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Analytical approximations for stick–slip vibration amplitudes
International Journal of Non-Linear Mechanics 38 (2003) 389–403
Analytical approximations for stick–slip vibration amplitudes Jon Juel Thomsena; ∗ , Alexander Fidlinb a Department
of Mechanical Engineering, Solid Mechanics, Technical University of Denmark Building 404, DK-2800 Lyngby, Denmark b Tennesseealee 35, D-76149 Karlsruhe, Germany
Abstract The classical “mass-on-moving-belt” model for describing friction-induced vibrations is considered, with a friction law describing friction forces that -rst decreases and then increases smoothly with relative interface speed. Approximate analytical expressions are derived for the conditions, the amplitudes, and the base frequencies of friction-induced stick–slip and pure-slip oscillations. For stick–slip oscillations, this is accomplished by using perturbation analysis for the -nite time interval of the stick phase, which is linked to the subsequent slip phase through conditions of continuity and periodicity. The results are illustrated and tested by time-series, phase plots and amplitude response diagrams, which compare very favorably with results obtained by numerical simulation of the equation of motion, as long as the di0erence in static and kinetic friction is not too large. ? 2002 Elsevier Science Ltd. All rights reserved. Keywords: Friction; Stick–slip; Self-excited vibrations
1. Introduction We consider the classical “mass-on-moving-belt” model for describing friction-induced vibrations. Taking the model for granted—we do not discuss its validity for mimicking real systems—the purpose is to contribute further understanding on what it predicts, i.e. to provide expressions and graphs illustrating the change of vibration character and amplitude with excitation speed. For a common friction law, with a kinetic coe6cient of friction that -rst decreases and then increases smoothly with sliding speed, we provide simple approximate expressions for the system state during each cycle, and for the stationary vibration amplitudes and base frequencies. This supplements the many works in this area who deals mainly with setting up criterions for the onset of such vibrations, considers only non-sticking motions, assumes piecewise linear friction laws, or rely solely on numerical simulation. Assuming small but -nite di0erences in static and kinetic friction, a perturbation approach is used to -rst set up expressions describing the occurrence, the stability, and amplitude characteristics of pure-slip motions, also showing that such motions are typically stable only for a narrow range of sliding speeds. ∗
Corresponding author. Fax: +45-4593-1475. E-mail addresses: [email protected] (J. Juel Thomsen), [email protected] (A. Fidlin). 0020-7462/03/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 0 - 7 4 6 2 ( 0 1 ) 0 0 0 7 3 - 7
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Below a certain excitation speed, stick–slip motions occur, for which we also provide simple analytical expressions. This is accomplished by using perturbation analysis for the stick phase, which is linked to the subsequent slip phase by requirements of continuity and periodicity. The results are illustrated and tested by showing time-series, phase plots and amplitude response diagrams, which compare very favorably to results obtained by numerical simulation of the equation of motion. Stick–slip vibrations show up in many kinds of engineering systems and everyday life, e.g. as sounds form a bowed violin, squeaking chalks and shoes, creaking doors, squealing tramways, chattering machine tools, and grating brakes—most of which are attributed to surfaces sliding with friction. Furthermore, many mechanical interfaces are characterized by a form of dry friction where the force–velocity curve has negative slope at low velocities. Initially, friction decreases as the contacting object starts to move, whereas at higher velocities the friction force increases again; in particular this characterizes surfaces with boundary lubrication. The initial negative slope corresponds to negative damping, and may thus cause oscillations that grow in amplitude, until a balance of dissipated and induced energy is attained, as pointed out already by Lord Rayleigh ([1], Vol. I, p. 212). Typically there are two phases of such oscillations: a stick phase with no slippage between parts and friction forces limited by static friction, followed by a slip phase with a somewhat lower friction force; though, on some conditions there is no stick phase. Numerous works has been devoted to the study of friction-induced oscillations. For ease of setup and interpretation an idealized physical system consisting of a mass sliding on a moving belt has been considered very often, as it will be in this present study. Panovko and Gubanova [2], for such a system with a friction characteristic having minimum coe6cient at velocity vm ; show that self-excited oscillations occur only when the belt velocity is lower than vm . Tondl [3], Nayfeh and Mook [4], and Mitropolskii and Nguyen [5] describe self-excited oscillations of the mass-on-belt system, presenting approximate expressions for the vibration amplitudes for the case, where there is no sticking between mass and belt. Popp [6] presents models and numerical and experimental results for four systems that are similar to the mass-on-moving-belt. Ibrahim [7,8] and McMillan [9] presents and discusses the basic mechanics of friction and friction models, and provides reviews on relevant literature. A very readable historical review on dry friction and stick–slip phenomena is given by Feeny et al. [10], and a large survey on friction literature until 1992 by Armstrong-HKelouvry et al. [11]. There are, however, very few works providing expressions for stationary stick–slip amplitudes. Armstrong-HKelouvry [12] performed a perturbation analysis for a system with Stribeck friction and frictional lag, predicting the onset of stick–slip for a robot arm. Gao et al. [13] derive analytical expressions for the change in position during the stick phase for systems that can be described by a linearized friction law, and a static friction coe6cient that increases with time. Elmer [14] discusses stick–slip and pure-slip oscillations of the mass-on-belt system with no damping and di0erent kinds of friction functions, provides analytical expressions for the transfer between stick–slip and pure-slip oscillations, and sketch typical local and global bifurcation scenarios. Thomsen [15] sets up approximate expressions for stick–slip oscillations of a friction slider, which are accurate for very small di0erences in static and kinetic friction. Much research seems concentrated in determining the onset of stick-vibrations in order to avoid these totally. However, stick–slip vibrations might be acceptable in applications, provided their amplitudes are su6ciently small. Therefore, simple expressions providing immediate insight into the inLuence of parameters on vibration amplitudes are believed to be useful. Since the equations of motion are non-linear and discontinuous in the highest derivatives, there are no straightforward standard procedures for calculating stationary amplitudes. However, indeed, it is possible in speci-c cases, as demonstrated in this paper. Also, it is sometimes possible to use perturbation methods for problems involving non-smooth functions and non-small terms, provided they are non-small only in a small time interval, as has been recently exempli-ed in Fidlin and Thomsen [16] and proven more generally in Fidlin [17]. Alternatively, the
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system can be considered as having a variable number of degrees of freedom (decreasing by one during times where a part sticks), as demonstrated in Fidlin [18] and Pechenev and Fidlin [19]. Section 2 of the paper de-nes the speci-c problem to be solved, and Section 3 provides solutions in the form of analytical expressions based on perturbation analysis. Section 4 then illustrates typical solutions, including amplitude diagrams showing how the stick–slip oscillation amplitude varies with excitation speed, and also compare to results obtained by using numerical simulation of the equation of motion. 2. The problem Fig. 1(a) shows the physical example system: a mass M on a belt that moves at constant speed Vb , to be termed the excitation speed below. The mass is a rigid body, at time t˜ positioned at X (t˜) in a -xed frame of coordinates. It is subjected to a normal static load F, linear spring-loading KX , damping force CdX=d t˜, and a friction force F (Vr ) where is the friction force as a function of the relative velocity, Vr = dX=d t˜ − Vb , and the static friction force is F s . The equation of motion is, in non-dimensional form: xO + 2x˙ + x + (x˙ − vb ) = 0 xO = 0;
x + 2vb 6 s
for
for x˙ = vb
x˙ = vb
(slip);
(stick);
where all parameters are positive, and variables and parameters are non-dimensionalized by X F C K Vb ; ; 2 = √ t = !0 t˜; !02 = ; x = ; L = ; vb = M L K !0 L KM
(1) (2)
(3)
where x˙ = d x=dt is the non-dimensional velocity of the mass at non-dimensional time t. Here lengths have been normalized by the characteristic length L and time by the linear natural frequency !0 of free oscillations of the mass when there is no damping and friction. For the friction function (vr ) we assume, as in e.g. Panovko and Gubanova [2] and Ibrahim [8], (vr ) = s sgn(vr ) − 1 vr + 3 vr3 ;
(4)
where vr = x˙ − vb is the (non-dimensional) relative velocity between mass and belt, and 1 ≡ 32 ( s − m )=vm ;
3 3 ≡ 12 ( s − m )=vm ;
(5)
Fig. 1. (a) The example system: a mass at position X on a belt that moves at constant speed Vb . (b) Friction function as given by (4) with s = 0:4; m = 0:25, and vm = 0:5.
