {\mathbf {0}}\\ {\mathbf {0}} &{} {\mathbf {0}} &{} {\mathbf {0}} &{} {\mathbf {M}}_{{\mathbf {s2}}} &{} {{\cdots }} &{} {\mathbf {0}} &{} \mathbf {0} &{} {\mathbf {0}}\\ {{\vdots }} &{} {{\vdots }} &{} {{\vdots }} &{} {{\vdots }} &{} {\mathbf {\ddots }} &{} {{\vdots }} &{} {{\vdots }} &{} {{\vdots }}\\ {\mathbf {0}} &{} {\mathbf {0}} &{} {\mathbf {0}} &{} {\mathbf {0}} &{} {{\cdots }} &{} {\mathbf {M}}_{{\mathbf {p1}}} &{} {\mathbf {0}} &{} {\mathbf {0}}\\ {\mathbf {0}} &{} {\mathbf {0}} &{} {\mathbf {0}} &{} {\mathbf {0}} &{} {{\cdots }} &{} {\mathbf {0}} &{} {\mathbf {M}}_{{\mathbf {p2}}} &{} \mathbf {0}\\ {\mathbf {0}} &{} {\mathbf {0}} &{} {\mathbf {0}} &{} {\mathbf {0}} &{} {{\cdots }} &{} {\mathbf {0}} &{} {\mathbf {0}} &{} {\mathbf {\ddots }}\\ \end{array} } \right] \, {\mathbf {B=}}\left[ {\begin{array}{*{20}c} {\mathbf {f}}_{{\mathbf {gm}}} &{} {\mathbf {f}}_{{\mathbf {dm}}} &{} {\mathbf {0}}\\ {\mathbf {f}}_{{\mathbf {gc}}} &{} {\mathbf {f}}_{{\mathbf {dc}}} &{} {\mathbf {0}}\\ {\mathbf {f}}_{{\mathbf {gs1}}} &{} {\mathbf {f}}_{{\mathbf {ds1}}} &{} {\mathbf {0}}\\ {\mathbf {f}}_{{\mathbf {gs2}}} &{} {\mathbf {f}}_{{\mathbf {ds2}}} &{} {\mathbf {0}}\\ {{\vdots }} &{} {{\vdots }} &{} {\mathbf {0}}\\ {\mathbf {f}}_{{\mathbf {gp1}}} &{} {\mathbf {0}} &{} \left[ {\begin{array}{*{20}c} {\mathbf {0}}\\ {\mathbf {1}}\\ \end{array} } \right] \\ {\mathbf {f}}_{{\mathbf {gp2}}} &{} {\mathbf {0}} &{} \left[ {\begin{array}{*{20}c} {\mathbf {0}}\\ {\mathbf {1}}\\ \end{array} } \right] \\ {{\vdots }} &{} {{\vdots }} &{} {{\vdots }}\\ \end{array} } \right] \\&{\mathbf {K=}}\left[ {\begin{array}{*{20}c} {\mathbf {K}}_{{\mathbf {m}}} &{} {\mathbf {K}}_{{\mathbf {mc}}} &{} {\mathbf {K}}_{{\mathbf {ms1}}} &{} {\mathbf {K}}_{{\mathbf {ms2}}} &{} {{\cdots }} &{} {\mathbf {0}} &{} {\mathbf {0}} &{} {\mathbf {0}}\\ {\mathbf {K}}_{{\mathbf {cm}}} &{} {\mathbf {K}}_{{\mathbf {c}}} &{} {\mathbf {K}}_{{\mathbf {cs1}}} &{} {\mathbf {K}}_{{\mathbf {cs2}}} &{} {{\cdots }} &{} \left[ {\begin{array}{*{20}c} {\mathbf {k}}_{{\mathbf {cp1}}}\left( t \right) &{} {\mathbf {0}}\\ \end{array} } \right] &{} \left[ {\begin{array}{*{20}c} {\mathbf {k}}_{{\mathbf {cp2}}}\left( t \right) &{} {\mathbf {0}}\\ \end{array} } \right] &{} {{\cdots }}\\ {\mathbf {K}}_{{\mathbf {s1m}}} &{} {\mathbf {K}}_{{\mathbf {s1c}}} &{} {\mathbf {K}}_{{\mathbf {s1}}} &{} {\mathbf {0}} &{} {{\cdots }} &{} {\mathbf {0}} &{} {\mathbf {0}} &{} {\mathbf {0}}\\ {\mathbf {K}}_{{\mathbf {s2m}}} &{} {\mathbf {K}}_{{\mathbf {s2c}}} &{} {\mathbf {0}} &{} {\mathbf {K}}_{{\mathbf {s2}}} &{} {{\cdots }} &{} {\mathbf {0}} &{} {\mathbf {0}} &{} {\mathbf {0}}\\ {{\vdots }} &{} {{\vdots }} &{} {{\vdots }} &{} {{\vdots }} &{} {\mathbf {\ddots }} &{} {{\vdots }} &{} {{\vdots }} &{} {{\vdots }}\\ {\mathbf {0}} &{} \left[ {\begin{array}{*{20}c} {\mathbf {k}}_{{\mathbf {p1c}}}{\mathbf {(}}t{\mathbf {)}}\\ {\mathbf {0}}\\ \end{array} } \right] &{} {\mathbf {0}} &{} {\mathbf {0}} &{} {{\cdots }} &{} {\mathbf {K}}_{{\mathbf {p1}}} &{} {\mathbf {0}} &{} {\mathbf {0}}\\ {\mathbf {0}} &{} \left[ {\begin{array}{*{20}c} {\mathbf {k}}_{{\mathbf {p2c}}}{\mathbf {(}}t{\mathbf {)}}\\ {\mathbf {0}}\\ \end{array} } \right] &{} {\mathbf {0}} &{} {\mathbf {0}} &{} {{\cdots }} &{} {\mathbf {0}} &{} {\mathbf {K}}_{{\mathbf {p2}}} &{} {\mathbf {0}}\\ {\mathbf {0}} &{} {{\vdots }} &{} {\mathbf {0}} &{} \mathbf {0} &{} {{\cdots }} &{} {\mathbf {0}} &{} {\mathbf {0}} &{} {\mathbf {\ddots }}\\ \end{array} } \right] \\&{\mathbf {C}}{\varvec{-K}}\approx 0 \\&{\mathbf {c}}_{{{\mathbf {pic}}}_{{\mathbf {2\times nc}}}} \end{aligned}$$

should be calculated as follows.

Parameter	Description
${\mathbf {M}}_{{{\mathbf {m}}}_{{\mathbf {nm\times nm}}}}$	$\left( {{{\varvec{\upalpha }} \mathbf {m}}}_{{\mathbf {nm\times 1}}}{\mathbf {.}}m_{ucdr}{\mathbf {I}}_{{\mathbf {1\times nm}}} \right) .\left( {{{{\varvec{\Phi }}\mathbf {m}(}}{\mathbf {x}}_{{\mathbf {drm}}}{\mathbf {,n)}}}_{{\mathbf {nm\times ndrm}}}\left( {{{\varvec{\Phi } \mathbf{m}(}}{\mathbf {x}}_{{\mathbf {drm}}}{\mathbf {,n)}}}_{{\mathbf {nm\times ndrm}}} \right) ^{{\mathbf {T}}} \right) {\mathbf {+}}$
	$\left( {{{\varvec{\upalpha }} \mathbf { m}}}_{{\mathbf {nm\times 1}}}{\mathbf {.}}m_{cst}{\mathbf {I}}_{{\mathbf {1\times nm}}} \right) .\left( {{{\varvec{\Phi } \mathbf{m}(}}{\mathbf {x}}_{{\mathbf {stcm}}}{\mathbf {,n)}}}_{{\mathbf {nm\times 2(nsp+1)}}}\left( {{{\varvec{\Phi } \mathbf{m}(}}{\mathbf {x}}_{{\mathbf {stcm}}}{\mathbf {,n)}}}_{{\mathbf {nm\times 2(nsp+1)}}} \right) ^{{\mathbf {T}}} \right) {\mathbf {+}}{\mathbf {I}}_{{\mathbf {nm\times nm}}}$
