Wave propagation and vibration of elastic rods with interfacial frictional slip

WAVE MOTION NORTH-HOLLAND

12 (1990)

513

513-526

WAVE PROPAGATION AND VIBRATION INTERFACIAL FRICTIONAL SLIP

OF ELASTIC

RODS

WITH

L.V. NIKITIN AND A.N. TYUREKHODGAEV Institute Received

of the Physics of the Earth, Moscow, USSR 29 November

1984, Revised

16 October

1989

Wave propagation and vibration of elastic rods interacting with their environment according to the Coulomb dry friction law, are studied. Exact solutions of the nonlinear problems of impact of semi-infinite or finite rods, with constant stress and velocity or by a rigid body are obtained. Problems of smooth loading and unloading of a semi-infinite rod are solved as well. A method of exact analytical solution of problems of vibration for steplike loading is developed. The cases of suddenly applied stress which is maintained constant afterwards or changes sign steplike with the period of free vibration, are studied. Continuously distributed and localized friction are considered. It is shown that extension of a zone of disturbances and dissipation in a system with friction, strictly depends on the history of loading.

Introduction

In many practical problems of dynamical loading of structures, damping is caused by a large dissipation of energy due to frictional contact at various places, such as joints or slipping interfaces. Such energy dissipation is called structural damping. In a quasi-statical approach this problem was studied systematically; a review of the relevant works may be found in the paper of Goodman [l] and the book of Panovko [2]. The effects of structural damping on vibrations of elastic systems have been studied in numerous papers. A review of many of them may be found in the monographs of Den-Hartog [3] and Pisarenko [4]. In those studies, losses due to structural damping as a rule were introduced from quasi-statical considerations or assumed a priori. Vibrations of a system with a finite number of degrees of freedom were generally considered. Structural damping with continuous parameters of elasticity and friction by exact analytical methods was studied by Nikitin [5,6,7] and by Nikitin and Tyurekhodgaev [8]. The approximate method of harmonic linearization (Bogolubov, Mitropolsky [lo]) was used by Koibin [9] for rod vibrations. From the mathematical point of view the problem of structural damping with continuous parameters can be reduced to solving an equation which differs from the well known wave equation only by a constant term with the sign of the velocity. The solution of boundary value problems for this equation is nontrivial, and exact solutions were not obtained up to recently. Even for cases when from physical considerations the direction of the velocity is known a priori, as a rule problems with unknown boundaries separating regions of rest and motion arise. Vibration problems for such cases are apparently not amenable to analytical solutions. Nevertheless their solutions for practically interesting cases were obtained.

Formulation

of the problem

Consider an elastic rod with homogeneous cross-section S. Let all or part of the rod surface with perimeter of cross-section L interact with the environment in accordance with the Coulomb dry friction 0165-8641/90/%03.50

@ 1990 - Elsevier

Science

Publishers

B.V. (North-Holland)

514

L. V. Nikirin, A. N. Tyurekhodgaev / Waves in rods with slip

law. In the case when slip takes place between rod and surrounding medium, the shear stress on the rod surface is equal to fN where N is the normal pressure on the rod, and f the coefficient of friction between the rod and the medium. motion. exceed

This stress acts always

In case of no relative the limiting

friction

Let us take the origin

sliding

the friction

in the direction

which is opposite

to the velocity

stress may take any value whose

magnitude

of rod

does not

stress.

of the x-axis at one of the ends of the rod and direct this axis along the rod. The

normal stress CTin the rod and the velocity equation of motion and Hooke’s law:

v of its sections

satisfy the set of equations

consisting

of the

au au au -=pg+Kq, ax dt=Eax. In terms of the displacement

u, equations

(1) (1) may be written

as

(2) p is the material density, E is Young’s modulus, a = (E/p)“’ is the sound velocity in the rod and fLN/S is assumed to be constant. The value of K in case of motion has the same sign as the velocity v. For v = 0 it may take any value from the range (-l,l). It should be noted that K must be determined in the process of solution. Due to the presence of K the system (1) is nonlinear. At the initial time, t = 0, the rod is assumed to be at rest and unstressed

Here q =

u = 0,

u = 0,

K =o;

t = 0.

(3)

Note that in general, due to the external friction, the stresses in the quiescent characteristics of the system (1) and relations among them have the forms dx = *a dt,

rod may be nonzero.

The

*da+apdv+Kqadt=O.

