ON THE ACOUSTICS AND FRICTION

Transcription

1 SISOM 2012 and Session of the Commission of Acoustics, Bucharest May ON THE ACOUSTICS AND FRICTION Veturia CHIROIU, Cristian RUGINĂ, Dan DUMITRIU, Ligia MUNTEANU Institute of Solid Mechanics, Romanian Academy, Bucharest This paper discusses the topic of acoustics related to friction in the sliding contact of solids. A remarkable review on the acoustics of friction was done by Akay in Starting from this review, the evolution of the friction stress at constant normal stress when the slip rate is changing, for a particular slip history, is discussed in this paper. The effect of the interface s roughness on the sound radiation from the friction induced excitations is analyzed. Key words: Acoustics, friction, sliding, contact, mode lock-in. 1. INTRODUCTION An important aspect in the field of the acoustics of friction is related to the continuous or transient unsteady contacts in solids [1]. Transient sliding is a source of squeaks and squeals in the violin sounds for example, while the continuous sliding can produce a various sound by the variation of the normal contact loading, as in the case of two pieces of sandpaper when rubbed against each other. The strong contact conditions expressed by nonlinear relations can lead to instabilities called mode lock-in, where the friction responds only to the fundamental frequency and its harmonics. This phenomenon depends on the normal load, sliding velocity and the contact geometry [2, 3]. This paper discusses the evolution of the friction stress at constant normal stress when the slip rate is changing, in the case of a particular slip history. Fermi, Pasta and Ulam studied in 1955 the problem of a dynamical system of n identical particles of unit mass on a fixed end line with forces acting between particles [4, 5]. The motion equation of the particle n is Q n = f ( Qn+ 1 Qn) f ( Qn Qn 1), (1.1) Q n where is the displacement of the particle with respect to the equilibrium position, and f(q) is the interaction force of the form or 2 f ( Q) = γ Q+α Q, (1.2) 3 f ( Q) = γ Q+β Q, (1.3) with γ a constant, and α, β chosen in such a way that the maximum displacement of Q nonlinear term is small. Using the McLaurin expansion f ( n+ a) = exp( a ) f( n) n, where n is now a continuous variable, the equation (1.1) becomes [2] Q= f[(exp( ) 1) Q] f[(1 exp( )) Q], n n caused by the

2 Veturia CHIROIU et al. 298 with Qtn (, ) = Q() t. Expanding the function f as a McLaurin series we obtain n Q 1 Q 1 Q Q Q= f (0) f (0) n 12 n 2! n n Q Q + f (0) ! n n This equation can be written as P l P 1 r P P P + 3 = f (0) l f (0) 2 l (1.4) x 12 x 2! x x P P + f (0) , 2 3! x x if we note x = nl and P= Ql r, where r is to be determined from the balance of the fourth derivative term with the nonlinear term. Using initial sine-wave data, integrating the equation (1.4) yields to the conclusion that the equipartition of energy criterion failed. The energy is kept in the initial vibration mode and a few nearby modes, without spreading in all the normal modes of vibrations. In addition, the motion of the dynamical system produces the mode lock-in phenomenon. The sound is radiated in the surrounding medium for the first, the second, or even the third natural frequency and its harmonics. 2. THE CONTACT MODEL A friction contact model between two bodies is proposed in this paper in the spirit of Yang, Chu and Menq [6], and Ruina [7]. Fig. 1 represents a contact with friction between two interfaces with an initial gap stress. The model consists from a massless elastic element characterized by two linear springs of lengths σ 0 l u and respectively lv, and moduli ku and respectively, kv,which account for the shear and normal stiffness properties, and a friction contact point which verifies the Coulomb friction law with the friction coefficient μ. The tangential relative displacement is denoted by u, the normal relative displacement by v while the tangential (slip) displacement of the contact point is denoted by δ. When the bodies are in contact, a tangential stick-slip motion it is possible to appear when the normal relative motion v becomes large we can have an intermittent separation of grains. The normal stress σ and the induced friction stress τ are given by σ 0 + kv v, when v σ0/ kv, σ= (2.1) 0, when v< σ0 / kv, τ = k ( u δ ). (2.2) u When the vibratory displacements are small, the contact interface sticks and the friction force is proportional to u with reference to δ, which is zero. The slip load always limits its magnitude ±μσ. When the friction force exceeds the positive value μσ, the contact interface starts to slip towards the positive u direction. The friction force remains equal to the varying slip load until the contact interface sticks again. Consider now the case of a corrugated interface (Fig.2).

