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Divergence and flutter instabilities of a non-conservative axial lattice under non-reciprocal interactions

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Abstract

Non-reciprocal interactions of discrete or continuous systems may induce surprising responses such as flutter instabilities. It is shown in this paper that a finite one-dimensional lattice under non-symmetrical elastic interactions may flutter for sufficiently strong unsymmetrical interactions. An exact solution is presented for the vibration of such one-dimensional lattices with direct and non-symmetrical elastic interactions. An internal force controlling the interactions is included in the model as an additional force for each mass, which acts proportionally to the elongation of a spring at its position. This non-conservative problem due to this circulatory interaction is solved from the resolution of a linear difference equation for this unsymmetrical repetitive lattice. It is possible to derive the exact eigenfrequency dependence with respect to the unsymmetrical interaction parameter, which plays the role of a bifurcation parameter. Divergence and flutter instabilities of this fixed–fixed non-conservative axial lattice under non-Hermitian interactions are theoretically predicted, from a direct approach or by solving the difference equation whatever the number of masses of the lattice. It is shown that the system may flutter for sufficiently strong unsymmetrical interactions, whatever the size of the system, for even or odd number of masses. However, divergence instability may arise in such a system only for even number of masses. The drastic change of response of the present system for odd or even number of particles is specific of the discrete nature of the dynamic system.

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The datasets generated and analyzed during the current study are available from the authors upon reasonable request.

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Author information

Authors and Affiliations

  1. Institut des Sciences de la Terre (ISTerre), CNRS UMR 5275, Université Grenoble Alpes, Grenoble, France

    Sina Massoumi

  2. School of Mechanical Engineering, Iran University of Science and Technology, Narmak, Tehran, 16846-13114, Iran

    Somaye Jamali Shakhlavi

  3. School of Engineering, Westlake University, Hangzhou, China

    Somaye Jamali Shakhlavi

  4. Centre de Recherche, IRDL – UBS – UMR CNRS 6027, Univ. Bretagne Sud, Rue de Saint Maudé, BP 92116, 56321, Lorient Cedex, France

    Noël Challamel

  5. Laboratoire de Mathématiques et Modélisation d’Evry, LaMME–UMR CNRS 8071, Univ. Evry – Université Paris-Saclay, 23 Bvd de France, 91037, Evry, France

    Jean Lerbet

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  1. Sina Massoumi
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Appendix: Analysis of the axial lattice composed of N elements

Appendix: Analysis of the axial lattice composed of N elements

We study the systems shown in Fig. 

Fig. 3
figure 3figure 3

An axial granular beam with (n − 1)-degree-of-freedom \((n = 2,3,...,11)\) of a fixed-fixed non-Hermitian lattice

Full size image

3, with n = 2, 3, …, 11- composed of n + 1 rigid particles with masses m connected by n + 2 massless springs k to each other (for the symmetrical interaction part). The flutter and the divergence load of the (n-1)-degree-of-freedom discrete non-Hermitian system can be analytically derived. Using Newton's second law, the matrix equation of motion in the absence of external forces for a fixed–fixed BCs is derived in the following general form,

$$ \begin{gathered} {\text{general}}\;{\text{form}} \Rightarrow (n - 1){\text{-degree-of-freedom}} \hfill \\ \left[ {\begin{array}{*{20}l} m \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & . \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & . \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & . \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & m \hfill \\ \end{array} } \right]_{n - 1 \times n - 1} \left\{ {\begin{array}{*{20}c} {\ddot{u}_{1} } \\ . \\ . \\ . \\ {\ddot{u}_{n - 1} } \\ \end{array} } \right\} + \left[ {\begin{array}{*{20}c} {2k - k_{CI} } & { - k} & 0 & 0 & 0 \\ {k_{CI} - k} & . & . & 0 & 0 \\ 0 & . & . & . & 0 \\ 0 & 0 & . & . & { - k} \\ 0 & 0 & 0 & {k_{CI} - k} & {2k - k_{CI} } \\ \end{array} } \right]_{n - 1 \times n - 1} \left\{ {\begin{array}{*{20}c} {u_{1} } \\ . \\ . \\ . \\ {u_{n - 1} } \\ \end{array} } \right\} = 0 \hfill \\ \end{gathered} $$
(40)