(4)

The wavefronts of strong discontinuity which may arise due to discontinuity of the boundary conditions coincide with the characteristics, and along them shocks of stress (a) and velocity (v) are related by conditions, following from (4): (g) = rap (v>;

x = fat + const.

(5)

If K is not changing in some region then the system (1) inside this region can be easily integrated the characteristics (4). The general solution of the equation (1) and (2) in this case may be written form

u = -i

2 ap+f*(P)+f*(a),

along in the

(64 (6b)

u=y(a-P)-Efi(P)+Ef;W.

v=-z(a+P)+afi(/l)+af;(n) where f, and f2 are arbitrary (Y= at+x,

functions

p=at-x.

and the following

characteristic

variables

a and p are adopted (7)

L. V. Nikitin, A. N. Tyurekhodgaev

In case of rest K must be found

from static equilibrium

/ Waves in rods with slip

between

friction

515

force and stress:

du dx=

(8)

Kq*

Let us now consider

1. Unidirectional

some particular

problems.

First we consider

cases of unidirectional

motion.

motions

1.1. Impact with constant stress of a semi-infinite Let at the end x = 0 of a semi-infinite a=--a,H(t);

rod

rod at time t = 0, a stress be applied

and then maintained

,x=0

constant (9)

where H(t) is the Heavyside unit step function. The sign of the velocity u in the region where motion has arisen is positive and the region of rest is separated from that of motion, at least for the initial stage of motion, by the characteristic x = at. Therefore K

should

be taken K =

Solution obtained

in the form

H(at-x).

in the region

(10) 1 of Fig. 1 bounded

with the help of equations C7= - cT(J+qx/2,

by the shock

front

x = at and the x-axis

can be easily

(5):

v = vg - qt/2p

(11)

where v. = uo/ up. At time tf = 2uo/ aq motion simultaneously vanishes everywhere, and the length of the disturbed part of the rod turns out to be equal to lf= 2uo/ q. Note that in a quasi-statical consideration the length of the disturbed part is only half of that value. After cessation of motion in the part of the rod, the residual stress can be found from (8) as a=-ao+qx/2. In the problem

(12)

being considered,

interaction

between

rod and environment

took place along the whole

length of the rod. However, in many practical cases such as a rod rotating in a bearing, overlapping panels, and so on, the zone of interaction is rather localized. In those cases friction forces may be assumed

5

4

at

Fig. 1. The region of motion (1) and rest.

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L. V. Nikitin, A. N. Tyurekhodgaeo

/ Waves in rods with slip

to be concentrated and can be represented by delta-functions. permits us to obtain the full pattern of motion for different semi-infinite

rod for which at the section

At this section cross-section.

there may appear The equation

x = x0 there is localized

a concentrated

of motion

This simplifies analysis and in many cases kinds of loading. As an example consider a

friction

frictional

contact

force with a limiting

may then be written

with the environment.

value of Q per unit of rod

in the form

$p$+Qd(x-x0) H(at-x). Consider

(13)

again the case when at the end of the rod x = 0 a stress -aoH(

to sections

of the rod x > x0 it is necessary

to meet the condition

t) is applied.

a0 > Q/2, otherwise

To transmit motion

motion would

be

the same as if the section x=x, were fixed. To solve the problem let us use the Laplace transform. In spite of the nonlinearity of the problem this is possible, since after the value of K has been predicted (10) the problem becomes linear. The Laplace-transformed solution of equation (13) which satisfies the initial condition (3) and the boundary conditions (9), and which is bounded at infinity, has the form 6 = - go e-pxln where p is the parameter The inverse

transform

Q e-2Pxo/’

sinh

of the Laplace

Px -+ a

Q H(x

transform

-x0)

e-P%/“

c&q

Pb-x0)

(14)

a

and a bar over a letter denotes

the Laplace

transform.

of (14) is

a=-a,H(at-x)+fQ[H(at-x-2x0)-H(at+x-2x,)] +iQH(x-x,)[H(at-x)+H(af+x-2x,)] Similarly

the following

expression

may be found

(15)

for the velocity:

apv=u,H(at-x)-iQ[H(at-x-2x,)+H(at-x+2x0)] -$QH(x-x,)[H(at-x)-H(at+x-2x,)]. The solution

for this case may also be found

for more complicated

(16) loadings.

