Accuracy of the Scaling Law for Experimental Natural Frequencies of Rectangular Thin Plates

Transcription

1 The 9th Conference of Mechanical Engineering Network of Thailand 9- October 005, Phuket, Thailand Accuracy of the caling Law for Experiental Natural Frequencies of Rectangular Thin Plates Anawat Na songkhla and Pairod inghatanadgid* epartent of Mechanical Engineering, Faculty of Engineering, Chulalongkorn University, Bangkok, 0330, Thailand *Tel: Fax: E-ail: Abstract A scaling law for vibration response of rectangular isotropic plates along with a siilarity requireent was derived and verified by experient ethod in this study. The scaling law was derived fro the governing equation of the proble, and verified with the closed for solution. Besides theoretical verification, the experient study was conducted on odel and prototype speciens. A nuber of nine aluinu rectangular plates with F boundary conditions were tested for natural frequencies using ipact test ethod. Acceleroeter and dynaic signal analyzer were eployed to easure and analyze the vibration response of the speciens. Natural frequencies of the first three vibration odes were obtained by transforing the acceleration in tie doain to that of the frequency doain. The natural frequencies of the odels were substituted into the scaling law to obtain the scaling natural frequencies of the prototypes, which were copared to the easured natural frequencies. Fro a total of 9 coparisons, the average percent discrepancy of the scaling natural frequencies is 0.5% with standard deviation of 6.%. Natural frequencies of the prototype deterined fro both approaches agree with each other very well. The accuracy in this study is notably better that of the siilar study on scaling law for buckling of plate. Thus, the derived scaling can be used in engineering applications, providing that the boundary conditions of the odel and prototype are identical. Keywords: caling Law, Natural Frequency, iilitude. Introduction The siilitude concept has been utilized in any engineering applications. It is very helpful for engineers to be able to replicate the behavior of the prototype using the appropriate scaled odel. The concept is also very powerful for probles with coplicated boundary conditions where analytical or nuerical solutions are not sufficiently accurate, if not ipossible. iilitude theory can be stated as []; the sufficient and necessary condition of siilitude between two systes is that the atheatical odel of the one be related by a bi-unique transforation to that of the other. If paraeters of the odel and prototype have such siilarity conditions, then the scaled replica can be built to duplicate the behaviors of the full-scaled syste, and the results fro the odel experients can be utilized to predict the behavior of the prototype. The siilitude theory have been applied to any probles in the field of structure engineering, especially vibration and buckling of plate probles which are in the interest of this study. iitses [] applied siilitude transforation to bending, buckling, and vibration of lainated plates. The derived scaling laws and appropriate siilarity requireents were successfully eployed to the probles. Rezaeepazhand et. al [3] deonstrated a procedure for deriving the scaling law for frequency response paraeter utilizing the closed for solution. Another approach of eploying the siilitude transforation for stability and vibration of lainated rectangular plate proble is presented in Ref.[-6]. In those studies, the siilitude transforation was applied to the governing equations of the proble directly. The advantage of this approach is that the solutions of the governing equations are not required. The obtained scaling laws were verified with the theoretical solution and found to be exact for coplete siilitude cases. Partial siilitude cases were also investigated and recoended. Beside theoretical work, the scaling law was also verified by experient ethod. Alanpitak [7] perfored buckling experient on coposite plates and shown that the buckling scaling law was accurate in ost of the odel-prototype pairs. The average percent discrepancy between scaling and experiental buckling loads was 5.9% with standard deviation of 8.7%. However, soe pairs of odel-prototype have percent discrepancy as high as 30% for coplete siilitude case. The unifority of the speciens and the buckling load identification