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22 Chapter 2. Existence results for discrete contact problems that no collisions with the obstacle will occur, so that a smooth solution is possible. To find the motion of a rigid truss we use equations of linear and angular momentum (see [16]). 13) (qi × fiapp + qi × r i ) i=1 where qi is the radius vector of node i from some given origin and × denotes the cross product. Let M be a diagonal matrix with the masses mi as diagonal elements. The rigid body motions N consist of translations and rotations, let us consider two vectors ex and exx in N that are defined by ˆ, exi = x ˆ × qi exx i =x ˆ is a unit vector in the x-direction.

We 33 34 Chapter 3. 5: Coordinate frame and node numbering of the cylinder. 9) + (wz b + wz wyy )(ezi · exx i ) yy xx + 2wz wyy (ezi · eyy i ) + (wxx wyy + bwyy )(ei · ei ) We prove that both expressions within square brackets are positive. Let us first consider the first square bracket. If a = wx , then |wx | ≤ R|wyy | and we obtain yy 2 2 x awx + wyy |eyy i,x | + (wx wyy + awyy )(ei,x · ei ) yy x = (wx exi + wyy eyy i,x ) · (wx ei + wyy ei,x ) ≥ 0 and if a = R|wyy |wx , |wx | then |wx | > R|wyy | so that yy x 2 2 |eyy awx + wyy i,x | + (wx wyy + awyy )(ei,x · ei ) yy 2 2 yy ≥ R|wx ||wyy | + |wyy |2 |eyy i,x | − |wx ||wyy ||ei,x | − R|wyy | |ei,x | yy = |wyy |(|wx | − |wyy ||eyy i,x |)(R − |ei,x |) ≥ 0 34 4 Cylinder in a cylindrical groove.

Yy yy yy I. Decompose eyy i in x and z-component, ei = ei,x + ei,z . We 33 34 Chapter 3. 5: Coordinate frame and node numbering of the cylinder. 9) + (wz b + wz wyy )(ezi · exx i ) yy xx + 2wz wyy (ezi · eyy i ) + (wxx wyy + bwyy )(ei · ei ) We prove that both expressions within square brackets are positive. Let us first consider the first square bracket. If a = wx , then |wx | ≤ R|wyy | and we obtain yy 2 2 x awx + wyy |eyy i,x | + (wx wyy + awyy )(ei,x · ei ) yy x = (wx exi + wyy eyy i,x ) · (wx ei + wyy ei,x ) ≥ 0 and if a = R|wyy |wx , |wx | then |wx | > R|wyy | so that yy x 2 2 |eyy awx + wyy i,x | + (wx wyy + awyy )(ei,x · ei ) yy 2 2 yy ≥ R|wx ||wyy | + |wyy |2 |eyy i,x | − |wx ||wyy ||ei,x | − R|wyy | |ei,x | yy = |wyy |(|wx | − |wyy ||eyy i,x |)(R − |ei,x |) ≥ 0 34 4 Cylinder in a cylindrical groove.

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