3 299 On the acoustics and friction Fig. 1. The contact interface model. Dieterich [8-11] have reported some experimental results regarding the slip instabilities which occur when the friction force is reduced during the sliding, or after a time dependent healing during stationary contact. Also, when the friction force becomes a decreasing function of the instantaneous slip rate, the instabilities can also occur during the accelerating slip. By idealizing the experiments, Dieterich has shown that for a unit of slip area, the friction stress τ can be determined by the normal stress σ and the slip rate δ=v. Other articles on the same topic are [12-22], Fig.2 presents a scheme for the variation of the friction stress τ at constant normal stress σ when the slip rate V is changed [7]. For δ=δ1 the slip rate is sharply increased from V1 to V2. The stress τ salts τ 1 to τ2. For δ=δ2 a sudden fall to V1 causes a sudden fall to τ 1. For δ =δ3the allure of the curve is the same as for δ=δ 1.. Fig. 2. The evolution of the friction stress when the slip rate is changing [7]. The transition between slip and stick depends on the tangential relative displacement u and on the variable normal load, which may decrease to reduce the slip load so that the occurrence of the transition can be postponed to some instant after the reversion of u. During the cycle of motion the contact normal load may vanish and cause the separation. In the displacement-force plane a hysteresis loop is appearing. The area of the loop gives the energy dissipated per cycle of motion. This is the damping capacity, which, when divided by the material volume, gives the per-unit-volume damping capacity. The stick-slip conditions are characterized by the friction stress τ and the slip rate V, which must be formulated in terms of the input relative displacements. We consider in the following ku ( u u0) +τ 0 for V = 0 (stick), τ= μσ=μσ 0 +μ kv v for V> 0 (positive slip), μσ = μσ 0 μ kv v for V < 0 (negative slip), (2.3)

4 Veturia CHIROIU et al. 300 where u0 and τ0 are the initial values of u and τ at the beginning of the stick state t = 0 : u(0) = u0, τ(0) =τ 0. From (2.1)-(2.3) we obtain 0, for stick, V = u μ( kv / ku) v, for positive slip, u +μ ( kv / ku) v, for negative slip. For the case of modulated interface between two bodies in contact, the contribution of the impulsive contact forces can be written as [1] m (2.4) F sin (( n x vt) / b) ( x n x), (2.5) ϒ = ϕ Π Δ δ Δ Akay n n n= 1 where n= integer( L/ Δx), m is the number of simultaneous contacts of length b traveling with a sliding velocity v. The function F represents the discrete impulsive point forces that appear during the motion at ϕ n n an angle with the axis of the beams. The corrugated micro-interface between two beams is shown in an exaggerated sizing, in Fig.3. Fig.3. Corrugated interface. 3. THE CANTILEVER BEAM The bending of a cantilever beam composed of two materials separated by a roughed interface S, can produce the mode lock-in phenomenon. This phenomenon is obtained by rubbing the materials along the length on the roughed interface S. We show further that the spectrum of the bending vibrations and radiated sound appears for the first, the second, or even the third or fourth natural frequencies of the beam and its harmonics. At the macro-scale the friction interacts with the vibrations setting up a feedback between the friction and vibrations into the rough interfaces [1]. The friction and the motion of the interface are strongly related and lead to the friction-induced vibrations. At the micro-scale the friction is a well known mechanism that converts kinetic energy into thermal energy. We suppose the modulation law of the corrugated interface is given by sin 2 πx / λ (Fig.3). This condition of coherency at the interface establishes that, at any location, the bodies in contact have the same in-plane lattice spacing. Away from the interface, the structure relaxes towards its unstrained condition if the strain is maximum. The forces acting at the corrugated interface have an impulsive character being defined as the rate of change of momentum. The bending vibrations of the beam are described by the equation 2 2 wxt (, ) wxt (, ) EI( x) A( x) 0, ρ + ϒ= 2 x x t where EI( x) is the bending stiffness ( E is the Young s modulus of elasticity and I( x) is the moment of inertia), ρax ( ) is the mass per unit length ( ρ is the density and Ax ( ) is the cross section area), and wxt (, ) is the transverse displacement, respectively. The third term in (3.1) represents the contribution of the impulsive contact forces per unit mass (3.1)

5 301 On the acoustics and friction 2πx ϒ= F ( kx vt ) sin dx, (3.2) λ S where k is the wave-number and v the sliding velocity v. The function F represents the impulsive point force developed during the motion of the beam. The term (3.2) is defined for a particular case of the viscous damping. The initial conditions are written as wx (,0) = w0 ( x), wxt (, ) t= 0= v0( x). t (3.3) The boundary conditions are written for a clamped beam wxt (, ) = 0, wxt (, ) = 0for x x = 0 and x = L. (3.4) The solutions of (3.1) show that the bending of the cantilever beam and the beam locking into a particular mode are characterized by stable vibrations. Let us consider two cantilever aluminum layers of length L = 28cm, with constant diameter d = 3cm, the Young s modulus E = 70GPa and the mass density 3 ρ=2700 kg/m, separated by a roughed interface of thickness h = 0.5cm. We suppose that the materials are rubbing along the length on the roughed interface S, and cannot be separated. The impulsive force is given by F ( kx vt) = 2exp[ 2 k( x x0) v( t t0)] δ( t to) δ( x xo). This force acts on the layers for short intervals of time during the rubbing. The frequency response of the cantilever beam is shown in Fig.4. The first natural frequencies are 19, 145, 360 and 745 Hz, respectively. We are interested to analyze how the beam responds to these natural frequencies. The beam response at three of its natural frequencies with corresponding harmonics is shown in Fig.5. We see that the mode lock-in is obtained to each of the natural frequencies, separately. In conclusion, we repeat what Akay said in [1]: the friction is the source of all imaginable types of waves within solids, the cause of music or noise, and the source of damping for resonant and unstable vibrations. Fig.4. The frequency response of the cantilever beam.