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where s is the coe6cient of static friction, and vm is the velocity corresponding to the minimum coe6cient m of kinetic friction, m 6 s ; and 1 ; 2 ¿ 0. Thus, the friction function satis-es | (0)| 6 s ; (vm ) = m ; (vm ) = 0; (−vr ) = − (vr ), and (−vr ) = (vr ), where = d =dvr . Fig. 1(b) depicts this function for typical parameter values (vm ; m ; s ) to be used in this study. As appears | | 6 s when the mass is at rest on the moving belt (vr = 0, stick phase), whereas when the mass starts sliding the friction forces initially decrease with increasing velocity (vr = 0, slip phase). This particular form of the friction law is not overly restricted; it resembles characteristic features of friction models in common use, e.g. see [11,20]. The so-called Stribeck curve, describing the friction-velocity relationship for systems with boundary lubrication, e.g. [12], also share the essential features of (4), though they di0er in detail. The problem to be solved below is to determine stable periodic solutions of (1) with (4). Closed form solutions are unavailable, due to the discontinuity and non-linearity of the friction function . However, for the important case of relatively small di0erence between static and kinetic friction coe6cient, we can employ perturbation analysis to set up approximate analytical expressions, and check the validity of results by using numerical simulation. 3. Solutions After having determined the di0erent types of motion that can occur, we solve for each of these, and then summarize the results in Section 3.4. 3.1. Types of motion According to (1) and (4), the mass has a static equilibrium at x = x, Q xQ = − (−vb ) = s − 1 vb + 3 vb3
(6)
since then x˙ = xO = 0. To study motions near this equilibrium we shift the origin by de-ning u(t) = x(t) − xQ
(7)
by which (1) and (2) transform into uO + u + h(u) ˙ =0 uO = 0;
(slip);
u + (−1 vb + 3 vb3 ) + 2vb 6 0
(8) for u˙ = vb (stick);
(9)
where h(u) ˙ ≡ 2u˙ + (u˙ − vb ) − (−vb ) = 2u˙ + s (1 + sgn(u˙ − vb )) + (−1 + 33 vb2 )u˙ − 33 vb u˙ 2 + 3 u˙ 3 :
(10)
Here 1 has been introduced merely as a book-keeping parameter, to indicate that the damping and the di0erence in static and kinetic friction coe6cient are assumed small; in the -nal results will be set to unity. The equilibrium u = u˙ = 0 of (8) corresponds to a state of steady sliding, with the mass being at rest and the belt sliding at constant velocity vb below it. This static equilibrium can be stable or unstable. If it is unstable, then stable periodic motion takes over; this is the only possibility, since generally the steady state must be either static equilibrium, periodic motion, or chaotic motion—and chaotic solutions cannot occur for a single second-order autonomous ordinary di0erential equation (e.g. [21]). Two di0erent kinds of periodic solutions to (8) are considered below: pure-slip oscillations, where u(t) ˙ ¡ vb at all times, i.e. the mass never catches up with the belt—and stick–slip oscillations where u(t) ˙ 6 vb ; i.e. the mass occasionally sticks to the belt.
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It is important to note that during stationary oscillations the velocity of the mass will never exceed that of the belt, i.e. u(t) ˙ 6 vb for t ¿ t0 where t0 is a -nite. This is so because the energy storing spring cannot accelerate the mass to a velocity exceeding the maximum velocity during the previous oscillation period, and the energy-providing belt cannot accelerate the mass to a velocity beyond its own. It is of course possible to start the system from a state with u˙ ¿ vb , however, viscous damping and dry friction will then drain energy until a stationary state is achieved with u˙ 6 vb . 3.2. Pure-slip oscillations With pure slip u˙ ¡ vb for all t, so that the discontinuity of the friction function is never met or crossed. For this case the function h in (10) can be written h(u) ˙ = h1 u˙ + h2 u˙ 2 + h3 u˙ 3
for u˙ ¡ vb ;
(11)
where h1 = 2 − 1 + 33 vb2 ;
h2 = − 33 vb ;
h3 = 3 :
(12)
The calculation of non-linear oscillation amplitudes follows Thomsen [15]; here we just give the main results to be used subsequently. Using standard averaging for solving (8), one -nds that u = A sin ;
u˙ = A cos ;
(t) ≡ t + (t);
(13)
where A(t) and (t) are solutions of the averaged equations A˙ = − 12 A(h1 + 34 h3 A2 );
A˙ = 0
(14)
for which there are two equilibriums: A trivial solution A(t) = 0; corresponding to the static equilibrium u(t) = 0, and a nontrivial solution given by (15) A() = A1 ≡ − 43 h1 =h3 ; () = 1 = constant for t → ∞ corresponding to periodic solutions u(t) = A1 sin(t + 1 ). As for the stability of solutions, one -nds that u = 0 is unstable when h1 ¡ 0; by (12) and (5) this condition becomes 4vm : (16) vb ¡ vb1 ≡ vm 1 − 3( s − m ) As appears, when there is no viscous damping ( = 0) the static equilibrium is unstable for excitation speeds lower than vm . Viscous damping stabilizes the equilibrium, and at su6ciently large damping, ¿ 34 ( s − m ), the static equilibrium is always stable. Periodic motions exist and are stable when h1 ¡ 0 and h3 ¿ 0. Since s ¿ m the latter requirement is automatically ful-lled. The amplitude A1 of the stable periodic motion is found by inserting into (15) and using (12) and (5) to give 4vm 2 − v2 ; = 2 vb1 vb0 ¡ vb ¡ vb1 ; (19) A1 = 2vm 1 − (vb =vm )2 − b 3( s − m ) where vb1 is the speed below which pure-slip oscillations -rst occur, as given by (16). This expression for A1 assumes pure slip, so the increase in amplitude for decreasing vb will cease when the mass starts ˙ = A1 (by (13) and (15)), it is found that sticking sticking to the belt, i.e. when max(u) ˙ = vb . With max(u) -rst occurs when A1 = vb . Inserting this into (19) and solving for vb we -nd that stick–slip oscillations
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occurs when vb ¡ vb0 where vb0 = 45 vb1 :
(20)
Hence, the range of belt velocities where pure-slip oscillations occur is rather small, its width (vb1 –vb0 ) being only 1 − 4=5 ≈ 10% of vb1 . It forms a transition zone to a wider range of belt velocities where stick–slip motions occur. When sticking just starts, the amplitude of oscillations is given by inserting vb = vb0 in (19), and then use (16) and (20) to -nd 4 4vm vm 1 − : (21) A1 |vb =vb0 = A1; max = vb0 = 5 3( s − m ) Hence, for vanishing damping the maximum amplitude grows linearly with the velocity vm of minimum kinetic friction. As appears from (13), the non-dimensional displacement amplitude equals the velocity amplitude; thus the velocity amplitude Av1 = A1 . 3.3. Stick–slip oscillations We here assume that vb ¡ vb0 , cf. (20), so that during part of an oscillation period the mass sticks to the belt. This case cannot be analyzed using the above averaging procedure for pure-slip oscillations, since the switch from slip to stick is accompanied by a discontinuous change in acceleration. However, it is possible to analyze the stick and the slip phases of the motion separately, and link the results together to obtain an approximate expression for one full oscillation period. 3.3.1. Slip phase During stick the mass moves with the belt, u˙ = vb . This continues until the force from the restoring spring and the damper has increased to the maximum static friction force, i.e. until the strict inequality in (2) or (9) is no longer satis-ed. We consider this the initial condition at time t = 0, where the stick phase ends and the slip phase begins (cf. Fig. 2), i.e. u(0) = − 2vb + (1 vb − 3 vb3 );
u(0) ˙ = vb ;
(slip starts):
(22)
Motions during the subsequent slip phase are then governed by (8). This equation is non-linear, so approximate methods are in need. However, during slip it is continuous in its highest derivatives, since u˙ ¡ vb during slip. Further, since the solution is only needed for the -nite time interval of the slip phase we can use a straightforward perturbation approach. Letting u(t) = u0 (t) + u1 (t);
t ∈ [0; ts ];
1;
(23)
we substitute into (8) and (22), balance terms of like powers of , insert (10), and obtain two new initial value problems for the determination of u0 and u1 : uO 0 + u0 = 0;
u0 (0) = 0;
uO 1 + u1 = − h(u˙ 0 );
u˙ 0 (0) = vb ;
u1 (0) = − 2vb + 1 vb − 3 vb3 ;
(24) u˙ 1 (0) = 0:
(25)
The solution to (24) for the zero-order approximation u0 is u0 = vb sin (t):
(26)
Inserting this and (10) into (25), the equation for the -rst-order correction u1 becomes uO 1 + u1 = 32 c3 − c1 cos (t) + 32 c3 cos (2t) − 14 c3 cos (3t);
(27)
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Fig. 2. Displacement u(t), velocity u(t), ˙ and acceleration u(t) O during one cycle of stick–slip oscillation. De-nitions of displacement and velocity amplitudes A0 and Av0 , switch time ts , and the times tm− ; tm+ ; tm of maximum absolute displacements and velocities.
