FE Analysis of a Vibrating Rigid Circular Piston in Water

Transcription

1 FE Analysis f a Vibrating Rigid Circular Pistn in Water K. Jagadeeshl and M. S. Vijaya2 I-Sr. Lecturer, 2 -Visiting Prfessr, Center fr Electrnic Materials and Devices Research, M. S. Raaiah Schl f Advanced Studies, Bangalre. Abstract Finite eleent analysis f vibrating pistns in water helps in the design f snar prjectrs fr naval applicatins. This paper reprts the results f finite eleent analysis f a vibrating circular pistn f radius 0.1 radiating acustic waves int water. Acustic finite eleents and acustic bundary eleents are used t del the water bdy. The finite eleent sftware tl PAFEC is used fr the analysis. The acustic pressure values alng the axis f the pistn as a junctin f distance fr the pistn and frequency are evaluated. The radiatin ipedance f the vibrating pistn is als evaluated. The results f FE analysis are cpared with the theretical values. Keywrds: Snar Prjectrs, Acustic FE, Far Field Pressure, and Radiatin Ipedance Nenclatures: Sybl Descriptin p Pressure P Fluid density r Far field distance a Radius f pistn k Wave nuber c Speed f sund t Tie (i) Angular frequency V Velcity A. Wave length Z Ipedance U Unifr velcity Abbreviatins: FEM Finite Eleent Methd BEM Bundary Eleent Methd Units Pa kg/3 /s rad/s /s N-s/ /s In this paper, a finite eleent tl PAFEC is used t analyze axial pressure, far field pressure and radiatin ipedance f a rigid circular pistn vibrating in water. The results f the FE analysis are cpared with the analytical results. 2. THEORY OF VRA TING CIRCULAR PISTON 2.1 Pressure alng the axis f the pistn Cnsider a circular pistn f radius a unted n a flat rigid baffle f infinite extent. Let the pistn vibrate with a unifr velcity U at a frequency (i), nral t the baffle. The velcity is expressed U = va exp(jojt) as (I) 1. INTRODUCTION The surface f the pistn is divided int infinitesial eleents, each f which acts like a siple surce. Acustic radiatin fr a vibrating rigid circular pistn is f interest in any applicatins, which include lud speakers, pen-ended rgan pipes and snar prjectrs. In snar prjectrs the pistn vibrates under water generating ultrasnic waves used fr ranging and detectin. Study fthe far field pressure prduced by the vibrating pistn and its radiatin ipedance helps in the design f acustic transducers fr snar applicatins. Michalakis A et al [1] have ade easureents f the pulses radiated by circular pistns in water and have cpared with the theretical results. X Zhang et al [2] have studied the radiatin ipedance f a circular pistn vibrating in a visc-elastic ediu fr applicatins in edical diagnsis equipent. Pierre N Gelat et al [3] have created an FE del fr the deterinatin f effective radiating area The pressure at a distance by r fr siple surce ds is given dp = P dq (2) AT where P is the density f the surrunding ediu, c is the velcity f the acustic wave in the ediu and dq is the surce strength given by uds. The pressure p at the pint P at a distance r fr the center f the pistn f radius a (Fig. I ) is btained by integrating ver the entire surface f the pistn: f physitherapy treatent heads. P(r,e,t) = jpc i f+,ej(ilji-kr'jds s (3) SAsTECH 97 Vl. VI, NO.1, April 2007

