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Purpose <p>The purpose of this work is to analyse the in-plane small oscillations of a slack catenary hanging from its endpoints, and calculate the mode shapes and frequencies. This work uses assumed mode expansion method. When the exact modes are used in this expansion, an indeterminate system of equations are obtained. This paper suggests a method to overcome that indeterminacy.</p> Method <p>The vertical displacement of the chain is expressed using an assumed mode (admissible functions) expansion. The horizontal displacement is calculated in terms of the vertical displacement using the condition of inextensibility. Lagrangian formulation is used to obtain the dynamic equations enforcing the horizontal fixity at the distal end as a scalar constraint.</p> Results <p>The frequencies and the mode shapes are obtained and compared with previously published experimental work and the match found is satisfactory.</p> Conclusion <p>A catenary, or an inextensible chain suspended from two endpoints, has the shape of a hyperbolic cosine at equilibrium. Due to pointwise inextensibility, the vertical and horizontal displacement components are related by a differential equation. An assumed mode solution for small oscillations of a slack catenary presents some challenges and has been missing from the literature. Here, starting with an assumed mode expansion for the vertical displacement, the horizontal displacement was obtained using the inextensibility condition. The horizontal fixity at the distal end was enforced using an additional scalar constraint. Subsequently, a Lagrangian formulation was used. One of the interesting aspects of this problem is that, in the Lagrangian formulation, the potential energy is linear in the generalized coordinates. Enforcement of the constraint through a Lagrange multiplier makes the oscillation frequencies determinate. Further, when these same modes, or any other assumed modes that satisfy the fixity constraint up to first order, are used to expand the vertical displacement, then degeneracy is encountered at first order. However, adding a small perturbation to that assumed mode, and carrying out the calculation to second order, gives the usual harmonic oscillator equation and a fully satisfactory solution.</p>

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Small In-Plane Oscillations of a Slack Catenary Using Assumed Modes

  • 期刊论文

  • 发表日期：

    2025年6月18日

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  • Bidhayak Goswami,
  • Indrasis Chakraborty,
  • Anindya Chatterjee

摘要

Purpose

The purpose of this work is to analyse the in-plane small oscillations of a slack catenary hanging from its endpoints, and calculate the mode shapes and frequencies. This work uses assumed mode expansion method. When the exact modes are used in this expansion, an indeterminate system of equations are obtained. This paper suggests a method to overcome that indeterminacy.

Method

The vertical displacement of the chain is expressed using an assumed mode (admissible functions) expansion. The horizontal displacement is calculated in terms of the vertical displacement using the condition of inextensibility. Lagrangian formulation is used to obtain the dynamic equations enforcing the horizontal fixity at the distal end as a scalar constraint.

Results

The frequencies and the mode shapes are obtained and compared with previously published experimental work and the match found is satisfactory.

Conclusion

A catenary, or an inextensible chain suspended from two endpoints, has the shape of a hyperbolic cosine at equilibrium. Due to pointwise inextensibility, the vertical and horizontal displacement components are related by a differential equation. An assumed mode solution for small oscillations of a slack catenary presents some challenges and has been missing from the literature. Here, starting with an assumed mode expansion for the vertical displacement, the horizontal displacement was obtained using the inextensibility condition. The horizontal fixity at the distal end was enforced using an additional scalar constraint. Subsequently, a Lagrangian formulation was used. One of the interesting aspects of this problem is that, in the Lagrangian formulation, the potential energy is linear in the generalized coordinates. Enforcement of the constraint through a Lagrange multiplier makes the oscillation frequencies determinate. Further, when these same modes, or any other assumed modes that satisfy the fixity constraint up to first order, are used to expand the vertical displacement, then degeneracy is encountered at first order. However, adding a small perturbation to that assumed mode, and carrying out the calculation to second order, gives the usual harmonic oscillator equation and a fully satisfactory solution.