Natural frequency split estimation for inextensional vibration of imperfect hemispherical shell

Journal of Sound and Vibration 330 (2011) 2094–2106

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Natural frequency split estimation for inextensional vibration of imperfect hemispherical shell Seong-Yoel Choi a, Ji-Hwan Kim b,n a

School of Mechanical and Aerospace Engineering, Seoul National University, San 56-1, Shinlim-dong, Kwanak-ku, Seoul 151-742, South Korea Institute of Advanced Aerospace Technology, School of Mechanical and Aerospace Engineering, Seoul National University, San 56-1, Shinlim-dong, Kwanak-ku, Seoul 151-742, South Korea

b

a r t i c l e in fo

abstract

Article history: Received 23 February 2010 Received in revised form 4 October 2010 Accepted 13 November 2010 Handling Editor: L.G. Tham Available online 22 December 2010

In this study, mathematical model of hemispherical shell is introduced using inextensional vibration mode shapes. Adopting energy equations, the natural frequency of the shell is determined by applying Rayleigh’s energy method. Further, the vibration for imperfect shell is investigated with point mass elements representing imperfections on the structures. Also, the effects are considered via energy relations, and the split amount of the natural frequencies can be determined. Finally, the influences of point mass are presented by explicit functions for the split of the natural frequency and shifting angle of mode orientation. Based on the proposed model of imperfect shell with multiple point masses, the structure can be expressed as an equivalent single mass model. & 2010 Elsevier Ltd. All rights reserved.

1. Introduction Hemispherical shell has been studied by numerous researchers for the various applications in engineering fields such as hemispherical resonator gyroscope (HRG), etc. Generally, the shell is not easy to determine the vibration mode shapes, thus a number of works have been performed up to now. Moreover, the influence of anomalies in the distribution of mass of the shell became one of the important issues due to the fact that the unbalance of the structure may develop error signals in HRG. A relatively simple analysis can be made by applying Rayleigh energy method for inextensional assumptions on the vibration behavior of thin shell. Zhuravlev [1] introduced the fundamental principle of resonating gyros using Foucault pendulum model. Also the mathematical model results in an inspiration of gyro model and makes it possible to control the resonating gyroscopes with drift estimation and calibration of imperfection factors. Especially, Kireenkov [2] estimated the natural frequency for designing vibratory gyros. Furthermore, Hwang [3] reported some experiments on the vibration of the model. Both the axisymmetrical and asymmetrical modes were excited, and the data were compared with the analytical results. de Souza and Croll [4] derived energy expressions of the shell and provided a picture of the nature of the shell’s resistance to linear vibrations. Also, Chang et al. [5] investigated modal precession of the shell by considering a small constant axial rotation. Chung and Lee [6] analyzed the vibration behavior of a nearly axisymmetric shell structure using a finite ring element. Saunders and Paslan [7] studied the inextensional vibration of a sphere–cone shell combination, and compared the theoretical results and experimental data of the natural frequencies. Lee et al. [8] performed analysis of the free vibration for jointed thin cylindrical-spherical shell structures using Rayleigh–Ritz method, and then the analytical results were compared with the modal test data and FEM results.

n

Corresponding author. Tel.: + 8228807383; fax: + 8228872662. E-mail address: [email protected] (J.-H. Kim).

