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

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.


Latin American Journal of Solids and
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.

PROBLEM STATEMENT
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, r e , and tangential strain, q e are related to radial displacement u , as Considering the polar orthotropy of the cylinder, the stress-strain relation can be expressed as where 11 C and 22 C are the stiffness modules in the radial and circumferential directions and 12 C is the material parameter that includes the Poisson's effect.Consider an element on the thick walled hollow cylinder bounded by lines ( , ) r q and ( , ) r dr d q q + + .Due to symmetry, the radial and hoop stresses remain constant along angular coordinate, i.e., [ ] 0 r s q ¶ ¶ = and [ ] 0 q s q ¶ ¶ = , and the shear stress component, rq t , must be zero.In this way, equilibrium equation in the radial direction gives, where u is the displacement component in radial direction that must be found in polar coordinate r and time t .Also r 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.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)),

= -=
(5) Aforementioned conditions can be expressed in terms of radial displacement, u , as follow where C C a = . The general form of the initial displacement and initial velocity conditions of can be satisfied by the following equation

THE SOLUTION METHOD
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: Latin American Journal of Solids and Structures 14 (2017) 1741-1751 In the above relation the role of 1 ( , ) u r t is to fully satisfy the boundary conditions stated in (6).
After determining 1 ( , ) u r t , obviously, the task of satisfying the actual initial conditions remains for 2 ( , ) u r t which will be explained later.Prior to construction of 1 ( , ) u 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: where N is the number of the basis functions to be used, i i T The magnitude of T may be determined by inspec- tion, i.e. in successive solutions, one can enlarge T until the final solution to u converges to a solu- tion for smaller time interval, Utilizing the method of separation of variables leads to express 1 ( , ) u r t in the following form: sin where 1, ( ) i u r is the solution to the following ordinary differential equation which comes from substi- tuting (11) in Equation (4).
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, ( )

c w
, and the second kind,

(
) . In other words, the solution can be stated as , Latin American Journal of Solids and Structures 14 (2017) 1741-1751 Finally, after doing some simplifications, 1 ( , ) u r t is therefore written as sin where 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 in (4) which yields: where A is a 2 2 ´ matrix depending on b, and C is a 2 1 ´ vector containing the coefficients of 3 c and 4 c .The components of b A are: To have non-trivial solution to (25), the determinant of A  is set zero.

b = A (27)
The above mentioned issue a non-standard eigenvalue problem which should be solved for b.
, 1,2,... for more details on the properties of Strum-Liouville problems), these coefficients have been determined by inserting ( 28) and ( 11) in ( 7) as follows In this way, the radial displacement of a thick-walled cylinder can now be written as:

RESULTS AND DISCUSSIONS
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 and 2 ka = .Moreover, the analytical solution in (31) has been computed using the first 100 terms of the series of 1 ( , ) u r t with 8 0 T = for the Fourier sin series expansion of ( ) 9), and the first 100 terms of the series of 2 ( , ) u r t , i.e.
1 0 0 M = .In Table 1, the first 15 sets of eigenvalues, j b , and ratio of the components of the related eigenvector, j C , have been provided for three types of material with 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, ( ) 1 cos(0.8), ( ) 1 The initial displacement and initial velocity conditions are also considered as follows:

CONCLUSIONS
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.

Figure 1 :
Figure 1: The geometry of thick walled hallow cylinder in polar coordinate.
13) The constant coefficients 1,i c and 2,i c in the above relation are determined by satisfaction of the radial stress boundary conditions at r a = constant.The solutions of the two separated ordinary differential equations in (22) for 2 ( ) u r and ( ) T t are respectively expressed as: determine the unknown coefficients of 3 c , 4 c and b, a characteristic problem must be formed by substituting (21) in the homogenous form of stress boundary conditions, i.e., 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 2, ( ) j u r with respect to weight function ( )

Figure 2 :
Figure 2: Variation of (a) ( , ) u a t and (b) ( , ) a t q s Figure 5 depicts the variations of ( , ) r r t s and

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

American Journal of Solids and Structures 14 (2017) 1741-1751
Baba and Keles (2015)(The variations of outer pressure was not considered in the mentioned reference, i.e.