where 15 vb 3 c1 ≡ 2vb − 1 vb + c3 = 2vb − ( s − m ) 4 2 vm 3 1 vb 3 : c3 ≡ 3 vb = ( s − m ) 2 vm
5 1− 4
vb vm
2
; (28)
The solution of this linear equation satisfying the initial conditions in (25) is 1 u1 = 32 c3 − 12 c1 t sin (t) + ( 55 32 c3 − c1 ) cos (t) − 2 c3 cos (2t) +
1 32 c3
cos (3t);
(29)
where the secular term t sin t is fully acceptable, since it remains bounded in the -nite time of slipping. Hence, by (23), (26), (29), and (5), motions during the slip phase are approximately given by 1 u(t) = vb sin (t) + [ 32 c3 − 12 c1 t sin (t) + ( 55 32 c3 − c1 ) cos (t) − 2 c3 cos (2t) +
+ O(2 );
t ∈ [0; ts ]
(slip phase);
1 32 c3
cos (3t)] (30)
where O(2 ) denote small terms. The corresponding velocities u˙ and accelerations uO are obtained simply by di0erentiation. We still need to determine ts , the time where slip stops and stick starts. This occurs after the mass has slipped back on the belt, and then has been accelerated forward by the spring and the friction, until
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the velocity of the mass again equals the excitation speed. Hence, we determine ts as the -rst solution ˙ a transcendental equation would have to be to u(t ˙ s ) = vb for ts ¿ 0. Using (30) directly to compute u, solved numerically to determine ts . The most consistent way to estimate its solution seems to be the asymptotic analysis used above. So the zero-order approximation according to (26) is vb cos ts = vb , with solution ts0 = 2 :
(31)
The perturbed Eq. (30) should then determine the -rst-order approximation for ts . However (31) is√ an extreme point of the cosine function, and so the accuracy of the -rst approximation will be O( ). Hence, for the typical values of friction di0erences used in this study, i.e. ( s − m )=vm = 0:3, the error in ts will be about 55%. This problem especially concerns ts , whereas similar approximations for the vibration amplitudes are fairly well (see below). If a better approximation for ts is in need, then one could either consider the higher-level approximations for the solutions of (8), or use the following approach: The simplest way is to notice, that the solution of (30) must be in the interval ts ∈ [ ; 2 ], and then use a Taylor-expansion of (30) near some point within this interval. Taking e.g. the center of the interval, one obtains 2 3 3 3 3 + uO ts − + O ts − : (32) u(t ˙ s ) = u˙ 2 2 2 2 Letting u(t ˙ s ) = vb and solving for ts yields an approximate solution ts for ts : 2 3 vb − u(3 =2) ˙ 3 ts = + + O ts − 2 u(3 =2) O 2
(33)
or, using (30) to compute u, ˙ and uO and omitting higher-order terms ts =
3 vb + (1=2)c1 − (13=8)c3 + : 2 vb − (3 =4)c1 − 2c3
(34)
This approximation is very accurate for typical cases, e.g. for the parameters given in the caption for Fig. 3 the error in ts as compared with numerical solutions is less than 0.3%, and is even lower for higher di0erences in friction. However, for lower levels of friction di0erence, the true value of ts approaches 2 , and expression (34) becomes more inaccurate. For such cases a more accurate approximation for ts can be obtained by Taylor expanding to second-order near t = 2 , which gives the approximation ts for ts : 2 c1 ts = 2 − + O(ts − 2 )3 (35) −vb + c1 + 27=32c3 for which the error is less than 1.4% for ( s − m )=vm ¡ 0:3, smaller for the lower friction di0erences, with other parameters as above. It does not seem possible to set up a single expression for ts , based on Taylor-expansion near any particular t, that will provide good approximations for a wide range of system parameters. In essence, the approximation ts is very accurate, except for situations where the stick phase is very short, for which instead the approximation ts is very accurate. What is needed is therefore to merge these solutions in a suitable manner, e.g. as in the following heuristically based expression, that favors ts unless the parameters corresponds to low levels of friction di0erence: ts = ts + e(vb −vb0 )= (ts − ts );
≡
4vm ; 3( s − m )
(36)
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where is identical to the constant under the radical in (16) ( ∈ [0; 1[ for stable stick–slip oscillations). This expression for ts gives approximations that, compared to numerical solutions, are in error by (typically much) less than 1% for all vb ¡ vb0 for ∈ [0:11; 1:00] (corresponding to ( s − m )=vm ∈ [0:07; 0:61] when (; vm ; m ; s ) are as for Fig. 3). It should be recalled that better numerical solutions for ts can always be obtained simply by solving the algebraic equation u(t ˙ s ) = vb numerically, with u˙ given by (30). 3.3.2. Stick phase Stick starts at t = ts , and then the mass just follows the belt, i.e. u˙ = vb ; uO = 0; u(t) = u(ts ) + vb (t − ts );
t ∈ ]ts ; T [;
(stick);
(37)
where u(ts ) is known by having applied (30) for the just completed phase of slip. The time t = T where stick ends is determined by applying the periodicity condition to (37), i.e. u(T ) = u(0), so that u(0) − u(ts ) ; (38) T = ts + vb where u(0) and u(ts ) is determined by (30). 3.3.3. Stick–slip vibration amplitude Considering velocity amplitudes during one stick–slip cycle, we note that during stick the velocity is constant, u˙ = vb , while during slip the velocity changes continuously with a maximum absolute value at O m ) = 0. Seeking an approximate solution, we let time t = tm de-ned by the solution to u(t tm = tm0 + tm1 ;
1:
(39)
Inserting this into (23) and Taylor-expanding for small , one -nds u(t O m ) = uO 0 (tm0 ) + (tm1 uO˙0 (tm0 ) + uO 1 (tm0 )) + O(2 ):
(40)
Balancing terms of like orders of magnitude it is found that u(t O m ) = 0 is approximately satis-ed when uO 0 (tm0 ) = 0;
uO 1 (tm0 ) = − tm1 uO˙0 (tm0 ):
(41)
By (26) the -rst equation is satis-ed by tm0 = . Inserting this and ((26), (29)) for (u0 ; u1 ) into the second equation, one can solve for tm1 and -nally insert into (39) to -nd (letting = 1): c1 + (73=32)c3 tm = − : (42) vb The velocity at this time is, by (23),(39), (26), (29), tm0 = , and Taylor expanding for 1: u(t ˙ m ) = u˙ 0 (tm0 ) + [tm1 uO 0 (tm0 ) + u˙ 1 (tm0 )] + O(2 ) = −vb + ( =2)c1 + O(2 ):
(43)
As a measure indicating the magnitude of oscillations, which are asymmetric with respect to u = 0, we de-ne the velocity amplitude Av0 of stick–slip oscillations as half the peak-to-peak velocity, i.e. ˙ m )); Av0 = 12 (vb − u(t
(44) 2
which on account of (43) and (28) becomes, ignoring terms of order and higher 2 3 vb
5 vb ( s − m ) Av0 = 1 − vb + 1− ; vb ¡ vb0 : 2 8 vm 4 vm
(45)
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The corresponding displacement amplitude A0 can be determined similarly, by calculating approximations to the times tm+ and tm− where u˙ = 0 (cf. Fig. 2); this yields 3 1 52 3 c3 ; u(tm− ) = − vb + 2c3 + c1 ; − c1 + tm− = 2 2 32 4 1 52 (46) tm+ = + c1 − c3 ; u(tm+ ) = vb + 2c3 − c1 : 2 2 32 4 De-ning the displacement amplitude A1 as half the peak-to-peak displacement, it becomes 1 A0 = (u(tm+ ) − u(tm− )) = vb − c1 2 2 2 3 vb 5 vb ( s − m ) = (1 − )vb + 1− ; vb ¡ vb0 : (47) 4 vm 4 vm One can show that this function has a maximum value at vb = vb∗ , vb for ( s − m )=vm 6 38 (1 − ( − 3)); 0 vb∗ = (1 − )vm 4 otherwise vm 15 1 + 3 =4( − ) s m
(48)
where vb0 is given by (20). Thus, if the di0erence in static and kinetic friction is not too large, the strongest stick–slip oscillations occur when the excitation speed reaches the value separating stick–slip oscillations from pure-slip oscillations, vb = vb0 . For larger friction di0erences the strongest oscillations occur at an excitation speed in the range vb ∈ ]0; vb0 [, as given by the second expression in (48). Since the analysis assumes friction di0erences that are small (but -nite), then the -rst case applies, so that we conclude that the strongest oscillations occur at vb = vb0 , hence 4 4vm A0 |vb =vb0 = A0; max = vb0 = vm 1 − = A1; max : (49) 5 3( s − m ) The last equality expresses that the predicted amplitude A1 of pure-slip oscillations equals the predicted amplitude A0 of stick–slip oscillations at the value of excitation speed separating these di0erent kinds of motion; this exact continuity is neither obvious nor required, since the two expressions were derived using approximate methods. It can be noted from (47) that at small values of the excitation speed, the stick–slip oscillation amplitude grows approximately linear with this speed 3 s − m vb for vb 1: (50) A0 ≈ 1 − + 4 vm Finally, since the velocity of the mass must change continuously with time, the maximum and minimum displacements of the mass must occur during the slip phase; they cannot occur during stick, because displacements here increase linearly with time until the time of slip. Hence, the amplitude A1 , determining displacements during the slip phase, also determines the oscillation amplitude of the complete stick–slip oscillation. 3.3.4. Stick–slip base frequency Since at t = T one cycle of slip and stick is completed, the base angular frequency of stick–slip oscillations is 2 vb 2 = : (51) !ss = T vb ts + u(0) − u(ts )
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This frequency is generally somewhat lower than the linear natural frequency of the system and the pure-slip oscillation frequency, !ss 6 !ps ≈ 1. 3.4. Summary of results For the system (1) – (2) with (4), or equivalently (8) – (9) with (10), the type of stationary motion depends on the excitation speed vb as follows, where vb0 and vb1 are given by (20) and (16), and all results holds approximately for small but -nite levels of vibration amplitudes, viscous damping , and relative friction di0erence ( s − m )=vm . For excitation speeds vb ¡ vb0 : Stick–slip oscillations, with stationary displacement and velocity amplitude A0 and Av0 , as given by (47) and (44), and base frequency !ss given by (51) (slightly less than the linear natural frequency of the system). For excitation speeds vb ∈ [vb0 ; vb1 ]: Pure-slip oscillations, with stationary displacement amplitude A1 (equal to the velocity amplitude Av1 ) as given by (19), and base frequency !ps = 1 (equal to the linear natural frequency of the system). For some parameters it may happen that vb0 = vb1 , so that pure-slip oscillations cannot occur at all. This occurs when the radical in (16) becomes negative, i.e. if viscous damping is su6ciently large, or the friction forces or di0erence between static and kinetic friction is su6ciently small. Q cf. (6). This corresponds to a For excitation speeds vb ¿ vb1 : Static equilibrium at position x(t) = x, state of steady sliding of the mass. 4. Illustration and testing of results 4.1. Oscillation cycles at di
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Fig. 3. One period of stick–slip displacements u(t) (top row), velocities u(t) ˙ (middle), and phase plane orbits u(u) ˙ (bottom row) for three values of excitation speed vb (a; b; c). (—– ) Analytical prediction (Eqs. (30) and (37)); (- - - - - -) numerical simulation of (1) with (4) and (7). Parameters: = 0:05; vm = 0:5; m = 0:25; s = 0:4, and (a) vb = 0:05, (b) vb = 0:25, (c) vb = 0:3944.
