Vibration Characteristics Oblate Shell "With and Without Framed Structure"

Transcription

1 Al-hwarizi Engineering Journal, Vol. 6, o. 2, PP 1-1 (21) Al-hwarizi Engineering Journal Viration Characteristics Olate Shell "With and Without Fraed Structure" adhi H. Ghlai* Hate R. Wasee** * Departent of echanical Engineering/ Dijlah University College ** Departent of echanical Engineering/ Engineering College/ University of Baghdad (Received 27 Octoer 29; accepted 28 April 21) Astract This paper presents the results of investigating the virational characteristics of olate dish with and without fraed structure. A finite eleent ethod, was applied to the dynaic analysis of olate spheroidal shell. Different types of eleents were considered in one diension and two diensions. It was found that the natural frequencies of olate shells had two types of ehavior against increasing the shell thickness and eccentricity, which are the erane ode and ending ode Since the erane odes natural frequencies tend to increase with the increasing the eccentricity of olate, while the ending odes natural frequencies decrease with the increasing the eccentricity till reach the optiu eccentricity. Experiental Validation tests have een carried out on the spheroid dish y using the odel analysis technique. eywords: Viration, structure. 1. Introduction When the large structure such as aircrafts, ridges, ships, vehicles and tall uilding eing constantly acted on y wave and otion, the resulting forces can introduced Virations at the resonance or repeated any ties which ay lead to structural failure. An essential requireent of an engineering structure is to sustain the loading applied to it during service life without failure. To insure that and to otain an optiu design of structure the dynaic characteristics ust e estalished. The dynaic characteristics of any structure are governed y its stiffness and ass properties. The ost iportant dynaic characteristic of the structure is the natural frequencies which are a function of aterial constants and diensions of structure. It is worthy to indicate the industrial applications and iportant of plates and shells structure. One of iportant used type of elastic thin shells which has a particular interest is the olate shells. The olate type of shell has any practical applications, one interesting application includes the protective shell used as the housing of the early-warning scanner of the airorne warning and control syste aircraft (AWACS). The study of the dynaic analysis of plates and shells has een treated y any investigators using different ethods. The olate spheroidal shells considered extensively have een restricted to few works, Benzes and Burgin [1] were solved the prole of the free viration of thin isotropic olate shells using Galerkin s ethod. Penzs [2] was extended this work to include thin orthotropic olate shells. Curved lades can e odeled approxiately y fact eleent [3]. Curved shell eleents ay provide a ore accurate facility for the finite eleent odeling of curved lades. The asic equations which descrie the ehavior of a thin elastic shell were originally derived y Loue [4]. pawsey [5] explained the asic proles coon to ost shell eleents, and which restrict ost eleents class of shells, either thin or thick, depending on the parent theory used for developing the eleent. Recently the concept of quasi coparison function has een introduced for the Reylegh Ritz discretization in self-adjoint eigen-value prole [6]. Baich [7] is studied the

2 adhi H. Ghlai Al-hwarizi Engineering Journal, Vol. 6, o. 2, PP 1-1 (21) staility and natural virations of shells with variale geoetry and echanical paraeters. 2. uerical Procedure General Large nuer of different theories has een derived y various Scientist, all purposing to static and dynaic analysis of shells. Differences aong these theories are due to various assuptions-shells theories ost incorporate ath flexural and extensional deforations. A structure, such as a dish, ay have zones with the variale thickness and construction when it is difficult to use one eleent type. If a indlin Facet eleent is eployed for a thin structure shear looking will occur and will lead to inaccurate results. The flat or facet shell eleent is the appropriate and easily eployed for curved shells. Two asic facet shell eleents are introduced. The irchoff facet eleent ased upon a coination of the 2-D plane stress eleent and the thin plate ending eleent. Thick facet eleent is ased upon a coination of the 2-D plane-stress eleent and the indlin plate eleent. Boundary conditions The dish is assued to e claped at two support legs and free at all other points. The oundary conditions for the claped point ay e written as follows: u v w at claped ends. xi i i i = at claped ends. yi zi Another type of oundary conditions were considered which are concerned with the half olate shell. This shell was claped at its peripheral ase. The three coponents of translation and three coponents of rotation of all nodes of the olate shell peripheral ase are assued equal to zero. Governing Equations The finite eleent procedure for the estiation of natural ode shape and natural frequency of olate dish can e forulated y using facet eleent and the principle of iniu potential energy theore. Express the total potential of the eleent in ters of nodal, displaceents, if the iniu total potential energy theore is used, then the equiliriu* equations for the cleent can e estalished. The total potential energy for the eleent can e defined as follows [8]:- X = U -W Where X is the total potential energy of the eleent, U is the strain energy of the eleent, W is the work done y external force F: W=F. Where is the nodal deforation. It can e deduced that t 2 t t ( B z B ) D( B z B dz L t 2 ) Then it can e proved that: L t B t B DB DB : hold for erane : hold for ending L : local stiffness atrix of the eleent Siilarly, ass atrix can e deduced that: where t D t D (1) (2) Where is the shape function. For thick facet eleent, the transverse shear strain was considered in addition to the erane and ending strain. The nodal dynaic equation for olate shell is ( t) C ( t) ( t) F( t)... and for undraped free viration the equation of otion ecoe. ( t) ( t) (3) Fig.(1) illustrated the flow chart of the aove equations. Fig.(2) shows the used esh of the olate dish. 2

