The problem of singularity in impacting systems
|
|
- Annis Cobb
- 5 years ago
- Views:
Transcription
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
Strange dynamics of bilinear oscillator close to grazing
Strange dynamics of bilinear oscillator close to grazing Ekaterina Pavlovskaia, James Ing, Soumitro Banerjee and Marian Wiercigroch Centre for Applied Dynamics Research, School of Engineering, King s College,
More informationDerivation of border-collision maps from limit cycle bifurcations
Derivation of border-collision maps from limit cycle bifurcations Alan Champneys Department of Engineering Mathematics, University of Bristol Mario di Bernardo, Chris Budd, Piotr Kowalczyk Gabor Licsko,...
More informationThe SD oscillator and its attractors
The SD oscillator and its attractors Qingjie Cao, Marian Wiercigroch, Ekaterina Pavlovskaia Celso Grebogi and J Michael T Thompson Centre for Applied Dynamics Research, Department of Engineering, University
More informationFast-Slow Scale Bifurcation in Higher Order Open Loop Current-Mode Controlled DC-DC Converters
Fast-Slow Scale Bifurcation in Higher Order Open oop urrent-mode ontrolled D-D onverters I. Daho*, D. Giaouris*, S. Banerjee**, B. Zahawi*, and V. Pickert* * School of Electrical, Electronic and omputer
More informationarxiv: v1 [nlin.cd] 7 Jun 2010
arxiv:006.236v [nlin.cd] 7 Jun 200 Simple models of bouncing ball dynamics and their comparison Andrzej Okninski, Bogus law Radziszewski 2 Physics Division, Department of Mechatronics and Mechanical Engineering
More informationModelling Research Group
Modelling Research Group The Importance of Noise in Dynamical Systems E. Staunton, P.T. Piiroinen November 20, 2015 Eoghan Staunton Modelling Research Group 2015 1 / 12 Introduction Historically mathematicians
More informationDynamical Analysis of a Hydraulic Pressure Relief Valve
Proceedings of the World Congress on Engineering 9 Vol II WCE 9, July -, 9, London, U.K. Dynamical Analysis of a Hydraulic Pressure Relief Valve Gábor Licskó Alan Champneys Csaba Hős Abstract A mathematical
More informationLyapunov exponent calculation of a two-degreeof-freedom vibro-impact system with symmetrical rigid stops
Chin. Phys. B Vol. 20 No. 4 (2011) 040505 Lyapunov exponent calculation of a two-degreeof-freedom vibro-impact system with symmetrical rigid stops Li Qun-Hong( ) and Tan Jie-Yan( ) College of Mathematics
More informationStability analysis of sliding-grazing phenomenon in dry-friction oscillator using TS fuzzy approach
Stability analysis of sliding-grazing phenomenon in dry-friction oscillator using TS fuzzy approach Kamyar Mehran School of Engineering University of Warwick Coventry, CV4 7LW, UK Email: k.mehran@warwick.ac.uk
More informationThis article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination.
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS 1 Complex Interaction Between Tori and Onset of Three-Frequency Quasi-Periodicity in a Current Mode Controlled Boost Converter Damian Giaouris,
More informationSzalai, R., & Osinga, H. M. (2007). Unstable manifolds of a limit cycle near grazing.
