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Vibration with “dynamic friction”
Journal of Sound and Vibration (1980) 69(3) 395-404
VIBRATION
WITH “DYNAMIC
FRICTION”
G. S. WHISTON Central Electricity Research Laboratories, Kelvin Avenue, Leatherhead KT22
7SE, England
(Received 17 July 1979, and in revised form 4 October 1979)
The free and harmonically forced vibration of an idealized straight be-amoscillating in its fundamental mode with frictional clamps at each end is modelled by using a singular differential equation. Locking and chatter phenomena are analyzed and slipping motions integrated for the free system and the continuously slipping steady state response to harmonic excitation is qualitatively discussed. 1. INTRODUCTION The motion of a simple linear oscillator under pure Coulomb friction is well understood
[l-3] both for free damped vibration and vibration under harmonic excitation. As with many types of motion usually discussed under the heading of discontinuous control systems [4], many interesting phenomena can occur which characterize this generally
discontinuous type of motion, particularly the locking and related chatter phenomena. In most systems of this type, the discontinuous nature of the motion is due to a discontinuous damping term. For example, in the Coulomb system the damping term changes sign as the sliding velocity passes through zero. It is the purpose of this paper to discuss a class of related dynamical systems which, although not previously discussed in the literature, do have an application in some engineering systems. Consider a beam disposed along the y-axis held in clamps through which the beam may slide with frictional damping (see Figure 1). If the beam is displaced perpendicular to its rest position at its mid-point, it will vibrate in its fundamental mode and sliding will take place at the clamp positions with frictional forces retarding the motion in the direction of sliding. The latter direction will vary smoothly with displacement. Here this type of friction is called “dynamic” since it varies with displacement and the motion of an idealized model of the above beam system will be presented below. Of course, the analysis will apply to other dynamically similar systems. In
tL.+LL Restoring
x-axis (a)
force = clastic+frktbn/
Restarinp
farce ~&tic-friction
(b)
Figure 1. (a) Idealiiationwed to modeldynamic friction for a beam vibrating in its fundamental mode whilst sliding through frictional clamps; (b) illustration of phase dependence of restoring force. 395 0022-460X/80/070395+
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the second section a discussion of the method of analysis is presented which is applied in section 3 to the linearized idealized beam system as an example and in section 4 the chatter phenomena associated with this type of friction are analyzed. Section 5 is concerned with the motion of the system under harmonic excitation for the continuous slipping solutions. It will be shown that, in general, two resonant frequencies exist whose separation is a function of frictional damping and that, of course, the two resonances merge as the damping becomes negligibly small.
2. METHOD
OF ANALYSIS
The following discussions will centre around oscillator equations of the form i+Pv2x=f(x,i),
(2.1)
where w is a scalar and f is a real valued function on the plane which is smooth on an open dense subspace U of the plane, representing a possibly singular dissipative force. The density of U means that any point of the plane R2 is arbitrarily close to U; hence so are, in particular, the complementary singular points, R2\U. Therefore any singular point of equation (2.1) is arbitrarily near a smooth point. It shall be further assumed that U can be split into the disjoint sum, (union), U = Uy=‘=,Ui, of open subspaces Vi such that f restricted to Vi, fi =fl Vi is smooth and of elementary form: that is, either linear or constant, fi(~, U) = six + PiU or fi (x, U) = yi for ai, pi and yi scalar and (x, U) a phase point in Vi. In this case, equation (2.1) is trivially integrable in each smooth subspace Vi. Moreover, each of the local differential equations i + W2X=fi(x, i),
l
(2.2)
is extendable to the whole of the plane since each fi is obviously extendable, Suppose that fi extends to E where A is of elementary form on the whole of the plane. Obviously EI Vi = fi so that the integral curves of fi coincide with the integral curves off in each Vi and hence by uniqueness, one has partial integrations of (2.1) in each smooth subspace. The method of integration of equation (2.1) is now obvious. Choose an initial point (x, V) in Uk say and then integrate equation (2.2) over the whole of the plane. It may be that the trajectory from (x, V) lies wholly within Uk, in which case the integration is complete. However, the trajectory of (2.2) may approach the boundary of Uk : that is, a singular point of equation (2.1). Suppose that the boundary point lies on the common boundary of Ukand Ufi If there exists an interior oriented trajectory of the global extension of equation (2.2) with i = j into Uj then the natural extension of the trajectory is into Uj along the integral curve from the boundary point as the initial point for the global extension of the jth equation. However, such an interior oriented trajectory may not exist and the boundary point must be regarded as a terminal or locking point of the integration of equation (2.1) from (x, 0). This will occur if the common boundaries are crossed with opposite orientation by the integrals of the globally extended equations. If an extension of equation (2.1) from U, into Uj is possible, the common boundary point of the trajectory can be called a “quasi-stationary” point if it lies on the x-axis. The trajectory will extend continuously across the boundary but not necessarily smoothly. These concepts are illustrated in Figure 2 where a hypothetical dynamical system is described by a singular differential equation whose singular points consist of the entire x and y axes and divide the plane into four open quadrants Vi where the functionsfi take four different forms. Consider the point ICI where the integral approaches the boundary point A. The trajectory cannot be extended through A into U, since oppositely oriented trajectories approach A. However, the integral from IC2 can be continued through B into VI, there being an accelerative discontinuity at that point.
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x-axis
Figure 2. Flow field of hypothetical singular dynamical system. The heavy line is a global integral from a point in U,.
