Paper received: 07.06.2009 Paper accepted: 25.09.2009
Homodyne Quadrature Laser Interferometer Applied for the Studies of Optodynamic Wave Propagation in a Rod
Tomaž Požar* - Janez Možina University of Ljubljana, Faculty of Mechanical Engineering, Slovenia
The localized Coulomb dry friction on the surface of a rod significantly alters the propagation of an optodynamic wave. The effects of frictional clamping on the rod's dynamics were measured with a homodyne quadrature laser interferometer at the rear end of the rod. The measurements are supported by a theoretical description of the discrete motion of the finite rod, which is based on a general result representing the interaction of a laser-ablation-induced stress wave with a frictional clamping location. © 2009 Journal of Mechanical Engineering. All rights reserved.
Keywords: optodynamics, homodyne quadrature laser interferometer, wave propagation, friction
0 INTRODUCTION
1 THEORY
In an earlier paper we described the formation of an optodynamic wave as well as its propagation and multiple reflections from both faces of an unclamped finite rod [1]. We observed that ablation of the front face of a rod using a nanosecond laser pulse induces a short stress wave - an optodynamic wave - that propagates and reverberates inside the rod. However, if the rod is locally clamped, the wave discretely transfers its initial linear momentum as it impinges on the clamping location multiple times. The momentum transfer to the clamping was determined by measuring the axial displacement of the rear end of the rod.
In order to monitor the motion of the rear end of the rod over its several micrometer long path, we had to develop an interferometric method based on quadrature detection [2]. This contactless displacement-measuring instrument, a homodyne quadrature laser interferometer (HQLI), enables high dynamic range, single shot measurements of abrupt high-amplitude ultrasound superimposed on moving objects with a sub-nanometer resolution.
This paper explains general effects on the rod's dynamics caused by the presence of the localized Coulomb dry friction. Such effects need to be taken into account in various fields, for example, ablative laser propulsion [3], split-Hopkinson pressure-bar (SHPB) experiments [4] dynamic behavior of frictional contacts [5], and friction studies on the nanoscale [6].
In order to determine what occurs to the incident compressional stress wave when it impinges on the clamping location, consider a semi-infinite rod, as illustrated in Fig. 1, setting L . The rod is circular, elastic, homogeneous and isotropic, has the length L, cross-section A, Young's modulus E, density p, and longitudinal acoustical impedance Z = pc0. A tp = 10-ns Nd-YAG pulse ablates its front face at x = 0 and induces a c0tp - long optodynamic wave propagating with the speed c0 =^E/p . The rod
is point-clamped at x = x0 with a variable contact normal force FN. Except at the clamping location, the rod has stress-free surfaces. Three displacements of interest are drawn: the end of
the rod, ue, the clamping position, uc, and the
*
center of mass, u .
Fig. 1. Basic parameters of the rod
To explain the interaction and the wave's dynamics theoretically we set a suitable linearized form of the elementary wave equation for a rod [7] and [8], including point-like, Coulomb dry-friction forces on the surface circumference at x = xn, as
d 2u (x, t) 1 d 2u( x, t) öx2 c2 dt2
(1)
+-F S( x - x0)S(t - xj c0). E
Here, the clamping is understood to be incompressible and localized, and, therefore, the friction forces are assumed to be concentrated and can be represented by a delta function. u(x,t) represents the axial displacement at any given point x and time t; v(x,t) is the corresponding particle velocity; and a(x,t) is the stress.
The passage of the wave under the clamping gives rise to a pressure impulse exerted by frictional forces IF =J PF (t)dt = pjA that
equals the momentum acquired by the clamping pc, divided by A. In this specific case, IF can also be expressed in terms of the coefficient of friction between the rod and the clamping, p, the normal force, Fn, and the time needed for the wave to cross the clamping location, tc, as IF = /uFNtjA .
The nanosecond laser ablation of the front face of the rod can be modeled by a boundary-loading condition [9] and [10], <r(0, t) = -IAS(t). Similar
to IF, IA =J PA (t) dt = p0/ A represents an
ablation pressure impulse that equals the momentum p0 obtained by the rod due to the laser-pulse ablation divided by A. The initial duration of the optodynamic wave, tc, equals [3] and [11] the laser-pulse duration, tp. If the dispersion [9] is neglected, tp = tc, otherwise
tp < tc. IA can be approximated by P0tp, where P0
describes the constant normal pressure acting evenly on the front plane of the rod during the laser ablation.