0022-460X/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.jsv.2010.11.014

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On the other hand, imperfections or uneven mass distributions are inevitable in the practical point of view. Also, the irregular masses may results in significant factors diminishing accuracy and vibratory performance of the structures. Dwivedy and Kar [9] studied a slender beam with an attached mass. Yoon and Kim [10] considered a concentrated mass effect on the spinning beam and analyzed the natural frequencies under various follower forces. Also, Bisegna and Caruso [11] investigated frequency split due to imperfection of circular ring. Using energy relations, Fox [12] proposed simpler analysis on frequency split for a circular ring with finite number of mass elements. Further, McWilliam et al. [13] considered random mass distributions and determined frequency splitting rule. Moreover, Rouke et al. [14] presented trimming theory in order to eliminate frequency splits for the circular ring. Meanwhile, other inspections on imperfection on the structure are presented by applying various methods. Rao et al. [15] applied the coupled displacement method to analyze the vibration of Timoshenko beam with a concentrated mass. Achong [16] dealt with mass loaded plates and shallow shells by the receptance method. Moreover, Alavandi [17] considered nonlinear characteristics of imperfect rectangular plates. In this paper, the natural frequency of hemispherical shell and frequency split due to imperfections of the point masses are presented. In order to determine the natural frequencies of the shell, the analysis are based on the Rayleigh energy method. Further, considering energy expressions for the shell with and without point masses yields mathematical model for frequency split due to mass distributions. Finally, the natural frequencies of the shell can be determined and parametric studies for various thicknesses to radius ratios are performed. 2. Formulations Using energy method, an analytical approach is applied to determine natural frequency of the model in this chapter. Also, an equivalent single mass model representing the imperfect shell with arbitrary located point masses is proposed. 2.1. Natural frequency Fig. 1 shows a model for a thin hemispherical shell with mean radius a and radial thickness h. The local displacement components are expressed by u, v, and w with respect to the directions of f, y and a, respectively. In addition, ‘C.M.’ denotes a concentrated mass (or a point mass) on the shell representing imperfection of the structure. The physical model of the shell is assumed to be isotropic and geometrically symmetric with free boundary conditions. Also, imperfection is considered with point mass element. The natural frequencies are determined by using energy expressions applied to Rayleigh method. First of all, the kinetic energy of the shell can be expressed as Z p=2 Z 2p 1 _ 2 Þ sin f dy df K0 ¼ arh ðu_ 2 þ v_ 2 þ w (1) 2 0 0 where K0 and r denote kinetic energy, and density of the shell, respectively. Also, the strain energy can be written as Z p=2 Z 2p Z h=2 1 U0 ¼ a ðs11 e11 þ s22 e22 þ s12 e12 Þ sin f dr dy df 2 0 0 h=2

(2)

where U0, sab, and eab denote strain energy, bending strains and stresses, respectively, and subscripts represent the direction for each term. Note that the terms with respect to radial direction are assumed to be negligible following shell theory. Additionally, the membrane strains are neglected based on the inextensional assumption of the shell as in Ref. [16].

Fig. 1. Coordinate system and model of hemispherical shell.

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Next, stress–strain relationship can be represented as

s11 ¼ s22 ¼

E ðe11 þ me22 Þ 1m2

E ðe22 þ me11 Þ, where e ¼ ak 1m2 E s12 ¼ e12 2ð1 þ mÞ

(3)

where E, m and k denote Young’s modulus, Poisson’s ratio and curvature of the middle surface of the shell, respectively. While, k terms are expressed in terms of displacement components u, v and w as in Ref. [18] 1 ðu wff Þ a2 f # 1 vy wyy k22 ¼ 2  þ cot fðuwf Þ a sin f sin2 f " ! # uy wfy 1 @ v wy k12 ¼ 2 sin f  þ @f sin f sin2 f sin f a

k11 ¼

"

(4)

where subscripts of the displacement terms denote derivatives with respect to corresponding directions. The energy expressions can be rewritten in terms of curvatures by substituting Eq. (3) into Eq. (2), and this means also can be defined by displacement terms. Then, the strain energy of the shell can be obtained as ZZZ 1 E 1 U0 ¼ (5) fk211 þ k222 þ 2mk11 k22 þ ð1mÞk212 gr sin f dr dy df 2 2 1m 2 Next, the imperfection is assumed to occur as a small additional portion in the kinetic energy only without any change of potential energy. This means that the total energies can be written as KT ¼ K0 þ KM , UT ¼ U0

(6)

where KM ¼

n X 1 i¼1

2

  _ 2M mi u_ 2M þ v_ 2M þ w

(7)

where subscripts T and M denote total energy of imperfect shell and energy of the point mass, respectively. Then, the natural frequency can be expressed using Rayleigh energy method. Based on Ref. [19], the inextentional vibration for hemispherical shell can be analyzed as follows. Determining the kinetic and potential energies, the displacement terms need to be expressed in terms of assumed mode shape functions as u ¼ Da sin f tann

f

v ¼ Da sin f tann

f

sinðnyÞ ei$0 t

2 2

cosðnyÞ ei$0 t

w ¼ Daðn þcos fÞ tann

f 2

sinðnyÞ ei$0 t

(8)

where D, n and o0 denote the magnitude of mode shape, mode number and natural frequency. Similarly, the displacement functions for point masses can be assumed as similar to the case without imperfection as in Ref. [19]: uM ¼ Da sin f tann

f 2

vM ¼ Da sin f tann

sinðnðyzÞÞ ei$n t

f 2

cosðnðyzÞÞ ei$n t

wM ¼ Daðn þ cos fÞ tann

f 2

sinðnðyzÞÞ ei$n t

(9)

where z denotes the shift angle for the orientation of corresponding mode shape. Moreover, the shift angle due to the imperfection is considered in circumferential directions only for the assumed mode shape has one nodal point. Thus, only (n, 0) mode are to investigate as in Ref. [6]. Then, the energies can be obtained by substituting Eqs. (8) and (9) into Eqs. (1), (5) and (7). Thus, the maximum kinetic and potential energy expressions are: K0max ¼

