# Kirchhoff’s Analogy between the Kapitza Pendulum Stability and Buckling of a Wavy Beam under Tensile Loading

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

## 3. Results and Discussion

^{2}> 2EI

_{0}/F.

^{2}under a tensile load F = −300 N. For h = 0.289 m, and Ω = 693 m

^{−1}, δ = −0.00025, ε = 0.05, which lie in the shaded stable region of the Ince–Strutt diagram. The corresponding solution to Equation (5) for boundary conditions $z\left(0\right)=0,{\left(\frac{dz}{d\kappa}\right)}_{\kappa =0}=0.1$ is shown in Figure 2a. In this case, the slope of the beam is a sinusoidal function. For the values of δ = −0.00035, ε = 0.05, which lie outside the shaded region, the solution to Equation (7) is shown in Figure 2b. In this case, the slope of the beam is an exponential function, implying buckled state.

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Ince–Strutt stability diagram for the Mathieu equation. The shaded region represents the domain of stable solutions for a tensile-loaded beam.

**Figure 2.**Solutions to the Mathieu equation for a tensile loaded beam with boundary conditions z = 0 at κ = 0 and dz/dκ = 0 at κ = 0. (

**a**) Stable solution for δ = −0.00025, ε = 0.05, and (

**b**) unstable solution for δ = −0.00035, ε = 0.05.

**Figure 3.**The deflection of a beam due to (

**a**) compressive load F (corresponding to the stable equilibrium of a regular pendulum), and (

**b**) tensile load F (corresponding to the unstable equilibrium of an inverted pendulum). (

**c**) The waviness of the beam would stabilize the equilibrium similarly to the vibrations stabilizing an inverted pendulum (reproduced with permission from [11]).

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**MDPI and ACS Style**

Ramachandran, R.; Nosonovsky, M.
Kirchhoff’s Analogy between the Kapitza Pendulum Stability and Buckling of a Wavy Beam under Tensile Loading. *Appl. Mech.* **2023**, *4*, 248-253.
https://doi.org/10.3390/applmech4010014

**AMA Style**

Ramachandran R, Nosonovsky M.
Kirchhoff’s Analogy between the Kapitza Pendulum Stability and Buckling of a Wavy Beam under Tensile Loading. *Applied Mechanics*. 2023; 4(1):248-253.
https://doi.org/10.3390/applmech4010014

**Chicago/Turabian Style**

Ramachandran, Rahul, and Michael Nosonovsky.
2023. "Kirchhoff’s Analogy between the Kapitza Pendulum Stability and Buckling of a Wavy Beam under Tensile Loading" *Applied Mechanics* 4, no. 1: 248-253.
https://doi.org/10.3390/applmech4010014