${\mathbf {M}}_{{{\mathbf {c}}}_{{\mathbf {nc\times nc}}}}$	$\left( {{{\varvec{\upalpha }} \mathbf { c}}}_{{\mathbf {nc\times 1}}}{\mathbf {.}}m_{lcdr}{\mathbf {\times }}{\mathbf {I}}_{{\mathbf {1\times nc}}} \right) .\left( {{{\varvec{\Phi } \mathbf{c}(}}{\mathbf {x}}_{{\mathbf {dr}}}{\mathbf {,n)}}}_{{\mathbf {nc\times ndr}}}\left( {{{\varvec{\Phi } \mathbf{c}(}}{\mathbf {x}}_{{\mathbf {dr}}}{\mathbf {,n)}}}_{{\mathbf {nc\times ndr}}} \right) ^{{\mathbf {T}}} \right) {\mathbf {+}}\left( {{{\varvec{\upalpha }} \mathbf { c}}}_{{\mathbf {nc\times 1}}}{\mathbf {.}}{{\mathbf {em}}}_{{\mathbf {re}}}{\mathbf {I}}_{{\mathbf {1\times nc}}} \right) .\left( {{{\varvec{\Phi } \mathbf{c}(}}{\mathbf {x}}_{{\mathbf {rec}}}{\mathbf {,n)}}}_{{\mathbf {nc\times nsp+1}}}\left( {{{\varvec{\Phi } \mathbf{c}(}}{\mathbf {x}}_{{\mathbf {rec}}}{\mathbf {,n)}}}_{{\mathbf {nc\times nsp+1}}} \right) ^{{\mathbf {T}}} \right) {\mathbf {+}}{\mathbf {I}}_{{\mathbf {nc\times nc}}}$
${\mathbf {M}}_{{{\mathbf {si}}}_{{\mathbf {ns\times ns}}}}$	$\left( {{{\varvec{\upalpha }} \mathbf { s}}}_{{\mathbf {ns\times 1}}}{\mathbf {.}}{\mathbf {m}}_{{\mathbf {ucdr}}}{\mathbf {I}}_{{\mathbf {1\times ns}}} \right) .\left( {{{\varvec{\Phi } \mathbf{s}(}}{\mathbf {x}}_{{\mathbf {drs}}}{\mathbf {,n)}}}_{{\mathbf {ns\times ndrs}}}\left( {{{\varvec{\Phi } \mathbf{s}(}}{\mathbf {x}}_{{\mathbf {drs}}}{\mathbf {,n)}}}_{{\mathbf {ns\times ndrs}}} \right) ^{{\mathbf {T}}} \right) {\mathbf {+}}\left( {{{\varvec{\upalpha }} \mathbf { s}}}_{{\mathbf {ns\times 1}}}{\mathbf {.}}{\mathbf {m}}_{{\mathbf {cst}}}{\mathbf {I}}_{{\mathbf {1\times ns}}} \right) .\left( {{{\varvec{\Phi } \mathbf{s}(}}{\mathbf {x}}_{{\mathbf {stcs}}}{\mathbf {,n)}}}_{{\mathbf {ns\times 2}}}\left( {{{\varvec{\Phi } \mathbf{s}(}}{\mathbf {x}}_{{\mathbf {stcs}}}{\mathbf {,n)}}}_{{\mathbf {ns\times 2}}} \right) ^{{\mathbf {T}}} \right) {\mathbf {+}}{\mathbf {I}}_{{\mathbf {ns\times ns}}}$
${\mathbf {M}}_{{{\mathbf {pi}}}_{{\mathbf {2\times 2}}}}$	$\left[ {\begin{array}{*{20}c} {\mathbf {m}}_{{\mathbf {p1i}}} &{} {\mathbf {0}} \\ {\mathbf {0}} &{} {\mathbf {m}}_{{\mathbf {p2i}}}\\ \end{array} } \right]$
${\mathbf {K}}_{{{\mathbf {m}}}_{{\mathbf {nm\times nm}}}}$	$\left( \left( {{{\varvec{\upalpha }} \mathbf { m}}}_{{\mathbf {nm\times 1}}}\left( {\mathbf {slkm.}}{\mathbf {k}}_{{\mathbf {drm}}} \right) \right) _{{\mathbf {nm\times ndrm}}}{\mathbf {.}}{{{\varvec{\Phi } \mathbf{m}}}\left( {\mathbf {x}}_{{\mathbf {drm}}}{\mathbf {,n}} \right) }_{{\mathbf {nm\times ndrm}}} \right) \left( {{{\varvec{\Phi } \mathbf{m}}}\left( {\mathbf {x}}_{{\mathbf {drm}}}{\mathbf {,n}} \right) }_{{\mathbf {nm\times ndrm}}} \right) ^{{\mathbf {T}}}$
