- Email: [email protected]
On the Analysis of Chatter in Mechanical Systems with Impacts
Available online at www.sciencedirect.com
ScienceDirect Procedia IUTAM 20 (2017) 18 – 25
24th International Congress of Theoretical and Applied Mechanics
On the Analysis of Chatter in Mechanical Systems with Impacts Harry Dankowicz∗, Erika Fotsch Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign, Urbana, Illinois, USA
Abstract In rigid-body mechanics, models that capture collisional contact as an instantaneous exchange of momentum may exhibit dynamics that include infinite sequences of impacts accumulating in finite time to a state of persistent contact, often referred to as chatter. In this paper, we review theoretical tools for the analysis of transient and steady-state behavior in the vicinity of critical periodic orbits for which chatter terminates at a point corresponding to the imminent release from persistent contact, and illustrate the application of this theory to a simplified model of a mechanical pressure relief valve. A general theory for single-degree-of-freedom impact oscillators, previously described in an unpublished manuscript by Nordmark and Kisitu 1 , is shown to yield both qualitative and quantitative agreement with model simulation results. The predicted bifurcation structure shows that the border orbit unfolds supercritically into a universal cascade of local attractors with nontrivial scaling relationships. c 2017 2016The TheAuthors. Authors. Published by Elsevier © Published by Elsevier B.V.B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of organizing committee of the 24th International Congress of Theoretical and Applied Mechanics. Peer-review under responsibility of organizing committee of the 24th International Congress of Theoretical and Applied Mechanics Keywords: Accumulation of impacts; Bifurcation analysis; One-dimensional map; Pressure relief valve; Universal cascade
1. Introduction In models of mechanical systems that account for persistent or intermittent contact with hard and rough obstacles, time histories of position and velocity coordinates are naturally partitioned into disjoint segments, governed by segment-specific force interactions, and characterized by termination and state reset conditions, for example associated with the onset or cessation of contact, or with transitions between relative stick and slip. In models of mechanical systems with dissipative impacts, there often exist open sets of initial conditions that lead to a phase of persistent contact following an infinite sequence of instantaneous collisions in finite time. Such a behavior of complete chatter corresponds to a singular contraction of state-space volume, as the subsequent dynamics remain confined to a co-dimension-two submanifold until a subsequent termination condition enables a release from contact. In dynamical systems models characterized by a three-dimensional state, for example, harmonically excited single-degree-of-freedom mechanical oscillators 2 , the release point is unique and independent of initial conditions. In such a model, forward dynamics at the critical transition between initial conditions that result in complete chatter and those that are followed by only finitely many collisions in every open interval of time may be understood in terms of a local analysis on a neighborhood of the release point. ∗
Corresponding author. E-mail address: [email protected]
2210-9838 © 2017 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of organizing committee of the 24th International Congress of Theoretical and Applied Mechanics doi:10.1016/j.piutam.2017.03.004
Harry Dankowicz and Erika Fotsch / Procedia IUTAM 20 (2017) 18 – 25
Fig. 1. A schematic representation of a pressure relief valve connected to a hydraulic system with effective compliance β and input flow rate q. Here, the time history for the displacement y1 of the valve plug relative to the valve seat reflects intervals of time during which the valve is open (y1 > 0) and closed (y1 = 0), respectively.
As shown in unpublished work by Nordmark and Kisitu 1 , such a local analysis predicts universal bifurcation scenarios, including cascades of smooth saddle-node and period-doubling bifurcations, intervals of robust chaos, and grazing bifurcations, associated with the parameter unfolding near a critical periodic orbit with complete chatter terminating at the unique point of release. In this paper, with permission, we provide a detailed review of their analysis and illustrate the predictions on a simplified model of a mechanical pressure relief valve 3,4 . 2. A pressure relief valve The dynamics of a pressure relief valve, used to ensure safe operation of a hydraulic system, may be modeled in terms of a three-dimensional state vector y, whose components represent the displacement of the valve plug relative to the valve seat, the velocity of the valve plug relative to the valve seat, and the (positive) input pressure to the valve, respectively. We say that the valve is open in any open time interval during which the displacement is positive, and closed in any open time interval during which the displacement equals zero. Under all operating conditions, changes to the input pressure are a result of fluid flow into the hydraulic system and fluid flow escaping through the valve. Similarly, the displacement dynamics of the valve plug are driven by the input pressure and resisted by a precompressed spring-damper suspension and intermittent contact between the plug and the seat. 2.1. Mathematical model In a simplified and suitably normalized model 3 (cf. Fig. 1), as long as the valve is open (and, thus, y1 > 0), the behavior of the pressure relief valve may be captured by