CEHMS Center for Energy Harvesting Materials and Systems

Transcription

1 CEHMS Center for Energy Harvesting Materials and Systems Shashank Priya, Director Phone: (540) Dennis Smith, Site Director Phone: (972) Center Website:

2 Vibration theory Introduction to basic vibration theory o Spring-mass-damper system o Mechanical resonance and mechanical damping o Base excitation o Intro to lumped mass modeling and distributed parameter modeling followed by case studies

3 Spring-Mass-Damper System Vibration is cause by the interaction of two different forces one related to position (stiffness) and one related to acceleration (mass). All real systems dissipate energy when they vibrate. To account for this we must consider damping. Spring Force Damping Force F = kx(t) F = cx (t) Force of vibrating mass F = mx (t) F=F 0 cosωt M k External applied force F = F 0 cos ωt x(t) c

4 Harmonic motion x 0 x(t) Slope here is v 0 Period T 2 n Amplitude A t f n n n rad/s 2 rad/cycle cycles n 2 s Maximum Velocity n A n 2 Hz (Inman, 2008)

5 Relationship between displacement, velocity and acceleration a v x Displacement x(t) Asin( n t ) Velocity x(t) n Acos( n t ) Acceleration x(t) n 2 Asin( n t ) A=1, ω n = Time (sec) (Inman, 2008)

6 Modeling methods: Deriving equation of motion o Force method o Free Body Diagram o Energy method o Conservation of Energy F=F 0 cosωt M x(t) k c Summing forces on the mass mx = kx cx + F 0 cos ωt F M kx(t) cx (t) Rearranging mx + cx + kx = F 0 cos ωt x(t) Dividing by the mass x + 2ζω n x + ω n 2 x = f 0 cos ωt Natural Frequency Damping Ratio Where ω n = k m, ζ = c 2mω n

7 Solving the differential equation of motion x + 2ζω n x + ω n 2 x = f 0 cos ωt Where ω n = k m, ζ = c 2mω n The solution is a combination of a homogenous and particular solutions Homogenous x + 2ζω n x + ω 2 n x = 0 subject to initial conditions of form x 0 = x 0, x 0 = v 0 Solution is of form x h t = Ae ζωnt sin(ω d t + φ) Where ω d = ω n 1 ζ 2 And the constants are of form A = x v 0 + ζω n x 0 ω d 2 φ = tan 1 x 0 ω d v 0 + ζω n x 0

8 Solving the differential equation of motion cont. x + 2ζω n x + ω n 2 x = f 0 cos ωt Particular For a harmonic forcing function of form f 0 cos ωt the particular solution is of the form x p t = X cos ωt θ Substitution in the equation of motion yields the particular solution x p t = f 0 ω n 2 ω ζω n ω 2 cos ωt tan 1 2ζω n ω ω n 2 ω 2

9 Solving the differential equation of motion cont. Therefore the total solution is x t = x v 0 + ζω n x 0 + ω d f 0 ω n 2 ω ζω n ω 2 e ζω nt sin(ω d t + tan 1 2 cos ωt tan 1 2ζω x 0 ω d v 0 + ζω n x 0 ) n ω ω n 2 ω 2 For large values of t, the homogeneous term approaches zero and the total solution becomes of form of the particular solution. Therefore the homogenous solutions is referred to the transient response and the particular solutions is the steady state response.

10 Concept of resonance In energy harvesting we often ignore the transient response and solely analyze the steady state solution x t = f 0 ω n 2 ω ζω n ω 2 cos ωt tan 1 2ζω n ω ω n 2 ω 2 Solution is the sum of two frequencies: the natural frequency of the mechanical system and the driving frequency of the applied force. What happens when ω n = ω?