Figure 3. Flow of Coulomb damped harmonic oscillator. x, = a/ w2.
in U1 whilst integrals from points in U2 will from points in U1 always remain attempt to cross into Us but are prevented from doing so by the oppositely oriented flow in U,. Note that, in general, accelerative discontinuities will occur as the integral curves traverse singular boundary points. As a physical example, take Coulomb friction where VI and U2 are the upper and lower half planes, and f(x, V) = a sgn (u) where a is a positive constant. The trajectory from any initial point in U, with Ix/> a/w' may be smoothly extended into Uz as can trajectories from points in Uz with Ix\> a/w*. However (see Figure 3) for 1x1G a/w* trajectories from tY2cannot cross into U1 since the trajectories in U1 with (x(~cu/ti* are exterior oriented. Quasi-stationary points are locking points if Ixl
Integrals
i+ w*.x = (Y(x,i) sgn (h(x, i)),
(2.3)
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G. S. WHISTON
where CYis a linear or constant real valued smooth function on the plane and h is a real valued function which might itself have a sign discontinuity. Let f-‘(6) be the subset of points with f(x, v) = 6; then if h is assumed to be smooth the function f(x, u) = cy(x, i) sgn (h(x, i)) is smooth on the open dense subspace U = R*\h-l(O) (the complement of h-‘(O)) and singular on the closed subspace h-‘(O), the set of points mapped into zero by h. U decomposes at U = U+ U U_ where Uk = h-‘(Rk) for R* the positive (resp. negative) real half lines. The equation reduces to the equations ,?+w2x=a(x,f)
in U+
and
i + w2x = -(Y(x, i)
in K.
If (Y is linear, both equations reduce to damped linear oscillator equations in their respective subspaces. Suppose that LY (x, U) = ax + bu ; then one obtains if-bi+(w*-a)x=O
in U+,
i+bi+(w*+a)x
=0
in U_.
Both equations extend smoothly to the plane and thus can be integrated by the local patching of solutions discussed above. The equations to be analyzed here correspond to a particular choice of h and will be discussed in the following section. 3. THE MODEL The motion of a straight beam or length of tubing held along the y-axis by loosely fitting clamps at Iyl = L, oscillating in its fundamental mode, can be roughly approximated as follows (see Figure 1). The beam of mass m and length greater than 2L is idealized by a massless, perfectly rigid pair of rods pinned at the centre. The mass of the beam is lumped at the pinned joint and its elasticity is represented by a spring of stiffness k from m (grounded at y = 0) extendable along the x-axis. The beam is free to slide through the supports with normal reactions N and coefficient of sliding friction CL.Suppose that at displacement x the tube slides through the supports with speed u ; then the energy E of the tube is dissipated at a rate k = -2 @N sgn (u)v. Equating this to the rate of change of E = mi2/2 + kx2/2 gives the equation i(mi+kx)=
-2@N sgn (0)~.
(3.1)
But t, is the rate of change of (x2 +L*)’ and one obtains ?J= xi/(x2 + L2)“*,
sgn (u) = sgn (x, It).
Therefore, upon cancelling the 1 term from equation (3.1), the following equation of motion is obtained: mi + kx = -2pN In practice,
sgn (x, lr)/(x2 + L2)l’*.
(3.2)
x/L is small and the above equation may be reduced to mi + kx = - (2pN/L)x
sgn (x, i).
(3.3)
This is the equation to be investigated in detail. Equation (3.3) may be written as i + wix = -_(yx sgn (xi),
(3.4)
where, of course, (Y= 2c~N/mL and wi = k/m. The smooth subspace of equation (3.4) consists of U+ = {(x, Y): xv > O}t and U_ = {(x, u): xu < 0). Thus U, is the disjoint sum of t {(x,v): P(x, u)}denotes the subspace of the plane of points (x, O) such that property P(x, v) is true: e.g., P(x, u) = XL)> 0.
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the first and third interior quadrants of the plane, whilst U_ is the disjoint sum of the second and fourth quadrants. The local reductions are x+(wt+cr)x
=0
in U+,
x+(wi-a)x=O
in U_.
(3.5)
The solutions of equations (3.5) are of course well known. If W; > (Y they represent harmonic motion of frequency (w: +a)“* in U, and (w; -cx)~‘* in U_. If however, wz < (Y,motion in U_ is hyperbolic. Plotted in phase space, the integrals of equations (3.5) represent segments of ellipses in their respective quadrants. The flow of the oscillator with wz = 11 and (Y= 1 is depicted in Figure 4. Note that flow extends smoothly to the whole
Figure 4. Integral curve of harmonic oscillator damped with dynamic friction N less than NC.
plane for wi > (Ysince the smooth subspace boundaries are always approached normally with exterior orientation along any integral curve originating in the interior of any quadrant. The integral curves wind smoothly into the origin with a well defined geometric amplitude decrement y: xi+1 = yxi for successive crossings of the x-axis on the right. y is given by [(wx - a)/( wi +a)]. To see this, suppose the initial condition is (x0, -E) for E > 0 and arbitrarily small. Then after traversing the segment of trajectory in the quadrant {(x, u): x > 0 and x < 0}, the u-axis will be approached arbitrarily closely near VI = -(wf~ --a)“*~~. Similarly with (--E, -VI) as initial point, the x-axis will be approached arbitrarily closely near xb = VJ(wi +a)l’*, after traversing the quadrant {(x, u): x < 0 and u < 0). Then from the initial point (XL,E), the third quadrant is traversed and the trajectory arrives at the positive u-axis arbitrarily near V2 = -xb(wi -(Y)“*. Traversing the fourth quadrant, the positive x-axis is re-approached with x1 = V2/(wE +(~)l’*. In short, xi =XO(W~-(Y)/(w~ +a). One can now consider the case wi < (Ywhen motion in U- is hyperbolic and associate the motion with locking phenomena. The flow of a system with w$ < a is depicted in Figure 5. Note that motion in U+ is still simple harmonic but that the flow cannot cross the x-axis from quadrant (1) into (2) or from (3) into (4). The resulting motions are described as follows. If released from initial conditions in quadrant (l), the system locks at a quasistationary point dependent upon the initial conditions. Initial points in quadrant (2) will either decay to a locking point on the positive x-axis or to a locking point on the negative x-axis if the initial velocity is high enough. Symmetrical comments apply to quadrants (3)
400
G. S. WHISTON
Figure 5. Flow of harmonic oscillator damped with dynamic friction with N greater than NC
and (4). Note that locking will occur from arbitrary initial conditions for W; < LYwithin half a revolution. This is to be contrasted to the case of pure Coulomb friction where, depending on the initial conditions, locking may occur only after many cycles of motion. To close this section the locking phenomenon may be interpreted in engineering terms. The geometric amplitude decrement is given by y = (wi -2pN/mL)/(wz
+2pN/ml).
