The problem of singularity in impacting systems

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1 The problem of singularity in impacting systems Soumitro Banerjee, Department of Physics Indian Institute of Science Education & Research, Kolkata, India The problem of singularity in impacting systems p. 1/5

2 Collaborators Soumya Kundu M.Tech student, IIT Kharagpur Yue Ma University of Tokushima, Japan Damian Giaouris Newcastle University, UK Marian Wiercigroch, Ekaterina Pavlovskaia, James Ing University of Aberdeen, U.K. The problem of singularity in impacting systems p. 2/5

3 Hard impact systems F ẋ ẋ Before impact After impact The problem of singularity in impacting systems p. 3/5

4 Soft impact systems k 1 M k F 0011 R R L 1 L 2 ẋ ẋ Before impact After impact The problem of singularity in impacting systems p. 4/5

5 Grazing zero velocity impact It has been found that the orbit abruptly loses stability at grazing. The state moves away from the periodic orbit and develops a much larger chaotic orbit. How to analyze the dynamics in such systems? The problem of singularity in impacting systems p. 5/5

6 Eperimental observations Note the narrow bands of chaos The problem of singularity in impacting systems p. 6/5

7 Poincaré maps for impacting systems F (0) (T) We have to obtain (T) = f((0)) n+1 = f( n ) A. B. Nordmark, Non-periodic motion caused by grazing incidence in an impact oscillator, Journal of Sound and Vibration, 145, , The problem of singularity in impacting systems p. 7/5

8 Description A general impacting system is given as d = F() if H() > 0 dt R() if H() = 0 with the switching manifold defined as Σ = {,H() = 0} The problem of singularity in impacting systems p. 8/5

9 Definitions The normal velocity v() is defined as the rate at which the trajectory approaches the impact boundary: v() := dh dt = H d dt = H F. The normal acceleration a() of the flow with respect to the boundary is The reset map is defined as a() := v()/ t = (H F) F. R() = + W()v(), where W is a smooth 2 1 matri. The problem of singularity in impacting systems p. 9/5

10 Grazing u The grazing point satisfies the conditions Σ * (0,0) v H( ) = 0 v( ) = 0 a( ) = a > 0 The scalar function H() is assumed to be well defined at =, i.e., H ( ) 0. The problem of singularity in impacting systems p. 10/5

11 Nordmark s approach T 0 τ 0 P 1 1 u * 2 3 P 2 Τ τ. u 0 Non impacting Grazing Impacting P s = P 2?? P 1 The problem of singularity in impacting systems p. 11/5

12 Zero-time discontinuity map 1 u * 3. u ZDM: 1 4. The problem of singularity in impacting systems p. 12/5

13 Nordmark s approach u T P 2 0 τ 0 P 1 1 * 2 3 Τ τ. u 0 Non impacting Grazing Impacting u * 3. u 5 P 1 = ϕ( 0,τ 0 ), P 2 = ϕ( 4,T τ 0 ), P s = P 2 ZDM P 1 The problem of singularity in impacting systems p. 13/5

14 Zero-time discontinuity map Nordmark showed that the form of the ZDM, ecluding higher order terms, is 4 = 1 W ( 2a )y where, W = W( ), and y = H min ( 1 ). 1 u 5 * 2 3 H min 1 ( ) v di Bernardo, Budd, Champneys and Kowalczyk, Piecewise-smooth Dynamical Systems: Theory and Applications, Springer, The problem of singularity in impacting systems p. 14/5

15 The Poincaré map In first order approimation, P 1 ZDM P 1 P 2 ZDM P 1 : N 1 : N 1 2a H min (N 1 )W : N 2 N 1 2a H min (N 1 )N 2 W where, N 1 := dp 1 d = 0, N 2 := dp 2 d =. Note that a square-root term H min remains in the epression for the function. This gives rise to square-root singularity. The problem of singularity in impacting systems p. 15/5

16 1D Map with sq-root singularity n+1 n n+1 = { an + µ if n 0 b n + µ if n 0 Square-root singularity. The problem of singularity in impacting systems p. 16/5

17 Constraints on the Jacobian A coordinate transformation yields J = ( tr(j) 1 det(j) 0 ) Transformed map close to a grazing orbit: [ n ] [ 1 ] [ n+1 ] J L y n + µ 0 for n 0 y n+1 = J R [ n y n ] + µ [ 1 0 ] for n 0 How do the trace and determinant change as a system is driven from non-impacting state to an impacting state? The problem of singularity in impacting systems p. 17/5

