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12.1: Undamped Mass-Spring System

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    7698
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    The subject of this chapter is free vibration of undamped mechanical systems, beginning with the one-degree-of-freedom (1-DOF) mass-spring system of Fig \(\PageIndex{1}\). We evaluated the \(m\)-\(k\) system more generally in Chapter 7, but we will re-visit it here using a different approach. Free vibration can arise from initial translation of the mass (relative to the static equilibrium position) and/or initial velocity, and also from excitation that has ceased, which might provide the mass with both initial translation and initial velocity at the instant when excitation ceases. However, it is easiest conceptually to visualize this motion as response to only non-zero initial translation relative to the static equilibrium position. Doing so sacrifices a little generality because we cannot also account for a non-zero initial velocity, but the essential characteristics of the free vibration are revealed by this approach. Consider an undamped mass-spring system with nonzero initial translation, \(y(0)=y_{0}\), and zero initial velocity, \(\dot{y}(0)=0\), but without forcing excitation, so that \(f_{y}(t)=0\). From Equation 7.1.2, the ODE of free vibration is

    \[m \ddot{y}+k y=f_{y}(t)=0\label{eqn:12.1} \]

    clipboard_e86fc1995ef403a18b2006d9ea6aaf6fe.png
    Figure \(\PageIndex{1}\)

    We now ask the questions: Can motion exist in the form \(y(t)=Y \cos \omega t\), and, if so, what are the constants \(Y\) and \(\omega\)? To find the answers, we substitute the presumed form of motion into ODE Equation \(\ref{eqn:12.1}\), perform the differentiations and algebra, and see where the process leads:

    \[m\left(-\omega^{2} Y \cos \omega t\right)+k Y \cos \omega t=\left(-\omega^{2} m+k\right) Y \cos \omega t=0 \nonumber \]

    This equation can be satisfied in general, for a non-trivial solution, only if

    \[\omega^{2}=k / m \equiv \omega_{n}^{2} \Rightarrow \omega=\omega_{n}=\sqrt{k / m} \nonumber \]

    Moreover, we can use the initial condition to find \(Y\):

    \[y(0)=y_{0}=Y \cos (\omega \times 0)=Y \nonumber \]

    \[\Rightarrow \quad y(t)=y_{0} \cos \omega_{n} t\label{eqn:12.2} \]

    This simple result was already derived in Equation 7.3.1; we re-derive it differently here just to establish the approach that will be useful next for higher-order systems.


    This page titled 12.1: Undamped Mass-Spring System is shared under a CC BY-NC 4.0 license and was authored, remixed, and/or curated by William L. Hallauer Jr. (Virginia Tech Libraries' Open Education Initiative) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.