Seminar: Vibrations and Structure-Borne Sound in Civil Engineering Theor and Applications Surve of Wave Tpes and Characteristics Xiuu Gao April 1 st, 2006 Abstract Mechanical waves are waves which propagate through a material medium (solid, liquid, or gas) at a wave speed which depends on the elastic and inertial properties of that medium. Often there will be energ transferring accompan with wave propogation. In this small report, we would like to introduce three different main wave tpes which are longitudinal waves, transverse waves and bending waves. The general procedure for our eploration will be to derive the governing differential wave equations with the help of kinematic, material and equilibrium equations for each individual wave tpe. Then we have a comparison of main characteristics for different wave tpes, particularl propagation velocit.
Contents Classification of wave tpes and their characteristics 1 2 3 1.1 1.2 2.1 2.2 3.1 3.2 Longitudinal Waves Pure longitudinal waves Quasi-longitudinal waves Transverse Waves Transverse plane waves Torsional waves Bending Waves Pure bending waves Corrected bending waves
1 Logitudinal Waves 1.1 Pure Longitudinal Waves Pure longitudinal waves can occur in solids, as well as in liquids and gases. This is defined as waves in which the direction of the particle displacements coincides with the direction of wave propagation. One can readil visualie such waves b studing the motion of two planes which in the undisturbed medium are parallel to each other and perpendicular to the direction of propagation. The kinematic relations are shown in the picture below: One plane is initiall at and displaced a distance ξ ; a second plane, which initiall is at a ξ distance d from the first, is displaced a distance ξ + d. The material whose initial length ξ was d thus eperience an etensional strain ε in the -direction, given b ε = (1) Such a strain is associated with a stress, here for the small deformation in relation to structureborne sound, Hooke s law holds. Then one can write σ = Dε (2) Or, b use of Eq. (1), ξ σ = D We can see from the picture above that the unbanlanced stress causes the element to accelerate, the corresponding equation of motion mabe written according to Newton s second law as σ ξ ( σ + d) σ = ρd = ρ (3) σ ξ It is convenient to describe the kinematics of a sound field in terms of velocit v B introducing the velocit, one ma rewrite Eq. (3) as σ v = ρ = ξ t (2a) (4) (5)
σ v One ma rewrite Eq. (2a), after differentiation with respect to time, as =D (6) We can observe that the relation between the two variables σ and v is such that the spatial derivative of the one is proportional to the time derivative of the other. So differentiate with respect to t or and combination of Eqs. (5) and (6) results in the wave eqution, D ( σ 2, v ) = ρ ( σ 2, v ) (7) We can have a look at the solution of this partial differential equation. For sinusoidal ecitation j Ω(t-/cL) 2 j Ω(t-/cL) 2 0 0 cl P = Pe the corresponding displacement v =v e jωt 0 jω Dv e ( ) = ρv e ( jω) c = L D ρ 0 j Ω(t-/cL) We can see that the velocit increases with increasing stiffness and decreases with increasing densit. 1.2 Qusai-Longitudinal Waves The previousl discussed pure longitudinal waves can occur onl in solids whose dimensions in all directions are much greater than the wave length. We would now have a derivation of the relationship between D (longitudinal stiffness of the material) and E (modulus of elasticit or Young s modulus). For the fist case, E is defined as the ratio of the stress to strain in the tension direction, which is under the condition of unconstrained cross section (cross-contraction phenomenon occurs). σ E = ( σ = σ = 0) (8) ε If the lateral contraction is constrained to ero, then there results a three-dimensional instead of a one-dimensional stress condition, because then there are produced the additional normal stresses σ and σ in the directions normal to the tension direction. These stresses reduce the displacement in the -direction. Poisson s effects are taken into consideration. Eε = σ - µ ( σ + σ ) Eε = σ - µ ( σ + σ ) (9) Eε = σ - µ ( σ + σ ) For the case where no cross-sectional contraction is permitted, namel for ε = ε = 0