3 adhi H. Ghlai Al-hwarizi Engineering Journal, Vol. 6, o. 2, PP 1-1 (21) Input Data Calculate atrix t 2 t t ( B z B ) D( B z B dz t 2 ), t B DB t B DB Calculate atrix t t D D Solve the equation ( t) ( t) to get the natural frequencies Fig.1. Flow Chart of the Finite Eleent Analysis. Fig.2. esh of the Olate Shell. 3

4 adhi H. Ghlai Al-hwarizi Engineering Journal, Vol. 6, o. 2, PP 1-1 (21) The standard solution of dynaic prole The ajor prole in structural analysis using the FE is the solution of a set of siultaneous algeraic equations. Iterative ethods are those which start with an initial approxiation and which y applying a suitale chosen algorith, Leads to successively etter approxiation. When the process converges good approxiate solution is expected to e otained. The accuracy and the rate of convergence of iterative ethods Vary with the algorith chosen, the coon approaches used for iterative solution are siple iteration ethod, suspace operation technique with the static and dynaic condensation, the asic steps of siple iteration and the suspace iteration algorith can e found in [9, 1 ] 3. Experiental odel Experiental values were otained for two odels which were constructed fro steel. The diensions for the two odels are given in tale(1). Basically, each one of the odels was constructed as follows. For each odel a flat sheet of plate was cut to proper diensions as shown in Fig.(3). After fraction of the circular plate on the ench of the pressing achine and y using a suitale press die oving up and down to ipact the circular plate several ties get on the dish end as shown in Fig.(3.). Repeat the pressing operation with another dies to otain the final half shell eer as shown in Fig.(3.c). Locate the position of legs and ake two holes in one half shell and then use the welding ethod to fix the two legs in its position. Fit, precisely the edges of the two half shell then welded together. The welding was ground flat on the outer surface of shell to otain on the two odels as shown in Fig.(3). Fig.(4) shows a lock diagra for different equipents and instruents used in testing the odels. The experiental work was carried out to easure the natural frequencies of the two olate dishes. The results of experiental and theoretical approaches as listed in tale (2) and tale (3). A natural frequency was distinguished y oserving the sharp increase in axiu aplitudes of the output signal and y the intensity of the acoustic tone eitted with phase angle. Tale 1, Specifications of Tested odels Paraeter odel (1) Values & units Radius of shell curvature (R) Radius of dish end (r) Height of dish end (h1) The extended edge height (h2) Total height of half shell (H) odulus of Elasticity ( E ) odel (2) Values & units GPa 28 GB 1 Density ( ) 785 kg\ kg\ 3 Poisons ratio ( ).3. D.3. D Tale 2, atural Frequencies (Hz) of Dish odel I ode Exp. of Frequencies Theo. Frequencies Discrepancy %

5 adhi H. Ghlai Al-hwarizi Engineering Journal, Vol. 6, o. 2, PP 1-1 (21) Circular Plate with Variale DS DS A- Circular Plate Olate Head With 5 Thickness R=65 TS R 65 B- Olate Head Shallow Head With 5 Thickness After Foring, R=65, r=75 TS R r 65 C- Shallow Head Farication and Processing Step of odel 65 5 R=65 WL 4*4*3 thick. 14*14*5 thick. Head 5,Efter Foring R=65,r= WL. 4*4*3 thick. 14*14*5 thick. Fig. 3. Types of odels. 5