The \(\left[ K \right]\) matrix is not symmetrical due to the effects of unsymmetrical internal forces of the control interactions. The axial natural frequency of the free vibration is computed from the determinant equation:

$$ \det \;\left( {\left[ K \right] - \omega^{2} \left[ M \right]} \right) = 0 $$
(41)

which can be equivalently presented for the fixed–fixed BCs, and with the assumptions \(\frac{k}{m} = \omega_{0}^{2}\), \(\frac{\omega }{{\omega_{0} }} = \Omega\) and \(\frac{{k_{CI} }}{k} = \alpha\) can be written as follow

$$ \begin{aligned} &(a)\;n = 2 \Rightarrow {\text{one-degree-of- freedom}} \\ &\Omega ^{2} + \alpha - 2 = 0 \\ &(b)\;n = 3 \Rightarrow {\text{two-degree-of-freedom}} \\ &\left| {\begin{array}{*{20}c} {2 - \alpha - \Omega ^{2} } & { - 1} \\ {\alpha - 1} & {2 - \alpha - \Omega ^{2} } \\ \end{array} } \right| = 0 \\ &(c)n = 4 \Rightarrow {\text{three-degree-of-freedom}} \\ &\left| {\begin{array}{*{20}c} {2 - \alpha - \Omega ^{2} } & { - 1} & 0 \\ {\alpha - 1} & {2 - \alpha - \Omega ^{2} } & { - 1} \\ 0 & {\alpha - 1} & {2 - \alpha - \Omega ^{2} } \\ \end{array} } \right| = 0 \\ &(d)n = 5 \Rightarrow {\text{four-degree-of-freedom}} \\ & \left| {\begin{array}{*{20}c} {2 - \alpha - \Omega ^{2} } & { - 1} & 0 & 0 \\ {\alpha - 1} & {2 - \alpha - \Omega ^{2} } & { - 1} & 0 \\ 0 & {\alpha - 1} & {2 - \alpha - \Omega ^{2} } & { - 1} \\ 0 & 0 & {\alpha - 1} & {2 - \alpha - \Omega ^{2} } \\ \end{array} } \right| = 0 \\ &(e)n = 6 \Rightarrow {\text{five-degree-of-freedom}} \\ &\left| {\begin{array}{*{20}c} {2 - \alpha - \Omega ^{2} } & { - 1} & 0 & 0 & 0 \\ {\alpha - 1} & {2 - \alpha - \Omega ^{2} } & { - 1} & 0 & 0 \\ 0 & {\alpha - 1} & {2 - \alpha - \Omega ^{2} } & { - 1} & 0 \\ 0 & 0 & {\alpha - 1} & {2 - \alpha - \Omega ^{2} } & { - 1} \\ 0 & 0 & 0 & {\alpha - 1} & {2 - \alpha - \Omega ^{2} } \\ \end{array} } \right| = 0 \\ &(f)\;n = 7 \Rightarrow {\text{sex-degree-of-freedom}} \\ &\left| {\begin{array}{*{20}c} {2 - \alpha - \Omega ^{2} } & { - 1} & 0 & 0 & 0 & 0 \\ {\alpha - 1} & {2 - \alpha - \Omega ^{2} } & { - 1} & 0 & 0 & 0 \\ 0 & {\alpha - 1} & {2 - \alpha - \Omega ^{2} } & { - 1} & 0 & 0 \\ 0 & 0 & {\alpha - 1} & {2 - \alpha - \Omega ^{2} } & { - 1} & 0 \\ 0 & 0 & 0 & {\alpha - 1} & {2 - \alpha - \Omega ^{2} } & { - 1} \\ 0 & 0 & 0 & 0 & {\alpha - 1} & {2 - \alpha - \Omega ^{2} } \\ \end{array} } \right| = 0 \\ &(g)\;n = 8 \Rightarrow {\text{seven-degree-of-freedom}} \\ &\left| {\begin{array}{*{20}c} {2 - \alpha - \Omega ^{2} } & { - 1} & 0 & 0 & 0 & 0 & 0 \\ {\alpha - 1} & {2 - \alpha - \Omega ^{2} } & { - 1} & 0 & 0 & 0 & 0 \\ 0 & {\alpha - 1} & {2 - \alpha - \Omega ^{2} } & { - 1} & 0 & 0 & 0 \\ 0 & 0 & {\alpha - 1} & {2 - \alpha - \Omega ^{2} } & { - 1} & 0 & 0 \\ 0 & 0 & 0 & {\alpha - 1} & {2 - \alpha - \Omega ^{2} } & { - 1} & 0 \\ 0 & 0 & 0 & 0 & {\alpha - 1} & {2 - \alpha - \Omega ^{2} } & { - 1} \\ 0 & 0 & 0 & 0 & 0 & {\alpha - 1} & {2 - \alpha - \Omega ^{2} } \\ \end{array} } \right| = 0 \\ &(h)\;n = 9 \Rightarrow {\text{eight-degree-of-freedom}} \\ &\left| {\begin{array}{*{20}c} {2 - \alpha - \Omega ^{2} } & { - 1} & 0 & 0 & 0 & 0 & 0 & 0 \\ {\alpha - 1} & {2 - \alpha - \Omega ^{2} } & { - 1} & 0 & 0 & 0 & 0 & 0 \\ 0 & {\alpha - 1} & {2 - \alpha - \Omega ^{2} } & { - 1} & 0 & 0 & 0 & 0 \\ 0 & 0 & {\alpha - 1} & {2 - \alpha - \Omega ^{2} } & { - 1} & 0 & 0 & 0 \\ 0 & 0 & 0 & {\alpha - 1} & {2 - \alpha - \Omega ^{2} } & { - 1} & 0 & 0 \\ 0 & 0 & 0 & 0 & {\alpha - 1} & {2 - \alpha - \Omega ^{2} } & { - 1} & 0 \\ 0 & 0 & 0 & 0 & 0 & {\alpha - 1} & {2 - \alpha - \Omega ^{2} } & { - 1} \\ 0 & 0 & 0 & 0 & 0 & 0 & {\alpha - 1} & {2 - \alpha - \Omega ^{2} } \\ \end{array} } \right| = 0 \\ &(i)\;n = 10 \Rightarrow {\text{nine-degree-of-freedom}} \\ &\left| {\begin{array}{*{20}c} {2 - \alpha - \Omega ^{2} } & { - 1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ {\alpha - 1} & {2 - \alpha - \Omega ^{2} } & { - 1} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & {\alpha - 1} & {2 - \alpha - \Omega ^{2} } & { - 1} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & {\alpha - 1} & {2 - \alpha - \Omega ^{2} } & { - 1} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & {\alpha - 1} & {2 - \alpha - \Omega ^{2} } & { - 1} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & {\alpha - 1} & {2 - \alpha - \Omega ^{2} } & { - 1} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & {\alpha - 1} & {2 - \alpha - \Omega ^{2} } & { - 1} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & {\alpha - 1} & {2 - \alpha - \Omega ^{2} } & { - 1} \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\alpha - 1} & {2 - \alpha - \Omega ^{2} } \\ \end{array} } \right| = 0 \\ &(j)\;n = 11 \Rightarrow {\text{ten-degree-of-freedom}} \\ &\left| {\begin{array}{*{20}c} {2 - \alpha - \Omega ^{2} } & { - 1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ {\alpha - 1} & {2 - \alpha - \Omega ^{2} } & { - 1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & {\alpha - 1} & {2 - \alpha - \Omega ^{2} } & { - 1} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & {\alpha - 1} & {2 - \alpha - \Omega ^{2} } & { - 1} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & {\alpha - 1} & {2 - \alpha - \Omega ^{2} } & { - 1} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & {\alpha - 1} & {2 - \alpha - \Omega ^{2} } & { - 1} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & {\alpha - 1} & {2 - \alpha - \Omega ^{2} } & { - 1} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & {\alpha - 1} & {2 - \alpha - \Omega ^{2} } & { - 1} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\alpha - 1} & {2 - \alpha - \Omega ^{2} } & { - 1} \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\alpha - 1} & {2 - \alpha - \Omega ^{2} } \\ \end{array} } \right| = 0 \\ \end{aligned} $$
(42)

