In this paper, the authors analyze the spectrum of a one-dimensional vibrating free rod under tension \((\tau>0)\) or compression \((\tau<0)\) governed by the differential equation \[ u^{(4)}-\tau u^{\prime\prime}=\mu u \] together with boundary conditions \[ u^{\prime\prime} (\pm1)=0, \qquad u^{\prime\prime\prime} (\pm1) - \tau u^{\prime} (\pm1)=0. \] Here, the eigenvalues \(\mu\) of the free rod depend on the tension parameter \(\tau\).
After an introduction, the authors establish properties of symmetry of the eigenfunctions, and introduce bijections of regions of the \((\tau,\mu)\)-plane and they analyze the eigenvalues in the upper half-plane and a super-parabolic region, respectively, by finding eigenvalue conditions for each region and then parameterizing the eigenvalue branches. They also discuss monotonicity, crossing properties, asymptotic growth of the eigenvalue branches, cascading and phantom lines in the spectrum. They analyze the eigenvalues in the sub-parabolic region. They find the eigenvalue conditions and describe the behavior of the two eigenvalue branches that lie in this region. In addition, they establish a result involving intersections of a family of parabolas with the eigenvalue branches. Moreover, they identify the crossings of the odd and even eigenvalue branches.