1. The system of Fig.

8.2 a is loaded by the periodic force

$$ F_1=F\cos (\omega t);\;\;\;F_2=F\sin (\omega t), $$

where the frequency

\(\omega \) is sufficiently slow for quasi-static conditions to apply.

Identify the times during each steady-state cycle that transitions occur between the various possible states, if the coefficient of friction

\(f\!=\!0.5\) and

$$ {\varvec{K}}=\left[ \begin{array}{cc}10&{}4\\ 4&{}10\end{array}\right] . $$

Would the behaviour be qualitatively different if the coefficient of friction were lower?

2. The system of Fig. 8.2 a is subjected to a force \({\varvec{F}}(t)\!=\!\{C_1t, C_2t\}\) starting at time \(t\!=\!0\) , where \(f\!>\!k_{11}/k_{21}\!>\!0\) and the constants \(C_1,C_2\) are defined such that \({\varvec{F}}\) moves through the multiple solution segment in Fig. 8.5 .

(i) Determine the quasi-static displacements v (t ), w (t ) under the assumption of backward slip.

(ii) Now set up the dynamic equations of motion for backward slip, including the inertia term \(-M\ddot{{\varvec{u}}}\) .

(iii) Show that the time-varying displacements from (i) satisfy the equations of motion, provided the mass has an appropriate initial velocity at time \(t\!=\!0\) , and find this velocity.

(iv) Show that if the initial velocity differs very slightly from the quasi-static value, this perturbation will then grow without limit. In other words, that the backward slip state is unstable.

3. The contact stiffness matrix (

8.13 ) for a two-node frictional system is defined such that

$$ {\varvec{A}}=\left[ \begin{array}{cc}10&{}3\\ 3&{}20\end{array}\right] ;\;\;\; {\varvec{B}}=\left[ \begin{array}{cc}3&{}10\\ 10&{}3\end{array}\right] ;\;\;\;{\varvec{C}}=\left[ \begin{array}{cc}20&{}10\\ 10&{}20\end{array}\right] . $$

Use Klarbring’s criterion [Sect.

8.3.4 ] to find the critical coefficient of friction above which the rate problem is ill-posed.

4. A two-node frictional system is defined by the contact stiffness matrix of Problem 8.3 . Find the minimum coefficient for which the system is capable of being wedged in a state with no external forces [\(p_i^w\!=\!q_i^w\!=\!0\) ] but non-zero nodal displacements. Notice that the critical state may involve both nodes in contact [\(w_1\!=\!w_2\!=\!0\) ], or only one node in contact.

5. Construct a diagram analogous to Fig. 8.9 for the three-dimensional case where node i is slipping in both solutions 1 and 2, but not in the same direction. Show that the contribution of this node to \(\dot{\mathcal {E}}\) is strictly negative under these conditions. Hence show that in the steady state, the direction of slip at slipping nodes is independent of initial conditions.

6. Figure

8.21 shows the extreme positions of the frictional constraints for a two-node system, so that the central white quadrilateral represents the safe shakedown space

\(\mathcal {S}\) . The periodic loading scenario causes the constraints to advance to and then retreat from these positions in the sequence I, III, II, IV, I, III .... Track the motion of the point

P from an arbitrary starting point outside

\(\mathcal {S}\) and hence partition this space into regions corresponding to the final position of

P lying on the lines

AB ,

BC ,

CD and

DA , or in one of the corners

A ,

B ,

C and

D . Also, explain which starting positions will involve the shakedown state being reached in one or two cycles, and which will involve an asymptotic approach to the steady state.

Fig. 8.21 The safe shakedown space

7. A cylindrical bar of diameter D and length \(L\,(\gg D)\) just fits inside a long cylindrical hole in a rigid body. The bar is now heated to a temperature T . If the coefficient of friction between the bar and the hole is f , find the distribution of contact pressure p and the mean axial stress \(\sigma _{zz}\) as functions of distance z along the bar. The material of the bar has Young’s modulus E , Poisson’s ratio \(\nu \) and coefficient of thermal expansion \(\alpha \) . Assume plane cross sections remain plane.

8. Figure

8.22 shows the geometry of a standard ‘pull-out’ test for fibrous composite materials, in which a tensile force

\(F_0\) is applied to the end of an exposed fibre and increased until failure occurs of the bond between the fibre and the matrix.

Fig. 8.22 The ‘pull-out’ test

An approximate model of the process is shown in Fig.

8.23 , where the fibre is represented by a rod (

a ) and the matrix by a pair of rods (

b ). The stiffnesses of the rods are denoted by

\(k_a, k_b\) respectively, defined through the relations

$$ e_{xx}=\frac{du_x}{dx}=\frac{F(x)}{k}, $$

where

\(u_x\) is the local displacement of the rod in the

x -direction and

F (

x ) is the tensile force in the rod at the point (

x ).

Fig. 8.23 Rod model of fibre pull-out

We make the further assumption that the transfer of force between the rods is achieved solely by friction, through a coefficient of friction, f , and that the rods are pressed together by a normal force w per unit length.

The force \(F_0\) is gradually increased from zero. Describe what happens and determine the location and magnitude of any regions of slip as functions of \(F_0\) . Also, give expressions for the axial forces in the matrix and the fibre as functions of x .

9. Suppose that each block in the discrete model of Fig. 8.15 has mass M . At time \(t\!=\!0\) , all the springs are unstretched, and a force \(F_0\!>\!fP\) is suddenly applied to the last block as shown in the figure. Find the displacement u of this block assuming that all the remaining blocks remain stationary. Hence determine the time \(t\!=\!t_1\) at which the second block will just start to move.

10. Suppose that the system of Fig.

8.17 can support a steady-state vibration such that

$$ u(t)=u_0+A\cos (\omega t). $$

Calculate (i) the energy absorbed in the damper and (ii) the work done by the friction force on the mass during one complete cycle

\(2\pi /\omega \) , assuming the friction coefficient is given by equation

where

\(f_0,f_1\) are positive constants. Hence show that such a state is possible if and only if

\(f_1\!=\!c/P\) and that in this case, the amplitude

A can take any value.

Repeat the calculation for the case where the friction coefficient is defined by the cubic relation

$$ f(V)=f(V_0)-f_1(V-V_0)+f_3(V-V_0)^3, $$

where

\(f_3\!>\!0\) . Show that the amplitude in the steady state is now determinate and find its value.

^{7} 11. A state-variable friction law for unidirectional sliding at speed

V is defined by the relations

$$ \dot{S}=|V|-\frac{S}{t_0};\;\;\;Q=f(S)P;\;\;\;f(S)=f_0+f_1\exp \left( -\frac{S}{L_0}\right) , $$

where

P ,

Q are the normal and frictional force respectively,

S is the state variable which has dimensions of length, and

\(t_0,L_0\) are constants with dimensions of time and length respectively.

(i) If the bodies slide at constant speed V , what will be the relation between the coefficient of friction Q / P and V .

(ii) If the sliding speed is held constant at \(V_0\) for some time, but then changed suddenly to \(V_1\) , how will the friction force vary during the ensuing transient.