smooth state-space trajectories governed by the vector field √ , (1) fopen (y) = y2 , −κy2 − y1 − δ + y3 , β q − y1 y3 where κ > 0 represents viscous damping in the suspension, δ > 0 represents the precompression of the spring suspension when the valve is closed, q > 0 is the flow rate into the hydraulic system, and β > 0 is the ratio of the system compliance to that of the spring suspension. When the valve is closed, the dynamics reduce to smooth state-space trajectories on the embedded submanifold y1 = y2 = 0, governed by the vector field fclosed (y) = 0, 0, βq . (2) The valve opens when y3 increases beyond δ, after which y1 again becomes positive. Such a transition is instantaneous in time and continuous in state, amounting only to a switch in governing vector field. In contrast, a transition from an open valve to a closed valve must involve at least one collision between the valve plug and the valve seat, here modeled as an instantaneous change in the velocity y2 → −ry2 in terms of a Newtonian coefficient of restitution r ∈ [0, 1). Indeed, except for the case of a perfectly plastic collision with r = 0, an infinite sequence of collisions, accumulating after a finite but nonzero elapsed time, must precede the closure of the
19
20
Harry Dankowicz and Erika Fotsch / Procedia IUTAM 20 (2017) 18 – 25
valve. Following convention in the impact-mechanics literature 2 , we refer to such a sequence as a case of complete (forward) chatter. If the initial condition (0, 0, δ) at the opening of the valve results in dynamics that pass through complete chatter after which the valve again closes, then this behavior must be periodic in time, since the valve again opens at (0, 0, δ). Such dynamics are only observed for very small (and industrially irrelevant) numerical values of q. 2.2. Near-chatter dynamics Following the general analysis of near-chatter dynamics in Nordmark and Kisitu 1 , consider initial conditions (0, y2 , y3 ) with y2 and δ − y3 small and positive. In this case, the valve remains open for a brief time ⎛ ⎞ ⎟⎟ 3 ⎜⎜⎜⎜ 8 2 (3) ⎜(δ − y3 ) − (δ − y3 ) − βqy2 ⎟⎟⎟⎠ + O y2 , (δ − y3 )2 , 2βq ⎝ 3 after which a collision between the plug and the seat results in an instantaneous jump to a new state of the same form. In terms of the pair of variables (motivated by the results of Budd and Dux 2 and adapted from the general treatment in Nordmark and Kisitu 1 ) βqy2 u , v δ − y3 , (4) (δ − y3 )2 and neglecting terms of O y2 , (δ − y3 )2 , the map from one such initial state to the next then takes the form (u, v) → (g(u, r), h(u)v) , where g(u, r) r
3s − 1 3s(1 − s) , h(u) , s 2 (1 − 3s)2
8 1 − u. 3
(5)
Since g(0, r) = 0, gu (0, r) = r, and g(·, r) is convex on [0, 1/3), it follows that there exists a unique nonzero value u = u∗ (r) on this interval for which g(u, r) = u. The fixed point (u∗ (r), 0) is a saddle point with eigenvalues λ(r) h(u∗ (r)) and 1/η(r) gu (u∗ (r), r), such that 0 < λ2 (r) < η(r) < λ(r) < 1 for r ∈ (0, 1), with unstable manifold v = 0 and stable manifold u = u∗ (r). Orbits on the stable manifold correspond to complete chatter sequences that terminate at the point (0, 0, δ) without resulting in the closure of the valve. Orbits based at initial conditions with u < u∗ (r) must converge to the family of degenerate fixed points (0, v) with v < 0 along the corresponding eigenvectors (1 − r, 2v). Each such orbit corresponds to a complete chatter sequence that accumulates on the corresponding point (0, 0, y3 ) with y3 < δ. In contrast, initial conditions with u > u∗ (r) result in at most a finite sequence of collisions, after which the valve remains open for an O(1) interval of time. Following Nordmark and Kisitu 1 , consider such an incomplete chatter sequence of n collisions for n 1 and let u0 > u∗ (r) denote the value of u after the last collision and suppose that u1 = g−1 (u0 , r), where √ (6u + r)(8u + 3r) − (2u + r) 3r(8u + 3r) −1 (6) g (u, r) 16(3u + r)2 is the inverse of g on the interval u ∈ [0, 1/3). Since g−1 is concave with slope η(r) at u = u∗ (r), it follows that ui+1 = g−1 (ui , r), i ≥ 1 ⇒ 0 < ui+1 − u∗ (r) < ui − u∗ (r) and 0 <
ui+1 − u∗ (r) ui − u∗ (r) , < ηi (r) ηi+1 (r)
(7)
where u1 ∈ (g−1 (3/8, r), 3/8). We conclude that ui − u∗ (r) i→∞ ηi (r)
un − u∗ (r) ≈ Q(u1 , r)ηn (r), Q(u1 , r) lim and, consequently,
βqy2,n − u∗ (r) ≈ Q(u1 , r)ηn (r) (δ − y3,n )2
for large n. By construction, Q (u1 , r) is an increasing function on (g−1 (3/8, r), 3/8) and Q g−1 (u1 , r) , r = Q(u1 , r)η(r).
(8)
(9)
(10)
21
Harry Dankowicz and Erika Fotsch / Procedia IUTAM 20 (2017) 18 – 25
1 P-u1 , r( 2 g-u1 , r( )
0.02
0.02
0.00
0.00
-0.01
-0.01
0.01 (x)
0.01
-0.02
-0.02 0.00
0.052 0.054 0.056 0.058 0.060 0.062
0.01
0.02
0.03
Q(u1 , r)
0.04
0.05
0.06
x
(g−1 (3/8, r), 3/8)
Fig. 2. (Left panel) The implicit relationship between g(u1 , r) − and Q(u1 , r) for r = 0.8 and u1 ∈ follows from the monotonicity of Q(u1 , r) with respect to u1 . (Right panel) The graph of the function Υ, here obtained for r = 0.8, represents the unique relationship between g(u1 , r) − 12 P2 (u1 , r)λ2n (r) and Q(u1 , r)ηn (r) for u1 ∈ (g−1 (3/8, r), 3/8) and across all nonnegative integers n. 1 2
P2 (u1 , r)
Each value of βqy2 /(δ − y3 )2 after the first collision of the incomplete chatter sequence thus identifies a unique integer n and unique value u1 ∈ (g−1 (3/8, r), 3/8). Similarly, let vi correspond to the value of v after each of the collisions in the incomplete chatter sequence, and assume that v0 > 0. By a similar analysis it follows that v0 ≈ vn P (u1 , r) λn (r), P (u1 , r)
∞
h(ui ) i=1
since 0< By construction,
λ(r)
,
ln h(ui+1 )/λ(r) h(ui+1 ) − λ(r) ui+1 − u∗ (r) < < η(r) < 1. < h(ui ) − λ(r) ui − u∗ (r) ln h(ui )/λ(r) P g−1 (u1 , r) , r = P (u1 , r) λ(r).