Eq. (1) can be solved using standard onesided Laplace-transformation methods [12]. The rod is initially undisturbed u( x,0) = 0 and at rest v( x, 0) = 0. Taking into account the initial and
boundary loading conditions and including the requirement u (x ^<x>, t) = 0, we obtained the
following solution of Eq. (1) in a dimensionless form composed of Heaviside unit-step functions. The solution for a displacement is presented in
terms of the forward- and backward-propagating waves.
u(x, %) =
i - 2h( x - x0)
H(% - x)
(2)
-— + — H(x - x0) 2 2
H(% + (x - 2x0)) + 1 H(% - (x + 2x0))
For the sake of compactness, the following dimensionless variables, designated by the tilde, are introduced:
x	t t
~ _ a % _ a _ a
a ~ T ' a . ~ j- I ' L	tL LIC0
% = IF =ßFNtc
u„
ZAu„

IJZ'
v„
= ■
tp <ya
p = Pa= P.
r a
(3)
P0 ZAu0 Ea Ea
^ I a]t p ' E E0 ZAu2/tp •
a can be substituted by any suitable symbol-string index or * in the superscript that denotes the center of mass. u0 is a measurable quantity [1] equal to one half of the displacement caused by the reflection of the wave from the rear end of the unclamped rod (see Fig. 3). p0 is the momentum carried by the primary wave and E0 is its total mechanical energy, consisting of equal contributions from the kinetic and elastic parts.
The first term on the right-hand side of Eq. (2) corresponds to the forward-propagating wave with its origin at x = 0. The value inside the square brackets is the displacement of the rod's cross-section once it has been traversed by the wave. In the region 0 < x < x0 it is termed the incident wave, and for x0 < x the transmitted
wave. The second term corresponds to the backward wave, starting at x = 2 x0. In the region where the value in the square brackets is nonzero, i.e., for 0 < x < x0, the wave corresponds to the reflected wave. The third term is of lesser
Table 1. Properties of the clamping and the waves before and after the interaction with a point-like frictional location in a concise lookup-table form_
location	no fiction I = 0 u v ä p	kinetic friction 0 < I < 2 u v a p	static friction I = 2 u v a p
incident wave transmitted w. reflected w. clamping loc.	11-11 11-11 0 0 0 0 1 0	11-11 1 -1/2 1 -1/2 -(1 -1/2) 1 -1/2 -1/2 -1/2 -1/2 -1/2 1 -1 /2 I	11-11 0 0 0 0 -1 -1 -1 -1 0 2
importance, and corresponds to the rebound of the reflected wave at the rod's free end. Note that the frictional term in Eq. (1) is simplified in such a manner that the friction acts only once, neglecting later reverberations of the reflected wave in the region 0 < X < X0.
Differentiating Eq. (2) with respect to t or X yields the corresponding particle velocity v and stress a, respectively. The expressions for v and a are composed of delta functions, which can be approximated by suitable rectangular pulses of a length equal to c0tp. It can also be inferred from Eq. (2) that I = 0 corresponds to no friction. If 0 < I < 2, slippage occurs and the friction is kinetic, and if I = 2, the friction is static, there is no slippage and only in this case does the frictional force drop from its maximum kinetic value in order to satisfy the condition IF = 2Ia . The values of u , v , a , and p for the
incident, transmitted, and reflected waves and for the clamping location in the three frictional regimes are gathered in Table 1. u is a displacement of the cross-section after the passage of the wave, v and a are the particle velocity and stress within a wave, and p is the momentum carried by the wave or the momentum transferred to the clamping. A negative value of a corresponds to a compression.
In the static friction case, the clamping location has identical effects on the wave propagation as a fixed rod-end termination, with its characteristic velocity reversal and double wave-momentum transmission to the rigid clamping. If the friction is kinetic, the incoming compressional wave is both transmitted and reflected as a compression. The clamping
receives the required momentum to satisfy the conservation-of-momentum law. This momentum is independent of the magnitude of the incoming wave if the friction is kinetic, but it is strongly dependent if the friction is static.
In the presence of slippage the energy is dissipated and the work is done by the non-conservative Coulomb dry-friction forces. The work equals the constant frictional force times the displacement of the clamping. This friction work WF = Wf/e0 = I (1 - 7/2) must also obey the work-energy theorem, and, therefore, be equal to the change of mechanical energies of the incident wave before, and the sum of the transmitted and reflected wave after, the interaction with the clamping. It achieves a maximum value when
I =1.
2 EXPERIMENTAL SETUP
The experimental setup is shown in Fig. 2. The optodynamic wave was induced by a Q-switched Nd:YAG operating at 1064 nm capable of producing 10-ns-long pulses with the maximum energy of 100 mJ. The excitation laser pulse was focused to a 0.5-mm-diameter spot on the center of the front plane of a steel (AISI: 304) solid cylindrical 12-cm-long and 2-mm-thick rod.
The effects of the clamping were measured at the rear end of the rod by monitoring the axial displacement of the rear cross-section. This displacement is caused by the reflections of friction-effected mechanical compressions and rarefactions. The out-of-plane displacement was measured with a homodyne quadrature laser interferometer whose operation is described in detail in [2].
