Analytical and Experimental Modal Analysis of Non Uniformly Ring-stiffened Cylindrical Shells

8/6/2019 Analytical and Experimental Modal Analysis of Non Uniformly Ring-stiffened Cylindrical Shells

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Arch Appl Mech (2006) 75: 177191DOI 10.1007/s00419-005-0429-y

O R I G I N A L

M. Bagheri A. A. Jafari

Analytical and experimental modal analysis of nonuniformlyring-stiffened cylindrical shells

Received: 19 February 2005 / Accepted: 22 August 2005 / Published online: 9 December 2005 Springer-Verlag 2005

Abstract In this research, the free vibration analysis of cylindrical shells with circumferential stiffeners, i.e.,rings with nonuniform stiffener eccentricity and unequal stiffener spacing, is investigated using analytical andexperimental methods. The Ritz method is applied in analytical solution, while stiffeners are treated as dis-crete elements. The polynomial functions are used for Ritz functions. The effects of nonuniformity of stiffenerdistribution on natural frequencies are considered for freefree boundary conditions. Results show that, atconstant stiffener mass, significant increments in natural frequencies can be achieved using nonuniform stiff-ener distribution. In experimental method, modal testing is performed to obtain modal parameters, includingnatural frequencies, mode shapes, and damping in each mode. Analytical results are compared with experimen-tal ones, showing good agreement. Because of insufficient experimental modal data for nonuniform stiffenerdistribution, the results of modal testing obtained in this study could be a useful reference for validating theaccuracy of other analytical and numerical methods for free vibration analysis.

Keywords Cylindrical shell Ring stiffener Free vibration Natural frequency Modal testing

1 Introduction

Ring-stiffened cylindrical shells are applied in many structures such as pressure vessels, submarine hulls, air-craft, launch vehicles, and offshore drilling rigs. The knowledge of these structures characteristics is necessaryto determine their structural integrity and fatigue life. The natural frequencies of vibrations are of special inter-est to aircraft and launch vehicle designers because of increasing use of sensitive electronic instrumentationand onboard computers and gyroscopes, which require vibration isolation from the main structure.

In the literature on this widely discussed subject, there are two main types of analysis, depending uponwhether the stiffening rings are treated by averaging their properties over the surface of the shell or by con-sidering them as discrete elements. When ring stiffeners of equal strength are closely and evenly spaced, the

stiffened shell can be modeled as an equivalent orthotropic shell. This is also called the smearing method.However, as the stiffener spacing increases or the vibration wavelength becomes smaller than the stiffenerspacing, the dynamic characteristics of a stiffened shell cannot be determined with high accuracy. Thus, for amore general model, the ring stiffeners must be treated as discrete elements. When modeled in this respect, it isadvantageous to use nonuniform eccentricity and unequally spaced and different materials for ring stiffeners.

M. Bagheri (B) A. A. JafariDepartment of Mechanical Engineering, K. N. Toosi University of Technology, P.O. Box 16765-3381, Tehran, IranE-mail: [email protected].: +98-21-77343300Fax: +98-21-77334338

M. BagheriDepartment of Aerospace Engineering, Shahid Sattari Air University, Tehran, Iran


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178 M. Bagheri, A. A. Jafari

The free vibration of stiffened cylindrical shells has been investigated since the 1950s by a number ofresearchers. Hopmann [3] investigated, analytically and experimentally, the free vibration of orthogonallystiffened cylindrical shells with simply supported ends. In this study smearing method for stiffeners is used inanalytical investigation. Mikulas and McElman [4] studied the free vibration of eccentrically stiffened simply