Szalai, R., & Osinga, H. M. (2007). Unstable manifolds of a limit cycle near grazing. Early version, also known as pre-print Link to publication record in Explore Bristol Research PDF-document University
More informationC.-H. Lamarque. University of Lyon/ENTPE/LGCB & LTDS UMR CNRS 5513
Nonlinear Dynamics of Smooth and Non-Smooth Systems with Application to Passive Controls 3rd Sperlonga Summer School on Mechanics and Engineering Sciences on Dynamics, Stability and Control of Flexible
More informationCommun Nonlinear Sci Numer Simulat
Commun Nonlinear Sci Numer Simulat 15 (21) 1358 1367 Contents lists available at ScienceDirect Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns Hard versus soft impacts
More informationTHE NONLINEAR behavior of the buck converter is a
ontrol of switching circuits using complete-cycle solution matrices Damian Giaouris, Member, IEEE, bdulmajed Elbkosh, Soumitro Banerjee, Senior Member, IEEE, Bashar Zahawi Senior Member, IEEE, and Volker
More informationControlling Chaos in a State-Dependent Nonlinear System
Electronic version of an article published as International Journal of Bifurcation and Chaos Vol. 12, No. 5, 2002, 1111-1119, DOI: 10.1142/S0218127402004942 World Scientific Publishing Company, https://www.worldscientific.com/worldscinet/ijbc
More informationCo-dimension-two Grazing Bifurcations in Single-degree-of-freedom Impact Oscillators
Co-dimension-two razing Bifurcations in Single-degree-of-freedom Impact Oscillators Phanikrishna Thota a Xiaopeng Zhao b and Harry Dankowicz c a Department of Engineering Science and Mechanics MC 219 Virginia
More informationSWITCHED dynamical systems are useful in a variety of engineering
1184 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS, VOL. 51, NO. 6, JUNE 2004 Bifurcation Analysis of Switched Dynamical Systems With Periodically Moving Borders Yue Ma, Hiroshi Kawakami,
More informationBifurcations in Nonsmooth Dynamical Systems
SIAM REVIEW Vol. 5, No. 4, pp. 629 7 c 28 Society for Industrial and Applied Mathematics Bifurcations in Nonsmooth Dynamical Systems Mario di Bernardo Chris J. Budd Alan R. Champneys Piotr Kowalczyk Arne
More informationVibrations: Second Order Systems with One Degree of Freedom, Free Response
Single Degree of Freedom System 1.003J/1.053J Dynamics and Control I, Spring 007 Professor Thomas Peacock 5//007 Lecture 0 Vibrations: Second Order Systems with One Degree of Freedom, Free Response Single
More informationModeling and Stability Analysis of Closed Loop Current-Mode Controlled Ćuk Converter using Takagi-Sugeno Fuzzy Approach
Modeling and Stability Analysis of Closed Loop Current-Mode Controlled Ću Converter using Taagi-Sugeno Fuzzy Approach Kamyar Mehran Damian Giaouris Bashar Zahawi School of Electrical, Electronic and Computer
More informationDynamics of Rotor Systems with Clearance and Weak Pedestals in Full Contact
Paper ID No: 23 Dynamics of Rotor Systems with Clearance and Weak Pedestals in Full Contact Dr. Magnus Karlberg 1, Dr. Martin Karlsson 2, Prof. Lennart Karlsson 3 and Ass. Prof. Mats Näsström 4 1 Department
More informationChaotic behavior analysis based on corner bifurcations
Author manuscript, published in "Nonlinear Analysis: Hybrid Systems 3 (29) 543-55" Chaotic behavior analysis based on corner bifurcations D. Benmerzouk a,, J.P. Barbot b, a University of Tlemcen,Department
More informationCommunications in Nonlinear Science and Numerical Simulation
Accepted Manuscript Periodic motions generated from non-autonomous grazing dynamics M.U. Akhmet, A. Kıvılcım PII: S17-574(17)342-4 DOI: 1.116/j.cnsns.217.2.2 Reference: CNSNS 419 To appear in: Communications
More informationGeneral Physics I. Lecture 12: Applications of Oscillatory Motion. Prof. WAN, Xin ( 万歆 )
General Physics I Lecture 1: Applications of Oscillatory Motion Prof. WAN, Xin ( 万歆 ) inwan@zju.edu.cn http://zimp.zju.edu.cn/~inwan/ Outline The pendulum Comparing simple harmonic motion and uniform circular
More informationLaboratory handout 5 Mode shapes and resonance
laboratory handouts, me 34 82 Laboratory handout 5 Mode shapes and resonance In this handout, material and assignments marked as optional can be skipped when preparing for the lab, but may provide a useful