	${\mathbf {+}}\left( \left( {{{\varvec{\upalpha }} \mathbf { m}}}_{{\mathbf {nm\times 1}}}{\mathbf {.}}{\mathbf {k}}_{{\mathbf {stc}}} \right) _{{\mathbf {nm\times 2(nsp+1)}}}{\mathbf {.}}{{{\varvec{\Phi } \mathbf{m}}}\left( {\mathbf {x}}_{{\mathbf {stcm}}}{\mathbf {,n}} \right) }_{{\mathbf {nm\times 2(nsp+1)}}} \right) \left( {{{\varvec{\Phi } \mathbf{m}}}\left( {\mathbf {x}}_{{\mathbf {stcm}}}{\mathbf {,n}} \right) }_{{\mathbf {nm\times 2(nsp+1)}}} \right) ^{{\mathbf {T}}}$
	${\mathbf {+}}\left( \left( {{{\varvec{\upalpha }} \mathbf { m}}}_{{\mathbf {nm\times 1}}}{\mathbf {.}}{\mathbf {k}}_{{\mathbf {su}}} \right) {\mathbf {.}}{{{\varvec{\Phi } \mathbf{m}}}\left( {\mathbf {x}}_{{\mathbf {rem}}}{\mathbf {,n}} \right) }_{{\mathbf {nm\times nsp+1}}} \right) \left( {{{\varvec{\Phi } \mathbf{m}}}\left( {\mathbf {x}}_{{\mathbf {rem}}}{\mathbf {,n}} \right) }_{{\mathbf {nm\times nsp+1}}} \right) ^{{\mathbf {T}}}{\mathbf {+}}{{{\varvec{\upomega }} }}^{{\mathbf {2}}}_{{{\mathbf {m}}}_{{\mathbf {nm\times nm}}}}$
${\mathbf {K}}_{{{\mathbf {c}}}_{{\mathbf {nc\times nc}}}}$	$\left( \left( {{{\varvec{\upalpha }} \mathbf { c}}}_{{\mathbf {nc\times 1}}}\left( {\mathbf {slk.}}{\mathbf {k}}_{{\mathbf {dr}}} \right) \right) _{{\mathbf {nc\times ndr}}}{\mathbf {.}}{{{\varvec{\Phi } \mathbf{c}}}\left( {\mathbf {x}}_{{\mathbf {drc}}}{\mathbf {,n}} \right) }_{{\mathbf {nc\times ndr}}} \right) \left( {{{\varvec{\Phi } \mathbf{c}}}\left( {\mathbf {x}}_{{\mathbf {drc}}}{\mathbf {,n}} \right) }_{{\mathbf {nc\times ndr}}} \right) ^{{\mathbf {T}}}{\mathbf {+}}\left( \left( {{{\varvec{\upalpha }} \mathbf { c}}}_{{\mathbf {nc\times 1}}}{\mathbf {.}}{\mathbf {k}}_{{\mathbf {re}}} \right) _{{\mathbf {nc\times nsp+1}}}{\mathbf {.}}{{{\varvec{\Phi } \mathbf{c}}}\left( {\mathbf {x}}_{{\mathbf {rec}}}{\mathbf {,n}} \right) }_{{\mathbf {nc\times nsp+1}}} \right) \left( {{{\varvec{\Phi } \mathbf{c}}}\left( {\mathbf {x}}_{{\mathbf {rec}}}{\mathbf {,n}} \right) }_{{\mathbf {nc\times nsp+1}}} \right) ^{{\mathbf {T}}}$
	${\mathbf {+}}\left( \left( {{{\varvec{\upalpha }} \mathbf { c}}}_{{\mathbf {nm\times 1}}}{\mathbf {.}}{\mathbf {k}}_{{\mathbf {c}}} \right) _{{\mathbf {nc\times npan}}}{\mathbf {.}}{{{\varvec{\Phi } \mathbf{c}}}\left( {\mathbf {x}}_{{\mathbf {pac}}}{\mathbf {(}}t{\mathbf {),n}} \right) }_{{\mathbf {nc\times npan}}} \right) \left( {{{\varvec{\Phi } \mathbf{c}}}\left( {\mathbf {x}}_{{\mathbf {pan}}}{\mathbf {(}}t{\mathbf {),n}} \right) }_{{\mathbf {nc\times npan}}} \right) ^{{\mathbf {T}}}{\mathbf {+}}{{{\varvec{\Omega }} }}^{{\mathbf {2}}}_{{{\mathbf {c}}}_{{\mathbf {nc\times nc}}}}$
