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Extension of Lagrangian–Hamiltonian mechanics for continuous systems—investigation of dynamics of a one-dimensional internally damped rotor driven through a dissipative coupling

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Abstract

In this paper, the extended Lagrangian formulation for a one-dimensional continuous system with gyroscopic coupling and non-conservative fields has been developed. Using this formulation, the dynamics of an internally and externally damped rotor driven through a dissipative coupling has been studied. The invariance of the extended or so-called umbra-Lagrangian density is obtained through an extension of Noether’s theorem. The rotor shaft is modeled as a Rayleigh beam. The dynamic behavior of the rotor shaft is obtained and validated through simulation studies. Results show an interesting phenomenon of limiting behavior of the rotor shaft with internal damping beyond certain threshold speeds which are obtained theoretically and affirmed by simulations. It is further observed that there is entrainment of whirling speeds at natural frequencies of the rotor shaft primarily depending on the damping ratio.

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Abbreviations

A n :

Amplitude of nth mode of the rotor

EI :

Rigidity of the continuous rotor

H * :

Umbra-Hamiltonian of the system

I d :

Rotary inertia of the rotor

I p :

Inertia of the beam through principal axis

L :

Lagrangian of the system

L * :

Umbra-Lagrangian of the system

R c :

Damping coefficient of dissipative coupling

V :

Infinitesimal generator of rotational SO(2) group

V t :

Real-time component of infinitesimal generator

V η :

Umbra-time component of infinitesimal generator

a :

Cross-sectional area of the rotor

n :

Mode number

p(η):

Umbra-time momentum

p(t):

Real-time momentum

q(t):

Generalized displacement in real time

q(η):

Generalized displacement in umbra-time

\(\dot{q}(t)\) :

Generalized velocity in real time

\(\dot{q}(\eta)\) :

Generalized velocity in umbra-time

s :

Angle or linear variables for rotational transformation or linear transformation

x i ( ):

Linear displacements in real time or umbra-time, where i=1,…,n

\(\dot{x}_{i}(\,)\) :

Linear velocity in real time or umbra-time, where i=1,…,n

t :

Real time, in s

Ω :

Excitation frequency, in rad/s

Ω n :

Natural frequency of the rotor shaft, in rad/s

η :

Umbra-time, in s

ω :

constant angular velocity

ω * :

Non-dimensional natural frequency \(=\frac{\dot{\theta}(t)}{\pi ^{2}\sqrt{EI/(\rho aL^{4})}}\)

θ( ):

Angular displacement in umbra-time or real time, in rad

\(\dot{\theta}(\,)\) :

Angular velocity of the shaft in umbra-time or real time in rad/s

ℒ:

Umbra-Lagrangian density

ρ :

Mass density of rotor shaft

μ a :

External damping of the beam

μ i :

Internal damping of the beam

γ * :

Damping ratio

u i (t):

Real displacement coordinates of beam

u i (η):

Umbra-displacement coordinates of beam

ℋ:

Umbra-Hamiltonian density

ℋ i ,ℋ e :

Interior and exterior umbra-Hamiltonian densities

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Authors and Affiliations

  1. Mechanical Engineering Department, Indian Institute of Technology, Kharagpur, 721302, India

    Amalendu Mukherjee, Vikas Rastogi & Anirvan Dasgupta

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  1. Amalendu Mukherjee
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  2. Vikas Rastogi
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  3. Anirvan Dasgupta
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Correspondence to Amalendu Mukherjee.

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Mukherjee, A., Rastogi, V. & Dasgupta, A. Extension of Lagrangian–Hamiltonian mechanics for continuous systems—investigation of dynamics of a one-dimensional internally damped rotor driven through a dissipative coupling. Nonlinear Dyn 58, 107–127 (2009). https://doi.org/10.1007/s11071-008-9464-x

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