2 x per, B,t) Cnsider an infinitesial area ds f the surface f the pistn. The tin f the eleent ds prduces at se ther eleent ds' at a distance r fr ds, an increental pressure dp. The ttal pressure at ds' is btained by a siilar integratin as in equatin (3) with r' replaced by r [5]. The ttal pressure p at ds' can be btained as: P-JPC- _. V II -e A. r j{aji-kr)d's s where r is the distance between ds and ds'. (7) Fig. Acustic pressure circular pistn f radius a p(r, e, t) at a pint P fr a The integral (eg.3) can be easily slved fr (a) pints alng the axis f the pistn (acustic axis) and fr (b) pints in the far field regin. The pressure aplitude alng the acustic axis f the pistn is btained as [1] T get the ttal frce fs n the pistn, the abve pressure p, ust be integrated ver the surface area ds'. That is, Is = fpds' The tw integrals, that is, ne ver ds and the ther ver ds', include bth the frces i.e., the frce n ds' resulting fr ds, and the frce n ds resulting fr ds'. The tw frces are equal fr acustic reciprcity. S the frce can be btained, by chsing the integratin liits apprpriately t include the frce between each pair f eleents nly nce and ultiplying it by tw [5]. (8) Fr far distances (;-) «1 and s the square rt ter abve siplifies t [I+~(; f} In additin, fr frequencies satisfying!... > ka (Le., fr distances r > 1l(l2 / A.), the a 2 pressure aplitude n the axis has the asypttic fr: The expressin fr radiatin ipedance is btained by dividing the frce by the velcity aplitude V. This gives where S = na2 and R1 and XI are special functins given by (9) Paxia/(r) = ~ pcv(;-)ka (5) 2.2 Radiatin ipedance The radiatin ipedance f a vibrating pistn in water is f special interest fr underwater acustics applicatins. The radiatin ipedance is defined as the ttal nral frce acting n the surface divided by the velcity f the vibrating surface. It is evaluated by dividing the pistn int a nuber f sall eleents and integrating ver the whle surface [4]: RJ XJ = 1_ 2J.(x) x = 2HJ(x) x Jl (x) is the Bessel functin and HI (x) Struve functin [4,6]. is the At lw frequencies (i.e., fr ka «1 ) the radiatin resistance and reactance can be apprxiated t: (10) Zr = V~S (6) dfs is the frce n the eleentary surface ds where the velcity is V. (II) SAsTECH 98 Vl. VI, N.1, April 2007

3 3. FINITE ELEMENT ANALYSIS Eklant Grups The sftware tl PAFEC is used fr the FE analysis f circular pistn vibrating in water. The circular pistn f radius 0.1 is deled using structural shell eleent. Only 30 segent f the circular pistn is deled. The water ediu surrunding the pistn is deled using acustic finite eleents and acustic bundary eleents. The frequency range f the analysis is IkHz t 30 khz. The 30 segent f the pistn is divided int eight eleents, which aunts t 4 eleents per wavelength at the highest frequency. The thickness f the acustic finite eleent is taken t be The acustic eleent used is 20-nded pressure based fluid finite eleent. The acustic bundary patches n tp f the acustic FE eleents are created n a face-t-face basis. Fr the bundary eleents, six-nded triangular and eight-nded quadrilateral patches are used. The rientatin f the bundary patches are such that the psitive nral pints ut f the fluid int the structure. The FE del f the pistn in water is shwn in Fig.2 (a) and (b). EIO<T1QI1! Grup' Fig.2 (b) The FE del f the pistn and the acustic BE patches The pistn is cnstrained in all degrees f freed except Uz i.e., it is peritted t vibrate nly in the nral directin. The bundary cnditins fr the acustic bundary eleent is selected t have a rigid bundary (i.e., (ap / an) = 0 ) in the planes y=0 and z=0. A unifr displaceent f 2 x 10-9in the Z directin is specified fr the circular pistn. The slutin type used fr the FE analysis is the CHIEF frulatin ethd [7]. 4. RESULTS AND DISCUSSIONS 4.1 Far field pressure The pressure at a distance f 1 fr the pistn n the axis is deterined as a functin f frequency. The pressure values calculated using equatin (4) and thse btained using PAFEC are tabulated in Table I. The graph f pressure as a functin f frequency is shwn in Fig (Hz) PAFEC Frequency Analytical Pressure distance Aplitude at 1 Fig 2 (a) The FE del f the pistn in water Table laxial pressure aplitude at a distance f 1 SA STECH 99 Vl. VI, NO.1, April 2007