The polynomial axial natural frequency relationship is extracted from the expansion of the above determinant in terms of as below

$$ \begin{aligned} &(a)\;n = 2 \Rightarrow {\text{one-degree-of-freedom}} \\ &\Omega ^{2} + \alpha - 2 = 0 \\ &(b)\;n = 3 \Rightarrow {\text{two-degree-of-freedom}} \\ &\left( {2 - \alpha - \Omega ^{2} } \right)^{2} + \alpha - 1 = 0 \\ &(c)\;n = 4 \Rightarrow {\text{three-degree-of-freedom}} \\ &\left( {2 - \alpha - \Omega ^{2} } \right)\left[ {\left( {2 - \alpha - \Omega ^{2} } \right)^{2} + 2\alpha - 2} \right] = 0 \\ &(d)\;n = 5 \Rightarrow {\text{four-degree-of-freedom}} \\ &\left[ \begin{gathered} \Omega ^{8} + \left( {4\alpha - 8} \right)\Omega ^{6} + \left( {6\alpha ^{2} - 21\alpha + 21} \right)\Omega ^{4} + \left( {4\alpha ^{3} - 18\alpha ^{2} + 30\alpha - 20} \right)\Omega ^{2} \hfill \\ + \left( {\alpha ^{4} - 5\alpha ^{3} + 10\alpha ^{2} - 10\alpha + 5} \right) \hfill \\ \end{gathered} \right] = 0 \\ &(e)\;n = 6 \Rightarrow {\text{five-degree-of-freedom}} \\ &\left[ \begin{gathered} \Omega ^{{10}} + \left( {5\alpha - 10} \right)\Omega ^{8} + \left( {10\alpha ^{2} - 36\alpha + 36} \right)\Omega ^{6} + \left( {10\alpha ^{3} - 48\alpha ^{2} + 84\alpha - 56} \right)\Omega ^{4} \hfill \\ + \left( {5\alpha ^{4} - 28\alpha ^{3} + 63\alpha ^{2} - 70\alpha + 35} \right)\Omega ^{2} + \left( {\alpha ^{5} - 6\alpha ^{4} + 15\alpha ^{3} - 20\alpha ^{2} + 15\alpha - 6} \right) \hfill \\ \end{gathered} \right] = 0 \\ &(f)\;n = 7 \Rightarrow {\text{six-degree-of-freedom}} \\ &\left[ \begin{gathered} \Omega ^{{12}} + (6\alpha - 12)\Omega ^{{10}} + (15\alpha ^{2} - 55\alpha + 55)\Omega ^{8} + (20\alpha ^{3} - 100\alpha ^{2} + 180\alpha - 120)\Omega ^{6} \hfill \\ + (15\alpha ^{4} - 90\alpha ^{3} + 216\alpha ^{2} - 252\alpha + 126)\Omega ^{4} + (6\alpha ^{5} - 40\alpha ^{4} + 112\alpha ^{3} - 168\alpha ^{2} + 140\alpha - 56)\Omega ^{2} \hfill \\ + (\alpha ^{6} - 7\alpha ^{5} + 21\alpha ^{4} - 35\alpha ^{3} + 35\alpha ^{2} - 21\alpha + 7) \hfill \\ \end{gathered} \right] = 0 \\ &(g)\;n = 8 \Rightarrow {\text{seven-degree-of-freedom}} \\ &\left[ \begin{gathered} - \Omega ^{{14}} + \left( {14 - 7\alpha } \right)\Omega ^{{12}} + \left( { - 21\alpha ^{2} + 78\alpha - 78} \right)\Omega ^{{10}} + \left( { - 35\alpha ^{3} + 180\alpha ^{2} - 330\alpha + 