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Modelling friction in the dynamics analysis of selected one-DOF spatial linkage mechanisms

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Abstract

A method of dynamics analysis of selected one-DOF spatial linkage mechanisms containing one spherical joint in their structure is presented in the paper. These mechanisms include neither redundant DOF nor passive constraints. In the procedure assumed here the spherical joint was treated as ideal, whereas friction can be taken into account in the remaining kinds of connections of links, i.e. in a rotational joint and a prismatic joint. The advanced LuGre model was used to take into account the phenomena of friction in the joints. The mechanism, in the form of a closed-loop kinematic chain, was divided by the cut-joint technique into two open-loop kinematic chains in the place of the spherical joint. Joint coordinates and homogeneous transformation matrices were used to describe the geometry of the system. Equations of the chains’ motion were derived using the formalism of Lagrange equations. Cut-joint constraints and joint forces, acting in the cutting place, were introduced to complete the equations of motion, thus a set of DAEs was obtained. To solve these equations, a procedure was applied based on double differentiation of the formulated constraint equations in relation to time. In order to determine the values of friction torque in a rotational joint and friction force in a prismatic joint, in each integrating step of the equations of motion, the joint forces and torques acting in these joints were calculated using the recursive Newton–Euler algorithm, which was taken from robotics. For the requirements of the method, models were developed of both a rotational joint and a prismatic joint. As an example, the dynamics analysis of a one-DOF RSUP spatial linkage mechanism, including rotational joints R, spherical joint S, universal joint U and prismatic joint P, is presented in the article.

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Abbreviations

\(c\) :

Kinematic chain index

\(g\) :

Acceleration of gravity

\(p\) :

Link index

\(a_{{}}^{(c,p)}\) :

Length of journal of rotational joint \((c,p)\)

\(\bar{a}_{{}}^{(c,p)}\) :

Spacing of friction rotational surfaces of journal of rotational joint \((c,p)\)

\(d^{(c,p)}\) :

Diameter of link \((c,p)\)

\(\left. {d_{\alpha }^{(c,p)} } \right|_{{\alpha \in \left\{ {A,B,C} \right\}}}\) :

Diameters of rotational friction surfaces A, B, C of rotational joint \((c,p)\)

\(l^{(c,p)}\) :

Length of link \((c,p)\)

\(m^{(c,p)}\) :

Mass of link \((c,p)\)

\(n_{b}^{(c)}\) :

Number of links in chain c

\(t_{dr,0}^{(1,1)}\) :

Parameter defining course of value of driving torque

\(t_{st}^{(1,1)}\) :

Starting time

\({\mathbf{f}}_{f}^{(c,p)}\) :

Friction force acting on slider of prismatic joint \((c,p)\)

\({\tilde{\mathbf{f}}}_{{C^{(c,p)} }}^{(c,p)}\) :

Force with opposite sense to inertial force of link \((c,p)\)

\({\tilde{\mathbf{f}}}_{{O^{(c,p)} }}^{(c,p)}\) :

Joint force exerted on link \((c,p)\) by link \((c,p - 1)\)

\({\tilde{\mathbf{n}}}_{{C^{(c,p)} }}^{(c,p)}\) :

Torque with opposite sense to inertial torque of link \((c,p)\)

\({\tilde{\mathbf{n}}}_{{O^{(c,p)} }}^{(c,p)}\) :

Joint torque exerted on link \((c,p)\) by link \((c,p - 1)\)

\({\tilde{\mathbf{r}}}_{A}^{(c,p)}\) :

Vector of position of point A of link \((c,p)\) in local coordinate system

\({\mathbf{r}}_{A}^{(c,p)}\) :

Vector of position of point A of link \((c,p)\) in global reference system \(\{ 1,0\}\)

\(\left( {\mathbf{r}} \right)_{\alpha }\) :

Component \(\alpha\) of vector \({\mathbf{r}}\)

\({\mathbf{t}}_{dr}^{(1,1)}\) :

Driving torque

\(\left. {{\mathbf{t}}_{f,\alpha }^{(c,p)} } \right|_{{\alpha \in \left\{ {A,B,C} \right\}}}\) :

Friction torques on rotational friction surfaces A, B, C of journal of rotational joint \((c,p)\)

\({\mathbf{t}}_{res}^{(1,1)}\) :

Resistance torque

\({\mathbf{H}}^{(c,p)}\) :

Inertial matrix \(4 \times 4\) of link \((c,p)\)

\({\mathbf{I}}_{{}}^{(c,p)}\) :

Inertial matrix \(3 \times 3\) of link \((c,p)\)