4 1.20 ria,!i e 300,35 Q ,35 200,35 100,35 0,35 +ANAl YIlCA Frequency Fig.3 Axial Pressure aplitude at a distance f 1 fr 4.2 Axial Pressure the pistn as a functin Hz f frequency The pressure aplitude alng the axis fthe pistn at a frequency f 25kHz, calculated using equatin (1) and thse evaluated using PAFEC are tabulated in Table 2 and the graph f the pressure aplitude as a functin f distance fr the pistn is shwn in Fig.4. PAFECAnalvtical Nralized Pressure at 25 khz ~ 1.00 :: 0.80 SAsTECH i::l~rl=b=ee ria (12) Fig. 4 Axial pressure aplitude as a functin f distance fr 4.3 Radiatin Ipedance the pistn Theretical values f radiatin ipedance calculated using equatin (11), are cpared with the values btained using PAFEC. In PAFEC the real and iaginary values f pressure are btained at each nde n the acustic finite eleent at the slid-fluid interface. The pressure values are assued t be cnstant ver a fixed radius. The average frce n the pistn is calculated by suing ver the average frces n annular rings fred between the FE ndes using the equatin [8], The radiatin resistance and reactance are deterined by dividing the real and iaginary average frces by the velcity. 95 The graph f the radiatin resistance and reactance as a functin f wave nuber (2ka) btained using PAFEC are cpared with the theretical values in Fig u ~ 0.8 'll D -+-Zr I(pAfEQ.E 0.6 l:: ~ 'ii 0.4 a: II- Zl.g(pAfEQ -+-Zr ljan.lytlcaq..... Zl.gjAn.lytlcaQ Fig.5 Radiatin resistance and reactance versus 2ka 2ka Table 2 Axial pressure aplitude at 25 khz The FE analysis values btained using PAFEC are fund t agree well with the theretical values fr all the paraeters up t axiu frequency f 30kHz. Fr still SAsTECH 100 Vl. VI, N.1, April 2007

5 higher frequencies the FE esh density ust be increased t get gd agreeent with thery. 5. CONCLUSIONS Acustic finite eleent analysis can be effectively used fr siulatin f vibrating pistns in water. The analysis can be extended t pistn arrays, which helps in the design f snar prjectrs fr naval applicatins. The cputatins help in the deterinatin f useful paraeters such as axial pressure, far field pressure, radiatin ipedance, directivity etc. fr ACKNOWLEDGEMENT The authrs are thankful t the Directr MSRSAS his kind encurageent. REFERENCES [1] Michalakis A. Averkin and Marie F. Hailtn, 'Measureents f finite aplitude pulses radiated by plane circular piatns in water', J. Acu. Sc. A., 1993, vlue 94, Issue 3, PP [2] X. Zhang and T J Rystn et ai, 'Radiatin ipedance f a finite circular pistn n a viscelastic half-space with applicatin t edical diagnsis', J. Acu. Sc. A., 2001,Vlue 109, Issue 2, PP [3] Pierre N. Gelat, Bajra Zeqiri and Mark Hdnett,' A finite-eleent del f the aperture ethd fr deterining the effective radiating area f physitherapy treatent heads',ultrasnics Vlue 43, Issue 5, March 2005, Pages [4] 0 Stansfield, 'Underwater Electracustic Transducers', First Editin, Bath university press, [5] L E Kinsler et al. 'Fundaental f Acustics', Furth Editin, Jhn Wiley & Sns, [6] W S Burdic, 'Underwater Acustic Syste Analysis', Secnd Editin, Peninsula Publishing, [7] PAFEC-FE User Manual, PAFEC Ltd., UK. [8] P.R. Sepanishen, 'The tie dependent frce & radiatin ipedance n a pistn in rigid infinite baffle', J. Acu. Sc. A., March 1971, vlue 49, Issue 3B, PP SAsTECH 101 Vl. VI, NO.1, April 2007