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Latin American Journal of Solids and Structures

versão impressa ISSN 1679-7817versão On-line ISSN 1679-7825

Lat. Am. j. solids struct. vol.14 no.9 Rio de Janeiro set. 2017 


An Analytical Time Domain Solution for the Forced Vibration Analysis of Thick-Walled Cylinders

Bashir Movahediana 

Meshkat Botshekana 

a Department of Civil Engineering, Isfahan University of Technology, Isfahan Iran.;


In this paper, we propose a time domain analytical solution for the forced vibration analysis of thick-walled hollow cylinders in presence of polar orthotropy. In this regard, solution of the governing equation is decomposed into two parts. The role of the first one is to satisfy boundary conditions utilizing the method of separation of variables besides of Fourier series expansion of the non-homogenous boundary conditions. The second part has been also expressed as the series of orthogonal characteristic functions with the aim of satisfaction of initial conditions. The proposed analytical solution has been implemented to evaluate the dynamic response of the cylinder in solution of some sample problems which are chosen from previous studies.

Keywords: Forced vibration; Thick-walled cylinders; Analytical time domain solution; Polar orthotropy


The increasing application of thick-walled cylinders subjected to dynamic inner pressure in diverse fields such as aerospace engineering, civil engineering and submarine structures has made these members of paramount importance. In this regards, so many studies have been done to compute time dependent responses of the both isotropic and anisotropic cylinders (Huang, 1969; Keles and Tutuncu, 2009; Shakeri et al. 2006; Baba and Keles, 2015; Ghannad and Gharooni, 2015). In most of these researches, the time dependency of the governing equation has been eliminated utilizing the Laplace transform (Huang, 1969; Keles and Tutuncu, 2009; Baba and Keles, 2015).

Recently, Baba and Keles (2015) proposed an analytical solution for the anisotropic hollow cylinders under the internal dynamic pressure in Laplace domain. They also employed the modified Durbin's numerical inversion to obtain solution in time. Since the transformation of the dynamic response from the Laplace domain to the time domain problems is associated with some difficulties, here the new idea is employed from previous studies (Shamsaei and Boroomand, 2011; Movahedian and Boroomand, 2014; Movahedian et al. 2013) to propose an analytical time domain solution for the governing differential equation of the mentioned problem. This solution enables us to estimate the dynamic responses of the cylinder, i.e. the radial and hoop stresses or radial displacement, with desirable accuracy directly in time.

The layout of the paper is as follows, in the next section, the model used for the dynamic analysis of orthotropic hollow cylinders is described and the governing equations are derived. In section 3, the superposition principle is employed to express the solution in terms of two parts. In section 4, the proposed solution is applied to cases which were studied by Baba and Keles (2015) in order to validate the study. In addition, a problem with non-homogenous initial conditions is included in this part. Finally, in section 5, the summary of the conclusions made throughout the paper are provided.


In this section, the governing differential equation of the vibration of thick-walled hollow cylinder in presence of polar orthotropy is derived. Utilizing the axisymmetric conditions, the radial strain, εr , and tangential strain, εθ are related to radial displacement u, as

εr=ur, εθ=ur. (1)

Considering the polar orthotropy of the cylinder, the stress-strain relation can be expressed as

[σrσθ]=[C11C12C12C22][εrεθ] (2)

where C 11 and C 22 are the stiffness modules in the radial and circumferential directions and C 12 is the material parameter that includes the Poisson's effect.

Consider an element on the thick walled hollow cylinder bounded by lines (r, θ) and (r + dr, θ + dθ) Due to symmetry, the radial and hoop stresses remain constant along angular coordinate, i.e., [σ r /∂θ] = 0 and [σ θ/∂θ ] = 0, and the shear stress component, τ , must be zero. In this way, equilibrium equation in the radial direction gives,

(hσr)r+h(σrσθ)r=ρh2u(r,t)t2 (3)

where u is the displacement component in radial direction that must be found in polar coordinate r and time t. Also ρ and h are the material density and element’s thickness. Figure 1 shows the geometry of thick walled hallow cylinder as well as stress component on the specified element in polar coordinate.

Figure 1: The geometry of thick walled hallow cylinder in polar coordinate. 

