One Dimensional (1D) and Two Dimensional (2D) Spring Mass Chains

Transcription

1 One Dimensional (1D) and Two Dimensional (2D) Spring Mass Chains Massimo Ruzzene D. Guggenheim School of Aerospace Engineering G. Woodruff School of Mechanical Engineering Georgia InsJtute of Technology Atlanta, GA Wave Propaga+on in Linear and Nonlinear Periodic Media: Analysis and Applica+ons June 21 25,

2 ConfiguraJon Coarse approximajon of a uniform rod: Rod is discrejzed into N elements of length a; Mass and sjffness distribujons are described as lumped parameters; LocaJon of n th mass: 2

3 Notes System under considerajon is the first, simplest example of a PERIODIC structure: Here obtained considering a dumb discrejzajon of a conjnuous rod; Can be considered as a simple academic exercise System inijally studied by Newton (1686) to calculate the speed of sound in air: Newton, Principia, Book II, System is used by John Bernoulli and son Daniel (1727) to demonstrate that a system of N masses is characterized by N modes of vibrajon and associated frequencies ConfiguraJon considered by Baden Powell (1841) to calculate the velocity of wave propagajon along one axis of a cubic lacce structure Results later corrected and expanded by Lord Kelvin (1881) Popular Lecture, Vol. I, p Detailed discussions can be found in: L. Brillouin, Wave Propaga+on In Periodic Structures, Dover C. Kihel, Introduc+on to Solid Sate Physics, 8th ed. John Wiley & Sons, Inc.,

4 Governing equajons & wave solujon System s behavior is governed by N equajons of the kind: Impose a harmonic solujon Impose a wave solujon where Under the assumpjon that no external forces are applied: Free wave propagajon 4

5 SubsJtute wave solujon at frequency ω in n th equajon: Dispersion relajons (1) Non trivial solujons Dispersion relajon (frequency wavenumber relajons): 5

6 :direct solujon ConJnuous rod vs. discrete: DiscreJzaJon process can be described in terms of FINITE DIFFERENCE formalism This approximajon is used in deriving equivalent conjnuum systems for discrete assemblies DiscreJzaJon causes the system to be dispersive Periodic/discrete ConJnuous 6

7 Dispersion relajon is PERIODIC in the wavenumber space: Notes k space is periodic of period 2 /a As a result, displacements are also periodic in the wavenumber space = 7

8 Result is due to the SAMPLING of a conjnuous system: Notes SpaJal sampling occurs at a frequency The result is an expression of the Sampling Theorem (Shannon) theorem, for a system sampled in space A single period of the wavenumber/frequency relajon for a periodic system is called: FIRST BRILLOUIN ZONE 8

9 :direct solujon First Brillouin Zone Irreducible Brillouin Zone 9

10 Analogy with Jme domain signals can be used to obtain a good guess about the NATURAL FREQUENCIES of a FINITE PERIODIC system with N masses (free free for simplicity): Finite system can be considered as a truncajon of an infinite one TruncaJon causes the system to be DISCRETE instead of conjnuous Notes where wavenumber resolujon is: and Discrete wavenumber values correspond to N values of frequencies Natural frequencies can be read directly on the dispersion curve, given the number of masses and boundary condijons 10

11 Natural frequencies (N=5) Is parallel with finite Jme signal completely true? Not quite. A factor 2 is missing!!!! 11

12 : inverse solujon AlternaJvely, the solujon of the dispersion relajon: can be found by imposing frequency: where The wave solujon to the governing equajon: should be expressed as follows AhenuaJon constant 12

13 Harmonic response of a finite system (N masses) AhenuaJon PropagaJon 13

14 Harmonic response of a finite system (N=100 masses) 14

15 Harmonic response of a finite system (N=100 masses) 15

16 Phase velocity: Group velocity: Wave speeds 16

17 Average Energy Average energy density: sum of average potenjal and kinejc energy of the unit cell: Average potenjal energy Average kinejc energy Total energy 17

18 Energy flow Energy flow from one cell to the next is the AVERAGE POWER flowing from one cell to the next (n) (n+1) Energy velocity: rate at which energy flows along the lacce The energy velocity equals the group velocity 18

19 Diatomic lacce System is representajve of: Bi material rod: E 1, 1 E 2, 2 NaCl crystal along one of lacce direcjons GaAs zincblende crystal: vibrajon of the (1 0 0) plane 19

20 Governing equajons for 2n and (2n+1) masses: Governing equajons Impose a solujon of the kind: This solujon describes waves propagajng only through parjcles (a) and (b). Wavelength and frequencies are the same, but the amplitudes of the two waves are not equal 20