supported cylindrical shells by averaging the stiffeners properties over the surface of the shells and foundthat the eccentricity could have significant effects on natural frequencies. Egle and Sewall [2] extended thisstudy with stiffeners treated as discrete elements. AL-Najafi and Warburton [1] considered the free vibration ofring-stiffened cylindrical shells using experimental and numerical methods for various boundary conditions. Atheoretical and experimental investigation into the vibration of axially loaded stiffened cylindrical shells wascarried out by Rosen and Singer [6] using Donell and Flugge theories. Mustafa and Ali [5] presented an energymethod for free vibration analysis of stiffened cylindrical shells. The analysis was performed considering theflexure and extension of the shell and the flexure, extension, and torsion of the stiffeners. Swaddiwudhiponget al. [9] presented free vibrations of cylindrical shells with rigid intermediate supports. An automated Ray-leighRitz method was adopted to evaluate the natural frequencies and the mode shapes.A special polynomialunified set of Ritz functions was used to span the displacement fields of various types and combinations ofend boundary conditions. Wang et al. [10] extended the Ritz method for solving free vibration problems ofcylindrical shells with varying ring stiffener distributions. The Ritz formulation also makes it possible forstiffener materials to be different from one another and from the material of the parent shell. Ruotolo [7]presented a comparison of some thin shell theories for dynamic analysis of stiffened cylinders while usingsmearing method for stiffeners. Few papers with experimental data have been published on this topic, and allsuch studies are related to evenly spaced stiffeners and uniform stiffener eccentricity.

In this study, freevibration analysis of ring-stiffened cylindricalshells with nonuniform stiffenerdistributionis performed using analytical and experimental methods. The effects of nonuniformity of stiffener distributionon the dynamic characteristics of stiffened shells are discussed, analytically. At constant stiffener mass, theimportance of stiffener distribution for increasing the natural frequencies is investigated. Moreover, modal test-ing is performed experimentally to obtain the modal characteristics of ring-stiffened cylindrical shells. Becauseof insufficient experimental modal data for nonuniform stiffener distribution, the results of modal testingobtained by the authors could be a useful reference for free vibration considerations of stiffened cylinders.

2 Analytical formulation

The cylindrical shell as shown in Fig. 1 is considered to be thin with uniform thickness h, radius R, lengthL, mass density , modulus of elasticity E, Poissons ratio , and shear modulus G = E/2(1 + ). The shellis circumferentially stiffened by N number of rings, which may be placed internally or externally. The ithring stiffener has a rectangular cross section with constant width bri and height of dri and is located at ai Lmeasured from the end of the shell. The spacing and height of the rings can be varied along the shell length.The ring stiffeners may be constructed from materials that are different both from one another and from theparent shell material. The ith stiffener properties are defined as mass density ri , modulus of elasticity Eri ,Poissons ratio ri , and shear modulus Gri .

2.1 Shell energy

Adopting Sanders [8] thin shell theory, the strain energy of stretching and bending of the aforementioned

cylindrical shell without stiffeners is expressed as:

U =

L0

20

Eh

2(1 2)

u

x

2+

1

R2

v

w

2+

2

R

u

x

v

w

+

1

2

v

x+

1

R

u

2

+Eh3

24(1 2)

2w

x2

2+

1

R4

2w

2+

v

2

+2

R2

2w

x2

2w

2+

v

+

2(1 )

R2

2w

x+

3

4

v

x

1

4R

u

2R d dx , (1)

where u, v, and w are displacements in the longitudinal, tangential, and radial directions, respectively, and xand are longitudinal and circumferential coordinates, respectively.


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Analytical and experimental modal analysis 179

Fig. 1 Ring-stiffened cylindrical shell with nonuniform stiffener distribution

Neglecting the effect of rotary inertia since the shell under consideration is thin, the kinetic energy of acylindrical shell without stiffeners will be:

T =1

2 hL

0

20

ut2

+v

t2

+w

t2

R d dx . (2)

2.2 Ring stiffener energy

In this analysis, the geometric characteristics and materials of the rings may be different from one another.Also, ring spacing and eccentricity can have nonuniform distributions.

