1. A body of mass M impinges with initial velocity \(V_0\) on a massless platform supported by a spring of stiffness k and a viscous damper of coefficient c . Find the exit (rebound) velocity of the mass \(V_1\) , and hence show that the coefficient of restitution \(V_1/V_0\) is independent of \(V_0\) .

2. Two homogeneous spheres of the same material (with properties \(E,\nu ,\rho \) ) and of radius \(R_1,R_2\) respectively approach each other with a normal relative velocity \(V_0\) . Write the equations of motion for each sphere and hence show that the duration of the impact is given by Eq. (20.15 ) with the substitutions (20.4 ).

3. Find an expression for the maximum contact radius \(a_{\max }\) in the Hertzian theory of impact and hence show that the criterion \(t_c\!\gg \! 2a_{\max }/c_2\) for the applicability of this theory is satisfied provided that \(V_0\!\ll \! c_2\) .

4. By comparing Eqs. (20.15 ), (20.2 ), determine the value of the stiffness k in a linear model of impact, if the approximation is to define the correct value for the impact period \(t_c\) . Use this value to determine the maximum displacement \(u_{\max }\) and force \(P_{\max }\) . Do the results agree with the Hertzian impact theory, and if not what is the nature and magnitude of the error?

5. Extend the analysis of Sect. 20.4.5 to the case where duration of the impact \(t_I\) lies in the range \(2t_0<t_I<3t_0\) . In particular, find the coefficient of restitution as a function of \(\lambda t_0\) and the range of values of \(\lambda t_0\) for which your solution applies.

6. Figure

20.19 shows a composite body comprising a rigid mass

M bonded to an elastic bar of length

L , cross-sectional area

A and elastic modulus

E . The body strikes a fixed rigid support when travelling to the left at speed

\(V_0\) . Describe the subsequent motion and determine the coefficient of restitution for cases where the duration of the impact is less than

\(4L/c_0\) , where

\(c_0\) is the plane stress wave speed. What restriction does this condition impose on the ratio

\(M/M_\mathrm{{bar}}\) ?

Fig. 20.19 An elastic bar with an end mass striking a rigid wall

7. The force F in Fig. 20.14 lies in the range \(Q_0/2\!<\!F\!<\!Q_0\) . Find the distance that the bar slips through the support in terms of \(F,A,E, Q_0\) .

8. The bar in Fig.

20.14 is loaded by a sinusoidal force

$$ F(t)=F_0\sin (\omega t)\;\;\;\;\;t>0, $$

where

\(F_0\!>\!Q_0\) . Describe the resulting wave propagation and determine the frictional energy dissipation per cycle in terms of

\(F_0,\omega ,A, c_0,E, Q_0\) . Assume that

\(\omega L\gg c_0\) and consider only the period

\(0<t<2L/c_0\) in which reflected waves have not had time to reach the support.

Fig. 20.20 An embedded elastic bar loaded by a transient end force

9. The pile driving problem is approximated in the form of the bar in Fig.

20.20 , with a frictional support defined by Eqs. (

20.94 ), (

20.95 ). The pile is loaded by impact of a mass on the free end, which can be approximated as generating the force

$$ F(t)=F_0\left[ H(t)-H(t-t_0)\right] , $$

where

\(c_0t_0\ll F_0/q_0\) . Determine the condition that must be satisfied if the further end of the bar

\(x\!=\!d\) is to slip during the initial wave propagation. Also, find the distance that the pile slips into the support at

O . Assume that

d is sufficiently large to ensure that the wave reflected from the end attenuates to zero before reaching

O .