Substituting definitions of (1) and (2), in the above relation leads to the following equation, (the element’s thickness has been removed from both sides of (3)),

(hσr)r+h(σrσθ)r=ρh2u(r,t)t2 (4)

in which, c=C11/ρ and n=C22/C11 is a non-dimensional parameter that indicates degree of anisotropy of material. If the variation of internal and external pressures are respectively expressed by functions PI (t) and PE (t), the governing boundary conditions at r = a and r = ka can be stated as follow

σr|r=a=PI(t), σr|r=ka=PE(t). (5)

Aforementioned conditions can be expressed in terms of radial displacement, u, as follow

[C11(ur+1αur)]r=a=PI(t), [C11(ur+1αur)]r=ka=PE(t), (6)

where α = C 11 /C 12. The general form of the initial displacement and initial velocity conditions of can be satisfied by the following equation

[u(r,t)]t=0=u0(r), [u˙(r,t)]t=0=u˙0(r). (7)


The aim here is to find the time domain analytical solution of the Equation (4) through employing Fourier’s series expansion of boundary conditions as well as defining a suitable characteristic problem to satisfy initial conditions. In this regard, we split the solution into two parts as follows:

u(r,t)=u1(r,t)+u2(r,t) (8)

In the above relation the role of u 1(r, t) is to fully satisfy the boundary conditions stated in (6). After determining u 1(r, t), obviously, the task of satisfying the actual initial conditions remains for u 2(r, t) which will be explained later. Prior to construction of u 1(r, t), one should expand the right hand sides of conditions in (6) in terms of Fourier series in time; i.e. Fourier sine series as:

PI(t)=i=1NAisinωit, Ai=2T0TPI(t)sinωitdt, (9)

PE(t)=i=1NBisinωit, Bi=2T0TPI(t)sinωitdt, (10)

where N is the number of the basis functions to be used, ωi = /T and Ί is the length of finite time interval (t ∊ [0, T] instead of t ∊[0, ∞)). The magnitude of Ί may be determined by inspection, i.e. in successive solutions, one can enlarge Ί until the final solution to u converges to a solution for smaller time interval, t ∊ [0, T 1}, T 1 < T. Utilizing the method of separation of variables leads to express u 1(r, t) in the following form:

u1(r,t)=i=1Nu¯1,isinωit (11)

where ū 1, i (r) is the solution to the following ordinary differential equation which comes from substituting (11) in Equation (4).

r2d2u¯1,idr2+rdu¯1,idr+(r2ωi2c2n2)u¯1,i=0. (12)

The aforementioned equation is known as the Bessel differential equation of order n. The solution of which can be expressed as the combination of the Bessel function of the first kind, Jn ( ωir/c ), and the second kind, Yn ( ωir/c ). In other words, the solution can be stated as

u¯1,i(r)=c1,iJn(ωir/c)+c2,iYn(ωir/c) (13)

The constant coefficients c 1, i and c 2, i in the above relation are determined by satisfaction of the radial stress boundary conditions at r = a and r =ka in (6) as

[C11(du¯1,idr+1αu¯1,ir)]r=a=Ai, (14)

[C11(du¯1,idr+1αu¯1,ir)]r=ka=Bi, (15)

Finally, after doing some simplifications, u 1(r, t) is therefore written as

u1(r,t)=i=1NcaαC11Yn(ωir/c)φ1,i+Jn(ωir/c)φ2,iφ3,i+φ4,isinωit (16)


φ1,i=ck(nα1)BiJn(ωia/c)+c(nα1)AiJn(ωika/c) ka α ωi[BiJn1(ωia/c)+AiJn1(ωika/c)] (17)

φ2,i=c(nα1)[kBiYn(ωia/c)+AiYn(ωika/c)] +ka α ωi[BiYn1(ωia/c)+AiJn1(ωika/c)] (18)

φ3,i=c3(nα1)2[Jn(ωika/c)Yn(ωia/c)Jn(ωia/c)Yn(ωika/c)] (19)

φ4,i=+aαωi{c(nα1)[Jn(ωika/c)Yn1(ωia/c)+kJn(ωia/c)Yn1(ωika/c) kJn1(ωika/c)Yn(ωia/c)+Jn1(ωia/c)Yn(ωika/c)] +kaαωi[ Jn1(ωika/c)Yn1(ωia/c)Jn1(ωika/c)Yn1(ωika/c)]} (20)