(11)
(12)
(13)
The trajectory based at (0, y2,0 , y3,0 ) = (0, u0 v20 , δ−v0 ) terminates at a point (y†1 , y†2 , y†3 ), with y†3 −δ small and positive, after an elapsed time τ
y†3 − y3,0 y3,0 − δ 2 βq 3 † βq ⇒ y†1 = y2,0 τ + τ + τ , y2 = y2,0 + (y3,0 − δ)τ + τ2 , βq 2 6 2
(14)
from which it follows that ⎞ ⎛ † ⎟⎟ 1 ⎜⎜⎜⎜ (y3 − δ)2 1 2 2 2n ≈ 2 2 ⎜⎝ + g(u1 , r) − vn P (u1 , r)λ (r)⎟⎟⎟⎠ (y†3 − δ), 6 2 βq ⎛ † ⎞ 2 ⎟⎟ 1 2 2 1 ⎜⎜⎜⎜ (y3 − δ) † 2n y2 = + g(u1 , r) − vn P (u1 , r)λ (r)⎟⎟⎟⎠ , ⎜⎝ βq 2 2
y†1
(15) (16)
where the expansion for y†1 is correct to O(v30 ). Notably, for 3/8 − u1 small and positive, g(u1 , r) = −3rs + O(s2 ) implying an infinite slope with respect to u1 in this limit. and Kisitu 1 , for any integer n and fixed r, by monotonicity we obtain an implicit dependence Following Nordmark 1 2 of g(u1 , r) − 2 P (u1 , r)λ2n (r) on Q(u1 , r)ηn (r) with infinite slope as u1 approaches 3/8 from below, and such that incrementing n by 1 results in a scaling of the former by λ2 (r) and of the latter by η(r). We graph this dependence in the left panel of Fig. 2 for n = 0, and let it define a universal function Υ across all non-negative integers n whose graph is shown in the right panel of Fig. 2. In particular, we note that lim x→0 Υ(x) = 0.
22
Harry Dankowicz and Erika Fotsch / Procedia IUTAM 20 (2017) 18 – 25
By (9), to lowest order, we may now express the map from (y2 , y3 ), after the first collision in the incomplete chatter sequence, to (y†1 , y†2 ) as † 1 (y†3 − δ)(δ − y∗3 )2 βq y2 − y∗2 δ − y∗3 y1 − y†∗ 1 , (17) ,B= 2 2 = BΥ C ,C= y3 − y∗3 βq(δ − y∗3 )2 y†2 − y†∗ βq (δ − y∗3 )3 2y∗2 2 †∗ ∗ ∗ where y†∗ 1 and y2 are the values obtained for initial conditions (y2 , y3 ) = (y2 , y3 ) that correspond to the limit n → ∞.
2.3. Bifurcation analysis Following Nordmark and Kisitu 1 , we refer to a periodic orbit that includes a complete chatter sequence terminating at (0, 0, δ) as a border orbit, as it separates trajectories with a complete chatter sequence that results in the closure of the valve from those with a incomplete chatter sequence without subsequent closure of the valve. When a border orbit exists, the map derived in the previous section may be composed with a linearization of the †∗ † † ∗ ∗ flow map, on a neighborhood of (y†∗ 1 , y2 , y3 ) on the section y3 = y3 to a neighborhood of (0, y2 , y3 ) on the section y1 = 0 corresponding to the state immediately following the first subsequent collision. This linearization may be expressed in terms of a constant matrix A ∈ R2×2 : † y2 − y∗2 y1 − y†∗ 1 =A † , (18) y3 − y∗3 y2 − y†∗ 2 which, when composed with (17), yields the first return map y2 − y∗2 y2 − y∗2 = AB Υ C . y3 − y∗3 n+1 y3 − y∗3 n
(19)
As noted by Nordmark and Kisitu 1 , it follows that the fixed point at (y2 , y3 ) = (y∗2 , y∗3 ) must be locally asymptotically stable, since 2 Υ(x)/|x|log λ (r)/ log η(r) = O(1) (20) as x → 0 and log λ2 (r)/ log η(r) > 1.2 for r ∈ [0, 1). We proceed to consider the case in which variables with a superscripted ∗ correspond to a particular value q = q∗ of the flow rate for which a border orbit exist in the pressure valve dynamics. In this case, equating the order of magnitudes of y − y∗ , q − q∗ and Υ, † y1 − y†∗ y2 − y∗2 ∗ 1 (21) = BΥ C + D (q − q ) + E (q − q∗ ), y3 − y∗3 y†2 − y†∗ 2 where B=
† (y†3 − δ)2 −2(y† − δ) βy∗2 1 βq∗ δ − y∗3 (y3 − δ)(δ − y∗3 )2 3 , D = , E = , C = . βq∗ (δ − y∗3 )2 −3βq∗ β2 q∗2 (δ − y∗3 )3 2y∗2 (δ − y∗3 )2 6β2 q∗3
Moreover, linearization along the border orbit yields † y2 − y∗2 y1 − y†∗ 1 = A + F (q − q∗ ), y3 − y∗3 y†2 − y†∗ 2 from which it follows that y2 − y∗2 y2 − y∗2 ∗ = AB Υ C + D (q − q ) + (AE + F) (q − q∗ ). y3 − y∗3 n+1 y3 − y∗3 n The dynamics for (y2 , y3 ) ≈ (y∗2 , y∗3 ) may thus be described by the one-dimensional map zn+1 = Υ CAB zn + (CAE + CF + D) (q − q∗ ) ,
(22)
(23)
(24)
(25)
23
Harry Dankowicz and Erika Fotsch / Procedia IUTAM 20 (2017) 18 – 25
Fig. 3. Brute-force bifurcation diagrams for the discrete dynamical sytem (25) with CAB = 1 and μ = (CAE + CF + D) (q − q∗ ). Decreasing values of μ correspond to increasing values of the number n of collisions in the incomplete chatter sequence. Increasing values of n correspond to decreasing magnitudes of the derivative of Υ (away from the u1 → 3/8 limit) and, consequently, changes to the steady-state dynamics compared to regions with smaller values of n. For each value of μ, the diagram shows 500 iterates of z following an initial transient of 500 iterates. Each diagram was obtained by increasing the value of μ from 0 and using the final iterate for the previous value of μ as the initial condition after incrementing μ.