appear below. Also, as the friction di0erence is decreased the approximations become more accurate; for example, if Fig. 3 is re-computed for ( s − m )=vm = 0:15 instead of 0.3, then the results from using analytical approximations is almost indistinguishable from numerical simulation (only Figs. 3(a) and (b) would apply in that case). 4.2. Displacement amplitude as a function of excitation speed Fig. 4(a) shows the predicted variation of displacement amplitude with excitation speed for typical parameters. Here A0 indicates the amplitude during stick–slip oscillations for vb ¡ vb0 according to (47), and A1 is the amplitude for pure-slip oscillations for vb ∈ [vb0 ; vb1 ] according to (19). As appears, when the excitation speed is increased from zero, stick–slip oscillations occur with increasing amplitude until, at vb = vb0 pure-slip oscillations take over. These prevail only in a narrow range of belt velocities, until vb1 , above which oscillations cease and steady slip becomes the stable type of motion. The agreement
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Fig. 4. (a) Amplitude A and (b) base frequency ! of stable periodic motions as a function of excitation speed vb . (—-) Analytical prediction (Eq. (19), (47), (51)); (- - - - - -) analytical prediction from Thomsen [15]; (O) Numerical simulation of (1) with (4). Parameters as for Fig. 3.
of the analytical predictions with numerical simulation (encircled points) is seen to be very good; this will hold true the more closely the actual parameters satis-es the assumptions underlying the analytical expressions, i.e. as long as the di0erence in static and kinetic friction, the amount of viscous damping, and the amplitudes are not too large. It also appears that the predictions for the stick–slip amplitudes A0 are much better than the low-order approximation (indicated in dashed line) given in [15]. Fig. 4(b) depicts the predicted variation of base frequency ! with excitation speed, showing a slight drop in frequency for the lower velocities. Seemingly, the agreement with numerical simulation is quite good; however, since the change in frequency is so small, the relative error in the prediction is somewhat larger than for the displacement and amplitude predictions. This is a consequence of the di6culty in predicting, with high accuracy, the end of the slip phase of a stick–slip cycle, based on an approximate expression for the displacements during that cycle. 4.3. The signi=cance of friction di
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Fig. 5. As Fig. 4(a), but for varying levels of relative friction di0erence ( s − m )=vm . (—-) Analytical prediction (Eq. (19), (47)); (; ×; ; ⊕; ♦) numerical simulation of (1) with (4) and, respectively ( s − m )=vm = (0:075; 0:099; 0:15; 0:3; 0:6). Other parameters: = 0:05; vm = 0:5; s = 0:4.
5. Conclusions The analytical expressions for stationary friction-induced oscillations produce predictions that agree favorably with numerical simulations, as long as the assumption of relatively small di0erences in static and kinetic friction is ful-lled. This holds for stick–slip oscillations, that occur for relatively small excitation speeds vb ¡ vb0 , and for pure-slip oscillations that occur for a narrow range of slightly higher speeds vb ∈ [vb0 ; vb1 ]. Beyond this range, according to the predictions and numerical simulations, the system exhibits steady sliding with no oscillations. The expressions provide insight into the inLuence of parameters on vibration characteristics such as amplitude and base frequency; features that can be measured in experiments and compared to assess the validity of the underlying friction model. Although results are given in terms of a speci-c friction model, the approach used is not tied to this. It can readily be applied to set up and test expressions for other friction models having the properties of being (1) discontinuous only at zero-valued relative velocity, and (2) su6ciently smooth (for non-zero relative velocities) to enable Taylor-expansion to an order high enough to include motion-restricting non-linearities (usually three). References [1] J.W.S. Rayleigh, The Theory of Sound, 2nd Edition, 1945 re-issue. Dover Publications, New York, 1877. [2] Y.G. Panovko, I.I. Gubanova, Stability and Oscillations of Elastic Systems; Paradoxes, Fallacies and New Concepts, Consultants Bureau, New York, 1965. [3] A. Tondl, Quenching of Self-Excited Vibrations, Elsevier, Amsterdam, 1991. [4] A.H. Nayfeh, D.T. Mook, Nonlinear Oscillations, Wiley, New York, 1979. [5] Y.A. Mitropolskii, V.D. Nguyen, Applied Asymptotic Methods in Nonlinear Oscillations, Kluwer, Dordrecht, 1997. [6] K. Popp, Some model problems showing stick–slip motion and chaos, Friction-Induced Vibration, Chatter, Squeal, and Chaos, ASME DE-Vol. 49, ASME Design Engineering Divison, New York, 1992, pp. 1–12. [7] R.A. Ibrahim, Friction-induced vibration, chatter, squeal, and chaos: part I—mechanics of friction, Friction-Induced Vibration, Chatter, Squeal, and Chaos, ASME DE-Vol. 49, ASME Design Engineering Divison, New York, 1992, pp. 107–121. [8] R.A. Ibrahim, Friction-induced vibration, chatter, squeal, and chaos: part II—dynamics and modeling, Friction-Induced Vibration, Chatter, Squeal, and Chaos, ASME DE-Vol. 49, ASME Design Engineering Divison, New York, 1992, pp. 123–138. [9] J. McMillan, A non-linear friction model for self-excited vibrations, J. Sound Vibration 205 (3) (1997) 323–335. [10] B. Feeny, ArdKeshir Guran, N. Hinrichs, K. Popp, A historical review on dry friction and stick–slip phenomena, ASME Appl. Mech. Rev. 51 (5) (1998) 321–341.
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[11] B. Armstrong-HKelouvry, P. Dupont, C. Canudas de Wit, A survey of models, analysis tools and compensation methods for the control of machines with friction, Automatica 30 (7) (1994) 1083–1138. [12] B. Armstrong-HKelouvry, A perturbation analysis of stick–slip, Friction-Induced Vibration, Chatter, Squeal, and Chaos, ASME DE-Vol. 49, ASME Design Engineering Divison, New York, 1992, pp. 41– 48. [13] C. Gao, D. Kuhlmann-Wilsdorf, D.D. Makel, The dynamic analysis of stick–slip motion, Wear 173 (1994) 1–12. [14] F.-J. Elmer, Nonlinear dynamics of dry friction, J. Phys. A—Math. General 30 (1997) 6057–6063. [15] J.J. Thomsen, Using fast vibrations to quench friction-induced oscillations, J. Sound Vibration 228 (5) (1999) 1079–1102. [16] A. Fidlin, J.J. Thomsen, Predicting vibration-induced displacement for a resonant friction slider, European J. Mech. A=Solids 20 (2001) 155–166. [17] A. Fidlin, On the asymptotic analysis of discontinuous systems, DCAMM Report 646, Technical University of Denmark, 2000. [18] A.Y. Fidlin, On averaging in systems with a variable number of degrees of freedom, J. Appl. Math. Mech. 55 (4) (1991) 507–510. [19] A.V. Pechenev, A.Y. Fidlin, Hierachy of resonant motions excited in a vibroimpact system with contact zones by an inertial source of limited power, Izv. AN SSSR. Mechanika Tverdogo Tela 27 (4) (1992) 46 –53. [20] N. Hinrichs, M. Oestreich, K. Popp, On the modelling of friction oscillators, J. Sound Vibration 216 (3) (1998) 435–459. [21] J.J. Thomsen, Vibrations and Stability, Order and Chaos, McGraw-Hill, London, 1997.