1.2. Impact of a semi-injinite rod with constant velocity Let the end of a semi-infinite v= v,H(t);

rod x = 0 start to move with constant

x=0

velocity

v. at time t = 0: (17)

Evidently the value of K in this problem is equal to +l everywhere in the region of motion. However the boundary x = x*(t) separating the regions of motion and rest is unknown and should be found as part of the solution. Initially the leading front coincides with the characteristic x = at, since it is a line of strong discontinuity. The solution in the region 1 bounded by the characteristic x = at and the axis x = 0 (see Fig. 2) can be found with the help of the general solution (6), the relations along the shock fronts (5) and the boundary conditions (17): (+ = - a0 + qx - aqt/2,

v = vo- qx/2ap.

(18)

The velocity along the leading front vanishes at x = lf= 2apvo/q. Therefore the solution (18) is valid only up to the characteristic x + at = 21f. Starting from the point x = lr, t = tf= If/ a, the leading front deviates from the characteristic x = at and proceeds along a path x = x*(t). In the region bounded by the

L. V. Nikitin, A. N. Tyurekhodgaev

/ Waves in rods with slip

517

Fig. 2. Regions of motion for the typical regions.

characteristics x + at = 21,, x - at = 21, and x = x+(t) the unknown functions f,, fiand x* are determined from conditions of continuity: + ml

=(+I,

(T;=0,

v:

=

x+at=21,,

v;;

v:=o;

(19)

x = x*1(t).

Here + marks regions adjoining the leading front and - the axis x = 0. These conditions region 1 of Fig. 2: a:=

v: = v,/2 - 3qx/8ap,

-3uo/2+9qx/8-3aqt/8,

x = x*l( t) = 21,/3+ ar/3.

give for the (20)

Solutions can be found in a similar manner for successive regions. The total region of motion is composed of pairs of the triangle-shaped regions adjoining the x-axis and the leading front. Having solutions in a few regions it is possible first to guess and then prove that solutions for the nth pair of regions have the forms that will be discussed next. For the triangular regions bounded by the characteristics x+at=n(n+l)z,

x-at=n(n-1)1,, and the axis x = 0 the solution is

cr, = - nuo+ qx - aqt/2n,

vi = v. - qx/2nap.

(21)

For the triangular regions bounded by the characteristics x+at=n(n+1)1,,

x-at=n(n+l)&

and the leading front the solution is

(22) The leading front consists of the straight line segments x=x*,=

at + n(n + l)Zf 2n+l

.

(23)

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L. V. Nikitin, A. N. Tyurekhodgaev

The envelope

of the piecewise

leading

/ Waves in rods with slip

front is the parabola

x = (a&) “*. t - n*&/a, n + 00, the asymptotic

For large f, when

a=-q[(atlJ”*-Xl, From velocity

form of both solutions

v = V,[l -X/(atl,)“*].

(25)

(25) it can be seen that the stress at fixed x grows proportionally approaches

the value

(24) (21) and (22) is the same

u,,. At the leading

to the square

front (24) both stress and velocity

root of t, and the

vanish.

1.3. Impact of a semi-injinite rod by a rigid body Let a rigid body of mass m which moves with velocity v,, strike a semi-infinite rod. At the end of the rod, x = 0, where impact takes place, the equation of motion of the mass m has the form dV

mG=

SW;

v=o; In the region

x=0

)

The boundary

K takes the value

let us again use the Laplace

is taken the same as in (lo), de z=ppO+q

(26)

t=o.

where the rod is in motion

form (10). To find the solution K

t > 0,

x = 0,

After transformation

stage

of equation

K

has the

(I), where

we obtain

ePpx’O,

condition

transform.

1 and at least in the initial

p,=E$.

(27)

(26) takes the form

mp(C-v,)=Sc?.

(28)

In the problem under consideration not only the length l,= 2aO/q, which is characteristic for damping due to friction, but also another length, Zi= m/pS, which is the ratio of the striking mass to the mass per unit of the rod’s length appears. This length is characteristic for damping due to inertia. The solution of equation (2) with the boundary condition (28) gives after inversion of the Laplace transform: (+ = 0,

v =o;

t
a=-~[(If-li)e-‘“‘-x)/‘i-x+l;], f This solution

(294 v=~[(lf-li)e-‘“‘-““‘~-al+~i];

f

is valid only up to a curve along which the velocity at-l. X=X,(t)=Ut+filn~ lf- li .

?>~/a.