The strain energy of the ith ring stiffener with the effects of stretching, biaxial bending, and wrapping isgiven by:

U ri =

2

0

Eri I zri

2

1

R + eri wr i

x+

1

R + eri

2uri

2 2

+Eri I xri

2

1

(R + eri )3 wr i +

1

R + eri

2wri

2 2

+Eri Ari

2

1

R + eri

vr i

wr i

+

Gri J ri

2

1

R + eri

2wri

x+

1

R + eri

uri

2d . (3)

The kinetic energy of the ith ring stiffener with the effects of triaxial translational inertia and rotary inertiaabout the x and z axes is given by:

T ri =1

2ri

20

Ari

uri

t

2+

vri

t

2+

wri

t

2

+(Ixri + I zri )

2wri

t x

2(R + eri )d, (4)


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where the second moments of areas I zri and I xri , cross-sectional area Ari , and torsional rigidity J ri are:

I zri =br 3i dri

12; I xri =

bri dr3i

12; Ari = bri dri;

J ri =1

3

1 192bri 5dri

n=1,3,5,..

1

n5tanh

ndri

2bri

br 3i dri , (5)and the eccentricity of the ring stiffener is expressed as:

eri = h + dri

2, (6)

where the + and signs represent external and internal stiffening, respectively.From geometrical considerations the relationships between the displacements (uri , vri , wri ) of the ith

stiffener and the displacements (u , v , w) of the shell at the position of the stiffener are given by:

uri = u + eriw

x,

vri = v

1 +eri

R

+

eri

R

w

,

wri = w. (7)

Substituting (5)(7) into (3) and (4), the ring stiffener energy can be written in the form of a shell middlesurface displacement.

Therefore, the energy functional of a ring-stiffened cylindrical shell can be written as:

F = U T +

Ni=1

(U ri T ri ) . (8)

The following functions are adopted to separate the spatial variables x and and the time variable t:

u(x,,t) = u(x) sin(n + t),

v(x,,t) = v(x) cos(n + t),

w(x,,t) = w(x) sin(n + t), (9)

where n is the number of circumferential waves and is the circular frequency of vibration.For generality and convenience, the following nondimensional terms are defined:

u =u

h; v =

v

h; w =

w

R; x =

x

L; =

R

L; =

h

R; eri =

eri

h

Eri =Eri

E; ri =

ri

; I zri =

I zri

Rh3; I xri =

I xri

Rh3; Ari =

Ari

h2; J ri =

J ri

Rh3

F = 2(1 2

)hRLE

F; 2 = (1 2

)R2

E2. (10)

Using (9) and (10), the nondimensional total energy functional can be expressed as:

F =

10

2 2

du

dx

2+ (nv + w)2 2

du

dx

(nv + w) +

1

2 2

dv

dx+ nu

2

+ 2

12

4

d2w

dx 2

2+ (n2w + nv)2

2 2

d2w

dx 2

(n2w + nv) + 2(1 ) 2

n

dw

dx+

3

4

dv

dx

n

4u

2


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2

2u2 + 2v2 + w2

dx

+

N

i=1

1 2

(1 + eri )3

Eri 2 Ur1i + Ur2i + Ur3i + Ur4i

2ri (1 + eri )Tr1i + Tr2i

, (11)

where

Ur1i = I zri

1 + eri n2 eri

dwdx

n2 u

2, (12-1)

Ur2i = I xri

1 n2

w2

, (12-2)

Ur3i = Ari(1 + eri )

2

n(1 + eri )v + (1 + n

2 eri )w

2

, (12-3)

Ur4i = J ri 12(1 + i )

n u + n dw

dx

2, (12-4)

and

Tr1i = Ari

u + 2eri

dw

dx

2+ ((1 + eri )v + eri nw)

2 + w2

, (13-1)

Tr2i = 2(Ixri + I zri )

dw

dx

2. (13-2)

2.3 Geometric boundary conditions

For free boundary conditions all of the displacement functions are nonzero:

u = 0; v = 0; w = 0. (14)

2.4 Ritz functions

In view of satisfying the foregoing geometric boundary conditions, the proposed Ritz functions for approxi-mating the displacements are:

u =

NS

j=1

pj xj1

(x)P

0u (1 x)P

1u =

NS

j=1

pj uj ,

v =

NS

j=1

qj xj1

(x)P0v (1 x)P1v = NS

j=1

qj vj ,

w =

NS

j=1

rj xj1

(x)P0w (1 x)P1w = NS

j=1

rj wj , (15)

where the powers of P are shown in Table 1. The superscripts ofP, i.e., 0 and 1, denote the cylindrical shellends at x = 0 and x = 1, respectively.

