# Transverse Vibrations of an Axially Travelling String

Conference paper
Part of the Lecture Notes in Mechanical Engineering book series (LNME)

## Abstract

In this paper, the transverse vibration of axially travelling string is analysed. The axial velocity of the string is periodically varying about an average value. Applying direct perturbation method (MMS), an analytical solution is found. An analysis of principal parametric resonances is carried out when changing frequency of the axial velocity is zero, close to zero and twice the natural frequency. Mathematical analysis is carried out to determine the stability and instability zones. The results show that instability occurs when changing frequency of the axial velocity is close to two times the natural frequency, whereas no instability occurs when changing frequency is close to zero. A case study of bandsaw is discussed. The stability and instability zones are plotted for the first five natural frequencies.

## Keywords

Axially travelling string Method of multiple scales (MMS) Parametric resonance Stability

## Notation

A

Cross-sectional area of the string

L

Length of the string

ρ

Mass density of the string

P

Tension force in the string

P0

Initial tension force in the string

κ

Pulley support parameter

v0

Dimensionless mean velocity of the string

$$y = \frac{{y^{*} }}{L}$$

Dimensionless transverse displacement of the string

$$x = \frac{{x^{*} }}{L}$$

Dimensionless spatial variable

$$t = (1/{\text{L}})\sqrt {(P_{0} /\rho A} )t^{ * }$$

Dimensionless time

$$\Omega ^{*} = \frac{1}{L}\sqrt {\frac{{P_{0} }}{\rho A}}\Omega$$

Dimensional frequency of velocity variation

$$v = v^{*} /\sqrt {P_{0} /\rho A}$$

Dimensionless axial velocity of the string

ε

Dimensionless parameter <<1

σ

Detuning parameter

ψn

Mode shapes of the travelling string

ωn

Natural frequency of the travelling string

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© Springer Nature Singapore Pte Ltd. 2020

## Authors and Affiliations

• Shashendra Kumar Sahoo
• 1
Email author
• H. C. Das
• 2
• L. N. Panda
• 3
1. 1.Department of Mechanical EngineeringInstitute of Technical Education and Research, SIKSHA ‘O’ ANUSANDHAN Deemed to be UniversityBhubaneswarIndia
2. 2.Department of Mechanical EngineeringN.I.TShillongIndia
3. 3.Department of Mechanical EngineeringC.E.T, BPUTBhubaneswarIndia