Vibration and sound radiation of submerged finite cylindrical shells with pre-stress

Theoretical solutions of vibration and sound radiation of submerged finite cylindrical shells with pre-stress were derived by using a modal summation method. Motion equations of cylindrical shells with pre-stress were established on the basis of Flügge theory. Additional impedance caused by pre-stress was added to the basic vibration equation. Pre-stress was expressed by uniform and trigonometric forms to obtain the sound radiation power, mean quadratic velocity and radiation efficiency of cylindrical shells. Numerical calculation was conducted to study the effects of tensile and compressive stresses, stress direction, value size and distribution on vibration and sound radiation of cylindrical shells. This study may provide a reference for controlling sound radiation of underwater vehicles.


Mathematical Modelling
Here, the cylindrical shell ( Fig. 1), whose thickness divided by radius is less than 5%, is thin with uniform thickness h, radius a, length L, mass density ρ p , modulus of elasticity E and Poisson's ratio ν. This shell is submerged in infinite water domain in which the sound velocity is c 0 , and the fluid density is ρ 0 . The axial, circumferential and radial displacements of the shell are represented by u, v and w, respectively.
Only the radial displacement w contributes to the sound field when the acoustic medium is surrounding the shell. The coupled vibration of the cylindrical shell in water is solved by the modal expansion method while ignor- The expansion coefficients are The modal equation is obtained by substituting Eqs (1) and (2) into the vibration equation of the shell, as shown as follows: The elements of the matrix are www.nature.com/scientificreports www.nature.com/scientificreports/ −Ω a k a n a a k an a a k a k a n k a a k a n k a a a n n k a a n k a n mn mn M mn mn where Z mn M is the mechanical impedance of the cylindrical shell, as shown as follows: The surface sound pressure caused by sound radiation load is: mn q qn qmn a where Z qmn a is the mutual radiation impedance between q n ( , ) and m n ( , ) order modes, as shown as follows: where Z K ( ) n a is the sound radiation impedance of K space, as shown as follows: The coupled vibration equation with sound radiation load is obtained as mn mn M q qn qmn a mn The sound radiation power of the shell is where S is the surface area of the shell. When = n 0 and ≥ n 1, ε = 1 n and ε = 2 n , respectively. The mean quadratic velocity is The radiation efficiency is

Effect of Pre-Stress
Liu and Chen 13 indicated that the vibration equation of cylindrical shell with pre-stress can be expressed as www.nature.com/scientificreports www.nature.com/scientificreports/ 2 is the bending stiffness; and C 1 , C 2 and C 3 represent the coupling terms of structural stress and vibration, whose specific expressions are as follows: where σ x r , σ ϕ r and τ ϕ x r are the axial positive, circumferential positive and shear stresses, respectively. The coupling terms (i.e. C 1 , C 2 and C 3 ) of stress vibration are added to the equations compared with the differential formula of the cylindrical shell without stress. When the cylindrical shell has no pre-stress, the terms C 1 , C 2 and C 3 are zero, and Eq. (16) is simplified as the classical motion differential formula of cylindrical shell. Substituting Eq. (1) into Eq. (16) and simplifying Eq. (14) by the trigonometric function's orthogonality yield the following:    . Different pre-stress forms lead to varying values. The results of two typical pre-stress distribution forms are presented as follows.
(1) Assume that the shear stress of the cylindrical shell is τ = ϕ 0 x r . Pre-stresses σ x r and σ ϕ r are uniformly distributed throughout the entire shell, which can be expressed as follows: Here, Eq. (21) can be obtained by the trigonometric function's orthogonality and substituting the pre-stress into K 1 , K 2 and K 3 .
mn is the coefficient matrix of the shell considering the pre-stresses. In comparison with the shell without pre-stress, the pre-stress influence matrix is added, as shown as follows: where σ g rx and σ ϕ g r are structural stress amplitudes, and g is a positive integer that represents the modal order corresponding to the stress.
The trigonometric function's orthogonality yields Equation (24) by substituting the pre-stresses into K 1 , K 2 and K 3 .
, Λ n is a diagonal matrix composed of the shell equation coefficients and R g is the pre-stress influence matrix.  , and the non-zero elements of R pq g can be obtained as follows.