11 X (db) Phase (rad) Concept of resonance In order to illustrate the concept of resonance we non-dimensionalize the response by factoring out ω 2 and dividing the magnitude by F 0 m X t = 1 1 r ζr 2 cos ωt tan 1 2ζr 1 r 2 where r = ω ω n z =0.01 z =0.1 z =0.3 z =0.5 z = r r (Inman, 2008)

12 Review Covered the following concepts o Introduction to spring-mass-damper system modeling o Mass, stiffness, damping and external force o Mechanical resonance and damping ratio We can now apply the basic concepts to model energy harvesters o Inductive Lumped mass parameter modeling of relative motion between magnet and coil under base excitation (Poulin et al., 2004) Correction factor (Oliver and Priya, 2009) o Piezoelectric Distributer parameter modeling of strain distribution in piezoelectric bimorph (Erturk and Inman, 2009)

13 Inductive Vibration Harvester For a vibration generator with relative motion between the coil and the magnet, the voltage induced in the coil can be expressed as the product of a flux linkage gradient and the velocity of movement according to Faraday s law of electromagnetic induction. V = dφ dt = dφ dx dx dt = NBAv = BLv alternatively V = v B dl = vbl (Marin et al., 2011)

14 Apply spring-mass-damper model System FBD m k( x y) c( x y) x(t) y(t) Summing the forces results in the following EOM mx + c x y + k x y = 0 The harvested power is dependent upon the relative motion between the coil and magnets (z = x y) therefore the equation can rewritten as mz + c z + k z = my

15 Modeling electrodynamics mz + c z + k z = my If we place a load resistance in series with the coil an additional force opposes the motion of the coil mz + c z + k z = my + Bli The above equation models the electrodynamics. In order to model the electrical system Kirchoffs voltage law is applied: U = Blz R e i L e di dt In order to solve the coupled equations for the electrodynamics we apply Laplace transform method and rearrange terms:

16 Solving for power generated Mechanical system Electrical system mz + c z + k z = my + Bli U = Blz R e i L e di dt ms 2 + fs + k X s = F s + BlI(s) U s = BlsX(s) R e I s L e si(s) ms 2 + fs + k X s ms + f + k s BlI(s) = F s sx s BlI(s) = F s I s = U s + BlsX(s) R e + L e s sx s = U s + Z e s I(s) Bl Z jω V 1 + Bl U + BlV 1 Z e (jω) = F I s = U s + BlsX(s) Z e (s) V 1 = Z ei + U Bl Z jω V 1 + BlU + Bl2 V 1 Z e (jω) = F

17 Solving for power generated Z jω V 1 + BlU + Bl2 V 1 Z e (jω) = F V 1 = Z ei + U Bl F + Bl Z e U = Bl 2 Z e Z e I + U Bl + Z jω Z ei + U Bl F + Bl Z e U = Bl 2 Z e + Z(jω) Z e I + U Bl I = U R L F + Bl Z e U = Bl 2 Z e + Z(jω) Z e U R L Bl + U Bl

18 Solving for power generated F + Bl Z e U = Bl 2 Z e + Z(jω) Z e U R L Bl + U Bl F + Bl U = Bl 2 + Z(jω) Z e Z e Z e R L Bl + 1 Bl U F = Bl 2 Z e + Z jω Z e BlR L + 1 Bl U Bl Z e U U F = 1 Bl 2 Z e + Z jω Z e BlR L + 1 Bl Bl Z e

19 Solving for power generated U F = 1 Bl 2 Z e + Z jω Z e BlR L + 1 Bl Bl Z e U F = 1 Bl 2 + Z jω Z e Z e Z e + R L BlR L Bl Z e U F = 1 Bl 2 + Z jω Z e Z e + R L Z e Bl Bl R L Z e BlR L Z e Z e U F = BlR L Z e Z e Bl 2 + Z jω Z e Z e + R L Z e Bl 2 R L Z e

20 Solving for power generated U F = BlR L Z e Bl 2 + Z jω Z e Z e + R L Bl 2 R L U F = BlR L Z e Bl 2 Z e + Z jω Z e Z e + Bl 2 R L + Z jω R L Z e Bl 2 R L U F = BlR L Z e Bl 2 Z e + Z jω Z e Z e + Z jω R L Z e U F = BlR L Bl 2 + Z jω Z e + Z jω R L