Wb)

vanishes:

A posteriori it is evident that application of the Laplace transform to this nonlinear problem was legitimate since lines t = const. intersect the curve v = 0 only once. Fig. 3 shows the curves x = XJ t) for different ratios If/&. Independently of the value of this ratio, the velocity along the characteristic x = at vanishes when x = lf. Therefore the solution (30) has no physical meaning beyond the characteristic x + at = 21,. For 1,= li all sections of the rod come to rest simultaneously at the moment t = rf. When f < If/fi < 00 the curve x = x,(t) is completely situated in the region of dependence

519

L. V; Nikifin, A. N. Tyurekhodgaev / Waves in rods with slip

f.0 -

4 Fig. 3. Boundary

between

regions

of motion

and rest for different

at& values of 1,/l,.

on the boundary conditions OAB (see Fig. 3) and the solution in the region bounded by the characteristic x = at, the curve x = x*(t) and the axis x = 0 is determined by the equations (29). When If/ Ii
u=~[(r,-li)(e-S”i-l)-fP];

(31)

f

and the unknown

curve a = a*(p) v =o;

u = 0,

along which the velocity

and the stress are zero:

a = a*(P).

(32)

The conditions (31), (32) permit us to determine the arbitrary functions yields the following form for the solution in the considered region:

UC-7

(lf-li)e-(~‘-x)/‘i+li_x-tliln f

v=F f

[

(If-li)e-

(nr-x)“~+ li- at +f li In

f2in the solution

(6). This

If- li (Ut+X)/2_li

[

fi and

1’

If- li (at+X)/2_Ii

1 ’

(33)

This solution is valid only up to a curve along which the velocity, determined by (33), vanishes. Fig. 3 shows such curves for lf/li equal to 0.4 and 0.32. The latter value corresponds to the case when the curve v = 0 goes through the point B. For smaller lf/li, equations (33) are valid up to the characteristic p = 21f. The solution to the right of this characteristic may be found similarly using the boundary condition (24) and equations (33) along the characteristic /3 = 21,. This solution is rather cumbersome and therefore is not written down here. Fig. 3 shows a curve along which the velocity is zero for lf/li = 0.1. The limiting case lf/ Ii= 0 is the problem considered in the previous section. 1.4. Impact of a jinite rod by a rigid body The previously obtained solutions are valid for a semi-infinite rod or for a rather long rod so that disturbances damp before a wave reaches the other end of the rod. Consider now a rod with finite length

520 1

L. V. Nikitin, A. N. Tyurekhodgaev

for which reflections

take place. The end x = 1 endures

of a mass m moving

impact

with velocity

v0

x = 1, t > 0,

-Su;

m$=

/ Waves in rods with slip

u=v,;

(34)

t =o.

x = 1,

The response of the surrounding medium at the end of the rod x = 0 is assumed to be of the rigid-plastic type. If the reactive stress there is less than some critical value R = a$, then the end x = 0 is at rest. In the opposite case the end moves under constant resisting stress. Since resistance appears only after the arrival

at the end x = 1, the boundary

of waves generated cr=-qH(at-1);

condition

at the end x = 0 is as follows

x=0.

(35)

The posed problem models a pile under impact loading. The sign of the velocity in this problem evidently is negative. The regions of rest and motion at least for the initial stage of motion are separated by the characteristic x + at = 1. Assume that this is true at least up to reflection from the other end. Then K is K =

The

problem

the velocity

-H(Ut+X-1).

(36)

under consideration

can be solved by the same methods

as above. In the region 1-x < at < 1+x

is

v=~[at-I,-(l~-li)e-‘a’+x-‘)‘,].

(37)

f

1+x < at < 31 -x

In the region

the velocity

is of the form

(38)

Ut+l-21i-2(1,-li)e-‘“““‘icosh~+~]. I

These equations

are valid only for those x and t for which the velocity

does not change

sign in the region

of dependence on the boundary conditions. Otherwise the solution has to be found taking into account cessation of motion in some region. Let us now find the displacement u0 of the end x = 0 which is the penetration of the pile per unit impact. Assume that the end x = 0 has come to rest at time t = to which satisfies the inequality l
(39)

This means that the end x = 0 has come to rest before the arrival If the velocity in the region of dependence on the boundary nonpositive

the displacement

The time to can be found

of the wave reflected from the end x = 1. conditions of the point x = 0, t = to is

u. is

from the transcendental

equation

(lf-li)e~~a~~~“~‘~-f(uto-l)+li-l+~=O. Due to the inequality

(39) the pertinent

parameters

(41) of the problem

satisfy the following

restrictions (42)

L. V. Nikifin, A. N. Tyurekhodgaev

/ Waves in rods with slip

521

Fig. 4 shows the domain of the parameters If/l, Ii/l and a,/aO where the inequalities (42) are met. It can be shown that for parameters from this domain, the velocity does not change sign in the region of dependence on the boundary conditions of point x = 0; t = t,,. In the opposite case the end x = 0 may also come to rest but then the leading front deviates from the characteristic at + x = 1, and the solution must be obtained by taking this fact into account. 1.5. Smooth loading of a semi-injinite

rod.

