Stability Analysis of Principal Parametric Resonance of Viscoelastic Pipes by Using Multiple Time Scales

Abstract

This study investigates the transverse vibrations taking place tensioned viscoelastic pipes conveying fluid with time-dependent velocity taking into account simple supports condition. The governing equation is derived from Newton’s second law, Boltzmann’s superposition principle, and the stress-strain relation given for Maxwell viscoelastic model. The time-dependent velocity is assumed to vary harmonically about mean velocity. This system experiences a Coriolis acceleration component which renders such systems gyroscopic. The equation of motion is solved using the multiple time scale method. Principal parametric resonance is investigated. Stability boundaries are determined analytically. It is demonstrated that instabilities occur when the frequency of velocity fluctuations is close to two times the natural frequency of the system with constant velocity or when the frequency is close to the sum of any two natural frequencies.

Keywords

• Viscoelastic
• Multiple time scale
• Stability
• Pipe

Stability analysis of principal parametric resonance of viscoelastic pipes by using multiple time scales

Abstract

This study investigates the transverse vibrations taking place tensioned viscoelastic pipes conveying fluid with time-dependent velocity taking into account simple supports condition. The governing equation is derived from Newton’s second law, Boltzmann’s superposition principle, and the stress-strain relation given for Maxwell viscoelastic model. The time-dependent velocity is assumed to vary harmonically about mean velocity. This system experiences a Coriolis acceleration component which renders such systems gyroscopic. The equation of motion is solved using the multiple time scale method. Principal parametric resonances is investigated. Stability boundaries are determined analytically. It is demonstrated that instabilities occur when the frequency of velocity fluctuations is close to two times the natural frequency of the system with constant velocity or when the frequency is close to the sum of any two natural frequencies.

Keywords

• viscoelastic
• multiple time scale
• pipe
• stability

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