More informationMath 221 Topics since the second exam
Laplace Transforms. Math 1 Topics since the second exam There is a whole different set of techniques for solving n-th order linear equations, which are based on the Laplace transform of a function. For
More informationLaboratory notes. Torsional Vibration Absorber
Titurus, Marsico & Wagg Torsional Vibration Absorber UoB/1-11, v1. Laboratory notes Torsional Vibration Absorber Contents 1 Objectives... Apparatus... 3 Theory... 3 3.1 Background information... 3 3. Undamped
More informationLaboratory Investigation of the Dynamics of the Inelastic Bouncing Ball
Laboratory Investigation of the Dynamics of the Inelastic Bouncing Ball Jacob Yunis Aerospace Systems Design Lab, Guggenheim School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, Georgia
More informationDynamics of Non-Smooth Systems
Dynamics of Non-Smooth Systems P Chandramouli Indian Institute of Technology Madras October 5, 2013 Mouli, IIT Madras Non-Smooth System Dynamics 1/39 Introduction Filippov Systems Behaviour around hyper-surface
More informationEffect of various periodic forces on Duffing oscillator
PRAMANA c Indian Academy of Sciences Vol. 67, No. 2 journal of August 2006 physics pp. 351 356 Effect of various periodic forces on Duffing oscillator V RAVICHANDRAN 1, V CHINNATHAMBI 1, and S RAJASEKAR
More informationChaos and R-L diode Circuit
Chaos and R-L diode Circuit Rabia Aslam Chaudary Roll no: 2012-10-0011 LUMS School of Science and Engineering Thursday, December 20, 2010 1 Abstract In this experiment, we will use an R-L diode circuit
More informationDynamics of structures
Dynamics of structures 1.2 Viscous damping Luc St-Pierre October 30, 2017 1 / 22 Summary so far We analysed the spring-mass system and found that its motion is governed by: mẍ(t) + kx(t) = 0 k y m x x
More informationLab 11 - Free, Damped, and Forced Oscillations
Lab 11 Free, Damped, and Forced Oscillations L11-1 Name Date Partners Lab 11 - Free, Damped, and Forced Oscillations OBJECTIVES To understand the free oscillations of a mass and spring. To understand how
More informationForced Mechanical Vibrations
Forced Mechanical Vibrations Today we use methods for solving nonhomogeneous second order linear differential equations to study the behavior of mechanical systems.. Forcing: Transient and Steady State
More informationNonlinear Oscillations and Chaos
CHAPTER 4 Nonlinear Oscillations and Chaos 4-. l l = l + d s d d l l = l + d m θ m (a) (b) (c) The unetended length of each spring is, as shown in (a). In order to attach the mass m, each spring must be
More informationAN ELECTRIC circuit containing a switch controlled by
878 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS II: ANALOG AND DIGITAL SIGNAL PROCESSING, VOL. 46, NO. 7, JULY 1999 Bifurcation of Switched Nonlinear Dynamical Systems Takuji Kousaka, Member, IEEE, Tetsushi
More informationAPPPHYS217 Tuesday 25 May 2010
APPPHYS7 Tuesday 5 May Our aim today is to take a brief tour of some topics in nonlinear dynamics. Some good references include: [Perko] Lawrence Perko Differential Equations and Dynamical Systems (Springer-Verlag
More informationDynamics of Structures: Theory and Analysis
1. Free vibrations 2. Forced vibrations 3. Transient response 4. Damping mechanisms Dynamics of Structures: Theory and Analysis Steen Krenk Technical University of Denmark 5. Modal analysis I: Basic idea
More informationModeling and Stability Analysis of Closed Loop Current-Mode Controlled Ćuk Converter using Takagi-Sugeno Fuzzy Approach
1 Modeling and Stability Analysis of Closed Loop Current-Mode Controlled Ću Converter using Taagi-Sugeno Fuzzy Approach Kamyar Mehran, Member IEEE, Damian Giaouris, Member IEEE, and Bashar Zahawi, Senior
More informationLecture 3 : Bifurcation Analysis
Lecture 3 : Bifurcation Analysis D. Sumpter & S.C. Nicolis October - December 2008 D. Sumpter & S.C. Nicolis General settings 4 basic bifurcations (as long as there is only one unstable mode!) steady state
More informationA Guide to linear dynamic analysis with Damping