1 2 2 2 o pD a rh 2 0

Z p=2 0

fðn þ cos fÞ2 þ2 sin2 fgsin f df

(10)

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U0max ¼

KMmax ¼

pD2 h3 E 2 2 n ðn 1Þ2 6ð1 þ mÞa2

Z p=2

sin3 f tan2n

0

f 2

2097

df

(11)

 o f n D2 mi o2n tan2n i ðn þ cos fi Þ2 sin2 ðnðyi zÞÞþ sin2 fi 2 2

X1 i

(12)

In Eq. (12), subscript ‘i’ denotes the ith point mass in total n point masses. Finally, the Lagrangian can be written as   L ¼ U0max  K0max þKMmax

(13)

Then, Rayleigh energy method is simply expressed as @L ¼0 @D

(14)

Thus, the natural frequencies of imperfect shell can be determined using Eq. (14) as oi31 2  n P h 2n 1 fi =2 ðn þ cos fi Þ2 sin2 ðnðyi zj ÞÞ þsin2 fi i mi tan 2 2 24 5 n o on ¼ o0 1 þ R p=2 2 2 2n f 1 a pr h tan ðn þ cos f Þ þ2 sin f f d f sin 0 2 2

ðj ¼ H,LÞ

(15)

where subscript j denotes the higher and lower split of the frequencies, and H and L represent higher and lower frequency, respectively. Further, z stands for the shift angle of mode orientation. Generally, the frequencies for H and L modes [11] are both lower than the corresponding frequencies for perfect model. Furthermore, the H-mode has very close magnitude relative to the frequencies for perfect model, while the L-mode shows considerable difference in the magnitude. From now on, they will be denoted by H-mode and L-mode frequency as in Ref. [11]. On the other hand, Eq. (12) can be rewritten more compact form as

o2n ¼ o20

1 ð1 þ eK Þ

where eK ¼

KMmax K0max

(16)

Using Rayleigh–Ritz type procedure as in Ref. [12], the shift angle can be determined as @o2n ¼0 @z

(17)

Applying Eq. (17) to Eq. (16) yields: 

@eK ¼0 ð1 þ eK Þ @z 1

2

(18)

In this equation, eK 51 is reasonable for sufficiently small point masses [10] Thus, Eq. (18) can be rewritten as @eK 1 @KMmax ¼ 2 ¼0 @z D on @z The shift angle of mode orientation is the angle obtained by substituting Eq. (12) into Eq. (19)  o 1X f n mi tan2n i 2nðn þ cos fi Þ2 sinðnðyi zÞÞ cosðnðyi zÞÞ ¼ 0  2 i 2

(19)

(20)

Using only an algebraic procedure, the more simple form is obtained as tanð2nzÞ ¼

X Ci sinð2nyi Þ i

Ci cosð2nyi Þ

o 1 f n where Ci ¼ mi tan2n i 2nðn þ cos fi Þ2 2 2

(21)

2.2. Equivalent model For practical models, the exact locations and the numbers of unbalancing masses are not easy to be detected. Thus, the structure requires estimation of the frequency split by using an equivalent model with a point mass. First of all, natural frequencies and mode shapes are to be determined, then the equivalent point mass can be determined by rearranging Eq. (18). Hence, the equation can be rewritten as   m A 1 o2n ¼ o20 1 þ 1 j ðj ¼ H,LÞ (22) B

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Table 1 Material properties of hemispherical shell. Symbol E

r m0 m1 a h

Quantity Young’s modulus Density Total weight Point mass weight Radius Radial thickness

Fig. 2. Normalized frequencies split with 0.1% of total weight single point mass.

Fig. 3. Normalized frequencies split.