21 Solving for power generated U = FR L Bl Z e + R L Z jω + Bl 2 p e = 1 R L U 2 2 p e = FR L Bl 2 2 Z e + R L Z jω + Bl 2 2 The above formulation over predicts the voltage generated for three reasons Magnetic field B which cuts coil is not constant over the face of the coil Portions of magnetic field cutting the coil do not contribute to voltage generation Velocity is not constant with length of the coil

22 Correction factor A alternative method for method for modeling the voltage is as follows: U e = z B dl z (y b = L beam )Φ T,eq Where Φ T,eq represents a more accurate approximation of the quantity Blz L coil U e = B(y c, 0 z c )z (y c, z c )dl cos(θ(y c, z c ) Assuming that the velocity is constant across the length of the coil the voltage and Φ T,eq can be written as : L coil U e = z B(y c, 0 L coil Φ T = B(y c, 0 z c ) dl cos(θ(y c, z c )) z c ) dl cos(θ(y c, z c ))

23 Correction factor for B and theta L coil Φ T = B(y c, z c ) dl cos(θ(y c, z c )) 0 Discretizing the coil volume the transformation factor can be rewritten as: Φ T B(y c, z c ) L coil cos θ y c, z c where L coil = L coil # of discrete volumes (Oliver and Priya, 2009)

24 Correction factor for velocity distribution Φ C 1 At resonance the deflection profile has a unique shape called the mode shape of vibration. The mode shape for cantilever beams with specific tip mass have been tabulated in (Laura et al., 1975). x b L beam = cos n i x b L beam cos n i +cosh n i sin n i +sinh n i sin n ix b L beam cosh n ix b L beam Φ T B(y c, z c ) L coil cos θ y c, z c Φ(y b ) + cos n i +cosh n i sin n i +sinh n i sinh n ix b L beam U e = z y b = L beam Φ T

25 Experimental verification Φ T B(y c, z c ) L coil cos θ y c, z c Φ(y b ) U e = z y b = L beam Φ T Replace Bl with Φ T U = FR L Φ T Z e + R L Z jω + Φ T 2 p e = FR L Φ T 2 2 Z e + R L Z jω + Φ T 2 2 (Marin et. al, 2011)

26 Piezoelectric vibration harvesting Bimorph In the previous example for inductive harvester the voltage is directly proportional to velocity. For piezoelectric harvester the voltage is directly proportional to the strain in the piezoelectric material. The bimorph piezoelectric harvester is typically modeled using distributed parameter modeling theory in order to calculate strain as a function of length along the beam (Erturk and Inman, 2009)

27 Distributed parameter modeling Lumped mass modeling predicts the response at a particular point as a function of time. Distributed parameter modeling predicts the response spatially as a function of time. Lumped Mass w (x,t) f (x,t) Distributed parameter h 1 M(x,t) f(x,t) M(x,t)+M x (x,t)dx x h 2 w(x,t) V(x,t) Q V(x,t)+V x (x,t)dx (Inman, 2008) dx A(x)= h 1 h 2 F = mx t M(x, t) = EI(x) 2 w(x, t) x 2 x x +dx

28 Distributed parameter modeling Lumped Mass Distributed parameter Displacement x(t) w(x, t) Forces Stiffness kx t Shear V x, t Moment M x, t EOM mx + kx = f(t) ρa 2 w(x, t) t 2 + EI 4 w(x, t) x 4 BC s x 0 = x 0, x 0 = v 0 w x, 0 = w 0 x, w t and w l, t = 0, w x = f(x, t) x, 0 = w t 0 l, t = 0 x Form of solution x t = A sin ωt + φ + X cos ωt w(x, t) = X x T(t)

