Abstract
Oscillatory or vibrational motion has applications in all of physics as well as in engineering. We begin with the one-dimensional harmonic oscillator, including friction and external excitation, and continue to study wave propagation along linear chains, which eventually leads us to normal mode analysis.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Stokes, Sir George Gabriel, British mathematician and physicist, *Skreen (County Sligo, Ireland) 13.8.1819, †Cambridge 1.2.1903.
- 2.
A justification for (6.8) in a special case is given on p. 276.
- 3.
Notice that if \( \vec {F} = - \frac{\partial U}{ \partial \vec {r}} \equiv -\vec \nabla U \), then \( \oint \vec {F} \cdot d \vec {s} = - \oint \vec {\nabla } U \cdot d \vec {s} = - \oint d U = 0 \).
- 4.
This means we take a snapshot, showing the oscillating chain at time \(t=0\).
- 5.
Everything we just discussed also applies to the \(\sin \)-part.
- 6.
The displacements \(u_s\) and \(v_s\) are real of course.
- 7.
Notice that \( \overline{\sin ^2 ( \ldots )} = \frac{\omega }{2 \pi } \int _0^{2 \pi /\omega } \sin ^2 (\omega t + c) dt \), where the quantity c is independent of t. The substitution \( x = \omega t + c \) yields
$$\begin{aligned} \overline{\sin ^2 ( \ldots )} = \frac{1}{2 \pi } \int _c^{2 \pi + c} \sin ^2 x d x = \frac{1}{2} \frac{1}{2 \pi } \int _c^{2 \pi + c} \underbrace{\left( \sin ^2 x + \cos ^2 x \right) }_{= 1} dx = \frac{1}{2} \;. \end{aligned}$$ - 8.
E.g., \(\frac{K^{bb}}{2} \delta b_1 \delta b_2\).
References
D.R. Lide (ed.), Handbook of Chemistry and Physics (CRC Press, Boca Raton, 2003)
W.D. Cornell, P. Cieplak, C.I. Bayly, I.R. Gould, K.M. Merz, D.M. Ferguson, D.C. Spellmeyer, T. Fox, J.W. Caldwell, P.A. Kollman, A second generation force field for the simulation of proteins, nucleic acids, and organic molecules. J. Am. Chem. Soc. 117, 5179 (1995)
U. Burkert, N.L. Allinger, Molecular Mechanics, vol. 177, ACS Monograph (American Chemical Society, Washington D.C., 1982)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this chapter
Cite this chapter
Hentschke, R. (2017). Small Oscillations. In: Classical Mechanics. Undergraduate Lecture Notes in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-48710-6_6
Download citation
DOI: https://doi.org/10.1007/978-3-319-48710-6_6
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-48709-0
Online ISBN: 978-3-319-48710-6
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)