# Elastic-wave propagation in a joined cylindrical-conical-cylindrical shell

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## Abstract

The problem of longitudinal impact of a thin finite-joined shell, consisting of a cylinder-truncated cone-cylinder, is analyzed both experimentally and analytically. the model analyzed is a 1/100-scale replica of a portion of the Apollo/Saturn V vehicle. Experimental results were obtained from a drop-test system. Longitudinal and circumferential strain pulses were monitored on each section of the joined shell. The velocity of the impacter ring prior to impact was measured and used as a boundary condition in the solution of the governing partial-differential equations. A “bending” theory, including transverse-shear, radial-inertia and rotary-inertia effects, was used to analyze the finite-joined shell. Appropriate transformation relations were developed at each of the joints between the cylinders and truncated cone. The results were then obtained by solving the governing equations numerically by the method of characteristics. Good agreement between analytical and experimental strain profiles was obtained.

## Keywords

Boundary Condition Mechanical Engineer Fluid Dynamics Governing Equation Circumferential Strain## Nomenclature

*c*_{p}plate velocity=(

*E*_{ p }/ρ)^{1/2}*c*_{s}shear velocity=

*k*(*G*/ρ)^{1/2}*E*Young’s modulus of elasticity

*E*_{p}\(E/(1 - v^2 )\)

*G*shear modulus of elasticity=

*E*/2 (1+ν)*h*thickness of shell

*k*^{2}shear-correction factor=0.87

*M*_{s}shell moment

*N*_{s},*Q*_{s}shell forces

*r*radial coordinate

*r*_{o}radius of midsurface of cone at the small truncated end

*R*radius of midsurface of the shell

*s*, θ, ξshell coordinates; meridional, circumferential and normal, respectively

*u*_{s},*u*_{ξ}meridional and normal displacements, respectively

*u, w*meridional and normal displacements of the centroidal surface, respectively

*V*velocity imparted to shell by ring

*x*axial distance

*ɛ*_{s}meridional strain

*ɛ*_{θ}circumferential strain

- ϕ
1/2 apex angle of cone

- τ
_{o} pulse duration

- ν
Poisson’s ratio

- η
centroidal distance=\(\frac{{h^2 \cos \phi }}{{12(r_0 + ssin\phi )}}\)

- Ψ
rotation about the centroidal surface

- λ
pulse length

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## References

- 1.
*Hu, W. C. L.*and*Raney, J. P.*, “*Experimental and Analytical Study of Vibrations of Joined Shells*,”*AIAA J.*,**5**(*5*),*976–980*(May*1967*).Google Scholar - 2.
*Heimann, J. H.*and*Kolsky, H.*, “*The Propagation of Elastic Waves in Thin Cylindrical Shells*,”*J. Mech. Phys. Solids*,**14**,*121–130*(*1966*).Google Scholar - 3.
*Mortimer, R. W., Rose, J. L.*and*Chou, P. C.*, “*Longitudinal Impact of Cylindrical Shells*,”Experimental Mechanics,**12**(*1*),*25–31*(*1972*).CrossRefGoogle Scholar - 4.
*Mortimer, R. W.*and*Blum, A.*, “*Transient Solutions of a Longitudinally Impacted Conical Shell*,”*J. Appl. Mech.*,**38**(*2*),*545–547*(June*1971*).Google Scholar - 5.
*Kenner, V. H., Goldsmith, W.*and*Sackman, J. L.*, “*Longitudinal Impact on a Hollow Cone*,”*J. Appl. Mech.*,**36**(*3*),*445–450*(Sept.*1969*).Google Scholar - 6.
*Mortimer, R. W., Rose, J. L.*and*Blum, A.*, “*Longitudinal Impact of Cylindrical Shells with Discontinuous Cross-Sectional Area*,”*J. Appl. Mech.*,**39**(*4*),*1005–1010*(Dec.*1972*).Google Scholar - 7.
*Ripperger, E. A. and Abramson, H. N., “Reflection and Transmission of Elastic Pulses in a Bar at a Discontinuity in Cross Section,” Proc. Midwestern Conf. on Solid Mechanics, 3rd ed., 135–145 (1957)*.Google Scholar - 8.
*Rader, D.*and*Mao, M.*, “*Amplification of Longitudinal Stress Pulses in Elastic Bars with an Intermediate Tapered Region*,”Experimental Mechanics,**12**(*2*),*90–94*(*1972*).Google Scholar - 9.
*Chiu, S. S.*, “*Longitudinal Elastic Waves in a Composite Bar with Conical Sections*,”*AIAA J.*,**10**(*3*),*273–275*(*1972*).Google Scholar - 10.
*Mortimer, R. W., Chou, P. C. and Kiesel, H., “A General Linear Theory of Thick Shells of Revolution,” DIT Report No. 340-3 (June 1969)*.Google Scholar - 11.
*Leadbetter, S. A., Leonard, H. W. and Brock, E. J., “Design and Fabrication Considerations for a 1/10 Scale Replica Model of the Apollo/Saturn V,” NASA TN D-4138 (Oct. 1967)*.Google Scholar - 12.
*Chou, P. C.*and*Mortimer, R. W.*, “*Solution of One-Dimensional Elastic Wave Problems by the Method of Characteristics*,”*J. Appl. Mech.*,**34**(*3*),*745–750*(Sept.*1967*)Google Scholar - 13.
*Mortimer, R. W. and Hoburg, J. F., “MCDIT 21-A Computer Code for One-Dimensional Elastic Wave Problems,” NASA CR-1306 (April 1969)*.Google Scholar