DIFFERENT SOUND WAVES THROUGH THE JUNCTION BETWEEN THE TWO FLOOR PLATES

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1 DIFFERENT SOUND WAVES THROUGH THE JUNCTION BETWEEN THE TWO FLOOR PLATES Vytautas J. Stauskis Vilnius Gediminas Technical University, Pylimo 6/Traku, LT, Vilnius, Lithuania. Abstract. It analyzes the sound waves of different amplitudes of cm thick floor concret plates coupled by and cm junction.. According to calculations, when the length of the junction is small, the amplitude of the longitudinal wave is increasing when frequency accelerates. When the length of the junction is increasing, the flexural, longitudinal, and distortional wave amplitude has a high resonant character. When the length of the junction increases times, the wave amplitude has a repeated every Hz resonance character. Traveling distortional waves have the maximum amplitude of junction.. Not flexural and longitudinal waves, but traveling distortional waves.have the maximum amplitude of floor. In a small jungtion length, the maximum amplitude in the frequency ranges between to 7 Hz. in a traveling distortional wave in a floor plate. When the junction length is large, all the wave amplitudes in floor plate have a repetitive. Keywords: flexural, longitidunal and distortional wave amplitude, modulus rigidity, wave velocity, wave number.. Introduction Sound energy spreads through plates not only directly but also through external walls. When reaching the construction, the direct sound waves create flexural, longitudinal and distortional waves. While they are traveling, transformation takes place in the construction junctions.: direct waves transform into flexural and lexural to longitudinal. Flexural waves radiate a sound wave energy into an adjacent room, and the direct sound waves carry energy structures. Flanking transmission indirect sound propagation paths of structure has for long been considered (Cremer 97, 98), (Budrin and Nikiforov 964). (Bhattacharya at al. 97) examined the different sound waves in a thin magnitude. cm thick aluminum panels, which junction plate. (Zaborov 965) made a very useful study of flanking transmission by a multyjunctionmodel, though he prevented exication of longitudinal waves. (Mead and Markus 98) examined flexural and longitudinal waves in jungtion. (Brunskog and Davidsson 4) sound propagation structures used by the finite element method. (Clasen and Langer 6) and (Kruppa ) examined a direct measurements of the structure-borne sound intensity has been tried earlier on thin-plate constructions in laboratory set-ups. (Gibbs 986) investigated wave type conversion for a various T- junctions. (Langley and Heron 99) described structure-borne sound transmission through various junctions at random incidence. The plates were either connected through a beam or connected directly forming a rigid line joint. (McColum and Cuschieri 99) analyzed vibration transmission through L junction evaluating inplane waves, schear and rotary inertia effects in the plates. (Craven and Gibbs 98) estimeted sound energy attenuation at the joints of buildings by using thin plate theory. (Hopkins ) applied SEA theory for calculation of joints and compared the obtained rezults similar to the calculated results obtained by finite elements method. (Mees 99, 996) analysed the problems related to sound transmission through the joints of ligh structures. (Zhang at all ) developed theoretical models for vibration transmission of ligh-heavy structures. (Stauskis and Mickaitis 4, 5) investigation the influence of geometrical and material properties were determined for the sound energy attenuation in the joints of buildings, and transmission losses for the in-line joint, for the L joint and the T joint. However, research is lacking in the sound wave transmission from one floor plate to another through a real junction.. Traveling of the sound energy through building structures Let s consider the case where the two concrete floor plates are coupled with a concret junction element. 9

