Introduction to Waves in Structures Mike Brennan UNESP, Ilha Solteira São Paulo Brazil
Waves in Structures Characteristics of wave motion Structural waves String Rod Beam Phase speed, group velocity Low and high frequency behaviour Wave transmission and impedance
Wave Motion
Wave Motion Wave motion Wave motion Particle motion Particle motion
Harmonic Wave Motion ( x, ) = A sin( ωt ± k ) or W ( x, t ) = A cos( ωt ± kx ) W t x
Harmonic Wave Motion ( x, ) = A sin( ωt ± k ) or W ( x, t ) = A cos( ωt ± kx ) W t x Fix position Fix time
Harmonic Wave Motion ( x, ) = A sin( ωt ± k ) W t x W W { } ( ) ( ), Im j ω t ± x t = A e kx ( t ) x, = Ae ω e ± j t jkx ( x ) = e jkx W A ( ) = e jkx W x A
The Wavenumber (propagating waves) time distance T ω = π T Temporal frequency k λ π = = λ ω c Spatial frequency (wavenumber) k is the phase change per unit distance Ae jkx
Superposition of Waves
Interference of Waves ( ) Wave motion A sin ωt kx Wave motion A sin ( ωt kx φ ) Wave motion W ( x, t ) ( x, t ) = sin( ωt x ) + sin( ω k φ ) W A k A t x φ W = A x ( x, t ) cos sin( ωt k )
Standing Waves Wave motion A sin( ωt kx ) Wave motion A sin( ω t + kx ) Standing wave W ( x, t ) ( x, t ) = sin( ωt x ) + A sin( ωt + kx ) W A k ( x, t ) sin( ωt ) cos( x ) W = A k
Beating Wave motion Wave motion A A sin sin ( ω t k x ) 1 1 ( ω t k x ) Wave motion W ( x, t ) ( x, t ) = sin( ω t k x ) + A sin( ω k ) W A t x 1 1 W ω ω k k ω + ω k + k A x x 1 1 1 1 ( x, t ) = cos t sin t
Waves in solids
Waves motion in a string
Wave motion in a string y + y dx x dy θ dx y T String segment T x x + dx ( sinθ ) ( sinθ ) df = T T y x + dx x
Wave motion in a string Applying the Taylor s series expansion gives f f x dx f x dx x ( + ) = ( ) + +... x ( T sin θ ) df = ( T sinθ ) + dx T x ( sinθ ) y x x If θ is small, then sin θ θ = y x, then y df = T dx y x Stiffness force
Wave motion in a string The inertia force in the y direction is given by df y = ρ dx L t where ρ L is the mass per unit length Applying Newton`s second law gives the wave equation y x y 1 y = c t 0 T where c = is the phase speed ρ L
Wave motion in a string c = T ρ L is a constant - the wave is non-dispersive Wave speed c
Free harmonic wave motion in a string Start with the wave equation y 1 y = 0 x c t Assume time harmonic motion, gives d y k y 0 dx This has a solution + = where k y() x = Ae + Be jkx jkx ω = c
Wave motion in a string - impedance y f y T x y = ( ) j t kx Ae ω boundary condition ( ) f = T sinθ Ty 0 y y y = = jkae xx y yɺ = = j ωae t ( ω ) j t kx ( ω ) j t kx Z f y = = yɺ ( 0) ρ c L which is real and hence damping-like
Wave motion in a rod
Wave motion in a rod u f x x + dx f f + x df bar element of area S and density ρ f S = E u x..(1) Applying Newton`s second law gives f f + dx f = ρsdx x t u ()
Wave motion in a rod Now from (1) f = ES x x which combines with () to give the wave equation u u 1 u = x c t 0 E where c = is the phase speed (constant with frequency) ρ
Wave motion in a rod - impedance f x u rod area S and density ρ u = ( ) j t kx Ae ω f = ESu ( 0) ( ) u = jkae ω uɺ = j ωae ω j t kx ( ) j t kx f Z = = S E uɺ ( 0) ρ which is real and hence damping-like
In-Plane Wave motion rod c = E ρ plate c = ρ E ( 1 ν ) ν = Poisson's ratio 3D solid c = E ( 1 ν ) ( 1+ )( 1 ) ρ ν ν
Shear Waves
Shear Waves The wave equation is given by where w 1 w = x c t 0 c = G ρ Shear modulus ( ) E = 1+ν G ν = Poisson's ratio E = Young's modulus G = shear modulus Note that torsional waves are also shear waves
Wave motion in a beam
Wave motion in a beam y w Beam segment of length dx crosssectional area S M M + dm V V + dv Summing forces V V V + dx = ρsdx x x t w (1)
V x Wave motion in a beam Summing moments about the any point on the right hand face From (1) and () w + ρs = t M M + dx M Vdx = x 0 V 0 () Bending stiffness M w = and M = EI x x Combining these equations gives the wave equation EI w w 4 + ρs = 0 4 x t
Wave speed in a beam Note that the wave equation is fourth order rather than second order. What is the phase speed? Is it dependent on frequency? First consider the equation for a single degree of freedom mass-spring-damper system d u ω u 0 n dt + = (Ordinary differential equation - time) Now consider the wave equation for a rod u 1 u = x c t 0 (Partial differential equation - space and time)