\({\tilde{\mathbf{R}}}^{(c,p)}\) :

Rotation matrix \(3 \times 3\) from local coordinate system of link \((c,p)\) to system of link \((c,p - 1)\)

\({\tilde{\mathbf{T}}}^{(c,p)}\) :

Transformation matrix \(4 \times 4\) from local coordinate system of link \((c,p)\) to system of link \((c,p - 1)\)

\({\mathbf{T}}^{(c,p)}\) :

Transformation matrix \(4 \times 4\) from local coordinate system of link \((c,p)\) to global reference system \(\{ 1,0\}\)

\(\left( {\mathbf{T}} \right)_{\alpha ,\beta }\) :

Element of matrix \({\mathbf{T}}\) being on intersection of row \(\alpha\) and column \(\beta\) \(\begin{aligned} {\mathbf{T}}_{i}^{(c,p)} & = \frac{{\partial {\mathbf{T}}^{(c,p)} }}{{\partial q_{i}^{(c,p)} }} \\ {\mathbf{T}}_{i,j}^{(c,p)} & = \frac{{\partial^{2} {\mathbf{T}}^{(c,p)} }}{{\partial q_{i}^{(c,p)} \partial q_{j}^{(c,p)} }} \\ \end{aligned}\)

\(\dot{\psi }_{0}^{(1,1)}\) :

Parameter defining course of value of resistance torque

\({\tilde{\varvec{\upomega }}}^{(c,p)}\) :

Angular velocity of link \((c,p)\) defined in local coordinate system

\({\dot{\tilde{\varvec{\upomega}}}}^{(c,p)}\) :

Angular acceleration of link \((c,p)\) defined in local coordinate system

\({\dot{\tilde{\varvec{\upsilon}}}}_{{C^{(c,p)} }}^{(c,p)}\) :

Acceleration of centre of mass \(C^{(c,p)}\) of link \((c,p)\) defined in local coordinate system

\({\dot{\tilde{\varvec{\upsilon}}}}_{{O^{(c,p)} }}^{(c,p)}\) :

Acceleration of origin of local coordinate system of link \((c,p)\)

DOF:

Degree(s)-of-freedom

DAEs:

Differential-algebraic equations

ODEs:

Ordinary differential equations

\(\left. {\mu_{\alpha }^{(c,p)} } \right|_{{\alpha \in \left\{ {A,B,C} \right\}}} ,\,\,\mu_{{}}^{(c,p)}\) :

Friction coefficient in rotational joint \((c,p)\) and prismatic joint \((c,p)\), respectively

\(\left. {\mu_{s,\alpha }^{(c,p)} } \right|_{{\alpha \in \left\{ {A,B,C} \right\}}} ,\,\,\mu_{s}^{(c,p)}\) :

Static (limiting) friction coefficient (exactly—maximal value of the friction coefficient)

\(\left. {\mu_{k,\alpha }^{(c,p)} } \right|_{{\alpha \in \left\{ {A,B,C} \right\}}} ,\,\,\mu_{k}^{(c,p)}\) :

Minimal value of the friction coefficient

\(\left. {\sigma_{0,\alpha }^{(c,p)} } \right|_{{\alpha \in \left\{ {A,B,C} \right\}}} ,\,\,\sigma_{0}^{(c,p)}\) :

Stiffness coefficient of the bristles

\(\left. {\sigma_{1,\alpha }^{(c,p)} } \right|_{{\alpha \in \left\{ {A,B,C} \right\}}} ,\,\,\sigma_{1}^{(c,p)}\) :

Damping coefficient of the bristles

\(\left. {\sigma_{2,\alpha }^{(c,p)} } \right|_{{\alpha \in \left\{ {A,B,C} \right\}}} ,\,\,\sigma_{2}^{(c,p)}\) :

Viscous friction (damping) coefficient

\(\left. {z_{\alpha }^{(c,p)} } \right|_{{\alpha \in \left\{ {A,B,C} \right\}}} ,\,\,z_{{}}^{(c,p)}\) :

Deflection of the bristles

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  1. Faculty of Mechanical Engineering and Computer Science, Department of Mechanics, University of Bielsko-Biala, Willowa 2, 43-309, Bielsko-Biała, Poland

    Andrzej Harlecki & Andrzej Urbaś

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  1. Andrzej Harlecki
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Harlecki, A., Urbaś, A. Modelling friction in the dynamics analysis of selected one-DOF spatial linkage mechanisms. Meccanica 52, 403–420 (2017). https://doi.org/10.1007/s11012-016-0390-6

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