29 Distributed parameter modeling Previous equations neglected the following: Relative motion Composite beam Damping The forced vibrations of a piezoelectric beam are given by (Erturk and Inman, 2009): EI 4 w rel (x, t) x 4 + c s I 5 w rel (x, t) x 4 t dδ x dδ x L +θu t dx dx + c a w rel (x, t) t + M b L 2 w rel (x, t) t 2 = M b L + M tδ(x L) 2 y(t) t 2

30 Distributed parameter modeling w rel x, t = Φ r (x)η r (t) Φ r x = C r cos λ rx b L cosh λ rx b L + ζ r sin λ rx b L sinh λ rx b L ζ r = sin λ r sinh λ r + λ r M t M b cos λ r cosh λ r λ r M t M b cos λ r cosh λ r sin λ r sinh λ r 1 + cos λ r cosh λ r + λ r M t M b cos λ r sinh λ r sin λ r cosh λ r = 0 0 L Φ r x b M b L Φ r x b dx b + Φ r L M t Φ r L = 1

31 Distributed parameter modeling w rel x, t = Φ r (x)η r (t) d 2 η r s (t) dt 2 dη s r (t) + 2ζ r ω r + ω 2 dt r η s r t + χ s r U t = M b d 2 y t L dt 2 Φ r x dx 0 L M t Φ r (L) d2 y(t) dt 2 Dynamics External Force Backward Coupling χ r = θ dφ r(x) dx x=l = 1 d 31 b s E 11 ah p 2 h s 4 h p + h s 2 2 dφr (x) dx x=l

32 Distributed parameter modeling Electrical system C p du(t) + au(t) dη s r (t) = i a dt 2R p t = κ r L dt C p = ε 33bL h p Coupling term κ r = d 31h pc b s 11 E 0 L d 2 Φ r (x) dx 2 dx = d 31h pc b s 11 E dφ r (x) dx x=l

33 Distributed parameter modeling d 2 η r s (t) dt 2 dη s r (t) + 2ζ r ω r + ω 2 dt r η s r t + χ s r U t = M b d 2 y t L dt 2 Φ r x dx 0 L M t Φ r (L) d2 y(t) dt 2 s 2 + 2ζ r ω r s + ω r 2 H s + χ r s U(s) = F s s + 2ζ r ω r + ω r 2 s sh s + χ r s U(s) = F s C p du(t) + au(t) a dt 2R L dη s r (t) = κ r dt C p a su(s) + a 2R L U(s) = κ r sh(s) s + 2ζ r ω r + ω r 2 s jω + 2ζ r ω r + ω r 2 jω C p su(s) + a U(s) aκ r 2κ r R L C p aκ r jωu(t) + a 2κ r R L U(t) + χ r s U(s) = F s + χ r s U(t) = f r (t)

34 Distributed parameter modeling Force terms f r t = M b L d 2 y t dt 2 Φ r x dx 0 L M t Φ r (L) d2 y(t) dt 2 F s = M b L s2 Y(s) L Φ r x dx 0 M t Φ r (L)s 2 Y(s) F s = s 2 Y s M b L f r (t) = jω 2 Y t M b L L Φ r x dx 0 L Φ r x dx 0 M t Φ r (L) M t Φ r (L) f r t = ω 2 Y t M b L L Φ r x dx 0 M t Φ r (L)

35 Distributed parameter modeling jω + 2ζ r ω r + ω r 2 jω C p aκ r jω + a 2κ r R L U(t) + χ r s U(t) = f r (t) jω + 2ζ r ω r + ω r 2 jω C p aκ r jω + a 2κ r R L + χ r s U(t) = f r (t) U(t) = jω + 2ζ r ω r + ω r 2 jω f r (t) C p aκ r jω + a 2κ r R L + χ r s U t = jωκ r M b L L 0 2ζ r ω r jω + ω r 2 ω 2 Φ r x dx M t Φ r (L) C p a + a 2R L + jωκ r χ r s ω 2 Y 0 e jωt