A Guide to linear dynamic analysis with Damping This guide starts from the applications of linear dynamic response and its role in FEA simulation. Fundamental concepts and principles will be introduced
More informationPersonalised Learning Checklists AQA Physics Paper 2
4.5.1 Forces and their interactions 4.5.2 Work done and energy AQA Physics (8463) from 2016 Topics P4.5. Forces Topic Student Checklist R A G Identify and describe scalar quantities and vector quantities
More informationMeasurement Techniques for Engineers. Motion and Vibration Measurement
Measurement Techniques for Engineers Motion and Vibration Measurement Introduction Quantities that may need to be measured are velocity, acceleration and vibration amplitude Quantities useful in predicting
More informationLinear and Nonlinear Oscillators (Lecture 2)
Linear and Nonlinear Oscillators (Lecture 2) January 25, 2016 7/441 Lecture outline A simple model of a linear oscillator lies in the foundation of many physical phenomena in accelerator dynamics. A typical
More informationDynamical behaviour of a controlled vibro-impact system
Vol 17 No 7, July 2008 c 2008 Chin. Phys. Soc. 1674-1056/2008/17(07)/2446-05 Chinese Physics B and IOP Publishing Ltd Dynamical behaviour of a controlled vibro-impact system Wang Liang( ), Xu Wei( ), and
More informationChaotifying 2-D piecewise linear maps via a piecewise linear controller function
Chaotifying 2-D piecewise linear maps via a piecewise linear controller function Zeraoulia Elhadj 1,J.C.Sprott 2 1 Department of Mathematics, University of Tébéssa, (12000), Algeria. E-mail: zeraoulia@mail.univ-tebessa.dz
More informationON THE ROBUSTNESS OF PERIODIC SOLUTIONS IN RELAY FEEDBACK SYSTEMS. Mario di Bernardo Λ Karl Henrik Johansson ΛΛ;1 Ulf Jönsson ΛΛΛ Francesco Vasca Λ
Copyright IFAC 1th Triennial World Congress, Barcelona, Spain ON THE ROBUSTNESS OF PERIODIC SOLUTIONS IN RELAY FEEDBACK SYSTEMS Mario di Bernardo Λ Karl Henrik Johansson ΛΛ;1 Ulf Jönsson ΛΛΛ Francesco
More informationThe Completely Inelastic Bouncing Ball
The Completely Inelastic Bouncing Ball November 2011 N. Arora, P. Gray, C. Rodesney, J. Yunis Completely Inelastic Bouncing Ball? Can there be an inelastic bouncing ball? The ball is constrained to move
More informationDriven Harmonic Oscillator
Driven Harmonic Oscillator Physics 6B Lab Experiment 1 APPARATUS Computer and interface Mechanical vibrator and spring holder Stands, etc. to hold vibrator Motion sensor C-209 spring Weight holder and
More informationLimitations of Bifurcation Diagrams in Boost Converter Steady-State Response Identification
Limitations of Bifurcation Diagrams in Boost Converter Steady-State Response Identification Željko Stojanović Zagreb University of Applied Sciences Department of Electrical Engineering Konavoska 2, 10000
More informationConference Paper Controlling Nonlinear Behavior in Current Mode Controlled Boost Converter Based on the Monodromy Matrix
Conference Papers in Engineering Volume 23, Article ID 8342, 7 pages http://dx.doi.org/./23/8342 Conference Paper Controlling Nonlinear Behavior in Current Mode Controlled Boost Converter Based on the
More informationAbstract: Complex responses observed in an experimental, nonlinear, moored structural
AN INDEPENDENT-FLOW-FIELD MODEL FOR A SDOF NONLINEAR STRUCTURAL SYSTEM, PART II: ANALYSIS OF COMPLEX RESPONSES Huan Lin e-mail: linh@engr.orst.edu Solomon C.S. Yim e-mail: solomon.yim@oregonstate.edu Ocean
More information2.034: Nonlinear Dynamics and Waves. Term Project: Nonlinear dynamics of piece-wise linear oscillators Mostafa Momen
2.034: Nonlinear Dynamics and Waves Term Project: Nonlinear dynamics of piece-wise linear oscillators Mostafa Momen May 2015 Massachusetts Institute of Technology 1 Nonlinear dynamics of piece-wise linear
More informationChapter 12. Recall that when a spring is stretched a distance x, it will pull back with a force given by: F = -kx
Chapter 1 Lecture Notes Chapter 1 Oscillatory Motion Recall that when a spring is stretched a distance x, it will pull back with a force given by: F = -kx When the mass is released, the spring will pull
More informationComplicated behavior of dynamical systems. Mathematical methods and computer experiments.
