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Pulley Element

Assumptions used for the cable-pulley element

  • The pulley element is a three node element

  • The cable is perfectly flexible and can only carry tension force.

  • The pulley is frictionless. The tension force in the cable segments is constant ( ).

The equilibrium condition of the element is .

  • Thermal effects are neglected.

  • The weight of the cable pulley is neglected

Element formulation

With the aforementioned assumptions, the cable-pulley element can be idealized as shown in next Figure.

The initial unstrained length of the cable is defined by: $$ |L_0|=|L_{01}|+|L_{02}| $$

The two strands P1P3 and P3P2 remain rectilinear. The vectors of the strands can be written as $$ |L_1|=\sqrt{\vec{U_1T}\vec{U_1}}~~~;~~~|L_2|=\sqrt{\vec{U_2T}\vec{U_2}} $$

$$ \vec{l_1}=\frac{\vec{U_1}}{|L_1|}~~~;~~~\vec{l_2}=\frac{\vec{U_2}}{|L_2|} $$

The cable is assumed to be linear elastic and the constitutive relation between the tension force \(T\) and cable strain can be expressed as:

\[ T= \begin{cases} EA\varepsilon + T_0 &\text{if } \varepsilon\ge0\\ 0 &\text{if } \varepsilon<0 \end{cases} \]

where \(EA\) is the axial stiffness of the cable. The strain \(\varepsilon\) must remain small, so that the cross-sectional area, \(A\), of the cable is unchanged. The strain is defined as: $$ \varepsilon=\frac{|L_1|+|L_2|-|L_0|}{|L_0|} $$

The internal forces are the forces that must be exerted on the nodes to keep it in its equilibrium configuration. These forces are defined by: $$ \vec{F_1}=T\vec{l_1}~;~\vec{F_2}=T\vec{l_2}~;~\vec{F_3}=-(\vec{F_1}+\vec{F_2}) $$

The stiffness matrix denoted \(K\) defines the linear relation between some increments of nodal displacements: $$ \delta U=\begin{bmatrix} \delta U_1 \ \delta U_2 \ \delta U_3 \ \end{bmatrix} $$

and they are related as : $$ \delta F=K\delta U $$

where \(K\) is the tangent stiffness matrix and \(\delta F\) is issued from [1], $$ \delta F=\lim\limits_{\eta\rarr0}\frac{d}{d\eta}F(U+\eta\delta U) $$

The final expression of K is: $$ K=\begin{bmatrix} K_{11} & K_{12} & -(K_{11}+K_{12}) \ K_{12}^T & K_{22} & -(K_{22}+K_{12}^T) \ -(K_{11}+K_{12}^T) & -(K_{22}+K_{12}) & -(K_{13}+K_{23}) \end{bmatrix} $$

where: $$ K_{11}=\biggl(\frac{EA}{|L_0|}-\frac{T}{|L_1|}\biggr)l_1l_1^T+\frac{T}{|L_1|}I_3 $$

\[ K_{22}=\biggl(\frac{EA}{|L_0|}-\frac{T}{|L_2|}\biggr)l_2l_2^T+\frac{T}{|L_2|}I_3 \]
\[ K_{12}=\biggl(\frac{EA}{|L_0|}\biggr)l_1l_2^T \]

where \(I_3\) is a 3x3 identity matrix.