36 Experimental Validation of Model Erturk and Inman, 2009

37 Review Covered the following concepts o Introduction to spring-mass-damper system modeling o Mass, stiffness, damping and external force o Mechanical resonance and damping ratio o Inductive Lumped mass parameter modeling of relative motion between magnet and coil under base excitation (Poulin et al., 2004) Correction factor (Oliver and Priya, 2009) o Piezoelectric Distributer parameter modeling of strain distribution in piezoelectric bimorph (Erturk and Inman, 2009) Next we will discuss the experimental methods used to validate the models

38 Typical Experimental Setup Marin et. al, 2011

39 Electrodynamic shakers which apply a force through a range of frequencies (harmonic or random inputs) Accelerometer - produce a signal proportional to acceleration at the point of attachment Laser vibrometers - produce a signal proportional to local velocity Measurement

40 Digital Acquisition Accelerometer and laser vibrometer output analog signals ex. x t and x (t) The analog signals can be sampled or digitized using a A/D converter Next, we use DFT to transform the digital time domain data to the frequency domain By computing Power Spectral Density and Cross Spectral Density we can create frequency response functions from which vibration data is extracted

41 Frequency response function 1 Autocorrelation : R xx ( ) lim T T T 0 x(t)x(t )d Power Spectral Density (PSD): S xx ( ) Crosscorrelation : R xf ( ) lim T T Cross Spectral Density: S xf ( ) 1 2 H jω DFT DFT T 0 x(t) f (t )d R xf ( )e j d = S xx(ω) S xf ω = 1 k mω 2 + cjω R xx ( )e j d Tells how fast x(t) is changing Tells how fast x(t) is changing relative to f(t) CPSD function in matlab Frequency response function H s = S xx(s) S xf s = 1 ms 2 + cs + k Transfer function γ 2 = S xf (ω) 2 S xx (ω)s ff (ω), 0 γ2 1 Coherence

42 Experimental Modal Analysis Oliver, J., Priya, S Design, Fabrication, and Modeling of a Four-bar Electromagnetic Vibration Power Generator, Journal of Intelligent Material Systems and Structures. Extract all system parameters through curve fitting of experimental results H s = S xx(s) S xf s = 1 ms 2 + cs + k = 1 s 2 + 2ζω n s + ω n 2 ζ = c 2 km ω n = k m

43 Other damping estimation methods Peak picking or 3 db method Logarithmic decrement method H( a ) H( b ) H( d ) 2 z b a 2 d

44 Power characterization Inductive Piezoelectric It has been proven, that the extraction of electrical power is maximized when ζ E = ζ M. Φ T i(t) R L,opt = R coil + Φ T 2 χ r s U t c m R L,opt = R s + 1 jωc

45 Vibration energy harvesting U = FR L Φ T Z e + R L Z jω + Φ T 2 U t = jωκ r M b L 2ζ r ω r jω + ω r 2 ω 2 L Φ r x dx M 0 t Φ r (L) C p a + a s + jωκ 2R r χ r L ω 2 Y 0 e jωt Φ T B(y c, z c ) L coil cos θ y c, z c Φ(y b ) Geometry and source parameters + Material parameters Harvester performance κ r = d 31h pc b s 11 E dφ r (x) dx x=l

46 Maximum power extraction for given geometry and source Maximum power conversion occurs at resonance Inductive high relative velocity Piezoelectric high relative strain (b) x(t) Tip mass C K y(t) Harvester Base Vibration Input mz + c z + k z = my

47 Maximum Power Extraction z t = k m ω2 ω c Tω m 2 Y sin ωt φ Maximum power conversion occurs at resonance at peak amplitude z t = 1 2ζ T 2 Y sin ω nt φ The maximum power dissipated into an electrical load consists of multiplying maximum applied force applied to the damper caused by the load itself times maximum relative velocity. Where P max = z max = c E z max z max ω n 2ζ T 2 Y