Complicated behavior of dynamical systems. Mathematical methods and computer experiments. Kuznetsov N.V. 1, Leonov G.A. 1, and Seledzhi S.M. 1 St.Petersburg State University Universitetsky pr. 28 198504
More informationSystem Parameter Identification for Uncertain Two Degree of Freedom Vibration System
System Parameter Identification for Uncertain Two Degree of Freedom Vibration System Hojong Lee and Yong Suk Kang Department of Mechanical Engineering, Virginia Tech 318 Randolph Hall, Blacksburg, VA,
More informationModeling and Experimentation: Mass-Spring-Damper System Dynamics
Modeling and Experimentation: Mass-Spring-Damper System Dynamics Prof. R.G. Longoria Department of Mechanical Engineering The University of Texas at Austin July 20, 2014 Overview 1 This lab is meant to
More informationPHYSICS 1 Simple Harmonic Motion
Advanced Placement PHYSICS 1 Simple Harmonic Motion Student 014-015 What I Absolutely Have to Know to Survive the AP* Exam Whenever the acceleration of an object is proportional to its displacement and
More informationThe reduction to invariant cones for nonsmooth systems. Tassilo Küpper Mathematical Institute University of Cologne
The reduction to invariant cones for nonsmooth systems Tassilo Küpper Mathematical Institute University of Cologne kuepper@math.uni-koeln.de International Workshop on Resonance Oscillations and Stability
More informationIntegrodifferential Hyperbolic Equations and its Application for 2-D Rotational Fluid Flows
Integrodifferential Hyperbolic Equations and its Application for 2-D Rotational Fluid Flows Alexander Chesnokov Lavrentyev Institute of Hydrodynamics Novosibirsk, Russia chesnokov@hydro.nsc.ru July 14,
More informationSubharmonic Oscillations and Chaos in Dynamic Atomic Force Microscopy
Subharmonic Oscillations and Chaos in Dynamic Atomic Force Microscopy John H. CANTRELL 1, Sean A. CANTRELL 2 1 NASA Langley Research Center, Hampton, Virginia 23681, USA 2 NLS Analytics, LLC, Glencoe,
More informationModeling the Duffing Equation with an Analog Computer
Modeling the Duffing Equation with an Analog Computer Matt Schmitthenner Physics Department, The College of Wooster, Wooster, Ohio 44691, USA (Dated: December 13, 2011) The goal was to model the Duffing
More informationNew approach to study the van der Pol equation for large damping
Electronic Journal of Qualitative Theor of Differential Equations 2018, No. 8, 1 10; https://doi.org/10.1422/ejqtde.2018.1.8 www.math.u-szeged.hu/ejqtde/ New approach to stud the van der Pol equation for
More informationChaotic Vibration and Design Criteria for Machine Systems with Clearance Connections
Proceeding of the Ninth World Congress of the heory of Machines and Mechanism, Sept. 1-3, 1995, Milan, Italy. Chaotic Vibration and Design Criteria for Machine Systems with Clearance Connections Pengyun
More informationSeminar 6: COUPLED HARMONIC OSCILLATORS
Seminar 6: COUPLED HARMONIC OSCILLATORS 1. Lagrangian Equations of Motion Let consider a system consisting of two harmonic oscillators that are coupled together. As a model, we will use two particles attached
More informationTHE GEOMETRY OF GENERIC SLIDING BIFURCATIONS
THE GEOMETRY OF GENERIC SLIDING BIFURCATIONS M. R. JEFFREY AND S. J. HOGAN Abstract. Using the singularity theory of scalar functions, we derive a classification of sliding bifurcations in piecewise-smooth
More informationNonsmooth systems: synchronization, sliding and other open problems
John Hogan Bristol Centre for Applied Nonlinear Mathematics, University of Bristol, England Nonsmooth systems: synchronization, sliding and other open problems 2 Nonsmooth Systems 3 What is a nonsmooth
More informationA Study of the Van der Pol Equation
A Study of the Van der Pol Equation Kai Zhe Tan, s1465711 September 16, 2016 Abstract The Van der Pol equation is famous for modelling biological systems as well as being a good model to study its multiple
More informationDYNAMIC ANALYSIS OF CANTILEVER BEAM
International Journal of Mechanical Engineering and Technology (IJMET) Volume 8, Issue 5, May 2017, pp. 1167 1173, Article ID: IJMET_08_05_122 Available online at http://www.iaeme.com/ijmet/issues.asp?jtype=ijmet&vtype=8&itype=5
More informationTime scales. Notes. Mixed Implicit/Explicit. Newmark Methods [ ] [ ] "!t = O 1 %
Notes Time scales! For using Pixie (the renderer) make sure you type use pixie first! Assignment 1 questions?! [work out]! For position dependence, characteristic time interval is "!t = O 1 % $ ' # K &!