ethod load are cited as the probable causes of the discrepancy. In this study, the scaling law for natural frequency of rectangular aluinu plates was derived and copared with the experient results. It is an objective of this study to investigate the accuracy and repeatability of the scaling law and vibration easureent, and copare to the previous buckling proble study by Alanpitak [7] where the standard deviation of the percent discrepancy of the scaling buckling load is quite high. o, this study is intended to confir that the scaling laws for structural probles are reliable and suitable to use in engineering application. Also, the experient result fro this study could indicate the accuracy of the natural

2 frequency easureent.. Natural frequency of plate The governing equation for vibration of isotropic rectangular thin plate can be written as [8]; W( x, y, t) W( x, y, t) W( x, y, t) x x y y ρ W( x, y, t) = 0 () t where W is the displaceent in the out-of-plane direction, ρ is the ass density of the specien, and is the plate bending stiffness. Assuing that the out-of-plane displaceent is separable as a function of position and tie, the governing equation is reduced to w w w ωρ w = 0 () x x y y where w is function of x and y only, i.e. w = w (x,y), and ω is the frequency of the vibration. The vibration equation, eq. (), can be solved if the boundary conditions of the plate are known. For siplesupported plates, the analytical closed for solution is C w C w C w C x C C x y C y w w w x x y y CCC ω ρ w ωρ w = 0 (5) C Coparing both equations, the vibration behavior of the odel and prototype are siilar if groups of the scaling factors in eq.(5) are all equal. This iplies that eq.(5) can be reduced to eq.() when the scaling factor groups are canceled out. Thus, the siilitude requireent is obtained as CC ω ρ x x y y = = = (6) C C C C C By assue that the odel and prototype have geoetric siilarity (C x = C y = C a = C b ), the siilarity requireent is siplified to, CCC ω ρ b C = (7) Eq.(7) is the siilitude invariant of the vibration of rectangular plates. This invariant can be reduced to the scaling law of plate natural frequency as, π a ωn = n a ρh b (3) b ρ ωp = ωc (8) b pρ p where ω n are natural frequency of the plate in Hz, a and b are plate width and length, respectively, h is specien thickness, and n are positive integer. 3. caling law for vibration of plate The scaling law for vibration of rectangular isotropic plates is derived fro the governing equation, eq.() by coparing the governing equation for the odel and prototype systes. Fro both equations, the siilitude invariant ter, which leads to the scaling law, is obtained. Let the variables of the prototype and their corresponding odel variables be related to each other as follows, x = Cx, y = Cy, w = Cw, = C, p x p y p w p ω = C ω,andρ = C ρ p ω p ρ where subscripted p refers to prototype and subscripted refers to odel, and C i are the scaling factors of the i paraeters. To derive the siilitude invariant, the governing equations of the odel and prototype are written as the following, w w w ω ρ x x y y w = 0 () In conclusion, the derived scaling law for vibration of plate is valid for a odel-prototype pair with coplete geoetric siilarity, i.e. C a = C b or both systes have the sae aspect ratio. The scaling law can be verified with the theoretical solution shown in the previous section, as shown in Table. Rectangular plates with b = 50 and aspect ratio of,.5, and are selected as odels, and used to predict the natural frequencies of the prototypes with b = 00 and 300. All plates are assued to be Al606-T6 with E = 68.9 GPa, v = 0.35, ρ = kg/ 3, and plate thickness h =. The natural frequencies of the odels deterined fro the analytical solution, eq.(3), are shown in colun. These natural frequencies are substituted into the scaling law to predict the scaling natural frequencies of the prototypes, as presented in the caling coluns. The scaling frequencies are verified with the theoretical solutions shown in colun and 6. It is confired that the natural frequencies deterined fro the scaling law and those fro the closed for solutions are identical. Therefore, the derived scaling law for natural frequency of rectangular plate is verified, theoretically. Next, the derived scaling law is validated by real easureents. The experient is perfored to deterine the accuracy and repeatability of the approach.