Analytical approximations for stick–slip vibration amplitudes Jon Juel Thomsena; ∗ , Alexander Fidlinb a Department
of Mechanical Engineering, Solid Mechanics, Technical University of Denmark Building 404, DK-2800 Lyngby, Denmark b Tennesseealee 35, D-76149 Karlsruhe, Germany
Abstract The classical “mass-on-moving-belt” model for describing friction-induced vibrations is considered, with a friction law describing friction forces that -rst decreases and then increases smoothly with relative interface speed. Approximate analytical expressions are derived for the conditions, the amplitudes, and the base frequencies of friction-induced stick–slip and pure-slip oscillations. For stick–slip oscillations, this is accomplished by using perturbation analysis for the -nite time interval of the stick phase, which is linked to the subsequent slip phase through conditions of continuity and periodicity. The results are illustrated and tested by time-series, phase plots and amplitude response diagrams, which compare very favorably with results obtained by numerical simulation of the equation of motion, as long as the di0erence in static and kinetic friction is not too large. ? 2002 Elsevier Science Ltd. All rights reserved. Keywords: Friction; Stick–slip; Self-excited vibrations
1. Introduction We consider the classical “mass-on-moving-belt” model for describing friction-induced vibrations. Taking the model for granted—we do not discuss its validity for mimicking real systems—the purpose is to contribute further understanding on what it predicts, i.e. to provide expressions and graphs illustrating the change of vibration character and amplitude with excitation speed. For a common friction law, with a kinetic coe6cient of friction that -rst decreases and then increases smoothly with sliding speed, we provide simple approximate expressions for the system state during each cycle, and for the stationary vibration amplitudes and base frequencies. This supplements the many works in this area who deals mainly with setting up criterions for the onset of such vibrations, considers only non-sticking motions, assumes piecewise linear friction laws, or rely solely on numerical simulation. Assuming small but -nite di0erences in static and kinetic friction, a perturbation approach is used to -rst set up expressions describing the occurrence, the stability, and amplitude characteristics of pure-slip motions, also showing that such motions are typically stable only for a narrow range of sliding speeds. ∗
Corresponding author. Fax: +45-4593-1475. E-mail addresses: [email protected] (J. Juel Thomsen), [email protected] (A. Fidlin). 0020-7462/03/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 0 - 7 4 6 2 ( 0 1 ) 0 0 0 7 3 - 7
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Below a certain excitation speed, stick–slip motions occur, for which we also provide simple analytical expressions. This is accomplished by using perturbation analysis for the stick phase, which is linked to the subsequent slip phase by requirements of continuity and periodicity. The results are illustrated and tested by showing time-series, phase plots and amplitude response diagrams, which compare very favorably to results obtained by numerical simulation of the equation of motion. Stick–slip vibrations show up in many kinds of engineering systems and everyday life, e.g. as sounds form a bowed violin, squeaking chalks and shoes, creaking doors, squealing tramways, chattering machine tools, and grating brakes—most of which are attributed to surfaces sliding with friction. Furthermore, many mechanical interfaces are characterized by a form of dry friction where the force–velocity curve has negative slope at low velocities. Initially, friction decreases as the contacting object starts to move, whereas at higher velocities the friction force increases again; in particular this characterizes surfaces with boundary lubrication. The initial negative slope corresponds to negative damping, and may thus cause oscillations that grow in amplitude, until a balance of dissipated and induced energy is attained, as pointed out already by Lord Rayleigh ([1], Vol. I, p. 212). Typically there are two phases of such oscillations: a stick phase with no slippage between parts and friction forces limited by static friction, followed by a slip phase with a somewhat lower friction force; though, on some conditions there is no stick phase. Numerous works has been devoted to the study of friction-induced oscillations. For ease of setup and interpretation an idealized physical system consisting of a mass sliding on a moving belt has been considered very often, as it will be in this present study. Panovko and Gubanova [2], for such a system with a friction characteristic having minimum coe6cient at velocity vm ; show that self-excited oscillations occur only when the belt velocity is lower than vm . Tondl [3], Nayfeh and Mook [4], and Mitropolskii and Nguyen [5] describe self-excited oscillations of the mass-on-belt system, presenting approximate expressions for the vibration amplitudes for the case, where there is no sticking between mass and belt. Popp [6] presents models and numerical and experimental results for four systems that are similar to the mass-on-moving-belt. Ibrahim [7,8] and McMillan [9] presents and discusses the basic mechanics of friction and friction models, and provides reviews on relevant literature. A very readable historical review on dry friction and stick–slip phenomena is given by Feeny et al. [10], and a large survey on friction literature until 1992 by Armstrong-HKelouvry et al. [11]. There are, however, very few works providing expressions for stationary stick–slip amplitudes. Armstrong-HKelouvry [12] performed a perturbation analysis for a system with Stribeck friction and frictional lag, predicting the onset of stick–slip for a robot arm. Gao et al. [13] derive analytical expressions for the change in position during the stick phase for systems that can be described by a linearized friction law, and a static friction coe6cient that increases with time. Elmer [14] discusses stick–slip and pure-slip oscillations of the mass-on-belt system with no damping and di0erent kinds of friction functions, provides analytical expressions for the transfer between stick–slip and pure-slip oscillations, and sketch typical local and global bifurcation scenarios. Thomsen [15] sets up approximate expressions for stick–slip oscillations of a friction slider, which are accurate for very small di0erences in static and kinetic friction. Much research seems concentrated in determining the onset of stick-vibrations in order to avoid these totally. However, stick–slip vibrations might be acceptable in applications, provided their amplitudes are su6ciently small. Therefore, simple expressions providing immediate insight into the inLuence of parameters on vibration amplitudes are believed to be useful. Since the equations of motion are non-linear and discontinuous in the highest derivatives, there are no straightforward standard procedures for calculating stationary amplitudes. However, indeed, it is possible in speci-c cases, as demonstrated in this paper. Also, it is sometimes possible to use perturbation methods for problems involving non-smooth functions and non-small terms, provided they are non-small only in a small time interval, as has been recently exempli-ed in Fidlin and Thomsen [16] and proven more generally in Fidlin [17]. Alternatively, the
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system can be considered as having a variable number of degrees of freedom (decreasing by one during times where a part sticks), as demonstrated in Fidlin [18] and Pechenev and Fidlin [19]. Section 2 of the paper de-nes the speci-c problem to be solved, and Section 3 provides solutions in the form of analytical expressions based on perturbation analysis. Section 4 then illustrates typical solutions, including amplitude diagrams showing how the stick–slip oscillation amplitude varies with excitation speed, and also compare to results obtained by using numerical simulation of the equation of motion. 2. The problem Fig. 1(a) shows the physical example system: a mass M on a belt that moves at constant speed Vb , to be termed the excitation speed below. The mass is a rigid body, at time t˜ positioned at X (t˜) in a -xed frame of coordinates. It is subjected to a normal static load F, linear spring-loading KX , damping force CdX=d t˜, and a friction force F (Vr ) where is the friction force as a function of the relative velocity, Vr = dX=d t˜ − Vb , and the static friction force is F s . The equation of motion is, in non-dimensional form: xO + 2x˙ + x + (x˙ − vb ) = 0 xO = 0;
x + 2vb 6 s
for
for x˙ = vb
x˙ = vb
(slip);
(stick);
where all parameters are positive, and variables and parameters are non-dimensionalized by X F C K Vb ; ; 2 = √ t = !0 t˜; !02 = ; x = ; L = ; vb = M L K !0 L KM
(1) (2)
(3)
where x˙ = d x=dt is the non-dimensional velocity of the mass at non-dimensional time t. Here lengths have been normalized by the characteristic length L and time by the linear natural frequency !0 of free oscillations of the mass when there is no damping and friction. For the friction function (vr ) we assume, as in e.g. Panovko and Gubanova [2] and Ibrahim [8], (vr ) = s sgn(vr ) − 1 vr + 3 vr3 ;
(4)
where vr = x˙ − vb is the (non-dimensional) relative velocity between mass and belt, and 1 ≡ 32 ( s − m )=vm ;
3 3 ≡ 12 ( s − m )=vm ;
(5)
Fig. 1. (a) The example system: a mass at position X on a belt that moves at constant speed Vb . (b) Friction function as given by (4) with s = 0:4; m = 0:25, and vm = 0:5.