2 Accepted that this element is the single-length plate. Computational scheme is shown in Fig. When the sound waves are spreading through the two floor plates through the junction, there will appear the longitudinal, flexural and distortional sound waves. Their amplitudes will be different. Sound energy, which is radiated from floor plate is dependent on wave amplitude. Fig. Coordinate system and positive direction of displacements.,, 4 and 5 floor plates; coupled junction Flexural, longitudinal, and distortional waves traveling through a connecting element in the bottom of the floor plate will create flexural, longitudinal, and distortional waves. Sound energy into the lower room will be mainly radiated by flexural waves. It is therefore important to know how the wave amplitude of all the mentioned waves change and which of them has a greatest influence. After equasion system is settled, the following sound wave amplitudes of different dependencies are received. If the sound spreads from the plate into the junction I, it will create other sound waves in the junction. Junction will appear in the direct sound wave, which is the amplitude G and reflected from the unit II of the direct wave, the amplitude of H. The traveling direct sound wave amplitude is equal to the junction ( js ) ( a j) G = s( a j) s(+ a j). () The reflected direct sound wave amplitude is equal to the junction ( js ) ( + a j) H = s( a j) + s( + a j) Ehp were: a = ; s = exp( jp L) ; s = exp( jpl) ; Dk E, h, p the junction material modulus of elasticity, its thickness and the longitudinal sound waves whale density into junction D floor plate stiffness () k - flexural wave number in plate. Eh D = ( µ ) ; E floor plate material elastic modulus; h floor plate thickness; j =. k m D = ( 4 ) ω flexural wave number in junction; ω p = longitudinal wave number in junction. cl Sound wave energy from one plane to another pass through the junction. Emanating from the floor plate flexural wave junction will create direct, flexural and distortional waves. The traveling flexural wave amplitude in junction r ( γ+ ) B = α γ + γ γ γ + γ { r } ( ) ( j) r 4 4 j( 4 4). () The reflected flexural wave amplitude in junction r ( γ j ) A = B r ( γ+ ). (4) The traveling distortional wave amplitude in junction cr ( r) ( r+ jr) C = B r ( γ+ ) The reflected distortional wave amplitude in junction can by expresed to equal r ( + j) F = B ( γ+ ) k were: r = exp( jkl) ; r = exp( jkl) ; k (5) (6) α= ; Dk γ=. Dk It is very important flexural wave amplitude of the floor plate, because it primarily radiation sound energy into the room below. Traveling flexural wave amplitude B 4 and B 5 in floor plate 4 and 5 B4 α α = j [ α ( j) [ α ( j) jc r( jc) r( c ) [ α (7) B 5 = jb 4 C 4 ( j). (8) Distortional wave amplitude 4 C and 5 C departing from junctions in the the plate 4 and 5 can by expresed to equal 9

3 C C α = [ α ( j) B 4 4 α = [ α ( j) B 5 5 (9) () Longitudinal wave amplitude G 4 and G5 departing from junctions in the plate 4 and 5 can by expresed to equal G = G = ( r A + r B + r C + F ) () 4 5 = exp( k L) were r. Computation results The way the sound energy will be exposed to radiation in the isolated room will depend on the overlay of different variations of speed and wave amplitude. It is therefore very important to know how the frequency and geometric parameters of the elements is flexural, distortional, and the longitudinal wave amplitude Fig shows the longitudinal waves propagating amplitude in junction. The directly reflected wave amplitude depends on the low frequency and its value is much lower. When the junction length increases times to cm, the reflected wave amplitude becomes very small and plays no role. The wave amplitude increases with increasing frequency uniformly. Repetitive resonanses appear at high frequencies. Variation of these amplitudes,is influenced only by floor plates and junction dimensions. Fig shows the evolution of different wave amplitude in junction , Series Series Series Series Series Series Series Series Series Series c) Series Series Fig. Longitudinal sound wave amplitude change in junction, depending on the frequency and length. the junction length L = cm the length L = cm. Floor thickness is cm. Series traveling longitudinal wave G ; Series reflected longitudinal wave H When the junction length is small, the traveling longitudinal wave amplitude increases synchronically to frequency acceleration. Starting with approximately 7 Hz, an amplitude already has the resonant character. Its value exceeds the value of the triggering amplitude. 9 Fig. Sound wave amplitude change in junction depending on the length and frequency. Floor thickness of cm the junction length L= cm. Series - flexural wave amplitude B 4 ; Series - amplitude C 4 ; Series B 5, and series 4- amplitude C 5. the junction length L = cm. Floor thickness of cm. Series traveling flexural wave B ; Series reflected flexural wave A ; c) the junction length L = cm. floor thickness of cm. Series traveling distortional wave C ; Series reflected distortional wave F