2µ One finds b adding the last two of Eqs. (9) that σ + σ = σ, which, after substitution 1 µ into the first of these equations, leads to 2 2µ Eε = σ(1 ) 1 µ Thus the longitudinal stiffness which was introduced in Eq. (2), depends on the material parameters E and µ according to the relation: E E(1 ) D σ = = = µ 2 ε 1 2 µ /(1 µ ) (1 + µ )(1 2 µ ) Clearl, D is alwas greater than E. The wave equation differs from the pure longitudinal wave onl in replacement of D b E: E ( σ 2, v ) = ρ ( σ 2, v ) (11) E The propagation velocit here is c L C = (12) ρ This value is smaller than the velocit of pure longitudinal waves. For µ = 0.3, the difference between these two velocities amounts to 16%, which is not entirel negligible. It thus is important to note which longitudinal wave is meant in an given situation. (10)
2 Transverse Waves 2.1 Transverse Plane Waves Solids do not onl resist changes in volume, the also resist changes in shape. This resistance to changes in shape comes about because, unlike a liquid or gas, a solid can support tangential stresses on an cutting plane, even with the material at rest. It is the shear stresses which make it possible for solids to eist in it s own shape. It is also because of shear stresses that transverse plane wave motions can occur in solids bodies, where the direction of propogation is perpendicular to the direction of the displacement. See below the kinematic relations: Because the transverse displacements of two planes which are a distance d apart differ b an η amount d, an element which originall was a rectangle with sides d and d is distorted into a parallelogram. The shear angle γ η = (13) The shear deformations alwas are associated with shear stresses τ and τ, where the first subscript indicates the ais normal to the plane on which the stress acts, and the second indicates the direction of the stress. Moment equlibrium of the element requires that the shear stresses on two perpendicular plane must be of equal magnitude. These stresses are proportional to the strain γ the produce, so that τ = τ = G γ (14)
η With the aid of Eq. (13), τ = G (14a) The constant of proportionalit G is known as the shear modulus which will be derived later. η The velocit in the -direction is associated with displacement as v = (15) t Differentiation of Eq. (14a) with respect to time as The newton s law relation is τ v = ρ τ v =G (16) (17) This procedure can be totall comparable with the procedure in derivation of pure longitudinal waves. Then there ields wave equation G ( τ 2,v ) = ρ ( τ 2,v ) (18) G From which one finds the propogation velocit is given b c T = (19) ρ G must be related to E b noticing the fact that normal stresses are alwas associated with shear stresses, and vice versa. We now have a derivation: Consider, for eample, the diagonal planes of a square whose edges are subject onl to shear stresses. Equlibrium of the forces demands that there act on the diagonal plane a compressive o σ or a tensile stress. 2τ cos 45 = τ = σ (20) o cos 45 σ (1 + µ ) From Eq. (9), we can find that ε = E
Strains are related to the angle γ as 1- ε tan(45 ) 1 / 2 o γ γ γ = ε = 1+ ε 2 1 + γ / If we combines these equations with τ = Gγ Then we obtains the desired relation between G and E: E G= (21) 2(1+ µ ) One ma observe that the shear modulus is alwas considerabl smaller than the modulus of elasticit E, and thus much smaller than the longitudinal stiffness D. For µ =0.3, the ratios have the values c T / cl = 0.535, and c T / cl = 0.620. 2.2 Torsional Waves If a narrow beam is subjected to a torque, suppose the beam ais coincides with the -ais, dχ We can see the relations between the angles that: γd = rd χ γ = r where χ represents d the angular displacement, in radians, from the equilibrium position. It is evident to obe that τ = Gγ = Gr χ (22) χ Torsional moment is defined as M = T (23) Where T represents the torsional stiffness of a rod with an annular cross-section. If one introduce the time-derivative of the angle of rotation χ, that is the angular velocit about the χ ais, ω = (24) t M ω Then one ma differentiate Eq. (23) with respect to time to obtain = T (25)