18 The soft impacting system k1 M 01 d k F 0011 R R L 1 L 2 The problem of singularity in impacting systems p. 18/5

19 The grazing orbit (2) (1) The problem of singularity in impacting systems p. 19/5

20 The Jacobian matri Using a suitable numerical technique, we can obtain the fied point, and the Jacobian: ( (T) v(t) ) = ( J11 J 12 J 21 J 22 ) ( (0) v(0) ) How do the parameters of the Jacobian matri change when a nonimpacting orbit changes to an impacting orbit? The problem of singularity in impacting systems p. 20/5

21 The Jacobian matri The eigenvalues of the fied point: at F m = (before grazing): ± i , at F m = (after grazing): , Abrupt change; no apparent pattern. The problem of singularity in impacting systems p. 21/5

22 The normal form We transform the Jacobian matri into the normal form [ n+1 v n+1 ] = [ τ 1 δ 0 ] [ n where τ is the trace and δ is the determinant of the Jacobian matri. v n ] The problem of singularity in impacting systems p. 22/5

23 The normal form For this system, [ ] [ ] Conclusion: The pattern of the jump of the eigenvalues becomes clear when epressed in the normal form. Determinant does not change instantaneously, but the trace does. Yue Ma, Manish Agarwal, and Soumitro Banerjee, Border Collision Bifurcations in a Soft Impact System, Physics Letters A, Vol. 354, No.4, 2006, pp The problem of singularity in impacting systems p. 23/5

24 Eperimental investigation Accelerometers Secondary Gap spring High/low pass filters Eddy current probe Parallel leaf springs Dynamic shaker Proimeter Charge amplifier Waveform generator Data acquisition The problem of singularity in impacting systems p. 24/5

25 Photograph of setup The problem of singularity in impacting systems p. 25/5

26 Eperimental technique Natural frequency 11Hz, damping 2.18 kg s 1, stiffness ratio of the secondary to primary springs: 13. The data was low pass filtered with a cutoff frequency of 100Hz before being amplified by the hardware. The data acquisition was performed using LABVIEW, with a scan rate of 1000 samples per second (approimately 150 per period of forcing). The data was smoothed with a Savitsky-Golay method, using a second order polynomial fit around 21 data points. Stroboscopic sampling was obtained by means of a linear interpolation between the closest data points. The problem of singularity in impacting systems p. 26/5

27 Obtaining the Jacobian Mathematically speaking, for the perturbation about a fied point of the stroboscopic map, v = P(v ), we can write, u n+1 = (v v ) n+1 = P where the derivative matri is the Jacobian. (v v ) n + O(u 2 n) v=v The eperimental procedure involved perturbing the state and observing its return to the periodic orbit. The problem of singularity in impacting systems p. 27/5

28 Observation of transients (a) 8 (b) v-v* 0-4 * * v-v* * * Before grazing After grazing The problem of singularity in impacting systems p. 28/5

29 Eperimental results (a) 1.0 (b) 2 Determinant Trace 1 0 (c) Amplitude (d) Amplitude Determinant Amplitude Trace Amplitude The problem of singularity in impacting systems p. 29/5

30 Conclusions from eperiment The determinant of the Jacobian remains unchanged by contact. In the case of the unprestressed spring the trace is clearly continuous across the grazing boundary. In the case of the prestressed spring, the trace, after grazing, is vastly changed and returns smoothly to a smaller value. This is consistent with a square-root singularity at the grazing point. Ing, Pavlovskaia, Wiercigroch, and Banerjee, Eperimental study of impact oscillator with one-sided elastic constraint, Philosophical Transactions of the Royal Society of London, Part A, Vol. 366, pp , The problem of singularity in impacting systems p. 30/5

31 Investigating the Jacobian u T P 2 0 τ 0 P 1 1 * 2 3 Τ τ. u 0 Non impacting Grazing Impacting u * 3. u 5 For a near-grazing impacting orbit, the Jacobian can be epressed as J = N 2 J ZDM N 1 The problem of singularity in impacting systems p. 31/5

32 Investigating the Jacobian Recall: The ZDM is given by 4 = 1 W ( 2a ) H min ( 1 ), a = normal acceleration calculated at the grazing point W = W matri of R() = + W()v(), calculated at H = H/, and H min = the lowest point the state would have reached if the switching manifold were not there. Jacobian of the ZDM: J ZDM = I + 2a W H 2 H min. The problem of singularity in impacting systems p. 32/5