Consider now smooth loading at the end x = 0 of a semi-infinite rod, so that (+= a”(t),

aO(O)=O;

x=0

(43)

where co(t) is a monotonic function. In this case the leading front p = /3.+(t) does not coincide with the characteristic x = at from the very beginning. At the leading front the following conditions hold (+= 0,

v = 0,

K=+l;

P

=

(44)

P*(t).

Here the sign of the velocity is chosen positive for definiteness, and the stress a”(t) is assumed to be negative. Substitution of the general solution (6) into the conditions (43), (44) gives the following set of functional equations for the unknown functions fi, f2 and &: f;(at)

-fi(at)

= -a”/

E,

4W(P,(~))

= q”,

4%(a)

= q&(a)

(45)

If the stress at the end x = 0 is linear in time a”(t) = -kt,

(46)

then the solution of (45) is straightforward:

fl,2=$$Y2+lv**Y19

P*(n) = [b2+ w2-

(47)

rla

where y = 2 k/ ap.

Fig. 4. Domain

of parameters

a,/~,,,

I/l, and I/J

for which the solution

given in the text is valid.

522

L. K Nikitin, A. N. Tyurekhodgaev

With the help of (47) the leading

front is found

/ Waves in rods with slip

in the form

(48) The stress and velocity

are

o=~[(yZ+1)‘%zytl,

“=~[(~~+1)%-yx]. zap

From (47) it is seen that if the rate of loading x = at. In the limiting

k increases,

(49) the leading

case k = 00 which takes place in the problems

front approaches

considered

the characteristic

before, this front coincides

with the characteristic.

2. Inverse motions and vibrations 2.1. Instantaneous unloading Until now, only unidirectional motions for which the direction of velocity is always the same and evident from the physical situation, were considered. Now consider the problem of instantaneous unloading at a moment t = t, of a rod which before was loaded at x = 0 by a steplike stress -u,+Y( t) as in Section 1. It is assumed that the length of the rod 1 exceeds the characteristic length lr so that l> If. It is obvious that unloading must produce inverse motion of the rod sections. The problem of loading was considered in Section 1. The solution for unloading can be found on the basis of the general solution from conditions along the wave fronts (5) and the boundary condition at x = 0. The solution will depend on the correlation between t, and cf. Let us consider t, < tf/2. The region of direct motion D+ is bounded as in the problem of loading by the leading front x = at, the axis x = 0, the vertical line t = tf and now by the unloading front x = a( t - t,) as well, see Fig. 5a. The solution in the region D+ is still given by eq. (11). Behind the unloading shock

Fig. 5. Regions

of direct and inverse

motion

and rest.

L. V. Nikitin, A. N. Tyurekhodgaev/ Waves in rods with slip

523

front the region of inverse motion D- appears. The solution in this region is determined by the conditions (5) along x = a( t - t,) and a = 0 at x = 0. It has the form Q = -4x/2,

v=q(t-2t,)/2p.

(50)

As follows from eq. (50) from the right this region is bounded by the vertical line t = 2t,. In the regions of rest Dy, 0: and Di, see Fig. 5a, are located to the right of the lines t = tr, x = a (t - t,) and t = 2 t, , and are separated from each other by the horizontal lines x = a( tr- t,) and x = at, : 0::

0::

w = -ao+qx/2,

u = -aqtJ2,

0::

cr = qx/2.

(51)

At t, = t,/2 the region 0: disappears and direct and inverse motions cease simultaneously at t = tr= 2t,. For t,/2 < t, <2t,/3, apart from D; a region of inverse motions 0; appears, see Fig. 5b. This region is bounded from one side by the unloading shock front and from the other side by the characteristic x + a ( t + t, - 2tr) = 0, which separates 0; from the region 0;. The conditions (5) along these fronts lead to the following solution in the region 0;: u=-vO/2+qx/8ap+aq(5t--7t,)/8p.

cr=-a,,/2-3qx/S+aq(t+t,)/8,

(52)

As is seen from eq. (52), the velocity vanishes along the line x+5at=a(7to+2t,).