For fixed beam parameters, y will approach zero as the support tightness N approaches the critical value N, = kL/2p. For N = N, the decrement is meaningless and is negative for N > N, : sufficient tightness will always prevent damped oscillations. Of course, for N > N,, it has been seen that some “overdamped” motion will occur but that locking will always occur in less than one half revolution for any initial condition. Suppose one finally considers equation (3.2), which is appropriate when x/L is of order 1. Locking will occur at a quasi-stationary point if the elastic force drops below the frictional force. One therefore seeks a phase point where the frictional force equals the elastic force at a quasi-stationary point: i.e., 2pNx/(x2 + L2)1’2 = kx. The solution is given by xc = {(2pN/k)2 - L2}“’ = (2,s/k)(N2 - N:)1’2. Clearly if N < N,, x, is imaginary and no locking will occur, damped vibration persisting indefinitely. However, if N > N, a real solution exists. In this case the interval -xc < x
4. CHATTER
PHENOMENA
Chattering is a steady state, usually relatively high frequency oscillation which can occur in many types of control system employing rapidly switching devices such as relays. Such devices have a small but finite reaction time to switching stimuli. The oscillations can also
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occur in systems governed by Coulomb friction [5] where the frictional force cannot instantaneously adjust to a change in direction of motion. One can model the switching inertia by the following discontinuous function: (z(x) = sgn (x -E sgn (i))
for E > 0.
Suppose that x is large and positive but decreasing, f < 0. Then a(x) will change sign when x + E = 0: i.e., x = -E. Similarly, when x is large and negative and f > 0, c(x) will change sign when x = +E. Thus u models inertia in the direction of change. Consider the beam system discussed previously, with imperfect switching. The equation of motion reduces to the form _?+ w;x = -axc+(xi) = -axff( V),
(4.1)
where V is the slipping velocity of the tube through the clamps. The switching lines of equation (4.1) are the rectangular hyperbolae ]xf] = E = 1VI, whereas those of equation (3.4) were the x and i axes. An integral of equation (4.1) from an initial point (x, -8) with S > E is shown in Figure 6. Motion at frequency (wi - 2pN/L) through U2 occurs and
u,
v-0
Iv--oxis
Figure 6. Chatter cycle of harmonic oscillator damped with dynamic friction with N < N,.
continues into Us until the trajectory meets the rectangular hyperbola when u switches and motion through U, into tY, occurs at frequency (wi + 2pN/L) until the branch of the hyperbola in U, is met when u switches again, and so on. Whilst the amplitude is greater than E, the trajectory of equation (4.1) closely resembles that of equation (3.4) from the same initial point, except for small accelerative discontinuities on crossing the switching hyperbolae. However, for amplitudes only slightly greater than E, the discontinuities begin to become important until, when the amplitude drops below E, a closed cycle of motion is attained, called the chatter cycle for this system. Such a cycle occurs because the trajectory no longer intersects any switching subspaces and will occur at frequency wi-2pN/L (since switching will occur for amplitude E for the higher eccentricity corresponding to wi + 2pNIL) at amplitude E.
402
G. S. WHISTON 5. HARMONIC
EXCITATION
In this section a discussion is presented of the steady state solutions for the harmonic response version of equation (3.4): mi+kx=-cuxsgn(x~)+&sin&?nt++).
(5.1)
Before doing so, and to establish the method, the analogous equations with viscous and Coulomb damping are briefly discussed. The integration of the linear smooth equation of a one-dimensional system with viscous damping, mf+kx+ni=Fosin(Ot+&),
(5.2)
proceeds in two stages. Since the equation is linear, the space of solutions is generated by linear combinations of kernel solutions where the right-hand side of equation (4.2) is replaced by zero and a particular integral xP = &I sin (fit + I$ - 4), where the magnification factor 9 and the phase lag I,+are the usual functions of m, k, 7) and R. Since the kernel solutions decay exponentially, the steady state solution is the harmonic response represented by the particular integral. The situation is much more complex in the case of Coulomb damping [ 1, 21. During slipping motion, equation (5.2) is replaced by mi+kx=-pNsgn(1)+F0sin(0t+4).
(5.3)
This equation is of course linear in U+ and U_, but here the “transients” can never be neglected since they are triggered at each quasi-stationary point. The response can never be harmonic and a family of steady state solutions exist with varying proportions of slipping and locked responses. For simplicity, attention is now restricted to continuous slipping responses at the excitation frequency. By assuming reflection symmetry in the x-axis one can characterize the response by calculating an expression for the phase lag of response to excitation and the response amplitude. The resulting expressions are x0= QFo={q2-(w~/f22) q5= tan-’
tan’ (1112)(~N/kF,)2}“2Fo,
{@2x0/
wOxc)
cot (G/2)).
The parameter 4 is the free magnification factor, x, is the locking displacement pN/k and $ = wow/w - woT/2 where T is the excitation period. As Den Hartog has pointed out, the above magnification factor is rather interesting. For large ratios of friction to excitation, Q is imaginary and the interpretation is, of course, that the system cannot be unlocked by insufficient excitation. For smaller friction to excitation ratios, Q is real but will always become infinite at resonance. To find steady state continuous slipping solution one can proceed in a manner analogous to the Coulomb friction case. The solution is assumed to be symmetric about the x-axis in phase space. One only has then to find a solution in, say, the lower half plane. In the quadrant {(x, v): x < 0 and u < 0)) equation (5.1) reduces to mf + kx = (2pN/L)x If w: = (k -2pN/L,/m) for k > 2pN/L, arbitrarily small E > 0) is
+Fo sin (at + 4).
a general solution starting from (x0, -E) (for
x =A cos (wlt)+B sin (wlt)+qlFosin
(f2t+4),
where the magnification factor q1 is {(w: - f2’)m}-‘: that is, 41 ={(w; -2~N/Lm)-02}-‘m-‘.