where substitution of the scalar
back in (24) yields
y2 − y∗2 = CAB z + (CAE + CF) (q − q∗ ) C y3 − y∗3
y2 − y∗2 y3 − y∗3
= AB zn+1 + (AE + F) (q − q∗ ),
(26)
(27)
n+1
from which discrete time histories for y2 and y3 may be reconstructed. Brute-force bifurcation diagrams for the discrete dynamical system in (25) with CAB = 1 are shown in Fig. 3 over several ranges of μ = (CAE + CF + D) (q − q∗ ) > 0. An enlargement from the upper left panel of Fig. 3 is shown in Fig. 4. 3. Numerical results We apply the bifurcation analysis in the previous section to the pressure-relief valve, for which a periodic orbit with complete chatter, terminating at (0, 0, δ) without a closing of the valve, may be found for β = 20, κ = 1.25, r = 0.8, δ = 10 and q = q∗ ≈ qc 0.020056. In this case, y∗2 ≈ 0.118256 and y∗3 ≈ 9.45248 corresponding to the velocity of the valve plug and the input pressure after the first collision following the completion of the chatter sequence. †∗ With y†3 = 10.1, we now find y†∗ 1 ≈ 0.001080 and y2 ≈ 0.011966, from which it follows that 0.186318 1.33805 −0.103297 B≈ ,C≈ , D ≈ 7.88954, E ≈ . (28) 0.747358 0.577996 −0.621515
24
Harry Dankowicz and Erika Fotsch / Procedia IUTAM 20 (2017) 18 – 25
Fig. 4. This enlargement of a region of the bifurcation diagram shown in the upper left panel of Fig. 3 provides a more detailed view of the bifurcation structure, including a period-doubling sequence to chaos and periodic windows. The catastrophic disappearance of a chaotic attractor toward the left of the diagram corresponds to a grazing bifurcation of an unstable periodic orbit (not shown) which connects to the stable periodic orbit at a saddle-node bifurcation (not shown).
Fig. 5. Bifurcation cascade of the steady-state response of the mechanical pressure relief valve (see Fotsch 4 ) near a critical parameter value q = q∗ ≈ qc 0.020056 for which a border orbit exists in the forward dynamics. Here, y2 denotes the valve plug velocity sampled at an intersection with the hyperplane y3 = 9.7 following the completion of each incomplete chatter sequence, and q is the input flow rate. An ad-hoc affine transformation is applied to the vertical axis to obtain a diagram in which the vertical scaling of the bifurcation structure is more readily apparent. In steady-state, the number of collisions in each incomplete chatter sequence increases as q decreases toward q∗ and equals 36 along the periodic orbits found near q = 0.02019.
The linearization about the corresponding trajectory then yields −11.2769 −0.667163 3.96147 A≈ ,F≈ 60.6417 3.55289 −16.6998
(29)
and, consequently, CAB ≈ 4.58679 and CAE + CF + D ≈ 0.754296. A brute-force bifurcation diagram of steady-state attractors sampled prior to the first collision at the intersection with a Poincar´e section at y3 = 9.7 is shown in Fig. 5. An enlargement for a smaller range of values of q is shown in Fig. 6. Both diagrams show more than a cursory resemblance to the bifurcation structure in the upper left panel of Fig. 3 and the enlargement in Fig. 4 (cf. similar diagrams in Alzate, Piiroinen, and di Bernardo 5 in a model of a cam-follower mechanism). 4. Concluding remarks In hybrid dynamical systems, e.g., mechanical systems with impacts and friction, degenerate interactions between system time histories and surfaces of discontinuity in the governing dynamics give rise to a unique set of phenomena that call for advances in theoretical and computational tools (e.g., di Bernardo et al. 6 and Nordmark and Piiroinen 7 ).
Harry Dankowicz and Erika Fotsch / Procedia IUTAM 20 (2017) 18 – 25
Fig. 6. An enlargement of the bifurcation structure in Fig. 5 shows strong resemblance to the universal predictions of the discrete dynamical system (25) in Fig. 4, with differences in orientation, affine dependence on q − q∗ , and number of collisions per incomplete chatter sequence captured by the numerical values of the coefficients CAB and CAE + CF + D, and the matrices GAB and G(AE + F) for an appropriately constructed matrix G corresponding to the reconstruction in (27) projected onto the y3 = 9.7 hyperplane.
For the phenomenology associated with forward-time chatter, a relatively mature theory provides insight and forms the foundation for theoretical bifurcation analysis and computational tools for simulation and parameter continuation (cf. Dankowicz and Schilder 8 but see also the bounce toolbox in Fotsch 4 ). In a class of rigid-contact models that account for frictional interactions during impact, the analysis in Nordmark, Dankowicz, and Champneys 9 shows the possibility also for reverse chatter, corresponding to a point of accumulation of infinitely many collisions in backward time. Such a degeneracy results in a paradoxical ambiguity in the forward dynamics, with infinitely many coexisting viable time histories. Rather than an opportunity for bifurcation analysis, as applied in this paper near a forward-time chatter orbit, the phenomenon of reverse chatter opens a line of inquiry into instabilities associated with multiple competing time scales that still awaits resolution. Acknowledgements This material is based upon work supported by the National Science Foundation under Grant No. 0855787. The authors gratefully acknowledge Alan Champneys for his hospitality and feedback on the pressure relief valve dynamics. The first author also wishes to acknowledge close to two decades of collaboration with Arne Nordmark. References 1. Nordmark, AB, Kisitu, RM. On Chattering Bifurcations in 1 DOF Impact Oscillator Models. Unpublished preprint, KTH Royal Institute of Technology, November 7, 2003. 2. Budd, C, Dux, F. Chattering and Related Behaviour in Impact Oscillators. Phil Trans R Soc Lond A 1994, 347:365–389. 3. H˝os, C, Champneys, AR. Grazing Bifurcations and Chatter in a Pressure Relief Valve Model. Physica D 2012, 241(22):2068–2076. 4. Fotsch, EL. Bifurcation Analysis Near the Cessation of Complete Chatter and Shilnikov Homoclinic Trajectories in a Pressure Relief Valve Model. M.Sc. thesis, University of Illinois at Urbana-Champaign, 2016. 5. Alzate, R, Piiroinen, PT, di Bernardo, M. From Complete to Incomplete Chattering: A Novel Route to Chaos in Impacting Cam-Follower Systems. Int J Bifur Chaos 2012, 22(5):1250102. 6. di Bernardo, M, Budd, C, Champneys, AR, Kowalczyk, P. Piecewise-smooth Dynamical Systems. Springer-Verlag London, 2008. 7. Nordmark, AB, Piiroinen PT. Simulation and Stability Analysis of Impacting Systems with Complete Chattering. Nonlinear Dyn 2009, 58:85– 106. 8. Dankowicz, H, Schilder, F. Recipes for Continuation. SIAM, 2013. 9. Nordmark, A, Dankowicz, H, Champneys, A. Friction-Induced Reverse Chatter in Rigid-Body Mechanisms with Impacts. IMA J Appl Math 2011, 76(1):85–119.