Numerical Calculation and Result Discussion
The numerical calculation program is conducted on the basis of the theorical equations. In order to verify the accuracy of the program, the acoustic power results of the cylindrical shell without pre-stress are compared with the results in ref. 4 Fig. 2    Influences of tensile and compressive pre-stresses. The uniform stresses of 300 MPa and −300 MPa, which often exist in the high-strength shipbuilding steel according to ref. 16 , are calculated as the basic stresses to apply to the cylindrical shell. The results are shown in Fig. 3. As shown in Fig. 3(a), pre-stress does not considerably change the sound radiation power because it alters the mechanical impedance of the shell and does not affect radiation impedance. In the low frequency band, the sound radiation mainly comes from the low-order modes. The mechanical impedance and sound radiation changes caused by pre-stress are small. However, the existence of pre-stress changes the resonance characteristics of the shell, thereby resulting in some changes in the position and height of the peak. tensile and compressive stresses exhibit different effects on structural vibration and sound radiation. In some frequency bands, compressive stress leads to considerable peaks of structural sound radiation curve. The peak value of sound radiation with compressive stress moves to a lower frequency than that with tensile stress. This occurrence is due to compressive and axial stresses that decrease and increase the stiffness and natural frequency of the shell, respectively. As shown in Fig. 3(b), the tensile pre-stress reduces the mean quadratic velocity of the shell, particularly in the low frequency band (0~500 Hz). This occurrence is because of the increase of the total mechanical impedance of the shell and the decrease of the vibration speed with pre-stress. According to Eq. (15), an inevitable increase in radiation efficiency occurs because the sound radiation power does not change considerably, and the vibration speed evidently decreases, as shown in Fig. 3(c). The mean quadratic velocity of the shell with tensile stress is considerably smaller than that with compressive stress due to the stiffness increase, thereby leading to the increase of radiation efficiency.

. A good agreement is shown in
Influences of axial and circumferential pre-stresses. The axial and circumferential uniform stresses are applied to the basic cylindrical shell. The stress value is 300 Mpa, and the results are shown in Fig. 4.
As depicted in Fig. 4(a), the circumferential and axial stresses demonstrate minimal effect on the sound acoustic power, which mainly change the extreme value at some frequencies. As shown in Fig. 4(b,c), relatively large changes are observed for the mean quadratic velocity and radiation efficiency. The circumferential stress exhibits a greater effect on the stiffness than the axial stress, which correspondingly changes the vibration characteristics of the shell. Simultaneously, the maximum mean quadratic velocity is generated for the mean quadratic velocity with circumferential stress due to the change of the mechanical impedance of the shell, which should be given attention during the structural design process.
Influence of pre-stress value size. The axial and circumferential uniform stresses of 100, 300 and 500 Mpa are applied.
As shown in Fig. 5, the pre-stress value size mainly changes the extreme value and makes the deviation of the sound radiation power curve. The influences of different pre-stress values on shell vibration and radiation efficiency vary in diverse frequency bands. This condition is due to the changes in the inherent characteristics of the cylindrical shell by pre-stress, including the natural frequencies and modes, thereby resulting in the changes of vibration and sound radiation properties. The change degree depends on the characteristics and pre-stress value of the shell.
Influence of pre-stress distribution. The trigonometric function is used to fit the pre-stress. The distribution forms of the pre-stress are shown in Fig. 6. The calculation results are shown in Fig. 7.
As shown in Fig. 7, different pre-stress distributions make diverse effects on the vibration and sound radiation of the cylindrical shell, thereby showing the effectiveness of the proposed method. The trigonometric function can effectively fit the pre-stress distribution and obtain accurate responses.

Conclusions
The theoretical solutions of vibration and sound radiation of submerged finite cylindrical shells with pre-stress are obtained on the basis of Flügge theory and modal summation method. The effects of tensile and compressive stresses, stress direction, value size and distribution on vibration and sound radiation of the cylindrical shell are studied by the numerical calculation.
The results show that additional impedance caused by pre-stress is added to the basic vibration equation, which changes the structural vibration and sound radiation. Pre-stress changes the position and height of the peak of sound radiation power. Compressive stress decreases structural stiffness, whereas tensile stress increases it. The mean quadratic velocity with tensile stress is smaller than that with compressive stress. Circumferential stress has a greater effect on the stiffness than the axial stress. The pre-stress value size and distribution may affect the structural responses differently. The effect of pre-stress should be considered during designing, manufacturing and accurately predicting vibration and sound radiation of underwater vehicles.

Data Availability
All data generated or analysed during this study are included in this published article.