220} \right)\Omega ^{8} \hfill \\ + \left( { - 35\alpha ^{4} + 220\alpha ^{3} - 550\alpha ^{2} + 660\alpha - 330} \right)\Omega ^{6} + \left( { - 21\alpha ^{5} + 150\alpha ^{4} - 450\alpha ^{3} + 720\alpha ^{2} - 630\alpha + 252} \right)\Omega ^{4} \hfill \\ + \left( { - 7\alpha ^{6} + 54\alpha ^{5} - 180\alpha ^{4} + 336\alpha ^{3} - 378\alpha ^{2} + 252\alpha - 84} \right)\Omega ^{2} + \left( { - \alpha ^{7} + 8\alpha ^{6} - 28\alpha ^{5} + 56\alpha ^{4} - 70\alpha ^{3} + 56\alpha ^{2} - 28\alpha + 8} \right) \hfill \\ \end{gathered} \right] = 0 \\ &(h)\;n = 9 \Rightarrow {\text{eight-degree-of-freedom}} \\ &\left[ \begin{gathered} \Omega ^{{16}} + \left( {8\alpha - 16} \right)\Omega ^{{14}} + \left( {28\alpha ^{2} - 105\alpha + 105} \right)\Omega ^{{12}} + \left( {56\alpha ^{3} - 294\alpha ^{2} + 546\alpha - 364} \right)\Omega ^{{10}} \hfill \\ + \left( {70\alpha ^{4} - 455\alpha ^{3} + 1170\alpha ^{2} - 1430\alpha + 715} \right)\Omega ^{8} + \left( {56\alpha ^{5} - 420\alpha ^{4} + 1320\alpha ^{3} - 2200\alpha ^{2} + 1980\alpha - 792} \right)\Omega ^{6} \hfill \\ + \left( {28\alpha ^{6} - 231\alpha ^{5} + 825\alpha ^{4} - 1650\alpha ^{3} + 1980\alpha ^{2} - 1386\alpha + 462} \right)\Omega ^{4} + \left( \begin{gathered} 8\alpha ^{7} - 70\alpha ^{6} + 270\alpha ^{5} - 600\alpha ^{4} \hfill \\ + 840\alpha ^{3} - 756\alpha ^{2} + 420\alpha - 120 \hfill \\ \end{gathered} \right)\Omega ^{2} \hfill \\ + \left( {\alpha ^{8} - 9\alpha ^{7} + 36\alpha ^{6} - 84\alpha ^{5} + 126\alpha ^{4} - 126\alpha ^{3} + 84\alpha ^{2} - 36\alpha + 9} \right) \hfill \\ \end{gathered} \right] = 0 \\ &(i)\;n = 10 \Rightarrow {\text{nine-degree-of-freedom}} \\ &\left[ \begin{gathered} - \Omega ^{{18}} + \left( {18 - 9\alpha } \right)\Omega ^{{16}} + \left( { - 36\alpha ^{2} + 136\alpha - 136} \right)\Omega ^{{14}} + \left( { - 84\alpha ^{3} + 448\alpha ^{2} - 840\alpha + 560} \right)\Omega ^{{12}} \hfill \\ + \left( { - 126\alpha ^{4} + 840\alpha ^{3} - 2205\alpha ^{2} + 2730\alpha - 1365} \right)\Omega ^{{10}} + \left( { - 126\alpha ^{5} + 980\alpha ^{4} - 3185\alpha ^{3} + 5460\alpha ^{2} - 5005\alpha + 2002} \right)\Omega ^{8} \hfill \\ + \left( { - 84\alpha ^{6} + 728\alpha ^{5} - 2730\alpha ^{4} + 5720\alpha ^{3} - 7150\alpha ^{2} + 5148\alpha - 1716} \right)\Omega ^{6} + \left( \begin{gathered} - 