Fig. 2. Schematic diagram of the experimental system
3 RESULTS AND DISCUSSION
So far we have considered the first transit of the optodynamic wave over the dry-frictional clamping. Following multiple reflections and transitions of the branching waves at the clamping location, according to the results summarized in Table 1, we obtained a discrete temporal evolution of the dynamics of a finite rod with a localized Coulomb dry friction. The measurement of the rear-end displacement of a laser-ablated rod that is point-clamped in the middle by a variable contact force is shown in Fig. 3.
Fig. 3. Axial displacement measured at the rear end of the 120-mm-long and 2-mm-thick steel rod as a function of time. Two displacement curves
are shown: the unclamped rod and the rod clamped at its middle by a normal force of 2.42 N. From the two measurements the friction parameter of the clamped rod can be estimated to
be I = 0.85
One of the conclusions of such reasoning is that once a wave within a finite rod is reflected at the fixture, it will not pass the clamping location again. It will echo back and forth, on either one side or the other of the clamping. The only process that enables the wave to pass over the clamping location again is the interference of two or more waves, simultaneously interacting with the clamping.
In the case of weak friction 0 < I << 1, the results are in accordance with the motion of a rigid body under the influence of constant deceleration due to the Coulomb friction. Even though the motion of the weakly clamped rod is discrete, its general envelope can be approximated by a smooth function. The rod slips over the clamping location by a terminal length
uca = l/1, and the time needed for this slippage is t f = 2/I. The rod's momentum and the
velocity of the center of mass both decrease linearly with time
p(t) = v(t) =
1 - ipf, 0 * % * %f
0,
tf < t
(4)
The displacement of the center of mass reaches its maximum value 1/1 quadratically
%*(%) = } v
t (1 - %/2%f ), 0 * t * tf
(5)
tf <t
t f is the time needed to achieve the total slippage or the characteristic time of the
momentum transfer from the rod to the clamping as a result of the localized Coulomb dry friction. It can be estimated indirectly according to the relationship tf = 2uc x by measuring the total
slippage. When the normal force equals the rod's weight, tf must be smaller than 2u0/(ugtp), where g is the acceleration due to gravity. This type of external friction was identified in the measurements of the geometrical dispersion effects on the optodynamic wave during its multiple transitions in a finite rod [9], where tf < 0.5 s and ucm< 0.2 mm were assessed
according to the above estimate.
A portion of the wave's momentum is also transferred to the surrounding atmosphere on each reflection at the front and rear ends of the rod at normal air pressure. This acoustic-impedance mismatch effect leads to an exponential transfer of the rod's momentum to the air in the form of sound. p(t) = exp(-t/f), where t = tLf = tL (Zsteel/2Zar). The characteristic time
of the momentum transfer from the rod to the surrounding air is t = 1.2 s for the rod in reference [9], and is of the same order as tf.
Normal force [N]
Fig. 4. Linear dependence of the frictional parameter I as a function of the normal force FN
We measured the characteristic non-dimensional parameter of the relative friction strength I using a homodyne quadrature laser interferometer in two distinct, contactless ways. In the weak-friction regime we deduced I from the total slippage displacement, which cannot be measured with compensated Michelson-type interferometers. On the other hand, when I is comparable to unity, it can be determined from the ratio of the first few discrete displacements of the rod's rear end, as shown in Fig. 3. The
experiments also confirm that I is proportional to the normal force (see Fig. 4).
Our analysis of the experimental conditions employs a simple representation of the ablation and friction using delta functions. Assuming a finite stress wave and a finite clamping contact, additional effects can be expected: the wave-shape, not just the amplitude, changes after the wave's reflection from the clamping, and the residual stress distribution remains under the contacting surfaces [13], thus the rod under the clamping is pre-stressed for the subsequent arrival of the wave. We can neglect the fact that the clamping is displaced to a new location once the slippage occurs, since in our experiments the slippage is very small compared to the length of the rod. Eq. (1) assumes that the cross-sections remain plane, that the waves are not leaking from the rod to the clamping, and that the Poisson effect does not enhance the normal force [14]. Apart from the dispersion, these are the limitations of the analysis that most likely contribute to the slight dissimilarity between the theoretical prognosis and the measured curve. Based on the above-stated restrictions, it can be concluded that our description of the laser-propelled rod-movement better explains the interaction of the wave with the clamping when the friction parameter I is smaller.
4	CONCLUSION
In conclusion, we have presented an experimentally supported description of the motion of an elastic rod that bears a localized propagating optodynamic wave and includes a stoppage mechanism in the form of the localized Coulomb dry friction. The introduced optodynamic method can also be extended to studies of nanoscale friction effects. The results can as well be applicable in a description of the structural damping of miniaturized SHPB systems.
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