4 When the junction length is small, the maximum magnitude of junction amplitudes has a distortional wave, which reaches its peak at 6 Hz and its amplitude is twice higher than that of the triggering amplitude. It s worth to note, that the traveling flexural and reflected distortional wave amplitudes are uniform and reach their maximum almost at 6 Hz. Completely different character of the amplitude variation is noticeable when the jungtion length increases times. The wave amplitudes have a repeated approximately every Hz resonance character. The maximum amplitude have traveling distortional waves. Their amplitude at repeated every Hz resonance frequencies varies from minimum. at 44 Hz to a maximum of.5 at 6 Hz. It shows that they greatly exceed the triggering wave amplitude. Traveling flexural wave amplitude also has resonances with maximum also repeated every Hz, and its value is between.9 and.5. Large amplitude is a reflected flexural wave. Reflected distortional wave amplitude varies from.85 to. at all repetitive resonances. The results are surprising in that it is not traveling flexural and longitudinal waves that have very high and the maximum amplitudes, but traveling distortional waves. So it is interesting how they respond to the wave formation of floor panels. Sound wave amplitude change in floor plate, depending on the junction length schow in Fig 4. Flexural waves in junction formed not the longitudinal wave plate, but flexural waves in this floor plate. In formula 5 there are members of the flexural wave numbers in floor plate, but there are no waves with the longitudinal wave numbers. Traveling flexural wave amplitude B 4 and B 5, depends on the longitudinal and flexural wave amplitude in junction. The same is in distortional waves C 4 and C 5 in floor plate. These waves occur where there is an active force. Moving further away from the triggering location they decrease by the exponent. When falling into the junction II traveling flexural and longitudinal waves in floor plates stimulate the amplitude B 4, B 5, C 4, C 5. Their biggest contribution is in radiation of sound energy. It will greatly depend on physical and geometrical parameters of a junction. If the dimensions of this element are much smaller than the floor plates, then the traveling flexural wave amplitude in floor B 4 and B 5, will depend on the longitudinal wave amplitude in the junction. Meanwhile, the flexural wave in junction will have a small influence on flexural wave emergence in floor panels B4 and B 5. Part of the energy carried by the longitudinal wave in junction elements is used to obtain flexural bands B4 and B5, and others to create a distortional wave amplitudes C 4 and C 5. Calculations show that for small junction length traveling distortional wave C 5 in floor plate 5. has the maximum amplitude of,5,8 in the frequency range from to 7 Hz. In floor plate 4 traveling distortional wave C 4 has smaller amplitudes and one c) Series Series Series Series4 Series Series Series Series Fig 4. Sound wave amplitude change in floor plates, depending on the junction length. Floor plates thickness is cm. the junction length of cm. Series traveling flexural wave B 4 in floor plate 4; Series traveling distortional wave C 4 in floor plate 4; Series traveling flexural wave B 5 in floor plate 5; Series 4 traveling distortional wave C 5 in floor plate 5; the junction length of cm. Series amplitude B ; Series - amplitude A ; c) the junction length of cm. Series -amplitude C, and series - amplitude F. maximum to the 4 Hz. Different results are obtained when the junction length is large. All wave amplitudes have a significant resonance repetitive character. The maximum amplitude of.6 at 6 Hz has a traveling distortional wave C 5 in the floor plate 5. However the traveling flexural wave amplitude B5 in floor plate 5 is greatly lower and ranges from. to.9 within the range of repetitive frequency resonances. Traveling flexural wave amplitude B4 in floor 4 with increasing frequency increases with repeated resonance to higher fre- 9