3 Table. The fundaental natural frequency in Hz for Al606-T6 speciens Aspect Model Prototype ratio b=50 b = 00 b = 300 Theory caling Theory caling Experient study To deterine the accuracy of the scaling law, rectangular aluinu thin speciens were prepared and easured for natural frequencies. The speciens were then classified as a odel or a prototype. The easured natural frequencies of the odels were used along with the scaling law to predict the natural frequencies of the prototypes. Then, the scaling natural frequencies were copared with the experiental natural frequencies of the prototype to find the accuracy of the scaling law. 3. Experient setup and speciens A total of nine speciens with the length and width of a and b, respectively, were tested in this study. The scheatic drawing of the specien diension and boundary condition are shown in Fig.. The specien boundary conditions are siple support, as shown by a dashed line in Fig., on three edges and free on one of the width edge. The specien aspect ratios (a/b) are,.5, and with specien noinal width b of 00, 50, and 300., respectively. The speciens are ounted in the experient setup and equipped with ipact haer and acceleroeter as shown in Fig.. The siply supported boundary condition is enforced by two stainless steel bars coupled on the specien. The support is cut in the inclined direction to for a knife-edge. These supports allow the specien to freely rotate, but restrain any outof-plane displaceent. The knife-edge supports are fixed with steel boxes by achine screws. There are additional achine screws used to push the support against the specien surface. The assebly of steel boxes and knifeedge supports was tested for natural frequency, also, to confir that their natural frequencies are not in the range of that of the speciens. The vibration test for natural frequency is deterined by ipact test. The ipact haer is used to excite the specien. The applied ipulse can be onitor by the dynaic signal analyzer. An acceleroeter is placed on the specien at a selected location to easure the plate response in ter of acceleration. It is recoended that the acceleroeter should not be on the node line of the vibration to avoid low response signal. If the node line is unknown or uncertain, ore than one easureent is recoended. In this study, five pretests were conducted to deterine a suitable acceleroeter s location. Measured accelerations fro the acceleroeter are collected by a dynaic signal analyzer and used to deterine the natural frequencies. 3. ata analysis The acceleration easured in tie doain is processed by a Fast Fourier Transfor (FFT) algorith using the dynaic signal analyzer. Fro the vibration response in frequency doain, the natural frequencies of the specien are identified fro the peak of the response. Theoretically, there are infinite nuber of natural frequency, however, only the first three odes are studied in this study. Fig.3 shows the vibration response easured in frequency doain obtained fro the data in tie doain for speciens with diension of 30 0 and 60 30c. The easured natural frequencies in Hz for the first three odes are 75, 79, 9 and 89, 0.5, 9, respectively. F b Fig. cheatic drawing of rectangular specien Ipact haer pecien a Knife-edge support Acceleroeter teel Box Fig. Experient setup with acceleroeter and ipact haer The experiental results siilar to those of Fig. 3 can be obtained fro an experient with excitation and acceleroeter locations located at particular positions. Ideally, the sae natural frequencies should be acquired no atter where the excitation and acceleroeter are located. In this study, a total of 5 easureents were perfored on each specien. An acceleroeter was placed on a selected point, while a total of 5 excitation points was systeatically varied to cover the whole specien area. The experiental natural frequency was deterined fro the average of each easureent. For a specific specien, natural frequencies fro each easureent are very siilar which confirs that the experient is very repeatable.

4 Response per excitation pecien 30 0 c Frequency Response per excitation pecien c Frequency Fig. 3. Vibration response in frequency doain. Experient result Fro the vibration response in the frequency doain, the first three natural frequencies of each specien are presented in Table. The speciens are classified into three groups of aspect ratios of,.5, and. It is noticed that the natural frequencies decrease when the specien is bigger. The speciens are assued to be a odel or a prototype and used to verify the scaling law, as shown in Table 3. Fro three speciens with aspect ratio of, three coparisons of scaling and easured natural frequencies can be ade. That is, as shown in colun and 3 of Table 3, a specien with 5 5 diension is set as a odel, and other two speciens are prototype. The other odel-prototype pair is specien 0 0 and specien The other two aspect ratios can also be copared in the sae approach. In Table 3, colun and colun 5 are the easured natural frequencies of the odel and prototype, respectively. The next colun labeled caling is the scaling natural frequency of the prototype. These scaling natural frequencies are deterined fro the scaling law shown in eq.