These forms of Ritz functions allow easy exact differentiation and integration. Also, by increasing thenumber of polynomial sentences NS, better convergence to exact solution can be achieved.


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Table 1 Powers ofP for Ritz functions

Boundary condition F

Pu 0Pv 0

Pw 0

2.5 Equations of motion

Applying the RayleighRitz method (minimization of nondimensional energy functional with respect to Ritzfunction coefficients), the equations of motion are derived as follows:

Fpj

= 0,

Fqj

= 0,

Frj

= 0

j = 1, 2, . . . , NS. (16)

Substituting (15) into (11) and then into (16) results in the following eigenvalue equation:[K] +

Ni=1

[Kri ] 2

[M] +

Ni=1

[Mri ]

{C} = {0} , (17)

where [K] and [M] are the stiffness and mass matrices of a cylindrical shell, respectively, and [Kri ] and [Mri ]arethe corresponding matrices of the ith ring stiffener.Also,{C} = {p1, . . . , pNS, q1, . . . , qNS, r1, . . . , rNS}

T

is the column vector of Ritz coefficients and 2 = (1 2)R 22/E is the nondimensional frequencyparameter.

3 Results and discussions

3.1 Analytical results for cylindrical shell with nonuniform ring spacing and eccentricity with freefreeboundary condition (FFBC)

Here, the effects of nonuniform ring spacing and nonuniform stiffener eccentricity distribution are considered,separately and simultaneously. The main purpose of this study is to determine whether it is possible to obtainhigher natural frequencies given a nonuniform stiffener distribution with constant stiffener mass. To this end,the M1 model with uniform stiffener distribution and freefree boundary condition (FFBC) is considered,which its properties are shown in Table 2. Uniform distribution is the case of evenly spaced stiffeners andstiffeners of equal depth. Some cases of nonuniform ring spacing and nonuniform eccentricity distributionsare shown in Fig. 2. The minimum stiffener depth is located at the two ends of the shell, and the maximumstiffener depth is located at the midsection of the shell length. The maximum and minimum stiffener depth isdetermined such that the mass and volume of stiffeners remain unchanged with respect to uniform distribution.

Table 2 Geometrical and material properties of stiffened shell

Characteristics (dimensions) Dimension values M1 model

Number of rings (N) 4Shell radius R (m) 0.0825Shell thickness h (m) 0.0025Shell length L (m) 0.2475Ring depth dr (m) 0.0037Ring width br (m) 0.002Modulus of elasticity E (Gpa) 201

Mass density (kg/m3) 7823Poissons ratio 0.3Stiffening type External


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Fig. 2 Nonuniform ring spacing and nonuniform eccentricity distribution

Assuming the same width for all stiffeners, the volume of rings for externally stiffened shell can be writtenas:

Vuniform = 2 N

R +

h + d0

2

bd0, (22-1)

Vnon-uniform = 2 b

N

=1

R + d1 +

h + d2

d

+ 2 N

R +

h + d1

2

bd1, (22-2)

where d1 represents the minimum stiffener depth and d0 is the stiffener depth in uniform distribution.Equating the volume of stiffeners in uniform and nonuniform distributions, a second-order equation

corresponding to D can be obtained as follows:

1N

D

R + d1 +h2

2 N=1

r 2 + 2N

D

R + d1 +h2

N=1

r

2

R + h2

(d0 d1)

R + d1 +h2

2

d20 d21

R + d1 +h2

2 = 0, (23)where D denotes the difference between the maximum and minimum depth of nonuniformly distributedstiffeners.