5 quencies. Traveling distortional wave amplitude C 4 in floor plate 4 is almost the same as traveling flexural wave amplitude B 5 in floor 5. Conclusions When the flooring junction is small, up to cm length, the traveling longitudinal wave amplitude with increasing frequency accelerates and reaches its maximum at 7 Hz. Reflected longitudinal wave amplitude slightly depends on the frequency and its value is much lower. When the junction length is small, the distortional wave has a maximum amplitude. It reaches its peak at 6 Hz and its amplitude is twice higher than the wave triggering amplitude. Traveling flexural and reflected distortional wave amplitude are uniform and reache their maximum almost at 6 Hz. When the length of the junction increases times amplitudes of all waves arehave a repetitive approximately every Hz resonance character. Traveling distortional waves have the maximum amplitude. Their amplitude every Hz has a repetitive resonance character. Traveling distortional wave amplitudes also have resonances with maximum repeated every Hz, and their value is between.9 and.5. Very large amplitudes are reflected flexural waves. References Cremer, L. 97. The propagation of structure-borne sound. Dep. Of Scientic and Industrial Research. Report N. Zaborov, V. I Sound insulation of double walls jointed at the edges, Soviet physics Acoustics : 5 4. Cremer, L.; Heckl, M.; Petersson, B. 98. Structure- Borne Sound, rd edition, ISBN Budrin, S. V.; Nikiforov, A. S Wave transmission through assorted plate joins. Soviet Physics Acoustics, vol. 9. Bhattacharya, M. C.; Mulholland, K. A.; Crocer, M. I. 97. Propagation and radiation of sound energy flanking transmission via structural junction, Building Acoustic : Mead, D.-J.; Markus, S. 98. Coupled flexural longitudinal wave motion in a periodic beam, Journal of sound and vibration 9(): 4. doi:.6/-46x(8)999- Kruppa, Per.. Measurement of structural intensity in building constructions. Acoust. Soc. Am. 7(6): Gibbs, B. M Mode coupling and energy partition of sound in a system of plate junctions, The Journal of Sound and Vibration 4(): 7 6. doi:.6/s-46x(86)85- Brunskog, J.; Davidsson, P. 4. Sound transmission of structures; a finite element approach with simplified room description, Acta Acustica United With Acustica 9(5): Clasen, D.; Langer, S. 6. Consideration of flanking transmission within numerical simulation of sound insulation, Acta Acustica United With Acustica 9(sup. ): 47. Walsh, S. J.; White, R. G.. Vibration power transmission in curved beams, The Journal of Sound and Vibration (): doi:.6/jsvi Craven, P. G.; Gibbs, B. M.98. Sound transmission and mode coupling at junctions of thin plates. Part I: representation of problem, The Journal of Sound and Vibration 77: doi:.6/s-46x(8)877- Hopkins, C.. Vibration transmission between coupled plates using finite element methods and statistical energy analysis. Part : Comparison of measured and predicted data for masonry walls with and without apertures, Applied Acoustics 64(): doi:.6/s-68x()6-8 Mess, P.; Vermeir, G.; Bosmans, I Vibration attenuation at junctions of joint connected plates, Acustica 8: 9 6. Mess, P.; Vermeir, G. 99. Structure borne sound transmission at elasticaly connected plates, The Journal of Sound and Vibration 66: doi:.6/jsvi.99.8 Vermeir, G.; Osipov, A Sound transmission in buildings with elastical layers at joints, Applied Acoustics 49(): 4 6. doi:.6/-68x(96)6- Zhang, X. M.; Fatemi, M.; Kinnick, R. R.; Greenleaf, J. F.. A model for vibration transmission of ligh-heavy structures, Applied Acoustics 64(): 7. doi:.6/s-68x()66-x Stauskis, V. J.; Mickaitis M. 4. Estimation of vibration transmission coefficient at asymetric junctions of buildings, in Abstract of the 8th Internationalconference Modern building materials, structures and techniques, May 9. Vilnius: Technika, Stauskis, V. J.; Mickaitis, M. 5. Vibration attenuation at asymetric cross-form joints of buildings, Journal of Civil Engineering and Management (): 9 5. Mickaitis, M.; Stauskis, V. J. 5. Vibration transmission through joints of walls and columns in framed buildings, Journal of Civil Engineering and Management (): Langley, R. S.; Heron, K. H.99. Elastic wave transmission through plate/beam junctions, The Journal of Sound and Vibration 4(): 4 5. doi:.6/-46x(9)995-w McColum, M. C.; Cuschieri, J. M.99. Bending and inplane wave transmission in thick connected plates using statistical energy analysis, Journal of Acoustical Society of America 88(): doi:./.44 94