36\alpha ^{7} + 336\alpha ^{6} - 1386\alpha ^{5} + 3300\alpha ^{4} \hfill \\ - 4950\alpha ^{3} + 4752\alpha ^{2} - 2772\alpha + 792 \hfill \\ \end{gathered} \right)\Omega ^{4} \hfill \\ + \left( { - 9\alpha ^{8} + 88\alpha ^{7} - 385\alpha ^{6} + 990\alpha ^{5} - 1650\alpha ^{4} + 1848\alpha ^{3} - 1386\alpha ^{2} + 660\alpha - 165} \right)\Omega ^{2} + \left( \begin{gathered} - \alpha ^{9} + 10\alpha ^{8} - 45\alpha ^{7} + 120\alpha ^{6} - 210\alpha ^{5} \hfill \\ + 252\alpha ^{4} - 210\alpha ^{3} + 120\alpha ^{2} - 45\alpha + 10 \hfill \\ \end{gathered} \right) \hfill \\ \end{gathered} \right] = 0 \\ &(j)\;n = 11 \Rightarrow {\text{ten-degree-of-freedom}} \\ &\left[ \begin{gathered} \Omega ^{{20}} + \left( {10\alpha - 20} \right)\Omega ^{{18}} + \left( {45\alpha ^{2} - 171\alpha + 171} \right)\Omega ^{{16}} + \left( {120\alpha ^{3} - 648\alpha ^{2} + 1224\alpha - 816} \right)\Omega ^{{14}} \hfill \\ + \left( {210\alpha ^{4} - 1428\alpha ^{3} + 3808\alpha ^{2} - 4760\alpha + 2380\Omega ^{{12}} } \right)\Omega ^{{12}} + \left( \begin{gathered} 252\alpha ^{5} - 2016\alpha ^{4} + 6720\alpha ^{3} - \hfill \\ 11760\alpha ^{2} + 10920\alpha - 4368 \hfill \\ \end{gathered} \right)\Omega ^{{10}} \hfill \\ + \left( {210\alpha ^{6} - 1890\alpha ^{5} + 7350\alpha ^{4} - 15925\alpha ^{3} + 20475\alpha ^{2} - 15015\alpha + 5005} \right)\Omega ^{8} + \left( \begin{gathered} 120\alpha ^{7} - 1176\alpha ^{6} + 5096\alpha ^{5} - 12740\alpha ^{4} + \hfill \\ 20020\alpha ^{3} - 20020\alpha ^{2} + 12012\alpha - 3432 \hfill \\ \end{gathered} \right)\Omega ^{6} \hfill \\ + \left( {45\alpha ^{8} - 468\alpha ^{7} + 2184\alpha ^{6} - 6006\alpha ^{5} + 10725\alpha ^{4} - 12870\alpha ^{3} + 10296\alpha ^{2} - 5148\alpha + 1287} \right)\Omega ^{4} \hfill \\ + \left( \begin{gathered} 10\alpha ^{9} - 108\alpha ^{8} + 528\alpha ^{7} - 1540\alpha ^{6} + 2970\alpha ^{5} - 3960\alpha ^{4} \hfill \\ + 3696\alpha ^{3} - 2376\alpha ^{2} + 990\alpha - 220 \hfill \\ \end{gathered} \right)\Omega ^{2} + \left( \begin{gathered} \alpha ^{{10}} - 11\alpha ^{9} + 55\alpha ^{8} - 165\alpha ^{7} + 330\alpha ^{6} - 462\alpha ^{5} \hfill \\ + 462\alpha ^{4} - 330\alpha ^{3} + 165\alpha ^{2} - 55\alpha + 11 \hfill \\ \end{gathered} \right) \hfill \\ \end{gathered} \right] = 0 \\ \end{aligned} $$
(43)