At this point, the second part of relation (8) must be determined. In this regard, the method of separation of variables is applied by substituting

u2(r,t)=u¯2(r)T(t) (21)

in (4) which yields:

r2d2u¯2(r)dr2+rdu¯2(r)drn2r2u¯2(r)u¯2(r)=1c2T(t)d2T(t)dt2=β2, (22)

where - β 2 is the separation constant. The solutions of the two separated ordinary differential equations in (22) for ū 2(r) and T(t) are respectively expressed as:

u¯2(r)=c3Jn(rβ)+c4Yn(rβ) (23)


T(t)=a sin cβt+b cos cβt (24)

In order to determine the unknown coefficients of c 3, c 4 and β, a characteristic problem must be formed by substituting (21) in the homogenous form of stress boundary conditions, i.e., PI (t) = PE (t) = 0 in (6), which results in following system of algebraic equations,

AβC=0 (25)

In the above relation, A β is a 2 × 2 matrix depending on β, and C is a 2×1 vector containing the coefficients of c 3 and c 4. The components of A β are:

Aβ=[aαβJn1(aβ)+(1nα)Jn(aβ)aαaαβYn1(aβ)+(1nα)Yn(aβ)aαkaαβJn1(kaβ)+(1nα)Jn(kaβ)kaαkaαβYn1(kaβ)+(1nα)Yn(kaβ)kaα]. (26)

To have non-trivial solution to (25), the determinant of A β is set zero.

|Aβ|=0. (27)

The above mentioned issue a non-standard eigenvalue problem which should be solved for β. Moreover, by substituting the roots of (27), i.e. βj , j = 1, 2, ... in (25), the corresponding null space C j, j = 1, 2, ...with components c 3, j and c 4, j , have been computed. In this way, the second part of the u(r, t) is therefore written as

u2(r,t)=j=1Mu¯2,j(r)×[ajsincβj t+bjcos cβj t], (28)

in which ū 2, j (r) = c 3,j Jn ( rβj ) + c 4, jYn ( rβj ) and ( aj , bj ) denote a set of new unknown coefficients to be determined by satisfying the initial displacement and velocity conditions in (7). Utilizing the orthogonality of the set ū 2, j (r) with respect to weight function w(r) = r, (See Hildebrand (1976), for more details on the properties of Strum-Liouville problems), these coefficients have been determined by inserting (28) and (11) in (7) as follows

aj=akaru¯2,j(r)[u˙0(r)i=1Nu¯1,i(r)ωi]drakacβjr[u¯2,j(r)]2dr, (29)

bj=akaru¯2,j(r)u0(r)drakar[u¯2,j(r)]2dr. (30)

In this way, the radial displacement of a thick-walled cylinder can now be written as:

u(r,t)=i=1NcaαC11Yn(ωir/c)φ1,i+Jn(ωir/c)φ2,iφ3,i+φ4,isin(ωit)+j=1M[c3,jJn(rβj)+c4,jYn(rβj)]×[ajsin(cβjt)+bjcos(cβjt)]. (31)


The proposed method has been used for both homogenous and nonhomogeneous initial displacement conditions. In the both cases the specifications of orthotropic hollow cylinder are taken form Baba and Keles (2015), as C 11 = 1, ρ = 1, a = 1 and ka = 2. Moreover, the analytical solution in (31) has been computed using the first 100 terms of the series of u 1(r, t) with T = 80 for the Fourier sin series expansion of PI (t) in (9), and the first 100 terms of the series of u 2(r, t), i.e. M = 100. In Table 1, the first 15 sets of eigenvalues, βj , and ratio of the components of the related eigenvector, C j , have been provided for three types of material with different degrees of anisotropy.

Table 1: Results of the first 15 sets of the non-standard eigenvalue problem in (25) for three anisotropy's types of the material. 