VIBRATION The
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initial conditions yield A and B A
=
XO@O
sin
“DYNAMIC
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FRICTION”
:
B = -i@Fo
(4),
cos @)/WI.
The compiete solution in this quadrant is therefore ~={~~-4~F~sin(~)}cos(w~t)-(~4~/w~)F~cos(~)sin(w~t)+4~F0sin(~t+~). The period T = 27r/0 also implies x(7r/20) =O, i(7r/20)= -V, which is, as yet, undetermined. One proceeds by obtaining a solution in the quadrant {(x, u): x < 0 and u < 0) with initial conditions (0, V) at time r/20. Equations which yield the steady state amplitudes x0 and relative phase r$ are obtained by imposing final conditions (-XC,,0) at t = r/R. In the second quadrant equation (4.1) reduces to mi + kx = -(2$V/L)x
+Fo sin (at + 4),
and 42=(w$-f12)-‘m-‘. where ws =(k+2pN/L)/m x(?r/20) = 0, I(2r/2R) = -V yields C and D:
Imposing the initial conditions
D = {- V + @Fo
C = 4#0 cos (4)~
sin (d)}/ w2.
The general solution with these conditions is therefore x = 42F0 cos (4) cos {wq(f - f0)}+{(--V+q2flF~
sin 4)/w2} sin {wz(t-f0))
+ 42Fo sin (0 + 4). Therefore 41= wl?r/2L! and 452= w22r/20. Then, imposingx(?r/LJ) and substituting for V yields -x0
=
= -x0 and i(rr/a)
{-42F0 cos ($2) - (flF04l/w2) cos ($1) sin (#z) + (&lFo/w2) +{(wl/w2)4lF0
=0
sin ($2)) cos (4)
sin (111)sin ($2)+42(~/wZ)FO sin (#Z)-42FO) sin (4)
- (W~/WZ)XO sin ($1) sin (+2), 0 ={w242FO sin (42) - fl4lF0 cos
(+l)
COs
(42) +@lFO cos (42) - 4zaFo) cos (4)
+{wl4lFO sin (4’1)cos (42) +42flF0 cos ($2)) sin (#)-xOw1 sin (#I) cos ($2). One may rewrite the above equations in the equivalent form (-“0”) =( ::,“:=1::
::,“:+‘::::PGt$
-(;zJ
(where the parameters al, bl, etc., are defined in the obvious way) in terms of the partial magnification factors 41 and 42 defined above. By gathering terms in x0 to the left-hand side one obtains a linear equation for the vector (cos 4, sin (4)) which yields the phase angle 4 and the amplitude x0 through the identity cos’ 12% +sin’ 4 = 1. The resulting equation for x0 is of the form xo= OF,= {(Aq:+Bq;+
@4z)/(a4:+@4;+
?‘414~)“~}Fo.
The new parameters introduced in this equation are related to the previous ones but here interest is in the variations of x0 as a function of the parameters 41 and 42. Writing 41= (f2’- w:)-‘, 42 = (a’- wz)-’ one obtains the equation {A(f2Z-w;)2+B(R2-w:)2+C(f+w:)(i22-w~)}F,, xo= (nZ-
w:)(fP--
w;)Ja(&-
w:)“+p(e-
w:y+y(fP-
wf)(&-
w;j
It is clear that xo will become infinitely large for 0 = w1 or w2 and that for 0 = w. it will
404
G. S.WHISTON
remain finite. The system therefore has resonances at the two damped natural frequencies present, occurring at w1 = (wi -2pN/Lm)“’ and w2 = (wi + 2pN/Lm)“2, and that the separation increases with damping until w1 = 0 when the continuous free slipping condition must be violated. The situation here differs both from that in the Coulomb friction case where the resonant frequency is the undamped natural frequency and that in the viscous damping case where the resonant frequency differs from both the damped and undamped natural frequencies. 6. CONCLUSIONS
A singular differential equation that could be used to model the dynamics of a beam supported by frictional clamps at either end has been investigated in both its autonomous and harmonically excited versions. The free vibration is either a semi-hyperbolic aperiodic decay to locking or an infinitely long decay to rest at the origin with a well-defined geometric amplitude decrement depending upon the clamp tightness. For sufficient clamping force locking can occur for any displacement in a long beam but sufficiently short beams show a locking behaviour similar to that of the pure Coulomb system. Similar calculations have been carried out for curved beams whose quantitative behaviour is somewhat different and these calculations may be discussed in a future paper. The chatter oscillations of the autonomous equations have also been analyzed. Solutions of the harmonically excited systems were investigated qualitatively for continuous slipping and it was shown that resonances occur at two damped frequencies associated with the system, the separation of the peaks being monotonically increasing with clamp force until the continuous slipping assumption is no longer valid. Of course, the two resonant frequencies may well be a function of the beam idealization rather than a real phenomenon. It might be interesting to check the occurrence of the twin peaks experimentally. REFERENCES 1. J. P. DEN HARTOG 1931 Transactions of the American Society of Mechanical Engineers 53, 107-l 13. Forced vibration with combined Coulomb and viscous friction. 2. K. MAGNLJS 1965 Vibrations.London: Blackie and Son limited. 3. M. S. HUNDAL 1979 Journal of Sound and Vibration 64,371-378. Response of a base excited
system with Coulomb and viscous friction.
4. I. FL~GGE-LOTZ 1968 Discontinuous and Optimal Control. New York: McGraw-Hill Book Company, Inc. 5. J. A. GLEN 1978 Bulletin of the Institute of Mathematics and its Applications, 151. The oscillations of a simple castor.