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where s is the coe6cient of static friction, and vm is the velocity corresponding to the minimum coe6cient m of kinetic friction, m 6 s ; and 1 ; 2 ¿ 0. Thus, the friction function satis-es | (0)| 6 s ; (vm ) = m ; (vm ) = 0; (−vr ) = − (vr ), and (−vr ) = (vr ), where = d =dvr . Fig. 1(b) depicts this function for typical parameter values (vm ; m ; s ) to be used in this study. As appears | | 6 s when the mass is at rest on the moving belt (vr = 0, stick phase), whereas when the mass starts sliding the friction forces initially decrease with increasing velocity (vr = 0, slip phase). This particular form of the friction law is not overly restricted; it resembles characteristic features of friction models in common use, e.g. see [11,20]. The so-called Stribeck curve, describing the friction-velocity relationship for systems with boundary lubrication, e.g. [12], also share the essential features of (4), though they di0er in detail. The problem to be solved below is to determine stable periodic solutions of (1) with (4). Closed form solutions are unavailable, due to the discontinuity and non-linearity of the friction function . However, for the important case of relatively small di0erence between static and kinetic friction coe6cient, we can employ perturbation analysis to set up approximate analytical expressions, and check the validity of results by using numerical simulation. 3. Solutions After having determined the di0erent types of motion that can occur, we solve for each of these, and then summarize the results in Section 3.4. 3.1. Types of motion According to (1) and (4), the mass has a static equilibrium at x = x, Q xQ = − (−vb ) = s − 1 vb + 3 vb3
(6)
since then x˙ = xO = 0. To study motions near this equilibrium we shift the origin by de-ning u(t) = x(t) − xQ
(7)
by which (1) and (2) transform into uO + u + h(u) ˙ =0 uO = 0;
(slip);
u + (−1 vb + 3 vb3 ) + 2vb 6 0
(8) for u˙ = vb (stick);
(9)
where h(u) ˙ ≡ 2u˙ + (u˙ − vb ) − (−vb ) = 2u˙ + s (1 + sgn(u˙ − vb )) + (−1 + 33 vb2 )u˙ − 33 vb u˙ 2 + 3 u˙ 3 :
(10)
Here 1 has been introduced merely as a book-keeping parameter, to indicate that the damping and the di0erence in static and kinetic friction coe6cient are assumed small; in the -nal results will be set to unity. The equilibrium u = u˙ = 0 of (8) corresponds to a state of steady sliding, with the mass being at rest and the belt sliding at constant velocity vb below it. This static equilibrium can be stable or unstable. If it is unstable, then stable periodic motion takes over; this is the only possibility, since generally the steady state must be either static equilibrium, periodic motion, or chaotic motion—and chaotic solutions cannot occur for a single second-order autonomous ordinary di0erential equation (e.g. [21]). Two di0erent kinds of periodic solutions to (8) are considered below: pure-slip oscillations, where u(t) ˙ ¡ vb at all times, i.e. the mass never catches up with the belt—and stick–slip oscillations where u(t) ˙ 6 vb ; i.e. the mass occasionally sticks to the belt.
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It is important to note that during stationary oscillations the velocity of the mass will never exceed that of the belt, i.e. u(t) ˙ 6 vb for t ¿ t0 where t0 is a -nite. This is so because the energy storing spring cannot accelerate the mass to a velocity exceeding the maximum velocity during the previous oscillation period, and the energy-providing belt cannot accelerate the mass to a velocity beyond its own. It is of course possible to start the system from a state with u˙ ¿ vb , however, viscous damping and dry friction will then drain energy until a stationary state is achieved with u˙ 6 vb . 3.2. Pure-slip oscillations With pure slip u˙ ¡ vb for all t, so that the discontinuity of the friction function is never met or crossed. For this case the function h in (10) can be written h(u) ˙ = h1 u˙ + h2 u˙ 2 + h3 u˙ 3
for u˙ ¡ vb ;
(11)
where h1 = 2 − 1 + 33 vb2 ;
h2 = − 33 vb ;
h3 = 3 :
(12)
The calculation of non-linear oscillation amplitudes follows Thomsen [15]; here we just give the main results to be used subsequently. Using standard averaging for solving (8), one -nds that u = A sin ;
u˙ = A cos ;
(t) ≡ t + (t);
(13)
where A(t) and (t) are solutions of the averaged equations A˙ = − 12 A(h1 + 34 h3 A2 );
A˙ = 0
(14)
for which there are two equilibriums: A trivial solution A(t) = 0; corresponding to the static equilibrium u(t) = 0, and a nontrivial solution given by (15) A() = A1 ≡ − 43 h1 =h3 ; () = 1 = constant for t → ∞ corresponding to periodic solutions u(t) = A1 sin(t + 1 ). As for the stability of solutions, one -nds that u = 0 is unstable when h1 ¡ 0; by (12) and (5) this condition becomes 4vm : (16) vb ¡ vb1 ≡ vm 1 − 3( s − m ) As appears, when there is no viscous damping ( = 0) the static equilibrium is unstable for excitation speeds lower than vm . Viscous damping stabilizes the equilibrium, and at su6ciently large damping, ¿ 34 ( s − m ), the static equilibrium is always stable. Periodic motions exist and are stable when h1 ¡ 0 and h3 ¿ 0. Since s ¿ m the latter requirement is automatically ful-lled. The amplitude A1 of the stable periodic motion is found by inserting into (15) and using (12) and (5) to give 4vm 2 − v2 ; = 2 vb1 vb0 ¡ vb ¡ vb1 ; (19) A1 = 2vm 1 − (vb =vm )2 − b 3( s − m ) where vb1 is the speed below which pure-slip oscillations -rst occur, as given by (16). This expression for A1 assumes pure slip, so the increase in amplitude for decreasing vb will cease when the mass starts ˙ = A1 (by (13) and (15)), it is found that sticking sticking to the belt, i.e. when max(u) ˙ = vb . With max(u) -rst occurs when A1 = vb . Inserting this into (19) and solving for vb we -nd that stick–slip oscillations
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occurs when vb ¡ vb0 where vb0 = 45 vb1 :
(20)
Hence, the range of belt velocities where pure-slip oscillations occur is rather small, its width (vb1 –vb0 ) being only 1 − 4=5 ≈ 10% of vb1 . It forms a transition zone to a wider range of belt velocities where stick–slip motions occur. When sticking just starts, the amplitude of oscillations is given by inserting vb = vb0 in (19), and then use (16) and (20) to -nd 4 4vm vm 1 − : (21) A1 |vb =vb0 = A1; max = vb0 = 5 3( s − m ) Hence, for vanishing damping the maximum amplitude grows linearly with the velocity vm of minimum kinetic friction. As appears from (13), the non-dimensional displacement amplitude equals the velocity amplitude; thus the velocity amplitude Av1 = A1 . 3.3. Stick–slip oscillations We here assume that vb ¡ vb0 , cf. (20), so that during part of an oscillation period the mass sticks to the belt. This case cannot be analyzed using the above averaging procedure for pure-slip oscillations, since the switch from slip to stick is accompanied by a discontinuous change in acceleration. However, it is possible to analyze the stick and the slip phases of the motion separately, and link the results together to obtain an approximate expression for one full oscillation period. 3.3.1. Slip phase During stick the mass moves with the belt, u˙ = vb . This continues until the force from the restoring spring and the damper has increased to the maximum static friction force, i.e. until the strict inequality in (2) or (9) is no longer satis-ed. We consider this the initial condition at time t = 0, where the stick phase ends and the slip phase begins (cf. Fig. 2), i.e. u(0) = − 2vb + (1 vb − 3 vb3 );
u(0) ˙ = vb ;
(slip starts):
(22)
Motions during the subsequent slip phase are then governed by (8). This equation is non-linear, so approximate methods are in need. However, during slip it is continuous in its highest derivatives, since u˙ ¡ vb during slip. Further, since the solution is only needed for the -nite time interval of the slip phase we can use a straightforward perturbation approach. Letting u(t) = u0 (t) + u1 (t);
t ∈ [0; ts ];
1;
(23)
we substitute into (8) and (22), balance terms of like powers of , insert (10), and obtain two new initial value problems for the determination of u0 and u1 : uO 0 + u0 = 0;
u0 (0) = 0;
uO 1 + u1 = − h(u˙ 0 );
u˙ 0 (0) = vb ;
u1 (0) = − 2vb + 1 vb − 3 vb3 ;
(24) u˙ 1 (0) = 0:
(25)
The solution to (24) for the zero-order approximation u0 is u0 = vb sin (t):
(26)
Inserting this and (10) into (25), the equation for the -rst-order correction u1 becomes uO 1 + u1 = 32 c3 − c1 cos (t) + 32 c3 cos (2t) − 14 c3 cos (3t);
(27)
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Fig. 2. Displacement u(t), velocity u(t), ˙ and acceleration u(t) O during one cycle of stick–slip oscillation. De-nitions of displacement and velocity amplitudes A0 and Av0 , switch time ts , and the times tm− ; tm+ ; tm of maximum absolute displacements and velocities.