More informationMechanics Departmental Exam Last updated November 2013
Mechanics Departmental Eam Last updated November 213 1. Two satellites are moving about each other in circular orbits under the influence of their mutual gravitational attractions. The satellites have
More informationChapter 14 Periodic Motion
Chapter 14 Periodic Motion 1 Describing Oscillation First, we want to describe the kinematical and dynamical quantities associated with Simple Harmonic Motion (SHM), for example, x, v x, a x, and F x.
More informationMathematical Modeling and response analysis of mechanical systems are the subjects of this chapter.
Chapter 3 Mechanical Systems A. Bazoune 3.1 INRODUCION Mathematical Modeling and response analysis of mechanical systems are the subjects of this chapter. 3. MECHANICAL ELEMENS Any mechanical system consists
More informationCh 6.4: Differential Equations with Discontinuous Forcing Functions
Ch 6.4: Differential Equations with Discontinuous Forcing Functions! In this section focus on examples of nonhomogeneous initial value problems in which the forcing function is discontinuous. Example 1:
More informationAutoparametric Resonance of Relaxation Oscillations
CHAPTER 4 Autoparametric Resonance of Relaation Oscillations A joint work with Ferdinand Verhulst. Has been submitted to journal. 4.. Introduction Autoparametric resonance plays an important part in nonlinear
More informationThis work is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 4.0 License.
University of Rhode Island DigitalCommons@URI Classical Dynamics Physics Course Materials 015 14. Oscillations Gerhard Müller University of Rhode Island, gmuller@uri.edu Creative Commons License This wor
More informationProblem 1: Lagrangians and Conserved Quantities. Consider the following action for a particle of mass m moving in one dimension
105A Practice Final Solutions March 13, 01 William Kelly Problem 1: Lagrangians and Conserved Quantities Consider the following action for a particle of mass m moving in one dimension S = dtl = mc dt 1
More informationON THE ARROW OF TIME. Y. Charles Li. Hong Yang
DISCRETE AND CONTINUOUS doi:10.3934/dcdss.2014.7.1287 DYNAMICAL SYSTEMS SERIES S Volume 7, Number 6, December 2014 pp. 1287 1303 ON THE ARROW OF TIME Y. Charles Li Department of Mathematics University
More informationMathematical Model of Forced Van Der Pol s Equation
Mathematical Model of Forced Van Der Pol s Equation TO Tsz Lok Wallace LEE Tsz Hong Homer December 9, Abstract This work is going to analyze the Forced Van Der Pol s Equation which is used to analyze the
More informationFirst-Order Low-Pass Filter
Filters, Cost Functions, and Controller Structures Robert Stengel Optimal Control and Estimation MAE 546 Princeton University, 218! Dynamic systems as low-pass filters! Frequency response of dynamic systems!
More informationDynamics of Structures
Dynamics of Structures Elements of structural dynamics Roberto Tomasi 11.05.2017 Roberto Tomasi Dynamics of Structures 11.05.2017 1 / 22 Overview 1 SDOF system SDOF system Equation of motion Response spectrum
More informationInfluence of friction coefficient on rubbing behavior of oil bearing rotor system
Influence of friction coefficient on rubbing behavior of oil bearing rotor system Changliang Tang 1, Jinfu ang 2, Dongjiang Han 3, Huan Lei 4, Long Hao 5, Tianyu Zhang 6 1, 2, 3, 4, 5 Institute of Engineering
More information1 Trajectory Generation
CS 685 notes, J. Košecká 1 Trajectory Generation The material for these notes has been adopted from: John J. Craig: Robotics: Mechanics and Control. This example assumes that we have a starting position
More informationChapter 23. Predicting Chaos The Shift Map and Symbolic Dynamics
Chapter 23 Predicting Chaos We have discussed methods for diagnosing chaos, but what about predicting the existence of chaos in a dynamical system. This is a much harder problem, and it seems that the
More informationHamiltonian Dynamics In The Theory of Abstraction
Hamiltonian Dynamics In The Theory of Abstraction Subhajit Ganguly. email: gangulysubhajit@indiatimes.com Abstract: This paper deals with fluid flow dynamics which may be Hamiltonian in nature and yet