48 Maximum Power Extraction Therefore P max = c E ω n Y 2ζ T 2 ω n Y 2ζ T 2 P max = c E ω n 2 Y 2 2ζ T 2 Substituting ζ E = c E 2mω n P max = 2ζ E mω n ω n 2 Y 2 4ζ T 2 The extraction of electrical power is maximized when ζ E = ζ M therefore ζ e = ζ T 2. P max = mω n 3 Y 2 4ζ T P max = ma2 4ζ T ω n

49 Vibration energy harvesting U = FR L Φ T Z e + R L Z jω + Φ T 2 U t = jωκ r M b L 2ζ r ω r jω + ω r 2 ω 2 L Φ r x dx M 0 t Φ r (L) C p a + a s + jωκ 2R r χ r L ω 2 Y 0 e jωt Φ T B(y c, z c ) L coil cos θ y c, z c Φ(y b ) Geometry and source parameters + Material parameters Harvester performance κ r = d 31h pc b s 11 E dφ r (x) dx x=l

50 Inductive coupling (magnets) Φ T B(y c, z c ) L coil cos θ y c, z c Φ(y b ) Neodymium Iron Boron Magnetic Field Strength koe Curie temperature F (310 C) Operating temperature F (80 C) Samarium Cobalt Magnetic Field Strength koe Curie temperature F (750 C) Operating temperature F (300 C)

51 Inductive coupling (coil) Φ T B(y c, z c ) L coil cos θ y c, z c Φ(y b ) fill factor = A wire L wire π r o 2 r i 2 t The fill factor of scramble wound coils will vary but a figure of 50 to 60% could be assumed.

52 P avg (mw) V avg (V) P avg (mw) V avg (V) Inductive coupling (coil) cont AWG (40 μm dia) P avg, Simulated V avg, Simulated Φ T B(y c, z c ) L coil cos θ y c, z c Φ(y b ) AWG (100 μm dia) P avg, Simulated Resistance (kohm) V avg, Simulated Resistance (kohm) Assuming fill factor remains constant decreasing wire size does not affect the power output Resistance (kohm) Resistance (kohm)

53 Piezoelectric coupling κ r = d 31h pc b s 11 E dφ r (x) dx x=l d 31 is the piezoelectric constant s 11 E is the elastic constant d 31 while s 11 E FOM on resonance k Q / s 2 E 31 m 11

54 Piezoelectric coupling (doping) Typical doping effect on piezoelectric properties Kind Dopants Effect Hardener Acceptor Ion Li 1+, K 1+, Fe 3+, Ni 2+, Co 2+, Mn 2+ FOM on resonance k Q / s Decrease in piezoelectric constant, d Decrease in elastic compliance, s Decrease in electromechanical coupling factor, k Increase in Q κ r = d 31h pc b s 11 E 2 E 31 m 11 dφ r (x) dx x=l Softener Donor Ion La 3+, Bi 3+, Nb 5+, W 6+, Ta 5+ Increase in piezoelectric constant, d Increase in elastic compliance, s Increase in electromechanical coupling factor, k Decrease in Q

55 Textured Ceramics

56 Vibration energy harvesting Geometry and source parameters Φ T B(y c, z c ) L coil cos θ y c, z c Φ(y b ) κ r = d 31h pc b s 11 E + Material parameters Harvester performance Which Mechanism? dφ r (x) dx x=l Size?

57 Inductive or Piezoelectric? (a) Inductive: U = N B t suggesting P V 2 Piezoelectric: A = N B t L2 = B t L3 U = σ j g ij t = ρag ij L 2 ma A g ij L = ρl3 a L 2 g ij L = suggesting P α V 4/3 Marin et. Al, 2011

58 Survey of state of art P max = ma2 4ζ T ω n Normalize by source terms to compare coupling as a function of size Marin et. Al, 2011

59 Harvester size? High frequency 17 mw 50 Hz 0.2 G ~ 70 cm 3 Low frequency 4 mw Hz 0.25 G ~ 6 cm 3 Marin et. Al, 2011