The above equations (a-j) are drawn in Fig. 

Fig. 4
figure 4

Variations of dimensionless frequencies versus the various values of \(\alpha\) in the n − 1-degree-of-freedom system for a fixed–fixed non-Hermitian lattice

Full size image

4 for a fixed–fixed non-Hermitian lattice. Based on the result obtained from the above equation, it is possible to prove the correctness of the general results calculated for divergence and flutter instabilities.

As explained earlier in the paper, flutter instability is calculated by setting \(\Delta = 0\) in the system characteristic equation. Hence, in order to better understand the concept of flutter instability and prove that, in Eq. (44) the results of the discriminant calculation, \({\text{discrim}}\left( {I_{n} ,x} \right),\) were added as follows to some of value n = 3,…, 7 and it was checked that this distinction disappeared, \({\text{eval}}\left\{ {{\text{discrim}}\left( {I_{n} ,x} \right),\alpha } \right\} = 0\) for \(\alpha_{{{\text{flutter}}}} = 1\).

$$ \begin{aligned} &{\text{ Supposing}}\;x = \Omega^{2} \\ &(I_{1} )\;n = 3 \Rightarrow {\text{two -degree-of-freedom}} \\ &\left( {2 - \alpha - x} \right)^{2} + \alpha - 1 = 0 \Rightarrow {\text{discrim}}\left( {I_{1} ,x} \right) = - 4\alpha + 4 \\ &(I_{2} )\;n = 4 \Rightarrow {\text{three -degree-of-freedom}} \\ &\left( {2 - \alpha - x} \right)\left[ {\left( {2 - \alpha - x} \right)^{2} + 2\alpha - 2} \right] = 0 \Rightarrow {\text{discrim}}\left( {I_{2} ,x} \right) = - 32\left( {\alpha - 1} \right)^{3} \\ &(I_{3} )\;n = 5 \Rightarrow {\text{four -degree-of-freedom}} \\ &\left[ \begin{gathered} x^{4} + \left( {4\alpha - 8} \right)x^{3} + \left( {6\alpha^{2} - 21\alpha + 21} \right)x^{2} + \left( {4\alpha^{3} - 18\alpha^{2} + 30\alpha - 20} \right)x \hfill \\ + \left( {\alpha^{4} - 5\alpha^{3} + 10\alpha^{2} - 10\alpha + 5} \right) \hfill \\ \end{gathered} \right] = 0 \\ &\Rightarrow {\text{discrim}}\left( {I_{3} ,x} \right) = \left( \begin{gathered} - 34560\alpha^{12} + {435456}\alpha^{11} - {2415744 }\alpha^{10} + {7888320}\alpha^{9} \hfill \\ - {17077536}\alpha^{8} + {26015328}\alpha^{7} - {28424624}\alpha^{6} + {21787680}\alpha^{5} \hfill \\ - {10927920}\alpha^{4} + {3100960}\alpha^{3} - {333840}\alpha^{2} - {13920}\alpha + {400} \hfill \\ \end{gathered} \right) \\ &\Rightarrow {\text{eval}}\left\{ {{\text{discrim}}\left( {I_{3} ,x} \right),\alpha = 1} \right\} = 0 \\ &(I_{4} )_{{}} n = 6 \Rightarrow {\text{five-degree-of-freedom}} \\ &\left[ \begin{gathered} x^{5} + \left( {5\alpha - 10} \right)x^{4} + \left( {10\alpha^{2} - 36\alpha + 36} \right)x^{3} + \left( {10\alpha^{3} - 