6 adhi H. Ghlai Al-hwarizi Engineering Journal, Vol. 6, o. 2, PP 1-1 (21) ultiplexer Lissitous Display Data Controller 2 Channel Aplifier Group Selector (2)B& 4391 (1)B& 286 Viration Exciter B&276 odel Recorder Force Transducer (1) Acceleroeter (2) B&234 HP. Coputer Analyzer Charge aplifier B&2635 Level Phase control Viration Exciter Control Fig. 4 Block Diagra of Dynaic easureent Syste. 6

7 atural Frequencies (Wn) in Hz atural Frequencies (Wn) in Hz adhi H. Ghlai Al-hwarizi Engineering Journal, Vol. 6, o. 2, PP 1-1 (21) Tale 3, atural Frequencies ( Hz) of Dish odel II ode Exp. of frequencies Theo. frequencies Discrepancy % optiu value of eccentricity, the natural frequencies will e decrease until the olate shell ecoes circular plate ( e = 1 ) as shown in Fig (1), while Fig (9) show the effect of shell thickness on the Von ises stresses of olate claped at its peripheral ase Theoretical results Zienkiewicz, plane stress eleent Experiental Results For theo. Results (=58, E=14, DOF=336) 4. Discussion and conclusion In the foregoing work it is deonstrated how the finite eleent ethod and experiental odel analysis, can e eployed to deterine the free viration frequencies and ode shapes of siplified representation of olate dish. The coparison of predicted and easured natural frequencies are written acceptale engineering agreeent for the syste tested. Two types of results are recorded in tales (1) and (3). Figure (5) and figure (6) show the variation of natural frequencies with the ode nuer of experiental and theoretical results. Figures (7) and (8) show the natural frequencies of virations as function of the shell thickness once of the eccentricity value e =.925 once again for half olate of eccentricity e =.585 claped at its peripheral ase for several odes. atural frequencies are seen to have two types of ehavior against increasing the shell thickness. One type, which is associated with the erane odes, reain unaffected y the thickness variations, which the other type, which is associated with the ending odes. Tends to increase with the thickness. The dynaic ehaviour of olate shells depend upon the coupling and uncoupling of erane odes and ending odes. It was found that when the eccentricies of olate increase the natural frequencies will e increase until reading an optiu value of eccentricities. after ; the ode Fig.5. Experiental Results of atural Frequencies versus ode for Olate Dish with (a=325, =92, ts=5, at: steel alloy) Dish with internal structure. Dish without internal structure. Zienkiewics, plane stress eleent for shell. To, ea ele. For ea. For dish without int. struct. (=58, E=14, DOF=336) For dish with int. struc. (=66, E=1196, DOF=3512) ode Fig.6. Theoretical Results of atural Frequencies versus ode for Olate Dish with & Without Internal Structure with (a=325, =55, ts=5, tp=3, at: steel). 7

8 atural Frequencies (Wn) in Hz Von issis stress in / atural Frequencies (Wn) in Hz Von issis stress in / adhi H. Ghlai Al-hwarizi Engineering Journal, Vol. 6, o. 2, PP 1-1 (21) 6 5 irchoff, plane stress eleent, =1 irchoff, plane stress eleent, =4 irchoff, plane stress eleent, =7 irchoff, plane stress eleent, =1 erane, plane stress eleent, =1 erane, plane stress eleent, =4 =121, E=228, DOF= First ode. Second ode. irchoff, plane stress eleent (=191, E=26, DOF=554) Thikness (t) in Thikness (ts) in. Fig.7. The Effect of Shell Thickness on the Frequencies of Half Olate Shell Claped at Peripheral Base with (a=185, =7, e=925, at: Aluinu). Fig.9. The Effect of Shell Thickness on the Von isses Stress of Olate Shell Claped at Peripheral Base with (a=185, =7, e=925, at: Aluinu) irchoff, plane stress eleent, =1 irchoff, plane stress eleent, =4 irchoff, plane stress eleent, =7 irchoff, plane stress eleent, =1 erane, plane stress eleent, =1 erane, plane stress eleent, =4 =121, E=228, DOF= First ode. Second ode. irchoff, plane stress eleent Thikness (t) in. Fig.8. The Effect of Shell Thickness on the Frequencies of Half Olate Shell Claped at Peripheral Base with (a=125, =15, e=585, at: Aluinu) Eccentricity (e) Fig.1. The Variation of Critical Von isses Stress of Olate Shell for Half Olate Spheroids Shell Claped at its Peripheral Base with (a=185,=7,e=925,at: Aluinu). 8