33 Forms of W and H At u = σ the reset map R is applied. The equation of the discontinuity boundary Σ is H() = H(u, u) = σ u Thus we have H u = 0, and hence, H = (h 1 0). Since the position of the mass just before the impact, u, is same as that just after the impact, u +, using the equation R() = + W()v() we can write W = (0 w 2 ) T ; w 2 : constant of restitution The problem of singularity in impacting systems p. 33/5

34 Character of the ZDM J ZDM = I + 2a W H 2 H min ( ) ( ) 1 0 2a 0 ( = h 1 0 H min w 2 ( ) ( ) 1 0 2a 0 0 = H min w 2 h 1 0 ( ) 1 0 =, α 1 ) where α = w 2h 1 2a 2 H min The problem of singularity in impacting systems p. 34/5

35 The determinant This gives J ZDM = 1, and hence J = N 2 J ZDM N 1 = N 2 N 1 Therefore the determinant does not contain the square root singularity. It should remain invariant in the immediate neighborhood of the grazing orbit. Away from the grazing condition, the saltation matri: S = R + [F(R( )) R R( )] H ( )F( H ( ) ) ( ) r 0 = r (1+r)(ku mg(t)) mv S = r 2 The problem of singularity in impacting systems p. 35/5

36 The determinant Therefore, as the change of a parameter drives the system through the grazing condition into impacting mode, the determinant varies continuously from the value before impact to r 2 times that value. 0.4 Determinant of the Jacobian Forcing function amplitude The problem of singularity in impacting systems p. 36/5

37 The trace In each smooth segment the equation for the oscillator can be epressed as d 2 u dt 2 + 2ζω du n dt + ω n 2 u = g(t) Consider an orbit p(t) = (u(t),v(t)) T and a perturbed orbit (p(t) + δp(t)). Then δu satisfies δü + 2ζω n δ u + ω n 2 δu = 0. The evolution of the perturbation follows ( d δu(t) dt δv(t) ) = ( 0 1 ω n 2 2ζω n ) ( δu(t) δv(t) ) The problem of singularity in impacting systems p. 37/5

38 The solution The solution of the above problem can be epressed as ( δu(τ) δv(τ) ) = N τ ( δu0 δv 0 ) where N τ = e ζω nτ ( cos(ω0 τ) + ζ 1 ζ 2 sin(ω 0τ) sin(ω 0 τ)/ω 0 1 ω 1 ζ 2 0 sin(ω 0 τ) cos(ω 0 τ) ζ 1 ζ 2 sin(ω 0τ) ) with ω 0 = ω n 1 ζ 2. The problem of singularity in impacting systems p. 38/5

39 The solution T 0 τ 0 P 1 u P 2 Τ τ 0 Non impacting Grazing Impacting 1 * 2 3. u ( ) N 1 = N τ0 = e ζω nτ n11 0 n 12 n 13 n 14 ( N 2 = N (T τ0 ) = e ζω n(t τ 0 ) n21 n 22 n 23 n 24 ) The problem of singularity in impacting systems p. 39/5

40 The Jacobian J = e ζω nt ( n21 n 22 n 23 n 24 )( 1 0 α 1 )( n11 n 12 n 13 n 14 ) = e ζω nt ( n 21 n 11 + n 22 n 13 + αn 22 n 11 n 23 n 12 + n 24 n 14 + αn 24 n 12 ) Tr(J) = e ζω nt {n 21 n 11 + n 22 n 13 + n 23 n 12 + n 24 n 14 + α(n 22 n 11 + n 24 n 12 )} The problem of singularity in impacting systems p. 40/5

41 The trace Since n 11 = cos(ω 0 τ 0 ) + ζ 1 ζ 2 sin(ω 0 τ 0 ) n 12 = sin(ω 0τ 0 ) ω 0 n 22 = sin{ω 0(T τ 0 )} ω 0 n 24 = cos{ω 0 (T τ 0 )} ζ 1 ζ 2 sin{ω 0 (T τ 0 )}, it follows that n 22 n 11 + n 24 n 12 = sin(ω 0T) ω 0 0, ω 0 mω forcing 2 The problem of singularity in impacting systems p. 41/5