VIBRATION
WITH “DYNAMIC
FRICTION”
G. S. WHISTON Central Electricity Research Laboratories, Kelvin Avenue, Leatherhead KT22
7SE, England
(Received 17 July 1979, and in revised form 4 October 1979)
The free and harmonically forced vibration of an idealized straight be-amoscillating in its fundamental mode with frictional clamps at each end is modelled by using a singular differential equation. Locking and chatter phenomena are analyzed and slipping motions integrated for the free system and the continuously slipping steady state response to harmonic excitation is qualitatively discussed. 1. INTRODUCTION The motion of a simple linear oscillator under pure Coulomb friction is well understood
[l-3] both for free damped vibration and vibration under harmonic excitation. As with many types of motion usually discussed under the heading of discontinuous control systems [4], many interesting phenomena can occur which characterize this generally
discontinuous type of motion, particularly the locking and related chatter phenomena. In most systems of this type, the discontinuous nature of the motion is due to a discontinuous damping term. For example, in the Coulomb system the damping term changes sign as the sliding velocity passes through zero. It is the purpose of this paper to discuss a class of related dynamical systems which, although not previously discussed in the literature, do have an application in some engineering systems. Consider a beam disposed along the y-axis held in clamps through which the beam may slide with frictional damping (see Figure 1). If the beam is displaced perpendicular to its rest position at its mid-point, it will vibrate in its fundamental mode and sliding will take place at the clamp positions with frictional forces retarding the motion in the direction of sliding. The latter direction will vary smoothly with displacement. Here this type of friction is called “dynamic” since it varies with displacement and the motion of an idealized model of the above beam system will be presented below. Of course, the analysis will apply to other dynamically similar systems. In
tL.+LL Restoring
x-axis (a)
force = clastic+frktbn/
Restarinp
farce ~&tic-friction
(b)
Figure 1. (a) Idealiiationwed to modeldynamic friction for a beam vibrating in its fundamental mode whilst sliding through frictional clamps; (b) illustration of phase dependence of restoring force. 395 0022-460X/80/070395+
10$02.00/O
@ 1980 Academic Press Inc. (London) Limited
396
G. S. WHISTON
the second section a discussion of the method of analysis is presented which is applied in section 3 to the linearized idealized beam system as an example and in section 4 the chatter phenomena associated with this type of friction are analyzed. Section 5 is concerned with the motion of the system under harmonic excitation for the continuous slipping solutions. It will be shown that, in general, two resonant frequencies exist whose separation is a function of frictional damping and that, of course, the two resonances merge as the damping becomes negligibly small.
2. METHOD
OF ANALYSIS
The following discussions will centre around oscillator equations of the form i+Pv2x=f(x,i),
(2.1)
where w is a scalar and f is a real valued function on the plane which is smooth on an open dense subspace U of the plane, representing a possibly singular dissipative force. The density of U means that any point of the plane R2 is arbitrarily close to U; hence so are, in particular, the complementary singular points, R2\U. Therefore any singular point of equation (2.1) is arbitrarily near a smooth point. It shall be further assumed that U can be split into the disjoint sum, (union), U = Uy=‘=,Ui, of open subspaces Vi such that f restricted to Vi, fi =fl Vi is smooth and of elementary form: that is, either linear or constant, fi(~, U) = six + PiU or fi (x, U) = yi for ai, pi and yi scalar and (x, U) a phase point in Vi. In this case, equation (2.1) is trivially integrable in each smooth subspace Vi. Moreover, each of the local differential equations i + W2X=fi(x, i),
l
(2.2)
is extendable to the whole of the plane since each fi is obviously extendable, Suppose that fi extends to E where A is of elementary form on the whole of the plane. Obviously EI Vi = fi so that the integral curves of fi coincide with the integral curves off in each Vi and hence by uniqueness, one has partial integrations of (2.1) in each smooth subspace. The method of integration of equation (2.1) is now obvious. Choose an initial point (x, V) in Uk say and then integrate equation (2.2) over the whole of the plane. It may be that the trajectory from (x, V) lies wholly within Uk, in which case the integration is complete. However, the trajectory of (2.2) may approach the boundary of Uk : that is, a singular point of equation (2.1). Suppose that the boundary point lies on the common boundary of Ukand Ufi If there exists an interior oriented trajectory of the global extension of equation (2.2) with i = j into Uj then the natural extension of the trajectory is into Uj along the integral curve from the boundary point as the initial point for the global extension of the jth equation. However, such an interior oriented trajectory may not exist and the boundary point must be regarded as a terminal or locking point of the integration of equation (2.1) from (x, 0). This will occur if the common boundaries are crossed with opposite orientation by the integrals of the globally extended equations. If an extension of equation (2.1) from U, into Uj is possible, the common boundary point of the trajectory can be called a “quasi-stationary” point if it lies on the x-axis. The trajectory will extend continuously across the boundary but not necessarily smoothly. These concepts are illustrated in Figure 2 where a hypothetical dynamical system is described by a singular differential equation whose singular points consist of the entire x and y axes and divide the plane into four open quadrants Vi where the functionsfi take four different forms. Consider the point ICI where the integral approaches the boundary point A. The trajectory cannot be extended through A into U, since oppositely oriented trajectories approach A. However, the integral from IC2 can be continued through B into VI, there being an accelerative discontinuity at that point.
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x-axis
Figure 2. Flow field of hypothetical singular dynamical system. The heavy line is a global integral from a point in U,.
