From the applied dynamics viewpoint, the conditions during the cycle are of less importance than those at exit. Two questions of particular interest are:
(i) how does the local angle of reflexion depend upon the local angle of incidence? and (ii) how does the answer to (i) differ from that obtained from the simple rigid body theory?
Figure 2 shows the value of $ at exit (=$a) as a function of $r for a homogeneous solid sphere with v = 0.3. The function Jls is a non-dimensional local angle of reflexion in the same sense that $I is a local angle of incidence (see Section 3.2). A positive angle of reflexion is one in which the tangential velocity retains the same sense.
The rigid body theory of impact agrees with the more exact theory for J/i > 4x, but predicts $s = 0 for $r < 4x. This is a reasonable approximation when J/l < 1 (see Fig. 2), but in the range 1 < $1 < 4x the error can be consi-
derable. The elastic recovery of the surfaces enables negative local angles of reflexion to be obtained, whereas the minimum value predicted from rigid body considerations is zero, corresponding to the condition of rolling at exit. It is also notable that gross slip occurs throughout the cycle at lower values of +I than those predicted by the simple theory, since tangential elastic recovery of the surfaces can maintain relative motion even when the sphere has been brought to rest.
From the applied dynamics viewpoint, the conditions during the cycle are of less importance than those at exit. Two questions of particular interest are:(i) how does the local angle of reflexion depend upon the local angle of incidence? and (ii) how does the answer to (i) differ from that obtained from the simple rigid body theory? Figure 2 shows the value of $ at exit (=$a) as a function of $r for a homogeneous solid sphere with v = 0.3. The function Jls is a non-dimensional local angle of reflexion in the same sense that $I is a local angle of incidence (see Section 3.2). A positive angle of reflexion is one in which the tangential velocity retains the same sense. The rigid body theory of impact agrees with the more exact theory for J/i > 4x, but predicts $s = 0 for $r < 4x. This is a reasonable approximation when J/l < 1 (see Fig. 2), but in the range 1 < $1 < 4x the error can be consi-derable. The elastic recovery of the surfaces enables negative local angles of reflexion to be obtained, whereas the minimum value predicted from rigid body considerations is zero, corresponding to the condition of rolling at exit. It is also notable that gross slip occurs throughout the cycle at lower values of +I than those predicted by the simple theory, since tangential elastic recovery of the surfaces can maintain relative motion even when the sphere has been brought to rest.
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