${\mathbf {K}}_{{{\mathbf {si}}}_{{\mathbf {ns\times ns}}}}$	$\left( \left( {{{\varvec{\upalpha }} \mathbf { s}}}_{{\mathbf {ns\times 1}}}\left( {\mathbf {slksi.}}{\mathbf {k}}_{{\mathbf {drsi}}} \right) \right) _{{\mathbf {ns\times ndrsi}}}{\mathbf {.}}{{{\varvec{\Phi } \mathbf{s}}}\left( {\mathbf {x}}_{{\mathbf {drs}}}{\mathbf {,n}} \right) }_{{\mathbf {ns\times ndrsi}}} \right) \left( {{{\varvec{\Phi } \mathbf{s}}}\left( {\mathbf {x}}_{{\mathbf {drs}}}{\mathbf {,n}} \right) }_{{\mathbf {ns\times ndrsi}}} \right) ^{{\mathbf {T}}}{\mathbf {+}}\left( \left( {{{\varvec{\upalpha }} \mathbf { s}}}_{{\mathbf {ns\times 1}}}{\mathbf {.}}{\mathbf {k}}_{{\mathbf {stc}}} \right) _{{\mathbf {ns\times 2}}}{\mathbf {.}}{{{\varvec{\Phi } \mathbf{s}}}\left( {\mathbf {x}}_{{\mathbf {stcs}}}{\mathbf {,n}} \right) }_{{\mathbf {ns\times 2}}} \right) \left( {{{\varvec{\Phi } \mathbf{s}}}\left( {\mathbf {x}}_{{\mathbf {stcs}}}{\mathbf {,n}} \right) }_{{\mathbf {ns\times 2}}} \right) ^{{\mathbf {T}}}{\mathbf {+}}{{{\varvec{\Omega }} }}^{{\mathbf {2}}}_{{{\mathbf {s}}}_{{\mathbf {ns\times ns}}}}$
${\mathbf {K}}_{{{\mathbf {mc}}}_{{\mathbf {nm\times nc}}}}$	$\left( {\left( {{{\varvec{\upalpha }} \mathbf { m}}}_{{\mathbf {nm\times 1}}}\left( {\mathbf {slkm.}}{\mathbf {k}}_{{\mathbf {drm}}} \right) \right) {\mathbf { }}}_{{\mathbf {nm\times ndrm}}}{\mathbf {.}}{{{\varvec{\Phi } \mathbf{m}}}\left( {\mathbf {x}}_{{\mathbf {drm}}}{\mathbf {,n}} \right) }_{{\mathbf {nm\times ndrm}}} \right) \left( {{{\varvec{\Phi } \mathbf{c}}}\left( {\mathbf {x}}_{{\mathbf {drm}}}{\mathbf {,n}} \right) }_{{\mathbf {nc\times ndrm}}} \right) ^{{\mathbf {T}}}$
${\mathbf {K}}_{{{\mathbf {cm}}}_{{\mathbf {nc\times nm}}}}$	$\left( {\left( {{{\varvec{\upalpha }} \mathbf { c}}}_{{\mathbf {nc\times 1}}}\left( {\mathbf {slkm.}}{\mathbf {k}}_{{\mathbf {drm}}} \right) \right) {\mathbf { }}}_{{\mathbf {nc\times ndrm}}}{\mathbf {.}}{{{\varvec{\Phi } \mathbf{c}}}\left( {\mathbf {x}}_{{\mathbf {drm}}}{\mathbf {,n}} \right) }_{{\mathbf {nc\times ndrm}}} \right) \left( {{{\varvec{\Phi } \mathbf{m}}}\left( {\mathbf {x}}_{{\mathbf {drm}}}{\mathbf {,n}} \right) }_{{\mathbf {nm\times ndrm}}} \right) ^{{\mathbf {T}}}$