48\alpha^{2} + 84\alpha - 56} \right)x^{2} \hfill \\ + \left( {5\alpha^{4} - 28\alpha^{3} + 63\alpha^{2} - 70\alpha + 35} \right)x + \left( {\alpha^{5} - 6\alpha^{4} + 15\alpha^{3} - 20\alpha^{2} + 15\alpha - 6} \right) \hfill \\ \end{gathered} \right] = 0 \\ &\Rightarrow {\text{discrim}}\left( {I_{4} ,x} \right) = \left( \begin{gathered} {6912}\alpha^{10} - {69120}\alpha^{9} + {311040}^{8} - {829440}\alpha^{7} + {1451520}\alpha^{6} \hfill \\ - {1741824}\alpha^{5} + {1451520}\alpha^{4} - {829440}\alpha^{3} + {311040}\alpha^{2} - {69120}\alpha + {6912} \hfill \\ \end{gathered} \right) \\ &\Rightarrow {\text{eval}}\left\{ {{\text{discrim}}\left( {I_{4} ,x} \right),\alpha = 1} \right\} = 0 \\ &(I_{5} )n = 7 \Rightarrow {\text{six-degree-of-freedom}} \\ &\left[ \begin{gathered} x^{6} + (6\alpha - 12)x^{5} + (15\alpha^{2} - 55\alpha + 55)x^{4} + (20\alpha^{3} - 100\alpha^{2} + 180\alpha - 120)x^{3} \hfill \\ + (15\alpha^{4} - 90\alpha^{3} + 216\alpha^{2} - 252\alpha + 126)x^{2} + (6\alpha^{5} - 40\alpha^{4} + 112\alpha^{3} - 168\alpha^{2} + 140\alpha - 56)x \hfill \\ + (\alpha^{6} - 7\alpha^{5} + 21\alpha^{4} - 35\alpha^{3} + 35\alpha^{2} - 21\alpha + 7) \hfill \\ \end{gathered} \right] = 0 \\ &\Rightarrow {\text{discrim}}\left( {I_{5} ,x} \right) = \left( \begin{gathered} - {153664}\alpha^{15} + {2304960}\alpha^{14} - {16134720}\alpha^{13} + {69917120}\alpha^{12} - {209751360}\alpha^{11} \hfill \\ + {461452992}\alpha^{10} - {769088320}\alpha^{9} + {988827840}\alpha^{8} - {988827840}\alpha^{7} + {769088320}\alpha^{6} \hfill \\ - {461452992}\alpha^{5} + {209751360}\alpha^{4} - {69917120}\alpha^{3} + {16134720}\alpha^{2} - {2304960}\alpha + {153664} \hfill \\ \end{gathered} \right) \\ &\Rightarrow {\text{eval}}\left\{ {{\text{discrim}}\left( {I_{5} ,x} \right),\alpha = 1} \right\} = 0 \\ \end{aligned} $$
(44)

The direct approach derived from the computation of the load–frequency curves based on the mass and the stiffness matrix is equivalent to the method based on the resolution of the difference equation with the n-1 branches of solutions:

$$ \Omega^{4} - 2\left( {2 - \alpha } \right)\Omega^{2} + \alpha^{2} + 4\left( {1 - \alpha } \right)\sin^{2} \frac{p\pi }{n} = 0 \Rightarrow {\text{for}}\;\;p \in \left\{ {1;2;...;n - 1} \right\} $$
(45)

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Massoumi, S., Shakhlavi, S.J., Challamel, N. et al. Divergence and flutter instabilities of a non-conservative axial lattice under non-reciprocal interactions. Arch Appl Mech 94, 187–203 (2024). https://doi.org/10.1007/s00419-023-02515-z

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