(53)

To the right of this line between the horizontal lines x=x,=a(t,+t,)/3

and

x=xz=2a(t,-2t,/3)

the motion ceases. The stress distribution in this region is w = -2u,/5-2qx/5+3aqt,/lO.

(54)

For 2t,/3 < t, G tf an additional region of inverse motion 0; appears, see Fig. 5c, bounded by the characteristic x = a( t + t, - 2tr) and the axis x = 0. The conditions of continuity of the solution along the characteristic, and the absence of stress at x = 0, lead to the following solution in the region D; : w = qx/4,

u = -v(J,,+3aq(t

- t,)/4.

(55)

As is seen from eq. (64) the velocity simultaneously vanishes at t = t3 = t,+2tJ3. After this motion ceases in the part of rod 0 < x < x3 = 2a( t, -2tJ3). The stress in the new region of rest is determined by (55). Finally for t, 2 tf the regions of inverse motion, D; and D;, and correspondingly the region of rest, Di, disappear. The solutions in the remaining regions, D+, D;, 0: and Dy, which take the forms shown in Fig. 5d have been determined before. Similarly, repeated loading and unloading as well as the case I < lr may be considered. Since these solutions are cumbersome, they are not given. 2.2. Free vibrations of the clamped rod Consider now the free vibrations of a rod with length 1 one end of which, x = 1, is clamped and at the other end, x = 0, a constant stress - a0 is applied at t = 0. Thus the boundary conditions are cr = -aoH( t),

x=0;

v=o ,

The difficulties of solving this nonlinear

x = 1.

(56)

problem consist of the dependence of K on the velocity which is unknown. However, in the case under consideration K may be established beforehand. The leading

524

L. V. Nikitin, A. N. Tyurekhodgaev

/ Waves in rods with slip

assumption is based on the fact that since the friction force is passive by its physical nature, the velocity cannot change sign. Therefore it should be expected in problems with steplike time loading in which wave fronts propagate along characteristics, that the distribution of velocity signs is the same as in the corresponding problems without friction. Let us check this guess. In the regions of motions O
-x+2nl
n=l,2,...

the velocity is zero in the absence of friction. For u=

n =

(57)

1, using eqs. (56) and (5), it is easy to obtain

Kq(l-x)/2ap.

(58)

By determination K is +l for a positive velocity and -1 for a negative one. Both these choices contradict eq. (58), and therefore in this region K must be taken equal to zero. Similarly a zero velocity is established for all regions (57), and the velocity signs in the remaining part of the region of motion are the same as in the problem without friction. Thus for problems with loading which varies steplike in time the sign of K coincides with the sign of the velocity for the corresponding problem without friction. This is the case up to the moment when the pulse has decayed so much that the leading front deviates from the characteristic. The final result is K =

-kE,

(-l)kH(at-x-2(k-1)1)+

f

(-l)kH(at+x-2kl).

(59)

k=l

Substitution of K from (59) into equation (1) reduces the initially nonlinear problem to a linear equation. It is then possible to use the Laplace transform. Transformation of equation (1) with K taken in the form (59) gives E=ppS+q

sinh-

p(l-x) a

cash @

E dfi

a’

(60)

dx=pa.

The solution of the system (60) satisfying the boundary conditions (56) is

apG = aa

a

cash p’ a

X

a

coshi! -j a

The inverse transformations of these rather huge expressions Consider firstly the regions of the type O
x+2(n+l)l
sinhP(w

coshP(l-4

sinhP(l-x)

a

a

cash@

a

1

(61)

.

are very simple in the concrete region. (62)

where n is a number of the wave fronts at-x = 2(n - l)J, at + x = 2nl. Numeration of direct fronts starts from the front x = at while inverse fronts start from x + at = 21. Let us present the expressions (61) in the form of an expansion in the running waves, that is expand cash-‘pi/a in terms of exp(-2pl/a). Then for any fixed time t or n the inverse transformation of the infinite sums gives only a finite number of nonzero terms, which can easily be established by counting

L. V. Nikitin, A. N. Tyurekhodgaev

direct and inverse waves participating stress:

