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\title{SOUND GENERATION UNDER DRY FRICTION CAUSED BY OSCILLATING LOAD}
\author{Vyacheslav A. Lazarev}
{Mechanical Engineering Research Institute\\
                Russian Academy of Sciences\\
                Belinskogo, 85, 603024 N. Novgorod, Russia}


A simple model of sound generation under dry friction, allowing for
oscillating loads perpendicular to sliding plane is considered. The rigid
body moves in $x$ direction at a constant speed $v$ along the homogeneous
plane $(z=0)$ above an elastic half space $z>0$. Process does not depend on
coordinate $y$. The body surface is periodically modulated with the period $%
2L$. The load and half space are separated by a small distance owing to the
static (for example, molecular) repulsion forces. We can come to the point,
describing processes at $z>0$ by the scalar equation

\begin{equation}
\label{Q1}\mu u_{xx}+\mu u_{xx}-\rho u_t=0
\end{equation}
with the following boundary condition at $z=0$

\begin{equation}
\label{Q2}\mu u_z=\exp (-i\Omega t)f(x-vt),
\end{equation}
where $\mu $ is the elasticity modulus of medium, $\rho $ is its density, $%
f(x-vt)$ is periodical function with the period $2L$ and $\Omega $ is the
circular frequency of the load oscillation. The expression (\ref{Q2})
describes the effect of the moving and simultaneously oscillating load on
the waves radiation power. The correlation between $\Omega $ and the period
of the roughness is not suppose. We may show that density of sound radiation
power, averaged over time and over the period of the roughness, in the
subsonic case $v<c=\sqrt{\mu /\rho }$ equals

\begin{equation}
\label{Q3}W=\sum\limits_{0\leq m\leq \frac{\Omega L}{\pi (c-v)}}\frac{\Theta
_mv+\Omega }{\mu Z_m}\left| A_m\right| ^2
\end{equation}
where $Z_m=c^-1\sqrt{(\Theta _mv+\Omega )^2-c^2\Theta _m^2},\;\Theta _m=%
\pi m/L,\; m=0,1,2,...$, $A_m$ is the Fourier coefficients of function $%
f(x-vt)$ for $m$ - th mode. For radiating load the $Z_m$ should be real. In
the subsonic case, if the load does not oscillate, wave process has surface
character and it is not accompanied by the sound radiation into the
elastic half space ($Z_m$ is imaginary).
Sound radiation of moving and simultaneously
oscillating load into the elastic half space
for $m\neq 0$ occurs at $v>v_m=c- \Omega/
{\Theta _m}$ ($Z_m$ is real), where $v_m$ is critical velocity, which less
sound velocity. The contribution of each mode in radiation power decreases
with velocity increase, tending to asymptote $ c \left| A_m\right|
^2/ \mu $. For $m=0$ , corresponding to the load oscillating without sliding,
the sound radiation occurs for arbitrary small $v$ and power of acoustic waves
does not depend on $v$ . As a result a series of maxima in dependence $%
W(v) $ at $v=v_m$ should be observed. Thus, it is established that in the
process of dry friction the oscillating load causes the radiation of
acoustic waves with a complicated and nonmonotonic velocity dependence.

Research is supported by the scientific program ''Russia University'' of
section ''Mechanics of Solids''.


\end{document}