Figure 3. Flow of Coulomb damped harmonic oscillator. x, = a/ w2.
in U1 whilst integrals from points in U2 will from points in U1 always remain attempt to cross into Us but are prevented from doing so by the oppositely oriented flow in U,. Note that, in general, accelerative discontinuities will occur as the integral curves traverse singular boundary points. As a physical example, take Coulomb friction where VI and U2 are the upper and lower half planes, and f(x, V) = a sgn (u) where a is a positive constant. The trajectory from any initial point in U, with Ix/> a/w' may be smoothly extended into Uz as can trajectories from points in Uz with Ix\> a/w*. However (see Figure 3) for 1x1G a/w* trajectories from tY2cannot cross into U1 since the trajectories in U1 with (x(~cu/ti* are exterior oriented. Quasi-stationary points are locking points if Ixl
Integrals
i+ w*.x = (Y(x,i) sgn (h(x, i)),
(2.3)
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where CYis a linear or constant real valued smooth function on the plane and h is a real valued function which might itself have a sign discontinuity. Let f-‘(6) be the subset of points with f(x, v) = 6; then if h is assumed to be smooth the function f(x, u) = cy(x, i) sgn (h(x, i)) is smooth on the open dense subspace U = R*\h-l(O) (the complement of h-‘(O)) and singular on the closed subspace h-‘(O), the set of points mapped into zero by h. U decomposes at U = U+ U U_ where Uk = h-‘(Rk) for R* the positive (resp. negative) real half lines. The equation reduces to the equations ,?+w2x=a(x,f)
in U+
and
i + w2x = -(Y(x, i)
in K.
If (Y is linear, both equations reduce to damped linear oscillator equations in their respective subspaces. Suppose that LY (x, U) = ax + bu ; then one obtains if-bi+(w*-a)x=O
in U+,
i+bi+(w*+a)x
=0
in U_.
Both equations extend smoothly to the plane and thus can be integrated by the local patching of solutions discussed above. The equations to be analyzed here correspond to a particular choice of h and will be discussed in the following section. 3. THE MODEL The motion of a straight beam or length of tubing held along the y-axis by loosely fitting clamps at Iyl = L, oscillating in its fundamental mode, can be roughly approximated as follows (see Figure 1). The beam of mass m and length greater than 2L is idealized by a massless, perfectly rigid pair of rods pinned at the centre. The mass of the beam is lumped at the pinned joint and its elasticity is represented by a spring of stiffness k from m (grounded at y = 0) extendable along the x-axis. The beam is free to slide through the supports with normal reactions N and coefficient of sliding friction CL.Suppose that at displacement x the tube slides through the supports with speed u ; then the energy E of the tube is dissipated at a rate k = -2 @N sgn (u)v. Equating this to the rate of change of E = mi2/2 + kx2/2 gives the equation i(mi+kx)=
-2@N sgn (0)~.
(3.1)
But t, is the rate of change of (x2 +L*)’ and one obtains ?J= xi/(x2 + L2)“*,
sgn (u) = sgn (x, It).
Therefore, upon cancelling the 1 term from equation (3.1), the following equation of motion is obtained: mi + kx = -2pN In practice,
sgn (x, lr)/(x2 + L2)l’*.
(3.2)
x/L is small and the above equation may be reduced to mi + kx = - (2pN/L)x
sgn (x, i).
(3.3)
This is the equation to be investigated in detail. Equation (3.3) may be written as i + wix = -_(yx sgn (xi),
(3.4)
where, of course, (Y= 2c~N/mL and wi = k/m. The smooth subspace of equation (3.4) consists of U+ = {(x, Y): xv > O}t and U_ = {(x, u): xu < 0). Thus U, is the disjoint sum of t {(x,v): P(x, u)}denotes the subspace of the plane of points (x, O) such that property P(x, v) is true: e.g., P(x, u) = XL)> 0.
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the first and third interior quadrants of the plane, whilst U_ is the disjoint sum of the second and fourth quadrants. The local reductions are x+(wt+cr)x
=0
in U+,
x+(wi-a)x=O
in U_.
(3.5)
The solutions of equations (3.5) are of course well known. If W; > (Y they represent harmonic motion of frequency (w: +a)“* in U, and (w; -cx)~‘* in U_. If however, wz < (Y,motion in U_ is hyperbolic. Plotted in phase space, the integrals of equations (3.5) represent segments of ellipses in their respective quadrants. The flow of the oscillator with wz = 11 and (Y= 1 is depicted in Figure 4. Note that flow extends smoothly to the whole
Figure 4. Integral curve of harmonic oscillator damped with dynamic friction N less than NC.
plane for wi > (Ysince the smooth subspace boundaries are always approached normally with exterior orientation along any integral curve originating in the interior of any quadrant. The integral curves wind smoothly into the origin with a well defined geometric amplitude decrement y: xi+1 = yxi for successive crossings of the x-axis on the right. y is given by [(wx - a)/( wi +a)]. To see this, suppose the initial condition is (x0, -E) for E > 0 and arbitrarily small. Then after traversing the segment of trajectory in the quadrant {(x, u): x > 0 and x < 0}, the u-axis will be approached arbitrarily closely near VI = -(wf~ --a)“*~~. Similarly with (--E, -VI) as initial point, the x-axis will be approached arbitrarily closely near xb = VJ(wi +a)l’*, after traversing the quadrant {(x, u): x < 0 and u < 0). Then from the initial point (XL,E), the third quadrant is traversed and the trajectory arrives at the positive u-axis arbitrarily near V2 = -xb(wi -(Y)“*. Traversing the fourth quadrant, the positive x-axis is re-approached with x1 = V2/(wE +(~)l’*. In short, xi =XO(W~-(Y)/(w~ +a). One can now consider the case wi < (Ywhen motion in U- is hyperbolic and associate the motion with locking phenomena. The flow of a system with w$ < a is depicted in Figure 5. Note that motion in U+ is still simple harmonic but that the flow cannot cross the x-axis from quadrant (1) into (2) or from (3) into (4). The resulting motions are described as follows. If released from initial conditions in quadrant (l), the system locks at a quasistationary point dependent upon the initial conditions. Initial points in quadrant (2) will either decay to a locking point on the positive x-axis or to a locking point on the negative x-axis if the initial velocity is high enough. Symmetrical comments apply to quadrants (3)
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G. S. WHISTON
Figure 5. Flow of harmonic oscillator damped with dynamic friction with N greater than NC
and (4). Note that locking will occur from arbitrary initial conditions for W; < LYwithin half a revolution. This is to be contrasted to the case of pure Coulomb friction where, depending on the initial conditions, locking may occur only after many cycles of motion. To close this section the locking phenomenon may be interpreted in engineering terms. The geometric amplitude decrement is given by y = (wi -2pN/mL)/(wz
+2pN/ml).