${\mathbf {K}}_{{\mathbf {msi}}}{\mathbf {K}}_{{{{\mathbf {msi}}}}_{{\mathbf {nm\times nsi}}}}$	$\left( {\left( {{{\varvec{\upalpha }} \mathbf { m}}}_{{\mathbf {nm\times 1}}}{\mathbf {k}}_{{\mathbf {stc}}} \right) {\mathbf { }}}_{{\mathbf {nm\times 2}}}{\mathbf {.}}{{{\varvec{\Phi } \mathbf{m}}}\left( {\mathbf {x}}_{{\mathbf {stcmi}}}{\mathbf {,n}} \right) }_{{\mathbf {nm\times 2}}} \right) \left( {{{\varvec{\Phi } \mathbf{s}}}\left( {\mathbf {x}}_{{\mathbf {stcs}}}{\mathbf {,n}} \right) }_{{\mathbf {ns\times 2}}} \right) ^{{\mathbf {T}}}$
${\mathbf {K}}_{{{\mathbf {sic}}}_{{\mathbf {ns\times nc}}}}$	$\left( {\left( {{{\varvec{\upalpha }} \mathbf { s}}}_{{\mathbf {ns\times 1}}}{\mathbf {k}}_{{\mathbf {stc}}} \right) {\mathbf { }}}_{{\mathbf {ns\times 2}}}{\mathbf {.}}{{{\varvec{\Phi } \mathbf{s}}}\left( {\mathbf {x}}_{{\mathbf {stcs}}}{\mathbf {,n}} \right) }_{{\mathbf {ns\times 2}}} \right) \left( {{{\varvec{\Phi } \mathbf{m}}}\left( {\mathbf {x}}_{{\mathbf {stcmi}}}{\mathbf {,n}} \right) }_{{\mathbf {ns\times 2}}} \right) ^{{\mathbf {T}}}$
${\mathbf {K}}_{{{\mathbf {csi}}}_{{\mathbf {nc\times ns}}}}$	$\left( {\left( {{{\varvec{\upalpha }} \mathbf { c}}}_{{\mathbf {nc\times 1}}}\left( {\mathbf {slksi.}}{\mathbf {k}}_{{\mathbf {drsi}}} \right) \right) {\mathbf { }}}_{{\mathbf {nc\times ndrsi}}}{\mathbf {.}}{{{\varvec{\Phi } \mathbf{c}}}\left( {\mathbf {x}}_{{\mathbf {drcsi}}}{\mathbf {,n}} \right) }_{{\mathbf {nc\times ndrsi}}} \right) \left( {{{\varvec{\Phi } \mathbf{s}}}\left( {\mathbf {x}}_{{\mathbf {drs}}}{\mathbf {,n}} \right) }_{{\mathbf {ns\times ndrsi}}} \right) ^{{\mathbf {T}}}$
${\mathbf {K}}_{{{\mathbf {sic}}}_{{\mathbf {ns\times nc}}}}$	$\left( {\left( {{{\varvec{\upalpha }} \mathbf { s}}}_{{\mathbf {ns\times 1}}}\left( {\mathbf {slksi.}}{\mathbf {k}}_{{\mathbf {drsi}}} \right) \right) {\mathbf { }}}_{{\mathbf {ns\times ndrsi}}}{\mathbf {.}}{{{\varvec{\Phi } \mathbf{s}}}\left( {\mathbf {x}}_{{\mathbf {drs}}}{\mathbf {,n}} \right) }_{{\mathbf {ns\times ndrsi}}} \right) \left( {{{\varvec{\Phi } \mathbf{c}}}\left( {\mathbf {x}}_{{\mathbf {drcsi}}}{\mathbf {,n}} \right) }_{{\mathbf {nc\times ndrsi}}} \right) ^{{\mathbf {T}}}$
${\mathbf {K}}_{{{\mathbf {cpi}}}_{{\mathbf {nc\times 2}}}}$	${{\mathbf {-}}\left( {\mathop {{{\varvec{\upalpha }} \mathbf{c}}}\limits ^{\rightarrow }}_{{\mathbf {nc\times 1}}}{\mathbf {.}}k_{c} \right) }_{{\mathbf {nc\times 1}}}.{{{{\varvec{\upvarphi }} \mathbf{c}}}\left( {\mathbf {x}}_{{\mathbf {paci}}}{\mathbf {(}}t{\mathbf {),n}} \right) }_{{\mathbf {nc\times 1}}}$