(8) using the easured natural frequencies of the odel. The scaling and experiental natural frequencies shown in colun 6 and 5 are copared to each other. The percent discrepancy of the scaling natural frequency is deterined according to, ω ω = (9) ω caling Exp. % is 00% Exp. The coparisons for vibration in ode and 3 are perfored siilar to those of ode, and shown in the second and third parts of the table. The average and standard deviation of percent discrepancy for each vibration ode are shown in the last two rows of Table 3. The average percent discrepancy in each ode is as high as.5% with the overall percent discrepancy of 0.5%. There is a case of coparison where percent discrepancy is as high as 8.7%. There is no significant difference in average percent discrepancy for each vibration ode. The overall standard deviations fro 7 coparisons of percent discrepancy are 6.%. The vibration ode has the highest deviation of percent discrepancy of 9.9% while the deviation of percent discrepancy decreases in the higher odes. This iplies that the easureent of natural of the high ode is accurate than that of the lower ode 5. iscussion and conclusion Copared to the scaling law for buckling of plate studied by Alanpitak [7], the scaling law for vibration response shows a better accuracy in prediction of natural frequency. The average percent discrepancy for vibration easureent is significantly better, i.e. 0.5% copared to 5.9%. iilarly, the standard deviation of the percent discrepancy for this vibration experient is lower than that of the buckling experient. Experient results of both buckling and vibration experient do not iply that the vibration scaling law is accurate than that of the buckling proble. The difference in accuracy found in both probles is probably initiated fro the difficulty in identifying the buckling point in buckling experient which is not found in vibration experient. The natural frequency deterined fro the vibration response in the frequency doain, as shown in Fig. 3, is ore accurate than the buckling load deterined fro the buckling experient. In conclusion, this research derives the scaling law for vibration response of rectangular thin plates. In addition to the scaling law, the siilitude requireents for two systes to behave siilarly are also obtained. The scaling law is verified with the theoretical solution and found that the scaling natural frequency is exactly atched with the closed for solution. The experient setup was built to accoodate the vibration experient. A set of nine rectangular aluinu plates was set in the test setup with siple support on three edges and free support on one edge. The specien was excited by ipact haer and easured for vibration response using an acceleroeter. The easured acceleration response in tie doain was then transfored to the data in frequency doain. The natural frequencies can be identified fro the peaks of the response in frequency doain. The experient results were used to verify the scaling law. It is found that the average discrepancy between the scaling and experiental natural frequencies

5 is 0.5% with 6.% standard deviation. This discrepancy is very uch lower than that of the buckling experient studied previously. This suggests that, in nature, the natural frequency can be easured with higher accuracy and repeatability. A better agreeent between the scaling and easured natural frequencies can be obtained if both systes have higher degree of siilarity. Table. Measured natural frequencies of the speciens F boundary conditions Aspect ratio.5 ize a b (c ) Mode Mode Mode Table 3. Natural frequencies deterined fro the scaling law copared with the experiental results Aspect ratio.5 Model Prototype ω of ode ω of ode ω ode 3 Model Prototype Model Prototype Model Prototype Exp. caling % is Exp. caling % is Exp. caling % is Avg. 0. Avg..5 Avg td. 9.9 td td..5 References [] z cs, Errin, iilitude and Modelling, Elsevier cientific Publishing Co., New York, 980. [] G.J. iitses, tructural siilitude for flat lainated surfaces, Coposite tructures, Vol.5, No., 00, pp [3] J. Rezaeepazhand, G.J. iitses, and J.H. tarnes, Jr, Use of scaled-down odels for predicting vibration response of lainated plates, Coposite tructures, Vol.30, No., 995, pp [] P. inghatanadgid, and V. Ungbhakorn, Buckling siilitude invariants of syetrically lainated plates subjected to biaxial loading, EM Annual Conference and Exposition, Milwaukee, WI, UA. June 0-, 00. [5] P. inghatanadgid, and V. Ungbhakorn, caling laws for vibration response of anti-syetrically lainated plates, tructural Engineering and Mechanics, Vol., No.3, 00, pp [6] V. Ungbhakorn, and P. inghatanadgid, iilitude invariants and scaling laws for buckling experients on anti-syetrically lainated plates subjected to biaxial loading, Coposite tructures, Vol.59, No., 003, pp [7]. Alanpitak, A verification of siilitude theory applied to a buckling proble of coposite plates by experient ethod, M.Eng. Thesis, Chulalongkorn University, Bangkok, Thailand, 005. [8].J. Goran, Free vibration analysis of rectangular plates, Elsevier, New York, 98.