For internally stiffened shell, the corresponding equation can be written as:

1

N

D

R d1 h2

2 N=1

r 2 +2

N

D

R d1 h2

N

=1

r

2

R h2

(d0 d1)R d1

h2

2 +

d20 d21

R d1 h2

2 = 0. (24)Selecting a value for d1, the value of D

can be determined by solving (23) or (24). The depth of eachstiffener drk would be obtained as follows:

r =dr

D=

(2a )

, N/2,

(2 (1 a )) , > N/2;

(25)

a = k

N + 1

, = 1, . . . , N; (26)


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Fig. 3 Natural frequency variations vs. depth ratio for equal ring spacing ( = 1) and m = 1. a n = 2. b n = 3. c n = 4. d n = 5

dr = r D

+ d1, (27)

where in Fig. 2 and in (25)(27) and represent, respectively, the order of variations of the eccentricitydistribution function and the ring spacing distribution function along the shell length.

For > 1, the stiffener concentration at the two ends of the shell is more than its middle. This means that

the ring spacing in the middle section of the shell is greater than the ring spacing at the two ends. On the otherhand, for < 1, the ring stiffeners are compressed in the midsection of the shell length. In this study, variesfrom 0.1 to 2. The case of = 1 denotes the case of evenly spaced ring stiffeners along the shell length. Also, varies from 0 to 2. The case of = 0 and = 1 denotes a uniform distribution of ring stiffeners along theshell length like the M1 model, as shown in Table 2.

Figure 3ad shows the variations of the natural vibration frequencies with respect to depth ratio (d1/d0),corresponding to circumferential waves n = 2, 3, 4, 5 and longitudinal wave m = 1, for different values of.Here, the ring spacing is uniform ( = 1), and only the effect of nonuniform eccentricity is considered. Themode associated to n = 1 and m = 1 is rigid body motion with natural frequency equal to zero, which isnot shown. In cylindrical shells with FFBC, the radial component of amplitude for different points is constantalong the shell length for m = 1.

It should be noted that reduction of the depth ratio increases the mass and stiffness in the midsection of theshell and decreases them at the two ends of the shell. Since the decrease in stiffness is more than the decrease

in mass at the two ends, the natural frequencies of various circumferential modes decrease, as can be seen inFig. 3ad. On the other hand, for higher depth ratios, an increment in natural frequencies is observed for thesemodes.

Figure 4ad shows the aforementioned results corresponding to m = 2. In this figure, the natural frequencyvariations are different for n = 2, 3 and n = 4, 5. In cylindrical shells with FFBC, the radial component ofamplitude for different points is a linear function ofx for m = 2 and one node is introduced in the midsectionof the shell length. Given a reduction in the depth ratio, the stiffness decrease is less than the mass decrease atthe two ends for n = 2, 3. Therefore, the natural frequencies for different values of increase, as can be seenin Fig. 3a,b.

Conversely, for n = 4, 5 reduction in the depth ratio, the stiffness decrease is greaterthan the mass decreaseat the two ends. Therefore, the natural frequencies for different values of decrease, as canbe seen in Fig. 3c,d.

Figures 5 and 6ad show the so-called results corresponding to m = 3. In these figures, the natural fre-quency variations are different for n = 1, 3, 4, 5 and n = 2. For m3, the radial component of amplitude for


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Fig. 4 Natural frequency variations vs. depth ratio for equal ring spacing ( = 1) and m = 2. a n = 2. b n = 3. c n = 4. d n = 5

different points is a characteristic curve, similar to that obtained for a freefree beam, with (m1) axial nodes.In m = 3, the first and the last rings are in axial nodes, and these stiffeners are not affected by the stiffness ofthe shell. At a constant stiffener mass, reducing the depth of the first and the last rings allow to increase thedepth of other stiffeners. Therefore, reduction of the depth ratio increases the natural frequencies for differentvalues of, as shown in Figs. 5 and 6a,c,d. The natural frequency variations for n = 2 (Fig. 6b) are different.Because the fundamental frequency of this shell occurred at n = 2, the behavior in this mode is irregular (seebelow).