${\mathbf {K}}_{{{\mathbf {pic}}}_{{\mathbf {2\times nc}}}}$	${-k}_{c}.\left( {{{\varvec{\Phi } \mathbf{c}}}\left( {\mathbf {x}}_{{\mathbf {paci}}}{\mathbf {(t),n}} \right) }_{{\mathbf {nc\times 1}}} \right) ^{{\mathbf {T}}}{-V_{p}C}_{c}.\left( {\varvec{{\acute{\Phi }}}{\mathbf {c}}\left( {\mathbf {x}}_{{\mathbf {paci}}}{\mathbf {(}}t{\mathbf {),n}} \right) }_{{\mathbf {nc\times 1}}} \right) ^{{\mathbf {T}}}$
${\mathbf {C}}_{{{\mathbf {pic}}}_{{\mathbf {2\times nc}}}}$	${-c}_{c}.\left( \left[ {{\varvec{\Phi } \mathbf{c}}}\left( {\mathbf {x}}_{{\mathbf {paci}}}{\mathbf {(}}t{\mathbf {),n}} \right) \right] _{{\mathbf {nc\times 1}}} \right) ^{{\mathbf {T}}}$
${\mathbf {K}}_{{{\mathbf {pi}}}_{{\mathbf {2\times 2}}}}$	$\left[ {\begin{array}{*{20}c} k_{p1i}+k_{c} &{} -k_{p1i}\\ {-k}_{p1i} &{} k_{p1i}+k_{p2i}\\ \end{array} } \right]$
${\mathbf {u}}_{{\mathbf {3\times 1}}}$	$\left[ {\begin{array}{*{20}c} \mathrm {g} &{} \mathrm {1} &{} f_{uplift}\\ \end{array} } \right] ^{{\mathbf {T}}}$
$f_{gm}$	$-{\rho A}_{m}.{{{\varvec{\upalpha }} \mathbf { m}}}_{nm\times 1}.\left( \int {{{\varvec{\Phi } }}{} \mathbf{m}\left( x,n \right) } dx \right) _{nm\times 1}$$-\left( m_{ucdr}.{{{\varvec{\upalpha }} \mathbf { m}}}_{nm\times 1} \right) . \left( {\mathbf {I}}_{1\times ndrm}\left( \left[ {{\varvec{\Phi } }}\mathbf {m(}\mathbf {x}_{\mathbf {drm}}\mathbf {,}\mathrm {n}\mathbf {)} \right] _{{\mathbf {nm\times ndrm}}} \right) ^{T} \right)$
${\mathbf {f}}_{{\mathbf {dm}}}$	${{{\varvec{\upalpha }} \mathbf { m}}}_{{\mathbf {nm\times 1}}}.{{\mathbf {(}} {{\mathbf {(}}{\mathbf {k}}_{{\mathbf {drm}}}{{.}}\left( {\mathbf {\Delta }}_{{\mathbf {m}}} {\mathbf {-}}{\mathbf {\Delta }}_{{\mathbf {cm}}} \right) {\mathbf {-}}{{\mathbf {f0}}}_{{\mathbf {drm}}} {\mathbf {)}}}_{{\mathbf {1\times ndrm}}}{\mathbf {\times }}\left( {{{\varvec{\Phi } \mathbf{m}(}}{\mathbf {x}}_{{\mathbf {drm}}}{\mathbf {,n)}}}_{{\mathbf {nm\times ndrm}}} \right) ^{{\mathbf {T}}}{\mathbf {)}}}^{{\mathbf {T}}}$
${\mathbf {f}}_{{\mathbf {gc}}}$	${\mathbf {-}}{{{{\varvec{\uprho }} \mathbf{A}}}}_{{\mathbf {c}}}.{{{\varvec{\upalpha }} \mathbf { c}}}_{{\mathbf {nc\times 1}}}. \left( \int {{{\varvec{\Phi } \mathbf{c}}}\left( {\mathbf {x,n}} \right) } {\mathbf {dx}} \right) _{{\mathbf {nc\times 1}}}$${\mathbf {-}}\left( m_{lcdr}{\mathbf {.}}{{{\varvec{\upalpha }} \mathbf { c}}}_{{\mathbf {nc\times 1}}} \right) .\left( {\mathbf {I}}_{{\mathbf {1\times ndr}}}{\mathbf {\times }}\left( {{{\varvec{\Phi } \mathbf{c}(}}{\mathbf {x}}_{{\mathbf {drc}}}{\mathbf {,n)}}}_{{\mathbf {nc\times ndr}}} \right) ^{{\mathbf {T}}} \right)$