210  109 N/M2 7800 kg/m3 6864 g Variable 100 mm 7 mm

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where  o 1 f n tan2n 1 ðn þcos f1 Þ2 sin2 ðnðy1 zj ÞÞ þ sin2 f1 2 2 Z p=2 o 1 fn ðn þ cos fÞ2 þ 2sin2 f sin f df B ¼ aprh tan2n 2 2 0

Aj ¼

(23)

Note that B is a constant and Aj stand for H- and L-mode, respectively. In addition, the equivalent model with single mass located at 901 meridian angle which denotes the bottom circumferential line of the shell. This means that mode shapes for meridian direction remain constant due to slight imperfections. As a result, any shell with slight imperfections can be represented by mathematically equivalent model. Then the two split frequencies can be stated as 

oH ¼ o20 1 þ

m1 AH B

1

  m1 AL 1 and oL ¼ o20 1 þ B

(24)

Rearranging the above equations with respect to equivalent mass m1

m1 ¼

  B o2H o2L o2L AL o2H AH

Fig. 4. Mode shapes for n= 2 (

L-mode,

(25)

H-mode).

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Fig. 5. Mode shapes of double masses for higher mode numbers.

Fig. 6. Frequency split of triple masses imperfection.

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2101

And then, the location of the point mass can be obtained by using Eq. (21) as tan ð2nzÞ ¼ tanð2ny1 Þ

(26)

In short, any structure with arbitrarily located point masses can be expressed as an equivalent model for a certain mode number. This means that the equivalent model for a specific mode number cannot represent for the other mode numbers. For example, each of estimated mass weight and location is different for each mode numbers. In this regard, one can only expect trimming process for one pre-determined mode number by single mass equivalent model.

Fig. 7. Mode shapes of triple masses for mode numbers.

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3. Results and discussions The results are summarized in this section for natural frequency split diagram with different mode numbers and point masses. Also, the parametric study shows the results for different radius and thickness of hemispherical shell. 3.1. Verification For the validation of this study, the point masses are neglected in Eq. (16). Then, the natural frequency of the perfect shell can be written as R p=2 sin3 f tan2n f=2df n2 ðn2 1Þ2 h2 E 0 o20 ¼ (27) R 3ð1 þ mÞra4 p=2 fðn þ cos fÞ2 þ2 sin2 fgsin f df 0

This result shows exactly same natural frequency expression for hemispherical shell in Ref. [19]. 3.2. Natural frequencies and mode shapes In this section, estimated frequencies and mode shapes for the proposed model are obtained. For convenience, the normalized natural frequency oNORM is defined as

oNORM ¼

o0 oH,L Don ¼ o0 o0

Fig. 8. Frequency split of quadruple masses.

Fig. 9. Mode shapes of quadruple masses for n= 4.

(28)

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Also, normalized mass mNORM is defined as mNORM ¼

mPOINT mSHELL

(29)

where mPOINT and mSHELL denote point mass and mass of the perfect shell, respectively. As a first example, single point mass effect on the vibration of the shell is investigated, and the point mass has 0.1% mass of the perfect shell. The material properties are listed in Table 1. Fig. 2 represents the split of natural frequencies at 01 of circumferential angle and 901 of meridian angle. The difference between H-mode frequency and perfect shell frequency is relatively small than L-mode. Also, split amount of H-mode decreases for the cases of large mode numbers, while increases for L-mode. Additionally, Fig. 3 includes the results of Fig. 2, and present direct relationships of mode number and point mass to frequency split simultaneously. The data estimates split of the frequencies due to small imperfections resulting in two distinct frequencies lower than the frequency of perfect shell. Next, vibration of the shell with multiple point masses is considered. First of all, identical double point masses located at opposite side of circumferential contour are investigated. Fig. 4 show mode shapes for n =2 with single and double masses indicated by small filled circles. The mode shape and the shift angle of the mode orientation is same as single mass case, but the split amount increases. All point masses are located on nodal points of H-mode shape for n = 2. Further, this condition Table 2 Natural frequencies of hemispherical shell without imperfection. a/h ratio

Natural frequency (Hz)

15 20 25 30 40

632 427 314 262 179

Fig. 10. Frequency split of different sizes for n= 2.