25
ScienceDirect Procedia IUTAM 20 (2017) 18 – 25
24th International Congress of Theoretical and Applied Mechanics
On the Analysis of Chatter in Mechanical Systems with Impacts Harry Dankowicz∗, Erika Fotsch Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign, Urbana, Illinois, USA
Abstract In rigid-body mechanics, models that capture collisional contact as an instantaneous exchange of momentum may exhibit dynamics that include infinite sequences of impacts accumulating in finite time to a state of persistent contact, often referred to as chatter. In this paper, we review theoretical tools for the analysis of transient and steady-state behavior in the vicinity of critical periodic orbits for which chatter terminates at a point corresponding to the imminent release from persistent contact, and illustrate the application of this theory to a simplified model of a mechanical pressure relief valve. A general theory for single-degree-of-freedom impact oscillators, previously described in an unpublished manuscript by Nordmark and Kisitu 1 , is shown to yield both qualitative and quantitative agreement with model simulation results. The predicted bifurcation structure shows that the border orbit unfolds supercritically into a universal cascade of local attractors with nontrivial scaling relationships. c 2017 2016The TheAuthors. Authors. Published by Elsevier © Published by Elsevier B.V.B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of organizing committee of the 24th International Congress of Theoretical and Applied Mechanics. Peer-review under responsibility of organizing committee of the 24th International Congress of Theoretical and Applied Mechanics Keywords: Accumulation of impacts; Bifurcation analysis; One-dimensional map; Pressure relief valve; Universal cascade
1. Introduction In models of mechanical systems that account for persistent or intermittent contact with hard and rough obstacles, time histories of position and velocity coordinates are naturally partitioned into disjoint segments, governed by segment-specific force interactions, and characterized by termination and state reset conditions, for example associated with the onset or cessation of contact, or with transitions between relative stick and slip. In models of mechanical systems with dissipative impacts, there often exist open sets of initial conditions that lead to a phase of persistent contact following an infinite sequence of instantaneous collisions in finite time. Such a behavior of complete chatter corresponds to a singular contraction of state-space volume, as the subsequent dynamics remain confined to a co-dimension-two submanifold until a subsequent termination condition enables a release from contact. In dynamical systems models characterized by a three-dimensional state, for example, harmonically excited single-degree-of-freedom mechanical oscillators 2 , the release point is unique and independent of initial conditions. In such a model, forward dynamics at the critical transition between initial conditions that result in complete chatter and those that are followed by only finitely many collisions in every open interval of time may be understood in terms of a local analysis on a neighborhood of the release point. ∗
Corresponding author. E-mail address: [email protected]
2210-9838 © 2017 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of organizing committee of the 24th International Congress of Theoretical and Applied Mechanics doi:10.1016/j.piutam.2017.03.004
Harry Dankowicz and Erika Fotsch / Procedia IUTAM 20 (2017) 18 – 25
Fig. 1. A schematic representation of a pressure relief valve connected to a hydraulic system with effective compliance β and input flow rate q. Here, the time history for the displacement y1 of the valve plug relative to the valve seat reflects intervals of time during which the valve is open (y1 > 0) and closed (y1 = 0), respectively.
As shown in unpublished work by Nordmark and Kisitu 1 , such a local analysis predicts universal bifurcation scenarios, including cascades of smooth saddle-node and period-doubling bifurcations, intervals of robust chaos, and grazing bifurcations, associated with the parameter unfolding near a critical periodic orbit with complete chatter terminating at the unique point of release. In this paper, with permission, we provide a detailed review of their analysis and illustrate the predictions on a simplified model of a mechanical pressure relief valve 3,4 . 2. A pressure relief valve The dynamics of a pressure relief valve, used to ensure safe operation of a hydraulic system, may be modeled in terms of a three-dimensional state vector y, whose components represent the displacement of the valve plug relative to the valve seat, the velocity of the valve plug relative to the valve seat, and the (positive) input pressure to the valve, respectively. We say that the valve is open in any open time interval during which the displacement is positive, and closed in any open time interval during which the displacement equals zero. Under all operating conditions, changes to the input pressure are a result of fluid flow into the hydraulic system and fluid flow escaping through the valve. Similarly, the displacement dynamics of the valve plug are driven by the input pressure and resisted by a precompressed spring-damper suspension and intermittent contact between the plug and the seat. 2.1. Mathematical model In a simplified and suitably normalized model 3 (cf. Fig. 1), as long as the valve is open (and, thus, y1 > 0), the behavior of the pressure relief valve may be captured by smooth state-space trajectories governed by the vector field √ , (1) fopen (y) = y2 , −κy2 − y1 − δ + y3 , β q − y1 y3 where κ > 0 represents viscous damping in the suspension, δ > 0 represents the precompression of the spring suspension when the valve is closed, q > 0 is the flow rate into the hydraulic system, and β > 0 is the ratio of the system compliance to that of the spring suspension. When the valve is closed, the dynamics reduce to smooth state-space trajectories on the embedded submanifold y1 = y2 = 0, governed by the vector field fclosed (y) = 0, 0, βq . (2) The valve opens when y3 increases beyond δ, after which y1 again becomes positive. Such a transition is instantaneous in time and continuous in state, amounting only to a switch in governing vector field. In contrast, a transition from an open valve to a closed valve must involve at least one collision between the valve plug and the valve seat, here modeled as an instantaneous change in the velocity y2 → −ry2 in terms of a Newtonian coefficient of restitution r ∈ [0, 1). Indeed, except for the case of a perfectly plastic collision with r = 0, an infinite sequence of collisions, accumulating after a finite but nonzero elapsed time, must precede the closure of the
19
20
Harry Dankowicz and Erika Fotsch / Procedia IUTAM 20 (2017) 18 – 25
valve. Following convention in the impact-mechanics literature 2 , we refer to such a sequence as a case of complete (forward) chatter. If the initial condition (0, 0, δ) at the opening of the valve results in dynamics that pass through complete chatter after which the valve again closes, then this behavior must be periodic in time, since the valve again opens at (0, 0, δ). Such dynamics are only observed for very small (and industrially irrelevant) numerical values of q. 2.2. Near-chatter dynamics Following the general analysis of near-chatter dynamics in Nordmark and Kisitu 1 , consider initial conditions (0, y2 , y3 ) with y2 and δ − y3 small and positive. In this case, the valve remains open for a brief time ⎛ ⎞ ⎟⎟ 3 ⎜⎜⎜⎜ 8 2 (3) ⎜(δ − y3 ) − (δ − y3 ) − βqy2 ⎟⎟⎟⎠ + O y2 , (δ − y3 )2 , 2βq ⎝ 3 after which a collision between the plug and the seat results in an instantaneous jump to a new state of the same form. In terms of the pair of variables (motivated by the results of Budd and Dux 2 and adapted from the general treatment in Nordmark and Kisitu 1 ) βqy2 u , v δ − y3 , (4) (δ − y3 )2 and neglecting terms of O y2 , (δ − y3 )2 , the map from one such initial state to the next then takes the form (u, v) → (g(u, r), h(u)v) , where g(u, r) r
3s − 1 3s(1 − s) , h(u) , s 2 (1 − 3s)2
8 1 − u. 3
(5)
Since g(0, r) = 0, gu (0, r) = r, and g(·, r) is convex on [0, 1/3), it follows that there exists a unique nonzero value u = u∗ (r) on this interval for which g(u, r) = u. The fixed point (u∗ (r), 0) is a saddle point with eigenvalues λ(r) h(u∗ (r)) and 1/η(r) gu (u∗ (r), r), such that 0 < λ2 (r) < η(r) < λ(r) < 1 for r ∈ (0, 1), with unstable manifold v = 0 and stable manifold u = u∗ (r). Orbits on the stable manifold correspond to complete chatter sequences that terminate at the point (0, 0, δ) without resulting in the closure of the valve. Orbits based at initial conditions with u < u∗ (r) must converge to the family of degenerate fixed points (0, v) with v < 0 along the corresponding eigenvectors (1 − r, 2v). Each such orbit corresponds to a complete chatter sequence that accumulates on the corresponding point (0, 0, y3 ) with y3 < δ. In contrast, initial conditions with u > u∗ (r) result in at most a finite sequence of collisions, after which the valve remains open for an O(1) interval of time. Following Nordmark and Kisitu 1 , consider such an incomplete chatter sequence of n collisions for n 1 and let u0 > u∗ (r) denote the value of u after the last collision and suppose that u1 = g−1 (u0 , r), where √ (6u + r)(8u + 3r) − (2u + r) 3r(8u + 3r) −1 (6) g (u, r) 16(3u + r)2 is the inverse of g on the interval u ∈ [0, 1/3). Since g−1 is concave with slope η(r) at u = u∗ (r), it follows that ui+1 = g−1 (ui , r), i ≥ 1 ⇒ 0 < ui+1 − u∗ (r) < ui − u∗ (r) and 0 <
ui+1 − u∗ (r) ui − u∗ (r) , < ηi (r) ηi+1 (r)
(7)
where u1 ∈ (g−1 (3/8, r), 3/8). We conclude that ui − u∗ (r) i→∞ ηi (r)
un − u∗ (r) ≈ Q(u1 , r)ηn (r), Q(u1 , r) lim and, consequently,
βqy2,n − u∗ (r) ≈ Q(u1 , r)ηn (r) (δ − y3,n )2
for large n. By construction, Q (u1 , r) is an increasing function on (g−1 (3/8, r), 3/8) and Q g−1 (u1 , r) , r = Q(u1 , r)η(r).
(8)
(9)
(10)
21
Harry Dankowicz and Erika Fotsch / Procedia IUTAM 20 (2017) 18 – 25
1 P-u1 , r( 2 g-u1 , r( )
0.02
0.02
0.00
0.00
-0.01
-0.01
0.01 (x)
0.01
-0.02
-0.02 0.00
0.052 0.054 0.056 0.058 0.060 0.062
0.01
0.02
0.03
Q(u1 , r)
0.04
0.05
0.06
x
(g−1 (3/8, r), 3/8)
Fig. 2. (Left panel) The implicit relationship between g(u1 , r) − and Q(u1 , r) for r = 0.8 and u1 ∈ follows from the monotonicity of Q(u1 , r) with respect to u1 . (Right panel) The graph of the function Υ, here obtained for r = 0.8, represents the unique relationship between g(u1 , r) − 12 P2 (u1 , r)λ2n (r) and Q(u1 , r)ηn (r) for u1 ∈ (g−1 (3/8, r), 3/8) and across all nonnegative integers n. 1 2
P2 (u1 , r)
Each value of βqy2 /(δ − y3 )2 after the first collision of the incomplete chatter sequence thus identifies a unique integer n and unique value u1 ∈ (g−1 (3/8, r), 3/8). Similarly, let vi correspond to the value of v after each of the collisions in the incomplete chatter sequence, and assume that v0 > 0. By a similar analysis it follows that v0 ≈ vn P (u1 , r) λn (r), P (u1 , r)
∞
h(ui ) i=1
since 0< By construction,
λ(r)
,
ln h(ui+1 )/λ(r) h(ui+1 ) − λ(r) ui+1 − u∗ (r) < < η(r) < 1. < h(ui ) − λ(r) ui − u∗ (r) ln h(ui )/λ(r) P g−1 (u1 , r) , r = P (u1 , r) λ(r).
