1.Determine the gap function \(g_0(x, y)\) for the problem in which body 1 is a cylinder of radius R with its axis aligned with the y-axis, and body 2 is a similar cylinder whose axis bisects the angle between the x- and y-axes. Assume \(x, y\ll \!R\).
2. In a discrete formulation of a frictionless contact problem, the normal forces \(P_i\)
at a set of nodes are related to the corresponding normal displacements \(u_j\)
in body 1 by the linear matrix equation
where Open image in new window
is the contact stiffness matrix
and can be assumed to be symmetric and positive definite.
Consider the case where body 2 is rigid and there are only two nodes (\(N\!=\!2\)). It is proposed to solve the discrete version of the contact problem (1.3) by (i) assuming both nodes are in contact, (ii) solving for \(P_1, P_2\) and (iii) changing the assumption to separation at any node for which \(P_i\!<\!0\). Find the condition(s) that must be satisfied by Open image in new window if the gap at any node so released is to be positive. Can you extend this argument to the case where \(N\!>\!2\)?
3. Two bodies with sinusoidal surfaces are defined by the gap functions
$$ g_1(x)=h_0[1+\cos (mx)]\;;\;\;\;g_2(x)=-h_0(1-\epsilon )[1+\cos (mx)], $$
where \(\epsilon \!\ll \!1\)
(1), so the mean slope is not small compared with unity. Determine the direction of the local normal to the surface of body 1 as a function of x
and hence use the protocol of Sect. 1.4
to determine the initial gap \(g_0\)
again as a function of x