Effect of Bending Rigidity and Nonlinear Strains on Free Vibration of Hemi-ellipsoidal Shell

This paper focuses on free vibration of hemi-ellipsoidal shells with the consideration of the bending rigidity

and nonlinear terms in strain energy. The appropriate form of the energy functional is formulated based

on the principle of virtual work and the fundamental form of surfaces. Natural frequencies and their

corresponding mode shapes are determined using the modified direct iteration method. The obtained

 results, which show a close agreement with previous research, are compared with those obtained based

on the membrane theory. The effect of the support condition, thickness, size ratio, and volume constraint

condition on frequency parameters and mode shapes is demonstrated. With the bending rigidity, shell

thickness has a significant impact on the frequency, especially in higher vibration modes and in shells with

a considerable thickness but the frequency parameter converges to that determined by using the

membrane theory while the reference radius-to-thickness ratio is increasing. In addition, accounting for

 the bending rigidity solves the issue of determining natural frequencies and mode shapes of the shells

 using the membrane theory without the volume constraint condition. The obtained results also indicate

 that the free vibration analysis with bending is essential for the hemi-ellipsoidal shell with a base radius-

 to-thickness ratio of less than 100, which gives over 2.84% difference compared with that of the shell

 derived by membrane theory, and this allows engineers to perform the analysis in more applications

This paper focuses on free vibration of hemi-ellipsoidal shells with the consideration of the bending rigidity

and nonlinear terms in strain energy. The appropriate form of the energy functional is formulated based

on the principle of virtual work and the fundamental form of surfaces. Natural frequencies and their

corresponding mode shapes are determined using the modified direct iteration method. The obtained

 results, which show a close agreement with previous research, are compared with those obtained based

on the membrane theory. The effect of the support condition, thickness, size ratio, and volume constraint

condition on frequency parameters and mode shapes is demonstrated. With the bending rigidity, shell

thickness has a significant impact on the frequency, especially in higher vibration modes and in shells with

a considerable thickness but the frequency parameter converges to that determined by using the

membrane theory while the reference radius-to-thickness ratio is increasing. In addition, accounting for

 the bending rigidity solves the issue of determining natural frequencies and mode shapes of the shells

 using the membrane theory without the volume constraint condition. The obtained results also indicate

 that the free vibration analysis with bending is essential for the hemi-ellipsoidal shell with a base radius-

 to-thickness ratio of less than 100, which gives over 2.84% difference compared with that of the shell

 derived by membrane theory, and this allows engineers to perform the analysis in more applications