More informationRandom Eigenvalue Problems in Structural Dynamics: An Experimental Investigation
Random Eigenvalue Problems in Structural Dynamics: An Experimental Investigation S. Adhikari, A. Srikantha Phani and D. A. Pape School of Engineering, Swansea University, Swansea, UK Email: S.Adhikari@swansea.ac.uk
More informationChapter 1. Introduction
Chapter 1 Introduction 1.1 What is Phase-Locked Loop? The phase-locked loop (PLL) is an electronic system which has numerous important applications. It consists of three elements forming a feedback loop:
More informationControl of a Chaotic Double Pendulum Using the OGY Method. Abstract
Control of a Chaotic Double Pendulum Using the OGY Method Matthew Cammack School of Physics Georgia Institute of Technology, Atlanta, GA 3033-0430, U.S.A (Dated: December 11, 003) Abstract This paper discusses
More informationNonlinear Dynamic Systems Homework 1
Nonlinear Dynamic Systems Homework 1 1. A particle of mass m is constrained to travel along the path shown in Figure 1, which is described by the following function yx = 5x + 1x 4, 1 where x is defined
More informationThe dynamics of a nonautonomous oscillator with friction memory
The dynamics of a nonautonomous oscillator with friction memory L. A. Igumnov 1, V. S. Metrikin 2, M. V. Zaytzev 3 1, 2 Research Institute for mechanics, National Research Lobachevsky State University
More informationNonlinear BEC Dynamics by Harmonic Modulation of s-wave Scattering Length
Nonlinear BEC Dynamics by Harmonic Modulation of s-wave Scattering Length I. Vidanović, A. Balaž, H. Al-Jibbouri 2, A. Pelster 3 Scientific Computing Laboratory, Institute of Physics Belgrade, Serbia 2
More informationAn Analysis of Nondifferentiable Models of and DPCM Systems From the Perspective of Noninvertible Map Theory
An Analysis of Nondifferentiable Models of and DPCM Systems From the Perspective of Noninvertible Map Theory INA TARALOVA-ROUX AND ORLA FEELY Department of Electronic and Electrical Engineering University
More informationThe Stick-Slip Vibration and Bifurcation of a Vibro-Impact System with Dry Friction
Send Orders for Reprints to reprints@benthamscience.ae 308 The Open Mechanical Engineering Journal, 2014, 8, 308-313 Open Access The Stick-Slip Vibration and Bifurcation of a Vibro-Impact System with Dry
More informationSlug-flow Control in Submarine Oil-risers using SMC Strategies
Slug-flow in Submarine Oil-risers using SMC Strategies Pagano, D. J. Plucenio, A. Traple, A. Departamento de Automação e Sistemas, Universidade Federal de Santa Catarina, 88-9 Florianópolis-SC, Brazil
More informationResponse of A Hard Duffing Oscillator to Harmonic Excitation An Overview
INDIN INSTITUTE OF TECHNOLOGY, KHRGPUR 710, DECEMBER 8-0, 00 1 Response of Hard Duffing Oscillator to Harmonic Excitation n Overview.K. Mallik Department of Mechanical Engineering Indian Institute of Technology
More informationOn the Dynamics of a n-d Piecewise Linear Map
EJTP 4, No. 14 2007 1 8 Electronic Journal of Theoretical Physics On the Dynamics of a n-d Piecewise Linear Map Zeraoulia Elhadj Department of Mathematics, University of Tébéssa, 12000, Algeria. Received
More informationLimit Cycles in High-Resolution Quantized Feedback Systems
Limit Cycles in High-Resolution Quantized Feedback Systems Li Hong Idris Lim School of Engineering University of Glasgow Glasgow, United Kingdom LiHonIdris.Lim@glasgow.ac.uk Ai Poh Loh Department of Electrical
More informationProblem Set Number 01, MIT (Winter-Spring 2018)
Problem Set Number 01, 18.377 MIT (Winter-Spring 2018) Rodolfo R. Rosales (MIT, Math. Dept., room 2-337, Cambridge, MA 02139) February 28, 2018 Due Thursday, March 8, 2018. Turn it in (by 3PM) at the Math.
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Exam Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) A 4.8-kg block attached to a spring executes simple harmonic motion on a frictionless
More informationLecture Notes for Math 251: ODE and PDE. Lecture 16: 3.8 Forced Vibrations Without Damping
Lecture Notes for Math 25: ODE and PDE. Lecture 6:.8 Forced Vibrations Without Damping Shawn D. Ryan Spring 202 Forced Vibrations Last Time: We studied non-forced vibrations with and without damping. We
More information