The stability boundaries of the Ziegler column are established, in a closed-form, for undamped and viscously damped conditions with equal and unequal damping in the joints. These boundaries are determined by the combined use of Routh–Hurwitz Stability Criterion and the root-locus plots to visualize the unique behavior of the dynamics of the Ziegler Column. Such an approach reveals clearly the reasons and the combination of the column design parameters that give rise to the observed and well-known phenomenon of the “Ziegler Paradox”. In that paradox, unequal dissipative damping forces in the joints induce a destabilizing effect even though the magnitude of these forces can be fairly small. The paradox has been reported in numerous studies indicating that this destabilizing effect is contrary to the common believe that damping is expected to generally have a stabilizing effect. For the undamped Ziegler column, it is found that the stability is achieved when the follower force F is less than 2.54 k with k denoting the equal stiffness of the springs in the joints. For Ziegler columns with equally damped joints, it is found that stability can be attained when the follower force F is less than <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="707_2025_4280_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="96" /> </InlineMediaObject> <EquationSource Format="TEX">$1.2c^{2} + 1.46k$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1.2</mn> <msup> <mi>c</mi> <mn>2</mn> </msup> <mo>+</mo> <mn>1.46</mn> <mi>k</mi> </mrow> </math></EquationSource> </InlineEquation> with c denoting the equal damping coefficient. But, columns with asymmetrical, or unequal, damping in the joints are found to be always unstable. It is envisioned that the use of the stability tools of the control systems theory enables a better understanding and visualization of the interactions of the design parameters that influence the column stability. Furthermore, these tools will further enhance the analysis of Ziegler columns with multi-degrees of freedom and with active/passive control capabilities.

On the Ziegler destabilization paradox

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发表日期：

2025年3月25日

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A. Baz

摘要

The stability boundaries of the Ziegler column are established, in a closed-form, for undamped and viscously damped conditions with equal and unequal damping in the joints. These boundaries are determined by the combined use of Routh–Hurwitz Stability Criterion and the root-locus plots to visualize the unique behavior of the dynamics of the Ziegler Column. Such an approach reveals clearly the reasons and the combination of the column design parameters that give rise to the observed and well-known phenomenon of the “Ziegler Paradox”. In that paradox, unequal dissipative damping forces in the joints induce a destabilizing effect even though the magnitude of these forces can be fairly small. The paradox has been reported in numerous studies indicating that this destabilizing effect is contrary to the common believe that damping is expected to generally have a stabilizing effect. For the undamped Ziegler column, it is found that the stability is achieved when the follower force F is less than 2.54 k with k denoting the equal stiffness of the springs in the joints. For Ziegler columns with equally damped joints, it is found that stability can be attained when the follower force F is less than $1.2c^{2} + 1.46k$ $1.2 c^{2} + 1.46 k$ with c denoting the equal damping coefficient. But, columns with asymmetrical, or unequal, damping in the joints are found to be always unstable. It is envisioned that the use of the stability tools of the control systems theory enables a better understanding and visualization of the interactions of the design parameters that influence the column stability. Furthermore, these tools will further enhance the analysis of Ziegler columns with multi-degrees of freedom and with active/passive control capabilities.