Sommario
Sono state calcolate le frequenze naturali delle vibrazioni laterali delle molle ad elica cilindrica con estremi incastrati, al variare della snellezza e del precarico. Il calcolo è stato effettuato assimilando la molla ad una trave a sezione costante, ed applicando a questa la teoria di Timoshenko.
La parte teorica del presente studio è una estensione del lavoro di Haringx sullo stesso argomento, non tanto perché in esso sono stati considerati modi di vibrare superiori al primo (problema questo trattato dall'Haringx), quanto per il fatto che una visione più generale del fenomeno delle vibrazioni laterali pone problemi di interpretazione dei risultati, non affrontati da questo Autore.
I valori calcolati delle frequenze naturali e le corrispondenti linee elastiche sono stati confrontati con quelli determinati sperimentalmente. Nel corso della sperimentazione è stato possibile visualizzare modi di vibrare caratteristici (Thickness-shear modes) previsti dalla teoria di Timoshenko, ma non osservabili nel caso delle travi.
Summary
In this paper we have calculated the natural frequencies of the transverse vibrations of cylindrical helical springs, with respect to the variation in slenderness and relative compression. The calculation was made by considering the spring as a constant-section beam, and by applying Timoshenko theory to the latter. The theoretical section of this paper is an extension of Haringx's work on the same subject, not so much because it studies natural modes higher than the first (a problem which was treated by Haringx), but rather because its more general approach to the transverse vibrations phenomenon raises problems of results' interpretation that this Author did not face. The calculated values of natural frequencies and the corresponding deflection curves have been compared with experimentally determined values. In the course of experimentation Thickness-shear modes were visualized, which Timoshenko theory takes into account, but which, as far as beams are concerned, are not detectable.
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Abbreviations
- A :
-
cross-section area
- c 1 :
-
bar velocity
- c c :
-
propagation velocity of a compression wave along the spring axis
- c s :
-
propagation velocity of the shear deformation
- c t :
-
propagation velocity of the torsion deformation
- D 0 :
-
diameter of the helix
- E :
-
Young's modulus
- G :
-
shear modulus
- I 0 :
-
polar area moment of inertia of the wire cross-section
- I 1 :
-
diametral area moment of inertia of the wire cross-section
- I :
-
area moment of inertia of the wire cross-section with respect to the neutral axis
- k′ :
-
section's shape factor
- l,l 0 :
-
spring lenght with and without relative compression
- M :
-
bending moment
- m :
-
mass per unit length
- N :
-
normal force
- n 0 :
-
number of spring's free coils
- P :
-
axial load
- r :
-
radius of gyration of the cross-section with respect to the axis of bending
- T :
-
shear force
- y :
-
transverse deflection of the abscissax section
- y b :
-
bending displacement of the abscissax section
- y s :
-
shear displacement of the abscissax section
- α,α 0 :
-
bending stiffness with and without relative compression
- β,β 0 :
-
shear stiffness with and without relative compression
- γ,γ 0 :
-
compression stiffness with and without relative compression
- μ :
-
wave number
- ν :
-
Poisson's ratio
- ξ :
-
relative compression under an axial loadP
- ρ :
-
material's density
- φ :
-
slope of the beam axis due to shear force
- ψ :
-
rotation of the abscissax section due to the bending moment
- ω a :
-
circular frequency of the first longitudinal mode for springs with clamped ends
References
WITTRICK W. H.,On elastic wave propagation in helical springs, Int. J. Mech. Sci., Vol. 8, Pergamon Press, 1966.
TIMOSHENKO S.,Vibration problems in engineering, D. Van Nostrand Co. Inc., New york, 1955.
HARINGX J. A.,On highly compressible helical springs and rubber rods, and their application for vibration-free mountings, Philips Res. Rep., Vol. 3, 1948, Vol. 4, 1949.
ANDERSON R. A.,Flexural vibrations in uniform beams according to the Timoshenko theory, J. Appl. Mech., Dec. 1953.
MIKLOWITZ J.,Flexural waves solution of coupled equations representing the more exact theory of bending, J. Appl. Mech., Dec. 1953.
MINDLIN R. D., DERESIEWICZ H.,Timoshenko's shear coefficient for flexural vibrations of beams, Proc. II U.S. Nat. Cong. Appl. Mech., 1954.
BOLEY B. A., CHAO C. C.,Some solutions of the Timoshenko beam equations, J. Appl. Mech., Dec. 1955.
HUANG T. C.,The effect of rotatory inertia and of shear deformation on the frequency and normal mode equations of uniform beam with simple end conditions, Trans. ASME, Dec. 1961.
CLARK S. K.,Dynamics of continuous elements, Prentice Hall Inc., 1972.
HERRMANN G.,R. D. Mindlin and applied Mechanics, Pergamon Press, 1973.
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Research sponsored by the National Research Committee (C.N.R.). Paper presented during the: Incontro di studio «Metodi classici e moderni per lo studio delle vibrazioni di sistemi dinamici», Università di Padova, maggio 1976.
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Guido, A.R., Della Pietra, L. & della Valle, S. Transverse vibrations of cylindrical helical springs. Meccanica 13, 90–108 (1978). https://doi.org/10.1007/BF02128537
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DOI: https://doi.org/10.1007/BF02128537