For fixed beam parameters, y will approach zero as the support tightness N approaches the critical value N, = kL/2p. For N = N, the decrement is meaningless and is negative for N > N, : sufficient tightness will always prevent damped oscillations. Of course, for N > N,, it has been seen that some “overdamped” motion will occur but that locking will always occur in less than one half revolution for any initial condition. Suppose one finally considers equation (3.2), which is appropriate when x/L is of order 1. Locking will occur at a quasi-stationary point if the elastic force drops below the frictional force. One therefore seeks a phase point where the frictional force equals the elastic force at a quasi-stationary point: i.e., 2pNx/(x2 + L2)1’2 = kx. The solution is given by xc = {(2pN/k)2 - L2}“’ = (2,s/k)(N2 - N:)1’2. Clearly if N < N,, x, is imaginary and no locking will occur, damped vibration persisting indefinitely. However, if N > N, a real solution exists. In this case the interval -xc < x
4. CHATTER
PHENOMENA
Chattering is a steady state, usually relatively high frequency oscillation which can occur in many types of control system employing rapidly switching devices such as relays. Such devices have a small but finite reaction time to switching stimuli. The oscillations can also
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occur in systems governed by Coulomb friction [5] where the frictional force cannot instantaneously adjust to a change in direction of motion. One can model the switching inertia by the following discontinuous function: (z(x) = sgn (x -E sgn (i))
for E > 0.
Suppose that x is large and positive but decreasing, f < 0. Then a(x) will change sign when x + E = 0: i.e., x = -E. Similarly, when x is large and negative and f > 0, c(x) will change sign when x = +E. Thus u models inertia in the direction of change. Consider the beam system discussed previously, with imperfect switching. The equation of motion reduces to the form _?+ w;x = -axc+(xi) = -axff( V),
(4.1)
where V is the slipping velocity of the tube through the clamps. The switching lines of equation (4.1) are the rectangular hyperbolae ]xf] = E = 1VI, whereas those of equation (3.4) were the x and i axes. An integral of equation (4.1) from an initial point (x, -8) with S > E is shown in Figure 6. Motion at frequency (wi - 2pN/L) through U2 occurs and
u,
v-0
Iv--oxis
Figure 6. Chatter cycle of harmonic oscillator damped with dynamic friction with N < N,.
continues into Us until the trajectory meets the rectangular hyperbola when u switches and motion through U, into tY, occurs at frequency (wi + 2pN/L) until the branch of the hyperbola in U, is met when u switches again, and so on. Whilst the amplitude is greater than E, the trajectory of equation (4.1) closely resembles that of equation (3.4) from the same initial point, except for small accelerative discontinuities on crossing the switching hyperbolae. However, for amplitudes only slightly greater than E, the discontinuities begin to become important until, when the amplitude drops below E, a closed cycle of motion is attained, called the chatter cycle for this system. Such a cycle occurs because the trajectory no longer intersects any switching subspaces and will occur at frequency wi-2pN/L (since switching will occur for amplitude E for the higher eccentricity corresponding to wi + 2pNIL) at amplitude E.
402
G. S. WHISTON 5. HARMONIC
EXCITATION
In this section a discussion is presented of the steady state solutions for the harmonic response version of equation (3.4): mi+kx=-cuxsgn(x~)+&sin&?nt++).
(5.1)
Before doing so, and to establish the method, the analogous equations with viscous and Coulomb damping are briefly discussed. The integration of the linear smooth equation of a one-dimensional system with viscous damping, mf+kx+ni=Fosin(Ot+&),
(5.2)
proceeds in two stages. Since the equation is linear, the space of solutions is generated by linear combinations of kernel solutions where the right-hand side of equation (4.2) is replaced by zero and a particular integral xP = &I sin (fit + I$ - 4), where the magnification factor 9 and the phase lag I,+are the usual functions of m, k, 7) and R. Since the kernel solutions decay exponentially, the steady state solution is the harmonic response represented by the particular integral. The situation is much more complex in the case of Coulomb damping [ 1, 21. During slipping motion, equation (5.2) is replaced by mi+kx=-pNsgn(1)+F0sin(0t+4).
(5.3)
This equation is of course linear in U+ and U_, but here the “transients” can never be neglected since they are triggered at each quasi-stationary point. The response can never be harmonic and a family of steady state solutions exist with varying proportions of slipping and locked responses. For simplicity, attention is now restricted to continuous slipping responses at the excitation frequency. By assuming reflection symmetry in the x-axis one can characterize the response by calculating an expression for the phase lag of response to excitation and the response amplitude. The resulting expressions are x0= QFo={q2-(w~/f22) q5= tan-’
tan’ (1112)(~N/kF,)2}“2Fo,
{@2x0/
wOxc)
cot (G/2)).
The parameter 4 is the free magnification factor, x, is the locking displacement pN/k and $ = wow/w - woT/2 where T is the excitation period. As Den Hartog has pointed out, the above magnification factor is rather interesting. For large ratios of friction to excitation, Q is imaginary and the interpretation is, of course, that the system cannot be unlocked by insufficient excitation. For smaller friction to excitation ratios, Q is real but will always become infinite at resonance. To find steady state continuous slipping solution one can proceed in a manner analogous to the Coulomb friction case. The solution is assumed to be symmetric about the x-axis in phase space. One only has then to find a solution in, say, the lower half plane. In the quadrant {(x, v): x < 0 and u < 0)) equation (5.1) reduces to mf + kx = (2pN/L)x If w: = (k -2pN/L,/m) for k > 2pN/L, arbitrarily small E > 0) is
+Fo sin (at + 4).
a general solution starting from (x0, -E) (for
x =A cos (wlt)+B sin (wlt)+qlFosin
(f2t+4),
where the magnification factor q1 is {(w: - f2’)m}-‘: that is, 41 ={(w; -2~N/Lm)-02}-‘m-‘.