n = 0.5, α = 3 n = 2, α = 2 n = 1, α = 2
j βj c3,j/ c4,j βj c3,j/ c4,j βj c3,j/ c4,j
1 0.25992 -0.91437 1.34969 -0.46413 0.61059 -0.92914
2 3.16786 -12.65974 3.45185 0.11203 3.20507 1.46431
3 6.29641 -25.17978 6.43419 -0.35032 6.31359 1.19987
4 9.43361 -37.73051 9.52479 -0.52919 9.44485 1.12786
5 12.573 -50.28905 12.6412 -0.62901 12.58137 1.0941
6 15.71327 -62.85071 15.76776 -0.69341 15.71994 1.07447
7 18.85398 -75.41394 18.89936 -0.73859 18.85953 1.06163
8 21.99494 -87.97806 22.03382 -0.77208 21.99969 1.05257
9 25.13606 -100.54275 25.17007 -0.79793 25.14021 1.04583
10 28.27728 -113.10781 28.30751 -0.81849 28.28097 1.04063
11 31.41858 -125.67314 31.44578 -0.83525 31.4219 1.03649
12 34.55993 -138.23865 34.58466 -0.84916 34.56295 1.03311
13 37.70132 -150.80431 37.72399 -0.8609 37.70409 1.03031
14 40.84274 -163.37007 40.86367 -0.87095 40.8453 1.02794
15 43.98419 -175.93592 44.00362 -0.87964 43.98656 1.02592

In the case of homogenous initial conditions, three different dynamic inner pressure functions were considered as PI (t) = 1, PI (t) = 1 - exp(0.8t) and PI (t) = 1 - cos(0.8t) to validate results with those stated in Baba and Keles (2015) (The variations of outer pressure was not considered in the mentioned reference, i.e. PE (t) = 0). In this regards, the variation of radial displacement, u(r, t), and hoop stress, σ θ (r, t) at r = a are illustrated in Figures 2 to 4. As can be seen, the results of in Baba and Keles (2015) and presented method follow similar trends in evaluation of both ur and σ θ .

Figure 2: Variation of (a) u(a, t) and (b) σ θ (a, t) due to PI (t) = 1 for three different degrees of anisotropy. 

Figure 3: Variation of (a) u(a, t) and (b) σ θ (a, t) due to PI (t) = 1 - exp(0.8t) for three different degrees of anisotropy. 

Figure 4: Variation of (a) u(a, t) and (b) σ θ (a, t) due to PI (t) = 1 - cos(0.8t) for three different degrees of anisotropy. 

As mentioned previously, the presented analytical solution is able to predict dynamic response of the hollow cylinder even in presence of nonhomogeneous initial and external pressure boundary conditions. In this regards, the second sample problem has been chosen to investigate the forced vibration of the orthotropic cylinder due to following variations of the inner and outer pressures,

PI(t)=1cos(0.8t), PE(t)=1 (32)

The initial displacement and initial velocity conditions are also considered as follows:

u0(r)=(ra)2(rk a)2, u˙0(r)=0 (33)

Figure 5 depicts the variations of σ r (r, t) and σ θ (r, t) within the thickness of the cylinder with n = 0.5 and α =3 at two time steps t = 2 and t = 5.

Figure 5: Variation of (a) σ r (r, 2), (b) σ r (r, 5), (c) σ r (r, 2) and (d) σ r (r, 5) within the cylinder’s thickness. 

In order to provide design criteria from the standpoint of fatigue of orthotropic cylinders, the variations of σ θ (a, t) and σ θ (ka, t) are illustrated in Figure 6 for three different degrees of anisotropy. As can be seen, increasing degrees of anisotropy, n, will result in an increase in the frequency of cylinder's response

Figure 6: Variation of (a) σ θ (a, t), (b) σ θ (ka, t) due to PI (t) = 1 - cos(0.8t) and PE (t) = 1 for three different degrees of anisotropy. 


In the present study, a semi analytical time domain solution has been proposed for the governing equation to the vibration of thick-walled hollow cylinder in the presence of polar orthotropy. The effects of different material properties and internal pressure variations on the dynamic responses of hollow cylinder have been investigated. The sufficient accuracy of the presented method has been also illustrated in comparison of the obtained results with those reported in Baba and Keles (2015). Finally, the superiorities of the presented solution can be listed as follows:

  • Employing the analytical solution, the dynamic response of the cylinder can be evaluated directly in time with no need to use any transformation such as inverse Laplace transform.

  • The proposed scheme can be used to evaluate dynamic response of polar orthotropic cylinders in presence of exterior pressure or non-homogenous initial conditions, which may be useful for designing purposes.

  • The presented method can be extended to evaluate transient response of the pipe conveying fluid due to internal and external temperature variations.


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Received: November 23, 2016; Revised: July 03, 2017; Accepted: July 03, 2017

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