42 The trace Therefore, when ω 0 mω forcing 2, the coeffcient of α in the epression of the trace must be a non-zero entity. Hence a square-root singularity must occur in the trace of the Jacobian. Corollary: The singularity must vanish for ω forcing = 2ω 0 m This implies that if the natural frequency of the oscillator is an even multiple of the forcing frequency, the singularity will disappear, and there will be no stretching of the phase space in the neighborhood of a grazing orbit. The problem of singularity in impacting systems p. 42/5

43 The vanishing singularity Position of the mass Forcing function amplitude m = 2.97 m = 3 Position of the mass (in m) Position of the mass (in m) Forcing function amplitude (in N) Forcing function amplitude (in N) m = 4 m = 5 The problem of singularity in impacting systems p. 43/5

44 Variation of the trace Trace of the Jacobian Forcing function amplitude (in N) Trace of the Jacobian Forcing function amplitude (in N) m =2.97 m =3 The problem of singularity in impacting systems p. 44/5

45 The 3/2 singularity Dankowicz and Nordmark showed that in systems with dry friction, the local map has 3/2 singularity. n+1 = { an + µ if n 0 a n b 3/2 n + µ if n 0 continuous at the border n = 0, there is no discontinuity in the slope. The problem of singularity in impacting systems p. 45/5

46 Map with 3/2 singularity n+1 n Even though the function is represented piecewise, it is continuous and smooth. The problem of singularity in impacting systems p. 46/5

47 The general classes (T) (0) 3/2 singularity (T) (0) Square root singularity The problem of singularity in impacting systems p. 47/5

48 The soft impacting system Impacting systems with cushion on the wall can belong to both the classes. What should be the nature of the map? What is the dependence of the nature of the map of the properties of the cushion? Ma, Ing, Banerjee, Wiercigroch, and Pavlovskaia, The nature of the normal form map for soft impacting systems, International Journal on Nonlinear Mechanics, vol. 43, pp , The problem of singularity in impacting systems p. 48/5

49 System-A k 1 M 0011 k F 0011 R L 1 L 2 (T) (0) determinant trace F F The problem of singularity in impacting systems p. 49/5

50 System-B k 1 M k F 0011 R R L 1 L 2 (T) (0) determinant trace R 2 = 0 R 2 = 0.1 R 2 = R 2 = 0 R 2 = 0.1 R 2 = F F The problem of singularity in impacting systems p. 50/5

51 System-C k 1 M d 0011 k F 0011 R L 1 L 2 (T) (0) determinant In( trace) d = d = F F The problem of singularity in impacting systems p. 51/5

52 System-D k 1 M d k F 0011 R 0011 R L 1 determinant d = 0 d = d = d = 0.01 d = 0.1 L F trace trace d = 0 d = d = d = 0.01 d = d = 0 d = d = d = 0.01 d = F F The problem of singularity in impacting systems p. 52/5

53 Conclusions The singularity appears only in the trace of the Jacobian, while the determinant does not change immediately at grazing. The singularity vanishes if ω forcing = 2ω 0 /m. For a soft impact system, the trace and the determinant follow a family of functions depending on the type of cushion. Can we develop a bifurcation theory for such a family of functions? The problem of singularity in impacting systems p. 53/5

54 Based on the papers 1. Soumitro Banerjee, James Ing, Ekaterina Pavlovskaia, Marian Wiercigroch, and Ramesh K. Reddy, Invisible Grazings and Dangerous Bifurcations in Impacting Systems: the Problem of Narrow-band Chaos accepted for publication in Physical Review E. 2. Yue Ma, James Ing, Soumitro Banerjee, Ekaterina Pavlovskaia, Marian Wiercigroch The nature of the normal form map for soft impacting systems International Journal on Nonlinear Mechanics, vol. 43, pp , James Ing, Ekaterina Pavlovskaia, Marian Wiercigroch, and Soumitro Banerjee, Eperimental study of impact oscillator with one-sided elastic constraint, Philosophical Transactions of the Royal Society of London, Part A, Vol. 366, pp , Rangoli Sharan and Soumitro Banerjee, Character of the map for switched dynamical systems for observations on the switching manifold, Physics Letters A (2008), doi: /j.physleta Yue Ma, Manish Agarwal, and Soumitro Banerjee, Border Collision Bifurcations in a Soft Impact System, Physics Letters A, Vol. 354, No.4, 2006, pp The problem of singularity in impacting systems p. 54/5

55 Thank you The problem of singularity in impacting systems p. 55/5

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