21 SubsJtuJng gives: Dispersion relajons In matrix form: CharacterisJc equajon SoluJon idenjfies TWO BRANCHES: 21

22 where: Dispersion relajons ACOUSTIC BRANCH OPTICAL BRANCH Both branches are PERIODIC in the wavenumber domain: 22

23 Direct solujon OPTICAL BRANCH ACOUSTIC BRANCH Band gap 23

24 Direct solujon Single mass system Two mass system Band gap disappears 24

25 Period of wavenumber/frequency domain is: First Brillouin zone Single mass system Two mass system The period of the dispersion relajon is always given by: where: Single mass system: Two mass system: 25

26 First Brillouin zone For any 1D periodic system, the frequency/wavenumber spectrum is periodic in the domain: where PropagaJon constant: First Brillouin zone: The definijon of the Brillouin zone can be used to define unequivocally the SPATIAL PERIOD of the system 26

27 : inverse solujon 27

28 Response 28

29 Response of a system of N=200 masses 29

30 Response of a system of N=200 masses 30

31 Spring Mass System Unit Cell: 6/7/10 M. Ruzzene Lecture 2 M. Ruzzene 31

32 EquaJon of harmonic mojon for mass n,m: Governing equajons & wave solujon Wave propagajon solujon: where 6/7/10 M. Ruzzene Lecture 2 M. Ruzzene 32

33 DirecJon of wave propagajon Note Wave front 6/7/10 M. Ruzzene Lecture 2 M. Ruzzene 33

34 Wave propagajon solujon Rewrite solujon as: and 6/7/10 M. Ruzzene Lecture 2 M. Ruzzene 34

35 SubsJtuJng in governing equajon leads to: Dispersion relajon 2D dispersion relajon Surface in the wavenumber domain 6/7/10 M. Ruzzene Lecture 2 M. Ruzzene 35

36 2D Dispersion relajon First Brioullin zone Irreducible Brillouin zone 6/7/10 M. Ruzzene Lecture 2 M. Ruzzene 36

37 2D Dispersion relajon 6/7/10 M. Ruzzene Lecture 2 M. Ruzzene 37

38 Group velocity According to definijon: Where Recall that: Velocity of energy flow equals the group velocity Energy flows in the direcjon corresponding to the group velocity 6/7/10 M. Ruzzene Lecture 2 M. Ruzzene 38

39 In this case: Group velocity Assume 6/7/10 M. Ruzzene Lecture 2 M. Ruzzene 39

40 Contour at a single frequency Contour of dispersion surface From dispersion relajons: DirecJon of energy flow at a given frequency and direcjon is perpendicular to isofrequency contour 6/7/10 M. Ruzzene Lecture 2 M. Ruzzene 40

41 Dispersion surface vs. group velocity 6/7/10 M. Ruzzene Lecture 2 M. Ruzzene 41

42 Dispersion surface vs. group velocity 6/7/10 M. Ruzzene Lecture 2 M. Ruzzene 42

43 Dispersion surface vs. group velocity 6/7/10 M. Ruzzene Lecture 2 M. Ruzzene 43

44 Dispersion surface vs. group velocity 6/7/10 M. Ruzzene Lecture 2 M. Ruzzene 44

45 Dispersion surface vs. group velocity 6/7/10 M. Ruzzene Lecture 2 M. Ruzzene 45

46 PropagaJon of waves is strongly direcjonal at specified frequency At those frequencies, waves propagate only in certain direcjons Notes BEAMING PHENOMENA For considered configurajon beaming is a very focused, but very narrow band phenomenon 6/7/10 M. Ruzzene Lecture 2 M. Ruzzene 46

47 Harmonic response of 40*40 lacce: Example: 2D spring mass lacce mass # mass # mass # mass # 6/7/10 M. Ruzzene Lecture 2 M. Ruzzene 47

48 Time domain response Time domain simulajons Input modulated sine burst at various frequencies e $!"*!"+!"'!"% f(t)!!!"%!!"'!!"+!!"*!$!!"# $ $"# % %"# & &"# ' '"# # t [s] ()$!!& 6/7/10 M. Ruzzene Lecture 2 M. Ruzzene 48

49 Time domain response 6/7/10 M. Ruzzene Lecture 2 M. Ruzzene 49

50 Anisotropic lacce 6/7/10 M. Ruzzene Lecture 2 M. Ruzzene 50

51 Anisotropic lacce 6/7/10 M. Ruzzene Lecture 2 M. Ruzzene 51

52 Anisotropic lacce 6/7/10 M. Ruzzene Lecture 2 M. Ruzzene 52

53 Forbidden propagajon zone Note Waves do not propagate along the x direcjon M. Ruzzene Lecture 2 6/7/10 53

54 Harmonic response of 40*40 lacce: Example: 2D spring mass lacce mass # mass # mass # mass # M. Ruzzene Lecture 2 6/7/10 54