Abstract
Let y 1 and y 2 denote the displacements of the bodies of mass m 1 and m 2 from their equilibrium positions, y 1 = 0 and y 2 = 0, respectively, where distances are measured in the downward direction. In these coordinates, y 1(t) and y 2(t) − y 1(t) represent the length the upper and lower springs are stretched at time t. There are two spring forces acting on the upper body. By Hooke’s law, the force of the upper spring is − k 1 y 1(t) while the force of the lower spring is given by k 2(y 2(t) − y 1(t)), where k 1 and k 2 are the respective spring constants. Newton’s law of motion then implies
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Notes
- 1.
In Chap. 9, we will study first order coupled systems in greater detail.
- 2.
There are other methods. For example, in the exercises, a nice Laplace transform approach will be developed. Theoretically, this is a much nicer approach. However, it is not necessarily computationally easier. In Chap. 9, we will consider systems of first order differential equations and show how (1) can fit in that context.
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© 2012 Springer Science+Business Media New York
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Adkins, W.A., Davidson, M.G. (2012). Linear Constant Coefficient Differential Equations. In: Ordinary Differential Equations. Undergraduate Texts in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-3618-8_4
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DOI: https://doi.org/10.1007/978-1-4614-3618-8_4
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4614-3617-1
Online ISBN: 978-1-4614-3618-8
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