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initial conditions yield A and B A
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XO@O
sin
“DYNAMIC
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:
B = -i@Fo
(4),
cos @)/WI.
The compiete solution in this quadrant is therefore ~={~~-4~F~sin(~)}cos(w~t)-(~4~/w~)F~cos(~)sin(w~t)+4~F0sin(~t+~). The period T = 27r/0 also implies x(7r/20) =O, i(7r/20)= -V, which is, as yet, undetermined. One proceeds by obtaining a solution in the quadrant {(x, u): x < 0 and u < 0) with initial conditions (0, V) at time r/20. Equations which yield the steady state amplitudes x0 and relative phase r$ are obtained by imposing final conditions (-XC,,0) at t = r/R. In the second quadrant equation (4.1) reduces to mi + kx = -(2$V/L)x
+Fo sin (at + 4),
and 42=(w$-f12)-‘m-‘. where ws =(k+2pN/L)/m x(?r/20) = 0, I(2r/2R) = -V yields C and D:
Imposing the initial conditions
D = {- V + @Fo
C = 4#0 cos (4)~
sin (d)}/ w2.
The general solution with these conditions is therefore x = 42F0 cos (4) cos {wq(f - f0)}+{(--V+q2flF~
sin 4)/w2} sin {wz(t-f0))
+ 42Fo sin (0 + 4). Therefore 41= wl?r/2L! and 452= w22r/20. Then, imposingx(?r/LJ) and substituting for V yields -x0
=
= -x0 and i(rr/a)
{-42F0 cos ($2) - (flF04l/w2) cos ($1) sin (#z) + (&lFo/w2) +{(wl/w2)4lF0
=0
sin ($2)) cos (4)
sin (111)sin ($2)+42(~/wZ)FO sin (#Z)-42FO) sin (4)
- (W~/WZ)XO sin ($1) sin (+2), 0 ={w242FO sin (42) - fl4lF0 cos
(+l)
COs
(42) +@lFO cos (42) - 4zaFo) cos (4)
+{wl4lFO sin (4’1)cos (42) +42flF0 cos ($2)) sin (#)-xOw1 sin (#I) cos ($2). One may rewrite the above equations in the equivalent form (-“0”) =( ::,“:=1::
::,“:+‘::::PGt$
-(;zJ
(where the parameters al, bl, etc., are defined in the obvious way) in terms of the partial magnification factors 41 and 42 defined above. By gathering terms in x0 to the left-hand side one obtains a linear equation for the vector (cos 4, sin (4)) which yields the phase angle 4 and the amplitude x0 through the identity cos’ 12% +sin’ 4 = 1. The resulting equation for x0 is of the form xo= OF,= {(Aq:+Bq;+
@4z)/(a4:+@4;+
?‘414~)“~}Fo.
The new parameters introduced in this equation are related to the previous ones but here interest is in the variations of x0 as a function of the parameters 41 and 42. Writing 41= (f2’- w:)-‘, 42 = (a’- wz)-’ one obtains the equation {A(f2Z-w;)2+B(R2-w:)2+C(f+w:)(i22-w~)}F,, xo= (nZ-
w:)(fP--
w;)Ja(&-
w:)“+p(e-
w:y+y(fP-
wf)(&-
w;j
It is clear that xo will become infinitely large for 0 = w1 or w2 and that for 0 = w. it will
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G. S.WHISTON
remain finite. The system therefore has resonances at the two damped natural frequencies present, occurring at w1 = (wi -2pN/Lm)“’ and w2 = (wi + 2pN/Lm)“2, and that the separation increases with damping until w1 = 0 when the continuous free slipping condition must be violated. The situation here differs both from that in the Coulomb friction case where the resonant frequency is the undamped natural frequency and that in the viscous damping case where the resonant frequency differs from both the damped and undamped natural frequencies. 6. CONCLUSIONS
A singular differential equation that could be used to model the dynamics of a beam supported by frictional clamps at either end has been investigated in both its autonomous and harmonically excited versions. The free vibration is either a semi-hyperbolic aperiodic decay to locking or an infinitely long decay to rest at the origin with a well-defined geometric amplitude decrement depending upon the clamp tightness. For sufficient clamping force locking can occur for any displacement in a long beam but sufficiently short beams show a locking behaviour similar to that of the pure Coulomb system. Similar calculations have been carried out for curved beams whose quantitative behaviour is somewhat different and these calculations may be discussed in a future paper. The chatter oscillations of the autonomous equations have also been analyzed. Solutions of the harmonically excited systems were investigated qualitatively for continuous slipping and it was shown that resonances occur at two damped frequencies associated with the system, the separation of the peaks being monotonically increasing with clamp force until the continuous slipping assumption is no longer valid. Of course, the two resonant frequencies may well be a function of the beam idealization rather than a real phenomenon. It might be interesting to check the occurrence of the twin peaks experimentally. REFERENCES 1. J. P. DEN HARTOG 1931 Transactions of the American Society of Mechanical Engineers 53, 107-l 13. Forced vibration with combined Coulomb and viscous friction. 2. K. MAGNLJS 1965 Vibrations.London: Blackie and Son limited. 3. M. S. HUNDAL 1979 Journal of Sound and Vibration 64,371-378. Response of a base excited
system with Coulomb and viscous friction.
4. I. FL~GGE-LOTZ 1968 Discontinuous and Optimal Control. New York: McGraw-Hill Book Company, Inc. 5. J. A. GLEN 1978 Bulletin of the Institute of Mathematics and its Applications, 151. The oscillations of a simple castor.