Wave speed in a beam Assume harmonic motion the wave equation becomes where k d u k u 0 dx + = (Ordinary differential equation - space) ω = c Now consider the wave equation for a beam 4 w ρs w + = x EI t 4 0 (4 th order Partial differential equation - space and time)
Wave speed in a beam Assume harmonic motion the wave equation becomes 4 d w 4 0 4 k w = dx (Ordinary differential equation - space) where k ω ρs = = c EI 1 4 ω 1 So c EI = ρs 1 4 ω 1 Note that the wave speed is dependent on frequency - it is dispersive
Group velocity Wave motion Wave motion A A sin sin ( ω t k x ) 1 1 ( ω t k x ) Carrier Velocity c g W ( x, t ) W ω ω k k ω + ω k + k A x x 1 1 1 1 ( x, t ) = cos t sin t c g ω ω ω k k k 1 = = 1 In the limit c g d ω = dk
Group velocity flexural waves The wavenumber is given by k ρs = EI 1 4 1 ω 1 EI ω = k ρs d ω EI = k c dk = ρs The group velocity is twice the phase velocity 1
Wave characteristics
Wave characteristics
Wave characteristics
Bending Waves motion in a beam Start with the wave equation EI w w 4 + ρs = 0 4 x t Assume time harmonic motion, gives 4 d w 4 0 4 k w = dx This has a solution where w() x = A e + A e + A e + A e Nearfield waves kx kx jkx jkx 1 3 4 ω k = c
Wave motion in a beam - impedance w x f w = A e e + A e e j ωt kx j ωt jkx 4 Beam of area S and density ρ 0 = EIw ( 0) Z f EI A = = + wɺ ( 0) 1/4 3/4 ( )( ρ ) 1/ ω (1 j ) f = EIw ( 0) which has a real part - damping-like and a positive imaginary part - mass-like
Bending Waves in a plate
Bending Waves in a plate Bending wave behaviour similar to the beam but in two dimensions The equation of motion is w w + h = t t 4 D ρ 0 where = + + x x y y 4 4 4 4 4 Assume time harmonic motion, gives 4 4 w k w = 0 and D = where 11 k Eh ( ν ) = 3 ω c
Bending Wavenumber in a plate The bending number for a plate is given by k 1ρ1 1 ( ν ) 4 1 = ω Eh The bending number for a rectangular beam is given by k 1ρ = Eh 1 4 ω 1
Bending Waves frequency limits Bending wave behaviour stems from the Euler- Bernoulli beam equation Neglects rotary inertia of the beam Assumes infinite shear stiffness Assumptions are ok for low frequencies, when λ > 10 depth of the beam
Material Properties Material Young s modulus E (N m - ) Density ρ (kg m -3 ) Poisson s ratio, ν Loss factor, η Mild Steel e11 7.8e3 0.8 0.0001 0.0006 Aluminium Alloy 7.1e10.7e3 0.33 0.0001 0.0006 Brass 10e10 8.5e3 0.36 < 0.001 Copper 1.5e10 8.9e3 0.35 0.0 Glass 6e10.4e3 0.4 0.0006 0.00 Cork 1..4e3 0.13 0.17 Rubber 1e6-1e9 1e3 0.4 0.5 0.1 Plywood 5.4e9 6e 0.01 Perspex 5.6e9 1.e3 0.4 0.03 Light Concrete 3.8e9 1.3e3 0.015 Brick 1.6e10 e3 0.015
Wavelength for 1 mm Euler-Bernoulli beam 10 3 Wavelength (mm) 10 10 1 10 1 10 10 3 10 4 10 5 Frequency (Hz)
Timoshenko beam y w Beam segment of length dx crosssectional area S M ρi M + dm V V + dv Equation of motion x E x κg x t κg t 4 4 4 I w E w I w I 1 S w + ρ + + ρ + ρ = 0 4 t 4
Timoshenko beam Assuming time harmonic motion Equation of motion the equation of motion becomes 4 S d w 4 S L 4 k k L B d w dx which leads to k k k + w = κ x κ S L B 0 - shear - Longitudinal - Bending k k k κ 4 κ 1 S 4 1 S = + k ± k + k L B L As ω 0, then k = k As ω, then k = k and k = k B S L
Wave motion in beams summary At low frequencies Torsional waves non-dispersive waves that propagate at the shear wave speed Longitudinal waves non-dispersive waves that propagate at the compressional wave speed Bending waves nearfield waves (do not propagate) and flexural waves that propagate at the bending wave speed At high frequencies Torsional waves, Longitudinal waves Flexural waves that propagate at (a) compressional wave speed (b) shear wave speed)
Reflection and Transmission of Waves Hard boundary Note the phase differences Soft boundary
Reflection and Transmission of Waves 1 High speed to low speed A Z Z r = A Z Z 1 1 1 A t = + 1 A Z Z 1 + 1 Low speed to high speed
Surface wave in a Solid Rayleigh wave Water wave
Summary Characteristics of wave motion Structural waves String Rod Beam Phase speed, group velocity Low and high frequency behaviour Wave transmission and impedance