where 15 vb 3 c1 ≡ 2vb − 1 vb + c3 = 2vb − ( s − m ) 4 2 vm 3 1 vb 3 : c3 ≡ 3 vb = ( s − m ) 2 vm
5 1− 4
vb vm
2
; (28)
The solution of this linear equation satisfying the initial conditions in (25) is 1 u1 = 32 c3 − 12 c1 t sin (t) + ( 55 32 c3 − c1 ) cos (t) − 2 c3 cos (2t) +
1 32 c3
cos (3t);
(29)
where the secular term t sin t is fully acceptable, since it remains bounded in the -nite time of slipping. Hence, by (23), (26), (29), and (5), motions during the slip phase are approximately given by 1 u(t) = vb sin (t) + [ 32 c3 − 12 c1 t sin (t) + ( 55 32 c3 − c1 ) cos (t) − 2 c3 cos (2t) +
+ O(2 );
t ∈ [0; ts ]
(slip phase);
1 32 c3
cos (3t)] (30)
where O(2 ) denote small terms. The corresponding velocities u˙ and accelerations uO are obtained simply by di0erentiation. We still need to determine ts , the time where slip stops and stick starts. This occurs after the mass has slipped back on the belt, and then has been accelerated forward by the spring and the friction, until
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the velocity of the mass again equals the excitation speed. Hence, we determine ts as the -rst solution ˙ a transcendental equation would have to be to u(t ˙ s ) = vb for ts ¿ 0. Using (30) directly to compute u, solved numerically to determine ts . The most consistent way to estimate its solution seems to be the asymptotic analysis used above. So the zero-order approximation according to (26) is vb cos ts = vb , with solution ts0 = 2 :
(31)
The perturbed Eq. (30) should then determine the -rst-order approximation for ts . However (31) is√ an extreme point of the cosine function, and so the accuracy of the -rst approximation will be O( ). Hence, for the typical values of friction di0erences used in this study, i.e. ( s − m )=vm = 0:3, the error in ts will be about 55%. This problem especially concerns ts , whereas similar approximations for the vibration amplitudes are fairly well (see below). If a better approximation for ts is in need, then one could either consider the higher-level approximations for the solutions of (8), or use the following approach: The simplest way is to notice, that the solution of (30) must be in the interval ts ∈ [ ; 2 ], and then use a Taylor-expansion of (30) near some point within this interval. Taking e.g. the center of the interval, one obtains 2 3 3 3 3 + uO ts − + O ts − : (32) u(t ˙ s ) = u˙ 2 2 2 2 Letting u(t ˙ s ) = vb and solving for ts yields an approximate solution ts for ts : 2 3 vb − u(3 =2) ˙ 3 ts = + + O ts − 2 u(3 =2) O 2
(33)
or, using (30) to compute u, ˙ and uO and omitting higher-order terms ts =
3 vb + (1=2)c1 − (13=8)c3 + : 2 vb − (3 =4)c1 − 2c3
(34)
This approximation is very accurate for typical cases, e.g. for the parameters given in the caption for Fig. 3 the error in ts as compared with numerical solutions is less than 0.3%, and is even lower for higher di0erences in friction. However, for lower levels of friction di0erence, the true value of ts approaches 2 , and expression (34) becomes more inaccurate. For such cases a more accurate approximation for ts can be obtained by Taylor expanding to second-order near t = 2 , which gives the approximation ts for ts : 2 c1 ts = 2 − + O(ts − 2 )3 (35) −vb + c1 + 27=32c3 for which the error is less than 1.4% for ( s − m )=vm ¡ 0:3, smaller for the lower friction di0erences, with other parameters as above. It does not seem possible to set up a single expression for ts , based on Taylor-expansion near any particular t, that will provide good approximations for a wide range of system parameters. In essence, the approximation ts is very accurate, except for situations where the stick phase is very short, for which instead the approximation ts is very accurate. What is needed is therefore to merge these solutions in a suitable manner, e.g. as in the following heuristically based expression, that favors ts unless the parameters corresponds to low levels of friction di0erence: ts = ts + e(vb −vb0 )= (ts − ts );
≡
4vm ; 3( s − m )
(36)
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where is identical to the constant under the radical in (16) ( ∈ [0; 1[ for stable stick–slip oscillations). This expression for ts gives approximations that, compared to numerical solutions, are in error by (typically much) less than 1% for all vb ¡ vb0 for ∈ [0:11; 1:00] (corresponding to ( s − m )=vm ∈ [0:07; 0:61] when (; vm ; m ; s ) are as for Fig. 3). It should be recalled that better numerical solutions for ts can always be obtained simply by solving the algebraic equation u(t ˙ s ) = vb numerically, with u˙ given by (30). 3.3.2. Stick phase Stick starts at t = ts , and then the mass just follows the belt, i.e. u˙ = vb ; uO = 0; u(t) = u(ts ) + vb (t − ts );
t ∈ ]ts ; T [;
(stick);
(37)
where u(ts ) is known by having applied (30) for the just completed phase of slip. The time t = T where stick ends is determined by applying the periodicity condition to (37), i.e. u(T ) = u(0), so that u(0) − u(ts ) ; (38) T = ts + vb where u(0) and u(ts ) is determined by (30). 3.3.3. Stick–slip vibration amplitude Considering velocity amplitudes during one stick–slip cycle, we note that during stick the velocity is constant, u˙ = vb , while during slip the velocity changes continuously with a maximum absolute value at O m ) = 0. Seeking an approximate solution, we let time t = tm de-ned by the solution to u(t tm = tm0 + tm1 ;
1:
(39)
Inserting this into (23) and Taylor-expanding for small , one -nds u(t O m ) = uO 0 (tm0 ) + (tm1 uO˙0 (tm0 ) + uO 1 (tm0 )) + O(2 ):
(40)
Balancing terms of like orders of magnitude it is found that u(t O m ) = 0 is approximately satis-ed when uO 0 (tm0 ) = 0;
uO 1 (tm0 ) = − tm1 uO˙0 (tm0 ):
(41)
By (26) the -rst equation is satis-ed by tm0 = . Inserting this and ((26), (29)) for (u0 ; u1 ) into the second equation, one can solve for tm1 and -nally insert into (39) to -nd (letting = 1): c1 + (73=32)c3 tm = − : (42) vb The velocity at this time is, by (23),(39), (26), (29), tm0 = , and Taylor expanding for 1: u(t ˙ m ) = u˙ 0 (tm0 ) + [tm1 uO 0 (tm0 ) + u˙ 1 (tm0 )] + O(2 ) = −vb + ( =2)c1 + O(2 ):
(43)
As a measure indicating the magnitude of oscillations, which are asymmetric with respect to u = 0, we de-ne the velocity amplitude Av0 of stick–slip oscillations as half the peak-to-peak velocity, i.e. ˙ m )); Av0 = 12 (vb − u(t
(44) 2
which on account of (43) and (28) becomes, ignoring terms of order and higher 2 3 vb
5 vb ( s − m ) Av0 = 1 − vb + 1− ; vb ¡ vb0 : 2 8 vm 4 vm
(45)
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The corresponding displacement amplitude A0 can be determined similarly, by calculating approximations to the times tm+ and tm− where u˙ = 0 (cf. Fig. 2); this yields 3 1 52 3 c3 ; u(tm− ) = − vb + 2c3 + c1 ; − c1 + tm− = 2 2 32 4 1 52 (46) tm+ = + c1 − c3 ; u(tm+ ) = vb + 2c3 − c1 : 2 2 32 4 De-ning the displacement amplitude A1 as half the peak-to-peak displacement, it becomes 1 A0 = (u(tm+ ) − u(tm− )) = vb − c1 2 2 2 3 vb 5 vb ( s − m ) = (1 − )vb + 1− ; vb ¡ vb0 : (47) 4 vm 4 vm One can show that this function has a maximum value at vb = vb∗ , vb for ( s − m )=vm 6 38 (1 − ( − 3)); 0 vb∗ = (1 − )vm 4 otherwise vm 15 1 + 3 =4( − ) s m
(48)
where vb0 is given by (20). Thus, if the di0erence in static and kinetic friction is not too large, the strongest stick–slip oscillations occur when the excitation speed reaches the value separating stick–slip oscillations from pure-slip oscillations, vb = vb0 . For larger friction di0erences the strongest oscillations occur at an excitation speed in the range vb ∈ ]0; vb0 [, as given by the second expression in (48). Since the analysis assumes friction di0erences that are small (but -nite), then the -rst case applies, so that we conclude that the strongest oscillations occur at vb = vb0 , hence 4 4vm A0 |vb =vb0 = A0; max = vb0 = vm 1 − = A1; max : (49) 5 3( s − m ) The last equality expresses that the predicted amplitude A1 of pure-slip oscillations equals the predicted amplitude A0 of stick–slip oscillations at the value of excitation speed separating these di0erent kinds of motion; this exact continuity is neither obvious nor required, since the two expressions were derived using approximate methods. It can be noted from (47) that at small values of the excitation speed, the stick–slip oscillation amplitude grows approximately linear with this speed 3 s − m vb for vb 1: (50) A0 ≈ 1 − + 4 vm Finally, since the velocity of the mass must change continuously with time, the maximum and minimum displacements of the mass must occur during the slip phase; they cannot occur during stick, because displacements here increase linearly with time until the time of slip. Hence, the amplitude A1 , determining displacements during the slip phase, also determines the oscillation amplitude of the complete stick–slip oscillation. 3.3.4. Stick–slip base frequency Since at t = T one cycle of slip and stick is completed, the base angular frequency of stick–slip oscillations is 2 vb 2 = : (51) !ss = T vb ts + u(0) − u(ts )
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This frequency is generally somewhat lower than the linear natural frequency of the system and the pure-slip oscillation frequency, !ss 6 !ps ≈ 1. 3.4. Summary of results For the system (1) – (2) with (4), or equivalently (8) – (9) with (10), the type of stationary motion depends on the excitation speed vb as follows, where vb0 and vb1 are given by (20) and (16), and all results holds approximately for small but -nite levels of vibration amplitudes, viscous damping , and relative friction di0erence ( s − m )=vm . For excitation speeds vb ¡ vb0 : Stick–slip oscillations, with stationary displacement and velocity amplitude A0 and Av0 , as given by (47) and (44), and base frequency !ss given by (51) (slightly less than the linear natural frequency of the system). For excitation speeds vb ∈ [vb0 ; vb1 ]: Pure-slip oscillations, with stationary displacement amplitude A1 (equal to the velocity amplitude Av1 ) as given by (19), and base frequency !ps = 1 (equal to the linear natural frequency of the system). For some parameters it may happen that vb0 = vb1 , so that pure-slip oscillations cannot occur at all. This occurs when the radical in (16) becomes negative, i.e. if viscous damping is su6ciently large, or the friction forces or di0erence between static and kinetic friction is su6ciently small. Q cf. (6). This corresponds to a For excitation speeds vb ¿ vb1 : Static equilibrium at position x(t) = x, state of steady sliding of the mass. 4. Illustration and testing of results 4.1. Oscillation cycles at di
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Fig. 3. One period of stick–slip displacements u(t) (top row), velocities u(t) ˙ (middle), and phase plane orbits u(u) ˙ (bottom row) for three values of excitation speed vb (a; b; c). (—– ) Analytical prediction (Eqs. (30) and (37)); (- - - - - -) numerical simulation of (1) with (4) and (7). Parameters: = 0:05; vm = 0:5; m = 0:25; s = 0:4, and (a) vb = 0:05, (b) vb = 0:25, (c) vb = 0:3944.