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should also be satisfied for higher mode numbers. Moreover, Fig. 5 presents results for higher mode numbers for double mass case, and the mode shapes in the figures satisfy the condition. As assumed mode shapes in Eq. (9), the shape of circumferential section of the shell is modeled by sinusoidal functions. Thus, the nodal point of H-mode mode shape always is located on 0 and (kp/n) rad, and kp/n= p is always satisfied with certain integer k and the mode number n. Therefore, the unbalancing effect always occurs for all higher mode numbers for single and double masses. The triple masses model is presented by locating masses on 01, 1201 and 2401, respectively. In Fig. 6, there are certain mode numbers without split of the frequencies. Especially, that mode numbers make Eq. (25) to be indeterminate with zero numerator and nominator simultaneously. It implies that for certain mode numbers, naturally balanced vibration can occur. The assumed mode shapes have nodal point at kp/n, so the mode numbers should satisfy following relationship (kp/n)=(2p/3). Furthermore, the mode numbers should satisfy Eq. (30) as n¼

3 k 2

(30)

In this equation, 32k should be an integer, therefore split of natural frequencies occurs for mode numbers which are multiples of 3. In Fig. 7, all the point masses are located on the nodal points of H-mode. Furthermore, higher mode split of frequencies are illustrated in Fig. 10. At last, effect of quadruple masses is to be investigated, and following equation can be induced as a general form in Eq. (30): kp 2p ¼ n N

(31)

where k, n and N denote arbitrary natural number, mode number and the number of point masses at circumferential line of the shell. Thus, only the mode numbers satisfying Eq. (31) will result in split of frequencies. Then, the mode numbers for split of frequency will be the multiples of 2 as shown in Fig. 8. Additionally, Fig. 9 presents mode shapes of quadruple case for n =4. Up to now, the imperfection effects are observed for different mode numbers and the number of point masses. Also, increasing the number of point masses makes the system to satisfy Eq. (31) in order to induce split of the frequencies.

Fig. 11. Normalized frequencies split for L-mode.

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3.3. Parametric studies This section deals the results for the model with different radius and thickness for single point mass. Firstly, the frequency split for different radius to thickness ratio (a/h) is investigated. Also, the certain range of a/h is selected as the bending deformation is dominant. Table 2 presents the frequency without imperfection, and frequency split for various a/h with a point mass 0.5% of the structure. For H- and L-mode frequency diagrams for different sizes of the shell are plotted in Fig. 10. As in the figures, both H- and L-mode frequencies have similar aspects in varying split amount of the model. Further, the

Fig. 12. Natural frequencies along a/h for the shell without imperfection.

Fig. 13. Frequency split due to a/h increment with constant total mass of the structure.

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increment of radius results in larger frequency split while the increment of thickness reduces split amount. This can be more easily apprehended by observing the results appeared in Fig. 11. The a/h increases by the change of thickness with constant radius as in Fig. 11(a). The effect of thickness variation seems to be more effective on the split amount than radius variation. The results shown in Fig. 11 have no further constraint in the physical model. However, the next results consider constant total mass of the structure. Thus, the a/h can be a considerable parameter for the shell, because thickness and radius become dependent to each other by constraint for total mass. For example, natural frequencies of the 5 kg shell without imperfection for various a/h are shown in Fig. 12. The diagrams for L- and H-mode are presented in Fig. 13(a) and (b), respectively. In the figures, the split amounts are displayed by Hz and split frequency decreases as a/h increases. The results reveal that the split amount decreases when a/h increases for constant mass of the structure. On the other hand, normalized split amount increases along increment of a/h for both L- and H-mode. Additionally, Fig. 13(c) and (d) demonstrate the same results as in Fig. 13(a) and (b) with different unit of the frequency axis. Percentage of the split amount with respect to the frequency of perfect shell is plotted in the figures. 4. Conclusions Mathematical model for hemispherical shell with multiple point masses is investigated. The shell is assumed to be in inextensional vibration. At first, the frequencies of perfect shell are verified by neglecting point masses in this work. Then, the split of natural frequencies are determined by using Rayleigh energy method. The study presented analytical model of imperfect hemispherical shell with free boundary conditions. As a result, the explicit expression for the natural frequencies is obtained. Using this result, any hemispherical shell with arbitrary located small point masses can be expressed as an equivalent single mass model. However, the equivalent model is only feasible to one pre-determined mode number. Moreover, the split of frequencies does not happen for certain mode numbers with conditions such as equidistantly located identical point masses. The criterion on split of natural frequency for previously stated case is presented in the study. The parametric investigation for the model is performed using radius to thickness ratio as a major parameter. The results for different radius and thickness within certain a/h show that the variation of thickness is more effective to the frequency split than variation of radius. Also, the model for different a/h with constant total mass is considered. From the results, larger amount of a/h yields lesser absolute value of split frequency with constant total mass of the structure. On the other hand, the percentage of the frequency split becomes larger when a/h increases.

Acknowledgement This work is financially supported by Korea Ministry of Land, Transport and Maritime Affairs as ‘‘Haneul Project’’. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]

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