Figure 7ad shows the variations of the natural frequencies of vibration versus the depth ratio, correspond-ing to n = 2 and m = 1, 2 for = 1 and different values of . The crossing of some natural frequency curvesis observed for higher depth ratios. Increasing , this crossing point occurs at higher depth ratios. Therefore,

Fig. 5 Natural frequency variations vs. depth ratio for equal ring spacing ( = 1) and m = 3, n = 1


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Fig. 6 Natural frequencies variations vs. depth ratio for equal ring spacing ( = 1) and m = 3. a n = 2. b n = 3. c n = 4.d n = 5

Fig. 7 Natural frequency variations vs. depth ratio for equal ring spacing ( = 1). a = 0.5. b = 1. c = 1.5. d = 2

if the depth ratio is increased, the fundamental frequency mode switches from m = 1, n = 2 to m = 2, n = 2and a sudden reduction in the fundamental frequency is observed. This case occurs in Fig. 7ad. Therefore, atconstant mass of total stiffeners, a higher fundamental frequency can be obtained by selecting a suitable valuefor d1/d0.


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Figure 8a,b shows the variations of the fundamental frequency vs. , corresponding to different values of for d1/d0 = 0.1, 0.5. The maximum fundamental frequency is obtained corresponding to = 2, = 2,and d1/d0 = 0.1.

3.2 Experimental results for nonuniformly ring-stiffened cylindrical shells

A model of a ring-stiffened cylindrical shell with unequal ring spacing and nonuniform eccentricity is con-sidered. Two approaches to this shell model with FFBC are applied for the modal analysis, namely, analyticaland experimental.

In the experimental method, modal testing is performed. To this end, an experimental stiffened shell modelwith dimensions as shown in Fig. 9 is machined from a thick steel pipe. It should be noted that both the ringstiffeners and the shell are made of the same material. The model is hung with rubber rope to simulate FFBC.The experimental testing equipment setup and FFBC simulation are shown in Fig. 10. The modal testing isperformed to obtain the modal characteristics of the stiffened cylindrical shell between 0 and 3200 Hz. Themodel is excited at predetermined points with an impact hammer in the radial direction of the shell. Then theresponse is measured using an accelerometer at a specified fixed point in the radial direction (roving hammer

and fixed response method).To avoid hiding of some modes and also to obtain accurate longitudinal and circumferential mode shapes,

the model is divided into 24 points in the circumferential direction and 12 points along the shell length (Fig. 11).The shell is excited by a hammer on these points. A piezoelectric accelerometer is attached to the model atpoint 1 by wax to measure the output acceleration. The analysis of frequency response functions (FRFs) isperformed using the STAR MODAL software. Outputs of this analysis are natural frequencies, mode shapes,and damping in each mode. Figure 12 shows the FRF of point 1, which is called the driving point. Table 3shows the obtained experimental results including natural frequencies and damping ratios for each mode. Intotal, 11 natural frequencies are found within that range with corresponding damping ratios and mode shapes.

Fig. 8 Fundamental frequency variations vs. parameter corresponding to different values. a d1/d0 = 0.1. b d1/d0 = 0.5

Fig. 9 Experimental model physical dimensions


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Fig. 10 Modal testing equipment setup and freefree boundary condition simulation

Fig. 11 Excitation points in modal testing

Table 4 shows the comparison of predicted analytical results of natural frequencies with obtained experi-

mental results of modal testing, which are in good agreement.Figures 1316 show some mode shapes of vibration obtained from analytical solution and modal testing.

In experimental mode shapes, dashed lines represent the undeformed shape.