appear below. Also, as the friction di0erence is decreased the approximations become more accurate; for example, if Fig. 3 is re-computed for ( s − m )=vm = 0:15 instead of 0.3, then the results from using analytical approximations is almost indistinguishable from numerical simulation (only Figs. 3(a) and (b) would apply in that case). 4.2. Displacement amplitude as a function of excitation speed Fig. 4(a) shows the predicted variation of displacement amplitude with excitation speed for typical parameters. Here A0 indicates the amplitude during stick–slip oscillations for vb ¡ vb0 according to (47), and A1 is the amplitude for pure-slip oscillations for vb ∈ [vb0 ; vb1 ] according to (19). As appears, when the excitation speed is increased from zero, stick–slip oscillations occur with increasing amplitude until, at vb = vb0 pure-slip oscillations take over. These prevail only in a narrow range of belt velocities, until vb1 , above which oscillations cease and steady slip becomes the stable type of motion. The agreement
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Fig. 4. (a) Amplitude A and (b) base frequency ! of stable periodic motions as a function of excitation speed vb . (—-) Analytical prediction (Eq. (19), (47), (51)); (- - - - - -) analytical prediction from Thomsen [15]; (O) Numerical simulation of (1) with (4). Parameters as for Fig. 3.
of the analytical predictions with numerical simulation (encircled points) is seen to be very good; this will hold true the more closely the actual parameters satis-es the assumptions underlying the analytical expressions, i.e. as long as the di0erence in static and kinetic friction, the amount of viscous damping, and the amplitudes are not too large. It also appears that the predictions for the stick–slip amplitudes A0 are much better than the low-order approximation (indicated in dashed line) given in [15]. Fig. 4(b) depicts the predicted variation of base frequency ! with excitation speed, showing a slight drop in frequency for the lower velocities. Seemingly, the agreement with numerical simulation is quite good; however, since the change in frequency is so small, the relative error in the prediction is somewhat larger than for the displacement and amplitude predictions. This is a consequence of the di6culty in predicting, with high accuracy, the end of the slip phase of a stick–slip cycle, based on an approximate expression for the displacements during that cycle. 4.3. The signi=cance of friction di
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Fig. 5. As Fig. 4(a), but for varying levels of relative friction di0erence ( s − m )=vm . (—-) Analytical prediction (Eq. (19), (47)); (; ×; ; ⊕; ♦) numerical simulation of (1) with (4) and, respectively ( s − m )=vm = (0:075; 0:099; 0:15; 0:3; 0:6). Other parameters: = 0:05; vm = 0:5; s = 0:4.
5. Conclusions The analytical expressions for stationary friction-induced oscillations produce predictions that agree favorably with numerical simulations, as long as the assumption of relatively small di0erences in static and kinetic friction is ful-lled. This holds for stick–slip oscillations, that occur for relatively small excitation speeds vb ¡ vb0 , and for pure-slip oscillations that occur for a narrow range of slightly higher speeds vb ∈ [vb0 ; vb1 ]. Beyond this range, according to the predictions and numerical simulations, the system exhibits steady sliding with no oscillations. The expressions provide insight into the inLuence of parameters on vibration characteristics such as amplitude and base frequency; features that can be measured in experiments and compared to assess the validity of the underlying friction model. Although results are given in terms of a speci-c friction model, the approach used is not tied to this. It can readily be applied to set up and test expressions for other friction models having the properties of being (1) discontinuous only at zero-valued relative velocity, and (2) su6ciently smooth (for non-zero relative velocities) to enable Taylor-expansion to an order high enough to include motion-restricting non-linearities (usually three). References [1] J.W.S. Rayleigh, The Theory of Sound, 2nd Edition, 1945 re-issue. Dover Publications, New York, 1877. [2] Y.G. Panovko, I.I. Gubanova, Stability and Oscillations of Elastic Systems; Paradoxes, Fallacies and New Concepts, Consultants Bureau, New York, 1965. [3] A. Tondl, Quenching of Self-Excited Vibrations, Elsevier, Amsterdam, 1991. [4] A.H. Nayfeh, D.T. Mook, Nonlinear Oscillations, Wiley, New York, 1979. [5] Y.A. Mitropolskii, V.D. Nguyen, Applied Asymptotic Methods in Nonlinear Oscillations, Kluwer, Dordrecht, 1997. [6] K. Popp, Some model problems showing stick–slip motion and chaos, Friction-Induced Vibration, Chatter, Squeal, and Chaos, ASME DE-Vol. 49, ASME Design Engineering Divison, New York, 1992, pp. 1–12. [7] R.A. Ibrahim, Friction-induced vibration, chatter, squeal, and chaos: part I—mechanics of friction, Friction-Induced Vibration, Chatter, Squeal, and Chaos, ASME DE-Vol. 49, ASME Design Engineering Divison, New York, 1992, pp. 107–121. [8] R.A. Ibrahim, Friction-induced vibration, chatter, squeal, and chaos: part II—dynamics and modeling, Friction-Induced Vibration, Chatter, Squeal, and Chaos, ASME DE-Vol. 49, ASME Design Engineering Divison, New York, 1992, pp. 123–138. [9] J. McMillan, A non-linear friction model for self-excited vibrations, J. Sound Vibration 205 (3) (1997) 323–335. [10] B. Feeny, ArdKeshir Guran, N. Hinrichs, K. Popp, A historical review on dry friction and stick–slip phenomena, ASME Appl. Mech. Rev. 51 (5) (1998) 321–341.
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[11] B. Armstrong-HKelouvry, P. Dupont, C. Canudas de Wit, A survey of models, analysis tools and compensation methods for the control of machines with friction, Automatica 30 (7) (1994) 1083–1138. [12] B. Armstrong-HKelouvry, A perturbation analysis of stick–slip, Friction-Induced Vibration, Chatter, Squeal, and Chaos, ASME DE-Vol. 49, ASME Design Engineering Divison, New York, 1992, pp. 41– 48. [13] C. Gao, D. Kuhlmann-Wilsdorf, D.D. Makel, The dynamic analysis of stick–slip motion, Wear 173 (1994) 1–12. [14] F.-J. Elmer, Nonlinear dynamics of dry friction, J. Phys. A—Math. General 30 (1997) 6057–6063. [15] J.J. Thomsen, Using fast vibrations to quench friction-induced oscillations, J. Sound Vibration 228 (5) (1999) 1079–1102. [16] A. Fidlin, J.J. Thomsen, Predicting vibration-induced displacement for a resonant friction slider, European J. Mech. A=Solids 20 (2001) 155–166. [17] A. Fidlin, On the asymptotic analysis of discontinuous systems, DCAMM Report 646, Technical University of Denmark, 2000. [18] A.Y. Fidlin, On averaging in systems with a variable number of degrees of freedom, J. Appl. Math. Mech. 55 (4) (1991) 507–510. [19] A.V. Pechenev, A.Y. Fidlin, Hierachy of resonant motions excited in a vibroimpact system with contact zones by an inertial source of limited power, Izv. AN SSSR. Mechanika Tverdogo Tela 27 (4) (1992) 46 –53. [20] N. Hinrichs, M. Oestreich, K. Popp, On the modelling of friction oscillators, J. Sound Vibration 216 (3) (1998) 435–459. [21] J.J. Thomsen, Vibrations and Stability, Order and Chaos, McGraw-Hill, London, 1997.