/ Waves in rods wirh slip

525

in the creation of motion. Let us demonstrate it for an example of

k=l

k=l n-1

-ql

n-1

(-l)kH(at+x-2(k+1)Z)-ql

C

1 (-l)kH(at-x-2(k+1)Z)

k=l

k=l n--l

-;qx

C (-l)kH(at+x-2(k+l)l)-;qx

f

k=l

(-l)kH(at-x-2(k-1)l)

k=l

-CQ-$(-1)“qx. Similarly the velocity in the corresponding

(63)

=

regions may be found as

(,,_41>.

v=(-I)“-’

2P

For the regions of type O
-x+2nl
(65)

where the velocity is zero, it is possible to obtain in the same way a=aJ(-l)“-l]-(-1)“nql.

(66)

The considered regions fully complete the region of motion. The solution is valid only up to the moment t = tr when, as is seen from eq. (64) motion ceases simultaneously everywhere in the rod. After cessation of motion there remain stresses in the rod. For O
=-noduo

(67)

2E

1 u(0.t)

0

4

Fig. 6. Time displacement

8

f.2

I

20

at&

of the end of the rod far constant stress.

526

L. V. Nikitin,

A. N. Tyurekhodgaev

/ Waves in rods with slip

where t’= (2(n - 1)1+x)/a, The equation the direct

(67) is obtained

wave reaches

t”=2(nl-x)/a. under

the clamped

front from the clamped

the assumption

end of the rod. If motion

end the expression

Note that in accordance with physical in the initial formulation of the problem If however

the expression

‘,‘; Dd = ail/2 + For quasi-static

loading

that the moment

for dissipation

would

be slightly

ceases

t = tr set in before after reflection

the last front of of the last direct

different.

intuition the limit of dissipation is nonuniform 0, then naturally there is not any dissipation

q =

(67) approaches

when q + 0. If in the system.

the limit when q+O, the result is

E. and ql/o,%

(68) 1, the dissipation

is (69)

As can be seen from a comparison of (68) and (69) the dependence of dissipation on the parameters of the system in these extreme cases of instantaneous and infinitely slow loading is essentially different and D, Q D,, since ql < co. This result can be easily explained. Indeed in the case of quasi-static loading with accuracy of ql/ao, work performed by the stress at the end of the rod is equal to a$/2E, and with the same accuracy is transferred to the stored elastic energy of the rod. Only a small part of the energy which is proportional to gl/u,, dissipates due to friction. In the case of instantaneous loading, the stress at the end performs twice as much work, but in the final state the stored elastic energy of the system is approximately the same as in the quasi-statical case. From this it follows that the dissipation in the system with friction strongly depends on the history of loading.

References Structural Damping, ASME, New York (1959). L.E. Goodman, “A review of progress in the analysis of interfacial damping”, Ya.G. Panovko, Influence o~Inrerna/ Friction on Vibrations of Hastic Systems, Fizmatgiz, Moscow (1960) (in Russian). J.P. Den-Hartog, Mechanical Vibrations, McGraw-Hill, New York (1965). G.S. Pisarenko, Vibrations of Mechanical Systems with Account of Imperfect Elasticity of Materials, Naukova dumka, Kiev (1970) (in Russian). [5] L.V. Nikitin, “Wave propagation in an elastic rod in the presence of dry friction”, Ingenerny Journal, Mechanika tverdogo tela 3 (l), 126-130 (1963) (in Russian). “The behaviour under load of an elastic rod buried in soil”, in: Problems of Rock [6] L.V. Nikitin and A.N. Tyurekhodgaev, Mechanics, Nauka, Alma-Ata, 314-321 (1966) (in Russian). [7] L.V. Nikitin, “Impact of a rigid body on an elastic rod with external dry friction”, Ingenerny Journal, Mechanika tverdogo tela 7 (2) 166-170 (1967) (in Russian). [g] L.V. Nikitin, “Longitudinal vibrations of elastic rods in the presence of dry friction”, Izvestiya AN SSSR, Mechanika tverdogo [l] [2] [3] [4]

tela 6, 137-145 (1978) (in Russian). Transactions [9] A.V. Koibin, “The dynamic problem of structural damping due to pure frictional interaction”, University 20 (5), 18-23 (in Russian). [lo] N.N. Bogolubov and Yu.A. Mitropolsky, Asymptotical Methods in the Theory of Nonlinear Vibration, Nauka,

of Petrozavodsky Moscow

(1974).