9 adhi H. Ghlai Al-hwarizi Engineering Journal, Vol. 6, o. 2, PP 1-1 (21) 5. References [1] Penzes, L., and Burgin, G. Free Viration of thin Isotropic olate Spheroidal shell, General Dynaics Report o. GPIC-BTD , [2] Penzes, L. Free virations of thin ortotropic olate Spheroidal shell, J.A coust Soc. Aer, Vol. 45, pp [3] Zienkiewicz, O.C. The Finite Eleent ethod, cgraw-hill ook Co.,1992. [4] Loue, A.E.H. A Treaties on the atheatical Theory of Elasticity, 4 th ed., Dover pulication, ew York, [5] Pawsey, S.F. The Analysis of oderately Thick to Thin shells y the finite eleent ethod, [6] Hagedran, p. The Rayleigh-Ritz ethod with Quasi-Coparison Functions in non selfadjoint proles, Tranaction of the ASE,Vol.115, pp (28-281),1993. [7] Baich D.V. andn horoshan staility and natural virations of shells with variale geoetric and echanical paraeters, International applied echanics, Vol. 37, o. 7, 21 [8] Daniea J. Inan Engineering Viration, third addition, Person Education Inc., Printing Hall, 28. [9] eyer, C. Solution of Linear Equations, state of the Art, ASCEJ. Of structure, DW. Vol 99, pp , [1] elosh, R.J, and Efficient Solution of Load Deflection Equations, ASCEJ. Of structure Div., Vol 95 pp ,

10 كاظم حمزة غليم مجلة الخىارزمي الهنذسية المجلذ 6 العذد 2 صفحة 1-1 )21( خواص االهتزازات لمصفائح المنبعجة بأضافة او عدم اضافة االطار الهيكمي كاظم حمزة غليم* حاتم رحيم وسمي** * لسى انه ذست ان كا ك ت/ كه ت دخهت اندايؼت االهه ت ** لسى انه ذست ان كا ك ت/ كه ت انه ذست/ خايؼت بغذاد الخالصة : ف هز انذساست تى تحه م خىاص االهتضاصاث نهظحى ان فهطحت بأستخذاو االطش انه كه ت او بذو ها. ح ث تى استخذاو طش مت انؼ اطش ان حذد ( )F.E. ف انتحه م ان ظشي نتحذ ذ انخىاص انذ اي ك ت نهزا ان ىع ي انظفائح وغانبا ياتستخذو هز انؼ اطش ف تحه م انخىاص ان كا ك ت نهظفائح ان بؼدت انكشو ت وشبت انكشو ت بأتداة واحذ او بأتداه. ح ث تى حساب انتشدداث انطب ؼ ت ن ىرج طف حت ي بؼدت طبما نحانت ه ص ادة س ك انظف حت واناليشكض ت نها يغ االخز ب ظش االػتباس تحذ ذ يا كافئ ف تهك انتشدداث ي سك اال ح اء وان سك انغشائ ح ث ا ان سىق انغشائ ت نالهتضاصاث انطب ؼ ت تضداد بض ادة انالستمشاس ت نظف ح ت ان بؼدت ب ا سىق اال ح اء تمم بض ادة اناليشكض ت نهظف حت وتست ش هز انحانت هكزا حتى ظم بانفحض انى انحانت ان ثان ت ناليشكض ت. وكزنك تى اخشاء تداسب ػ ه ت نفحض االهتضاصاث ػهى ىرج تى تظ ؼت ن ثم طح دواس نطائشاث اال زال ان بكش )) االواكس (( ح ث تى انحظىل ػهى تائح يثان ت تى يطابمتها يغ انتحه الث ان ظش ت. 1