Comlemented b the equation ω M = θ We finall reach the wave equation: T ( M, ) (, ) (27) 2 ω = θ M 2 ω Here θ represents the mass moment of inertia. T Propogation velocit in this case is ct Ι = (28) θ Now we have a discuss of the value of T and (26) θ. For rotational smmetric cross-section (like Π 4 4 Π 4 4 circular or annular cross-section). T = G( rα ri ) while θ = ρ ( rα ri ). Here r α and r i represents the outside and inside radii of the cross-section respectivel. c TΙ = G ρ (28a) If one consider a rectangle cross section instead, we make the cross section narrower in the mean time keep the area constant. We note that ecomes smaller and smaller as the heightto-width ratio becomes larger and larger. h b c c TI T c TI 1 1.5 2 3 6 10 0.92 0.85 0.74 0.56 0.32 0.19 For large values of h/b greater than 6, we can approimatel evaluate 3 3 ρ( bh + hb ) ρbh θ = 12 12 3 and 3 ρb h T 2b T. Then cti = = c 3 θ h We can see from above that the torsional velocit could be far less than transverse velocit. T (28b)
3 Bending Waves 3.1 Pure Bending Waves Bending waves are b far the most important for sound radiation because of the rather large lateral deflections associated with them. Bending wave differs largel from both longitudinal waves and transverse waves. It must be represented b 4 field variables instead of 2. Also the boundar conditions are more comple. Four filed variables are: 1) transverse velocit v 2) angular velocit ω 3) bending moment M Z 4) shear force F The lateral displacement η and the rotation of a cross section through the small angle β are η related b the approimate epression β = (29) β v Differentiate with respect to time then leads to ω = = (30) The rate of change of angular velocit with distance is equal to the time-wise rate of change of the curvature, ω v η = = ( ) (31) As is shown in elmentar strength of materials M 2 η B = 2 (32) M Combination of Eqs. (31) and (32) results in ω = B (33)
Application of the newton s law in verticle direction, one ma write F v F F + d = md ( ) (34) which reduces to F v = m (34a) M Equlibrium of an element ma be written M ( M + d) Fd = 0 (35) Which can be reduced to M = F (35a) From the sstem of Eqs. (30) (33) (34a) (35a), one obtains the equation for bending waves: 4 2 B ( v,,, ) (,,, ) 4 ω M F = m v 2 ω M F (36) B The propagation velocit here is represented b cb = m Ω (37) Now we give some remark on c B. It is obvious that the velocit is proportional to the input impulse frequenc. In this case, when we have a infinit frequenc impulse acting on it, the bending wave propagation velocit could also turn to infinit. This is in contradiction with the conclusion that longitudinal wave propagation velocit must be the largest one. So we need to modif the model to make sense. 3.2 Corrected Bending Waves The previousl discussed pure bending wave model needs to be modified in 2 aspects in order to be appl to a more generous condition.
1) We need to take into consideration of deformations which are caused b shear stresses acting on the cross section. That is the Timoshenko beam theor. Eq. (29) needs to be written as η = β + γ which results in the modification of Eq. (30) into v γ = ω + t. Substitute τ F F γ = = = into the previousl equation will finall G GS K generates v 1 F = ω (30a) K 2) We need to add the previousl omitted rotational inertia term when derive Eq. (35). M ω M ( M + d) Fd = Iρd results in M = ω F + θ (35b) From the sstem of Eqs. (30a) (33) (34a) (35b), one obtains the equation for bending waves: 4 2 4 4 B v v θ B v θ v + [ + ] + = 0 (38) 4 2 4 m m K K The first 2 terms correspond to the differential equation for the pure bending waves, the other 3 terms represent the corrections. θ m will occur if rotational inertia is considered. B K will occur if shear deformation is considered. The last term correspond to higher order correction. The propagation velocit for corrected bending wave is B c = BΠ m Ω h λ 2 (1 3.6( ) ) (39) References Cremer, Lothar, Heckl, Manfred. Structure-borne sound. Structural vibrations and sound radiation at audio frequencies. Published b Springer 1973.