4 Conclusions

The free vibration of ring-stiffened cylindrical shells with nonuniform eccentricity and unequal ring spacingfor FFBC is considered with analytical and experimental approaches. In the analytical method, the Ritz methodis used and stiffeners are treated as discrete elements. Some new natural frequency results for various ordersof eccentricity distribution function and ring spacing distribution function are presented. At constant stiffenermass, selecting the best stiffener distribution parameter values can increase natural frequencies significantly.Therefore, it is important to study the optimal distribution of ring stiffeners for further enhancement of naturalfrequencies.


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Fig. 12 Driving point frequency response functions (FRF)

Table 3 Modal testing results including natural frequencies and damping for each mode

Mode Frequency (Hz) Damping (Hz) Damping (%)

1 326.283 0.6868 0.21052 347.7278 4.1598 1.19623 877.7521 1.2028 0.13704 920.8615 1.0412 0.11315 1597.2084 7.0756 0.44306 1697.7704 2.1497 0.12667 2295.2971 12.9315 0.56348 2384.1682 6.7627 0.28369 2516.2180 4.4320 0.176110 2642.9446 2.0020 0.0757

11 3066.7891 4.4320 0.1445

Table 4 Comparison of experimental and analytical results

Mode number Experimental results (Hz) Analytical results (Hz) Difference (%)

m n

1 1 02 326.3 323.8 0.763 920.9 911.3 14 1703.4 1700 0.25 2643 2584 2.236 3639

2 1 22 347.7 324.3 6.733 877.7 874 0.424 1601.9 1622 1.255 2516 2543 16 3639

3 1 64722 37153 2384 2357 1.134 2295 2252.6 1.855 3066 3036 0.986 4144


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Fig. 13 Extracted mode shape. a Analytical. b Modal testing m = 2, n = 2




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Moreover, modal testing is performed experimentally, using fixed response approach, and modal param-eters are obtained from the test data. A comparison of the results of these methods showed good agreement.The results show that the presented analytical method has enough accuracy and is useful for free vibrationanalysis. Since the modal testing data for nonuniformly ring-stiffened cylindrical shells are not available, theseobtained data could be very useful for other researchers in this subject.

References

1. AL-Najafi, A.M.J., Warburton, G.B.: Free vibration of ring stiffened cylindrical shells. J Sound Vib 13(1), 925 (1970)2. Egle, D.M., Sewall, J.L.: Analysis of free vibration of orthogonally stiffened cylindrical shells with stiffeners treated as

discrete elements. AIAA J 6(3), 518526 (1968)3. Hoppmann, W.H.: Some characteristics of the flexural vibrations of orthogonally stiffened cylindrical shells. J Acoust Soc

Am 30, 7782 (1958)4. Mikulas, M.M., McElman, J.A.: On the free vibration of eccentrically stiffened cylindrical shells and plates. NASA TN-D

3010 (1965)5. Mustafa, B.A.J., Ali, R.: An energy method for free vibration analysis of stiffened circular cylindrical shells. Comput Struct

32(2), 335363 (1989)6. Rosen, A., Singer, J.: Vibrations of axially loaded stiffened cylindrical shells. J Sound Vib 34(3), 357378 (1974)7. Ruotolo, R.: A comparison of some thin shell theories used for the dynamic analysis of stiffened cylinders. J Sound Vib

243(5), 847860 (2001)8. Sanders, J.L.: An improved first-approximation theory for thin shells. NASA TR R-24 (1959)9. Swaddiwudhipong, S., Tian, J., Wang, C.M.: Vibrations of cylindrical shells with intermediate supports. J Sound Vib 187(1),

6993 (1995)10. Wang, C.M., Swaddiwudhipong, S., Tian, J.: Ritz method for vibration analysis of cylindrical shells with ring stiffeners. J

Eng Mech 123, 134142 (1997)

Analytical and Experimental Modal Analysis of Non Uniformly Ring-stiffened Cylindrical Shells

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