Abstract
Experimental evidence is presented to verify the steady-state solution for a thin circular arch, pinned at the ends, and subjected to symmetrical and unsymmetrical support excitation. The steady-state solutions consist of a series of the free modes of vibration. It is shown how these solutions are developed when both supports of the arch are moving simultaneously and in phase with one another. The unsymmetrical case, where only one support is moving, is also considered.
The arches chosen for testing had a radius-to-thickness ratio of 121 to 179. The arch-opening half-angles varied from 90 to 125 deg. The arches were vibrated on an electrodynamic-shaker table. Dynamic arch amplitudes were measured using a specially designed micrometer probe. Comparison of theory with experiment was considered good; the average error in prediction of resonant frequencies was less than three percent. For the firced excitation, the modal shapes agreed quite closely with that predicted by theory.
It was found that experimental arches were quite sensitive to variations in the arch radius and that, in general, for all arches tested, the degree of agreement between theory and experiment was more sensitive to changes in the opening half-angle rather than theR/H value. A further observation was that, for some poorly constructed arches, it was found that out-of-plane vibrations occurred at approximately 16 times the fundamental flexural frequency.
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Abbreviations
- C ni :
-
constants determined from the boundary conditions
- D i (t):
-
time-dependent function
- E :
-
modulus of elasticity
- E i :
-
constant for symmetrical excitation
- F i :
-
constant for unsymmetrical excitation
- g(t) :
-
function describing support displacement
- G :
-
amplitude of support movement
- H :
-
arch thickness
- I :
-
second moment of area of cross section
- k :
-
radius of gyration of cross section
- m :
-
arch mass per unit length
- P R (ϕ,t):
-
external radial load per unit arch length
- P T (ϕ,t):
-
external tangential load per unit arch length
- R :
-
radius of curvature
- t :
-
time
- u(ϕ,t) :
-
radial displacement
- v(ϕ,t) :
-
tangential displacement
- α:
-
arch-opening half-angle
- ϕ:
-
angular coordinate
- \(\lambda _n \) :
-
frequency parameter
- \(\omega _n \) :
-
nth natural frequency
- Ω:
-
frequency of support excitation
- η(t):
-
time-dependent function
References
Lamb, H., “On the Flexure and the Vibrations of a Curved Bar”,Lond. Math. Soc. Proc.,1 (19),365–376 (1887).
Lang, T. E., “Vibration of Thin Circular Rings, Part I, Solutions for Modal Characteristics and Forced Excitation”,Tech. Rep. 32–261, J.P.L., Caltech, (1962).
Lang, T. E., “Vibration of Thin Circular Rings, Part II, Modal Functions and Eigenvalues of Constrained Semicircular Rings”,Tech. Rep. 32–261, J.P.L., Caltech, (1962).
Bellow, D. G. andSemeniuk, A., “Symmetrical and Unsymmetrical Forced Excitation of Thin Circular Arches”,Int. J. Mech. Sci.,14 (3),185–195 (1972).
Volterra, E. andMorell, J. D., “Lowest Natural Frequenies of Elastic Hinged Arcs”,Acous. Soc. of Amer.,32,1787–1790 (1962).
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Bellow, D.G., Seminiuk, A. Experimental behavior of thin circular arches subjected to forced excitation. Experimental Mechanics 12, 489–495 (1972). https://doi.org/10.1007/BF02320744
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DOI: https://doi.org/10.1007/BF02320744