${\mathbf {f}}_{{\mathbf {dc}}}$	${\mathbf {-}}{{{\varvec{\upalpha }} \mathbf { c}}}_{{\mathbf {nc\times 1}}}. {{\mathbf {(}}{{\mathbf {(}}{\mathbf {k}}_{{\mathbf {drm}}}{\mathbf {.}}\left( {\mathbf {\Delta }}_{{\mathbf {m}}}{\mathbf {-}}{\mathbf {\Delta }}_{{\mathbf {cm}}} \right) {\mathbf {-}}{{\mathbf {f0}}}_{{\mathbf {drm}}}{\mathbf {)}}}_{{\mathbf {1\times ndrm}}}{\mathbf {\times }}\left( {{{\varvec{\Phi } \mathbf{c}(}}{\mathbf {x}}_{{\mathbf {drm}}}{\mathbf {,n)}}}_{{\mathbf {nc\times ndrm}}} \right) ^{{\mathbf {T}}}{\mathbf {)}}}^{{\mathbf {T}}}$${\mathbf {-}}{{{\varvec{\upalpha }} \mathbf { c}}}_{{\mathbf {nc\times 1}}}.{{\mathbf {(}}{{\mathbf {(}}{\mathbf {k}}_{{\mathbf {drs}}}{\mathbf {.}}\left( {\mathbf {\Delta }}_{{\mathbf {s}}}{\mathbf {-}}{\mathbf {\Delta }}_{{\mathbf {cs}}} \right) {\mathbf {-}}{{\mathbf {f0}}}_{{\mathbf {drs}}}{\mathbf {)}}}_{{\mathbf {1\times ndrs}}}{\mathbf {\times }}\left( {{{\varvec{\Phi } \mathbf{c}(}}{\mathbf {x}}_{{\mathbf {drs}}}{\mathbf {,n)}}}_{{\mathbf {nc\times ndrs}}} \right) ^{{\mathbf {T}}}{\mathbf {)}}}^{{\mathbf {T}}}$
${\mathbf {f}}_{{\mathbf {gsi}}}$	${\mathbf {-}}{\rho A}_{s}.{{{\varvec{\upalpha }} \mathbf { s}}}_{{\mathbf {ns\times 1}}}.\left( \int \left[ {{\varvec{\Phi } \mathbf{s}}}\left( {\mathbf {x,n}} \right) \right] {\mathbf {dx}} \right) _{{\mathbf {ns\times 1}}}$${\mathbf {-}}\left( {\mathbf {m}}_{{\mathbf {ucdr}}}{\mathbf {.}}{{{\varvec{\upalpha }} \mathbf { s}}}_{{\mathbf {ns\times 1}}} \right) .\left( {\mathbf {I}}_{{\mathbf {1\times ndrs}}}{\mathbf {\times }}\left( {{{\varvec{\Phi } \mathbf{s}(}}{\mathbf {x}}_{{\mathbf {drs}}}{\mathbf {,n)}}}_{{\mathbf {ns\times ndrs}}} \right) ^{{\mathbf {T}}} \right)$
${\mathbf {f}}_{{\mathbf {ds}}}$	${{{\varvec{\upalpha }} \mathbf { s}}}_{{\mathbf {ns\times 1}}}.{{\mathbf {(}}{{\mathbf {(}}{\mathbf {k}}_{{\mathbf {drs}}}{\mathbf {.}}\left( {\mathbf {\Delta }}_{{\mathbf {s}}}{\mathbf {-}}{\mathbf {\Delta }}_{{\mathbf {cs}}} \right) {\mathbf {-}}{{\mathbf {f0}}}_{{\mathbf {drs}}}{\mathbf {)}}}_{{\mathbf {1\times ndrs}}}{\mathbf {\times }}\left( {{{\varvec{\Phi } \mathbf{s}(}}{\mathbf {x}}_{{\mathbf {drs}}}{\mathbf {,n)}}}_{{\mathbf {ns\times ndrs}}} \right) ^{{\mathbf {T}}}{\mathbf {)}}}^{{\mathbf {T}}}$
${\mathbf {f}}_{{\mathbf {gpi}}}$	$\left[ {\begin{array}{*{20}c} {\mathbf {m}}_{{\mathbf {p1i}}} &{} {\mathbf {m}}_{{\mathbf {p2i}}}\\ \end{array} } \right] ^{{\mathbf {T}}}$