6 Veturia CHIROIU et al. 302 Fig.5. The beam responses at four natural frequencies with corresponding harmonics. REFERENCES 1. AKAY, A., Acoustics of friction, Journal of Acoustical Society of America, 111(4), , MOŞNEGUŢU, V., CHIROIU, V., On the dynamics of systems with friction, Proceedings of the Romanian Academy, Series A: Mathematics, Physics, Technical Sciences, Information Science, 11(1), 63 68, BADEA, T., CHIROIU, V., MUNTEANU, L., DONESCU, Şt., A Preisach model of hysteretic ehaviour of nonlinear mesoscopic elastic materials, Revue Roumaine des Sciences Techniques, Série de Mecaniqué Appliquée, 47(1-6), 83 96, DODD, R.K., EILBECK, J.C., GIBBON, J.D., MORRIS, H.C., Solitons and Nonlinear Wave Equations, Academic Press, London, NewYork, MUNTEANU, L., DONESCU, Şt., Introduction to Soliton Theory: Applications to Mechanics, Book Series Fundamental Theories of Physics, vol.143, Kluwer Academic Publishers, Dordrecht, Boston (Springer Netherlands) 2004 (new edition New York: Springer, 2005). 6. YANG, B. D., CHU, M. L., MENQ, C.H., Stick-slip-separation analysis and nonlinear stiffness and damping characterization of friction contacts having variable normal load, Journal of Sound and Vibrations, 210)(4), , RUINA, A., Slip instability and state variable friction laws, Journal of Geophysical Research, 88(B12) , DIETERICH, J.H., Time-dependent friction in rocks, Journal of Geophysical Research, 77, , DIETERICH, J.H., Time-dependent friction and the mechanics of stick-slip, Pure Applied of Geophysics, 116, , DIETERICH, J.H., Modeling of rock friction. 1. Experimental results and constitutive equations, Journal of Geophysical Research, 84, , DIETERICH, J.H., Modeling of rock friction. 2. Simulation of preseismic slip, Journal of Geophysical Research, 84, , MUNTEANU, L., CHIROIU, V., On the three-dimensional spherical acoustic cloaking, New Journal of Physics, 13(8), 1 12, DONESCU, Şt., MUNTEANU, L., MOŞNEGUŢU, V., The acoustic of the stick-slip phenomenon, Revue Roumaine des Sciences Techniques Série de Mécanique Appliquée 56(2), MUNTEANU, L., CHIROIU, V., On the dynamics of locally resonant sonic composites, European Journal of Mechanics- A/Solids, 29(5), , MOŞNEGUŢU, V., MUNTEANU, L., CHIROIU, V., On a follower force friction model for discs, Computer and Experimental Simulations in Engineering and Science (CESES), 1, TEODORESCU, P.P., CHIROIU, V., MUNTEANU, L., DELSANTO, P.P., GLIOZZZI, A., On the nanoacoustic auxetic panels, A 8-a Conferinţa Internaţională Acustică. Vibraţii. Acţiuni seismice. Sisteme inteligente de protecţie. Univ. Eftimie Murgu, Reşita, Octombrie 2-25, MUNTEANU, L., On the intermodal interaction of waves, Advanced Technology for acoustics and music, A Series of Reference Books and Textbooks 9th International Conference on Acoustics & Music: Theory & Applications (AMTA 08) Bucharest, 17 22, TEODORESCU, P.P., MUNTEANU, L., On the solitons and nonlinear wave equations, Advanced Technology for acoustics and music, A Series of Reference Books and Textbooks 9th International Conference on Acoustics & Music: Theory & Applications (AMTA 08) Bucharest, 41 46, MUNTEANU, L., MARIN, O., MOŞNEGUŢU, V., On the nonlinear stick-slip friction. Part I: Theoretical odelling, Proceedings of the Annual Symposium of the Institute of Solid Mechanics, Publishing House of the Romanian Academy (ed.t.sireteanu), 21 26, TEODORESCU, P.P., CHIROIU, V., MUNTEANU, L., Damping across the length scales, International Symposium Energy dissipation, acoustical processes, vibrations and seismology, Nov. 14, ŞTIUCĂ, P., CHIROIU, V., MUNTEANU, L., CHIROIU, C., On the acoustic solitons propagating in a crystalline lattice, Proceedings of the Institute of Solid Mechanics, Publishing House of the Romanian Academy (ed. T. Sireteanu), , MUNTEANU, L., Characterisation of chiral Cosserat solids by acoustical propagation techniques, Tenth International Congress on Sound and Vibration, 7-10 July, Stockholm, Sweden, , 2003.