(11)
(12)
(13)
The trajectory based at (0, y2,0 , y3,0 ) = (0, u0 v20 , δ−v0 ) terminates at a point (y†1 , y†2 , y†3 ), with y†3 −δ small and positive, after an elapsed time τ
y†3 − y3,0 y3,0 − δ 2 βq 3 † βq ⇒ y†1 = y2,0 τ + τ + τ , y2 = y2,0 + (y3,0 − δ)τ + τ2 , βq 2 6 2
(14)
from which it follows that ⎞ ⎛ † ⎟⎟ 1 ⎜⎜⎜⎜ (y3 − δ)2 1 2 2 2n ≈ 2 2 ⎜⎝ + g(u1 , r) − vn P (u1 , r)λ (r)⎟⎟⎟⎠ (y†3 − δ), 6 2 βq ⎛ † ⎞ 2 ⎟⎟ 1 2 2 1 ⎜⎜⎜⎜ (y3 − δ) † 2n y2 = + g(u1 , r) − vn P (u1 , r)λ (r)⎟⎟⎟⎠ , ⎜⎝ βq 2 2
y†1
(15) (16)
where the expansion for y†1 is correct to O(v30 ). Notably, for 3/8 − u1 small and positive, g(u1 , r) = −3rs + O(s2 ) implying an infinite slope with respect to u1 in this limit. and Kisitu 1 , for any integer n and fixed r, by monotonicity we obtain an implicit dependence Following Nordmark 1 2 of g(u1 , r) − 2 P (u1 , r)λ2n (r) on Q(u1 , r)ηn (r) with infinite slope as u1 approaches 3/8 from below, and such that incrementing n by 1 results in a scaling of the former by λ2 (r) and of the latter by η(r). We graph this dependence in the left panel of Fig. 2 for n = 0, and let it define a universal function Υ across all non-negative integers n whose graph is shown in the right panel of Fig. 2. In particular, we note that lim x→0 Υ(x) = 0.
22
Harry Dankowicz and Erika Fotsch / Procedia IUTAM 20 (2017) 18 – 25
By (9), to lowest order, we may now express the map from (y2 , y3 ), after the first collision in the incomplete chatter sequence, to (y†1 , y†2 ) as † 1 (y†3 − δ)(δ − y∗3 )2 βq y2 − y∗2 δ − y∗3 y1 − y†∗ 1 , (17) ,B= 2 2 = BΥ C ,C= y3 − y∗3 βq(δ − y∗3 )2 y†2 − y†∗ βq (δ − y∗3 )3 2y∗2 2 †∗ ∗ ∗ where y†∗ 1 and y2 are the values obtained for initial conditions (y2 , y3 ) = (y2 , y3 ) that correspond to the limit n → ∞.
2.3. Bifurcation analysis Following Nordmark and Kisitu 1 , we refer to a periodic orbit that includes a complete chatter sequence terminating at (0, 0, δ) as a border orbit, as it separates trajectories with a complete chatter sequence that results in the closure of the valve from those with a incomplete chatter sequence without subsequent closure of the valve. When a border orbit exists, the map derived in the previous section may be composed with a linearization of the †∗ † † ∗ ∗ flow map, on a neighborhood of (y†∗ 1 , y2 , y3 ) on the section y3 = y3 to a neighborhood of (0, y2 , y3 ) on the section y1 = 0 corresponding to the state immediately following the first subsequent collision. This linearization may be expressed in terms of a constant matrix A ∈ R2×2 : † y2 − y∗2 y1 − y†∗ 1 =A † , (18) y3 − y∗3 y2 − y†∗ 2 which, when composed with (17), yields the first return map y2 − y∗2 y2 − y∗2 = AB Υ C . y3 − y∗3 n+1 y3 − y∗3 n
(19)
As noted by Nordmark and Kisitu 1 , it follows that the fixed point at (y2 , y3 ) = (y∗2 , y∗3 ) must be locally asymptotically stable, since 2 Υ(x)/|x|log λ (r)/ log η(r) = O(1) (20) as x → 0 and log λ2 (r)/ log η(r) > 1.2 for r ∈ [0, 1). We proceed to consider the case in which variables with a superscripted ∗ correspond to a particular value q = q∗ of the flow rate for which a border orbit exist in the pressure valve dynamics. In this case, equating the order of magnitudes of y − y∗ , q − q∗ and Υ, † y1 − y†∗ y2 − y∗2 ∗ 1 (21) = BΥ C + D (q − q ) + E (q − q∗ ), y3 − y∗3 y†2 − y†∗ 2 where B=
† (y†3 − δ)2 −2(y† − δ) βy∗2 1 βq∗ δ − y∗3 (y3 − δ)(δ − y∗3 )2 3 , D = , E = , C = . βq∗ (δ − y∗3 )2 −3βq∗ β2 q∗2 (δ − y∗3 )3 2y∗2 (δ − y∗3 )2 6β2 q∗3
Moreover, linearization along the border orbit yields † y2 − y∗2 y1 − y†∗ 1 = A + F (q − q∗ ), y3 − y∗3 y†2 − y†∗ 2 from which it follows that y2 − y∗2 y2 − y∗2 ∗ = AB Υ C + D (q − q ) + (AE + F) (q − q∗ ). y3 − y∗3 n+1 y3 − y∗3 n The dynamics for (y2 , y3 ) ≈ (y∗2 , y∗3 ) may thus be described by the one-dimensional map zn+1 = Υ CAB zn + (CAE + CF + D) (q − q∗ ) ,
(22)
(23)
(24)
(25)
23
Harry Dankowicz and Erika Fotsch / Procedia IUTAM 20 (2017) 18 – 25
Fig. 3. Brute-force bifurcation diagrams for the discrete dynamical sytem (25) with CAB = 1 and μ = (CAE + CF + D) (q − q∗ ). Decreasing values of μ correspond to increasing values of the number n of collisions in the incomplete chatter sequence. Increasing values of n correspond to decreasing magnitudes of the derivative of Υ (away from the u1 → 3/8 limit) and, consequently, changes to the steady-state dynamics compared to regions with smaller values of n. For each value of μ, the diagram shows 500 iterates of z following an initial transient of 500 iterates. Each diagram was obtained by increasing the value of μ from 0 and using the final iterate for the previous value of μ as the initial condition after incrementing μ.
where substitution of the scalar
back in (24) yields
y2 − y∗2 = CAB z + (CAE + CF) (q − q∗ ) C y3 − y∗3
y2 − y∗2 y3 − y∗3
= AB zn+1 + (AE + F) (q − q∗ ),
(26)
(27)
n+1
from which discrete time histories for y2 and y3 may be reconstructed. Brute-force bifurcation diagrams for the discrete dynamical system in (25) with CAB = 1 are shown in Fig. 3 over several ranges of μ = (CAE + CF + D) (q − q∗ ) > 0. An enlargement from the upper left panel of Fig. 3 is shown in Fig. 4. 3. Numerical results We apply the bifurcation analysis in the previous section to the pressure-relief valve, for which a periodic orbit with complete chatter, terminating at (0, 0, δ) without a closing of the valve, may be found for β = 20, κ = 1.25, r = 0.8, δ = 10 and q = q∗ ≈ qc 0.020056. In this case, y∗2 ≈ 0.118256 and y∗3 ≈ 9.45248 corresponding to the velocity of the valve plug and the input pressure after the first collision following the completion of the chatter sequence. †∗ With y†3 = 10.1, we now find y†∗ 1 ≈ 0.001080 and y2 ≈ 0.011966, from which it follows that 0.186318 1.33805 −0.103297 B≈ ,C≈ , D ≈ 7.88954, E ≈ . (28) 0.747358 0.577996 −0.621515
24
Harry Dankowicz and Erika Fotsch / Procedia IUTAM 20 (2017) 18 – 25
Fig. 4. This enlargement of a region of the bifurcation diagram shown in the upper left panel of Fig. 3 provides a more detailed view of the bifurcation structure, including a period-doubling sequence to chaos and periodic windows. The catastrophic disappearance of a chaotic attractor toward the left of the diagram corresponds to a grazing bifurcation of an unstable periodic orbit (not shown) which connects to the stable periodic orbit at a saddle-node bifurcation (not shown).
Fig. 5. Bifurcation cascade of the steady-state response of the mechanical pressure relief valve (see Fotsch 4 ) near a critical parameter value q = q∗ ≈ qc 0.020056 for which a border orbit exists in the forward dynamics. Here, y2 denotes the valve plug velocity sampled at an intersection with the hyperplane y3 = 9.7 following the completion of each incomplete chatter sequence, and q is the input flow rate. An ad-hoc affine transformation is applied to the vertical axis to obtain a diagram in which the vertical scaling of the bifurcation structure is more readily apparent. In steady-state, the number of collisions in each incomplete chatter sequence increases as q decreases toward q∗ and equals 36 along the periodic orbits found near q = 0.02019.
The linearization about the corresponding trajectory then yields −11.2769 −0.667163 3.96147 A≈ ,F≈ 60.6417 3.55289 −16.6998
(29)
and, consequently, CAB ≈ 4.58679 and CAE + CF + D ≈ 0.754296. A brute-force bifurcation diagram of steady-state attractors sampled prior to the first collision at the intersection with a Poincar´e section at y3 = 9.7 is shown in Fig. 5. An enlargement for a smaller range of values of q is shown in Fig. 6. Both diagrams show more than a cursory resemblance to the bifurcation structure in the upper left panel of Fig. 3 and the enlargement in Fig. 4 (cf. similar diagrams in Alzate, Piiroinen, and di Bernardo 5 in a model of a cam-follower mechanism). 4. Concluding remarks In hybrid dynamical systems, e.g., mechanical systems with impacts and friction, degenerate interactions between system time histories and surfaces of discontinuity in the governing dynamics give rise to a unique set of phenomena that call for advances in theoretical and computational tools (e.g., di Bernardo et al. 6 and Nordmark and Piiroinen 7 ).
Harry Dankowicz and Erika Fotsch / Procedia IUTAM 20 (2017) 18 – 25
Fig. 6. An enlargement of the bifurcation structure in Fig. 5 shows strong resemblance to the universal predictions of the discrete dynamical system (25) in Fig. 4, with differences in orientation, affine dependence on q − q∗ , and number of collisions per incomplete chatter sequence captured by the numerical values of the coefficients CAB and CAE + CF + D, and the matrices GAB and G(AE + F) for an appropriately constructed matrix G corresponding to the reconstruction in (27) projected onto the y3 = 9.7 hyperplane.
For the phenomenology associated with forward-time chatter, a relatively mature theory provides insight and forms the foundation for theoretical bifurcation analysis and computational tools for simulation and parameter continuation (cf. Dankowicz and Schilder 8 but see also the bounce toolbox in Fotsch 4 ). In a class of rigid-contact models that account for frictional interactions during impact, the analysis in Nordmark, Dankowicz, and Champneys 9 shows the possibility also for reverse chatter, corresponding to a point of accumulation of infinitely many collisions in backward time. Such a degeneracy results in a paradoxical ambiguity in the forward dynamics, with infinitely many coexisting viable time histories. Rather than an opportunity for bifurcation analysis, as applied in this paper near a forward-time chatter orbit, the phenomenon of reverse chatter opens a line of inquiry into instabilities associated with multiple competing time scales that still awaits resolution. Acknowledgements This material is based upon work supported by the National Science Foundation under Grant No. 0855787. The authors gratefully acknowledge Alan Champneys for his hospitality and feedback on the pressure relief valve dynamics. The first author also wishes to acknowledge close to two decades of collaboration with Arne Nordmark. References 1. Nordmark, AB, Kisitu, RM. On Chattering Bifurcations in 1 DOF Impact Oscillator Models. Unpublished preprint, KTH Royal Institute of Technology, November 7, 2003. 2. Budd, C, Dux, F. Chattering and Related Behaviour in Impact Oscillators. Phil Trans R Soc Lond A 1994, 347:365–389. 3. H˝os, C, Champneys, AR. Grazing Bifurcations and Chatter in a Pressure Relief Valve Model. Physica D 2012, 241(22):2068–2076. 4. Fotsch, EL. Bifurcation Analysis Near the Cessation of Complete Chatter and Shilnikov Homoclinic Trajectories in a Pressure Relief Valve Model. M.Sc. thesis, University of Illinois at Urbana-Champaign, 2016. 5. Alzate, R, Piiroinen, PT, di Bernardo, M. From Complete to Incomplete Chattering: A Novel Route to Chaos in Impacting Cam-Follower Systems. Int J Bifur Chaos 2012, 22(5):1250102. 6. di Bernardo, M, Budd, C, Champneys, AR, Kowalczyk, P. Piecewise-smooth Dynamical Systems. Springer-Verlag London, 2008. 7. Nordmark, AB, Piiroinen PT. Simulation and Stability Analysis of Impacting Systems with Complete Chattering. Nonlinear Dyn 2009, 58:85– 106. 8. Dankowicz, H, Schilder, F. Recipes for Continuation. SIAM, 2013. 9. Nordmark, A, Dankowicz, H, Champneys, A. Friction-Induced Reverse Chatter in Rigid-Body Mechanisms with Impacts. IMA J Appl Math 2011, 76(1):85–119.
25