Abstract: In the present paper, a large turbo-generator shaft train including a heavy-duty gas turbine engine, a coupling, and a generator is established. The method of analysis is based on finite element simplified model for lateral and torsional vibration calculation. The basic elements of rotor are the shafts and the disks which are represented as circular cross section flexible beams and rigid body elements, respectively. For more accurate results, the gyroscopic effect and bearing dynamics coefficients and function of rotation are taken into account, and for the influence of shear effect, rotor has been modeled in the form of Timoshenko beam. Lateral critical speeds, critical speed map, damped mode shapes, Campbell diagram, zones of instability, amplitudes, phase angles response due to synchronous forces of excitation and amplification factor are calculated. Also, in the present paper, the effect of imbalanced rotor and effects of changing in internal force and temperature are studied.
Abstract: Singular value decomposition based optimisation of
geometric design parameters of a 5-speed gearbox is studied. During
the optimisation, a four-degree-of freedom torsional vibration model
of the pinion gear-wheel gear system is obtained and the minimum
singular value of the transfer matrix is considered as the objective
functions. The computational cost of the associated singular value
problems is quite low for the objective function, because it is only
necessary to compute the largest and smallest singular values (μmax
and μmin) that can be achieved by using selective eigenvalue solvers;
the other singular values are not needed. The design parameters are
optimised under several constraints that include bending stress,
contact stress and constant distance between gear centres. Thus, by
optimising the geometric parameters of the gearbox such as, the
module, number of teeth and face width it is possible to obtain a
light-weight-gearbox structure. It is concluded that the all optimised
geometric design parameters also satisfy all constraints.
Abstract: This paper presents the scaling laws that provide the
criteria of geometry and dynamic similitude between the full-size
rotor-shaft system and its scale model, and can be used to predict the
torsional vibration characteristics of the full-size rotor-shaft system by
manipulating the corresponding data of its scale model. The scaling
factors, which play fundamental roles in predicting the geometry and
dynamic relationships between the full-size rotor-shaft system and its
scale model, for torsional free vibration problems between scale and
full-size rotor-shaft systems are firstly obtained from the equation of
motion of torsional free vibration. Then, the scaling factor of external
force (i.e., torque) required for the torsional forced vibration problems
is determined based on the Newton’s second law. Numerical results
show that the torsional free and forced vibration characteristics of a
full-size rotor-shaft system can be accurately predicted from those of
its scale models by using the foregoing scaling factors. For this reason,
it is believed that the presented approach will be significant for
investigating the relevant phenomenon in the scale model tests.
Abstract: This paper presents a 4-DOF nonlinear model of a
cracked de Laval rotor-stator system derived based on Energy
Principles. The model has been used to simulate coupled torsionallateral
response of the faulty system with multiple parametric
excitations; rotor-stator-rub, a breathing transverse crack, eccentric
mass and an axial force. Nonlinearity of a “breathing” crack is
incorporated in the model using a simple hinge mechanism suitable
for a shallow crack. Response of the system while passing via its
critical speed with intermittent rotor-stator rub is analyzed. Effects of
eccentricity with phase and acceleration are investigated. Features of
crack, rub and eccentricity in vibration response are explored for
condition monitoring. The presence of a crack and rub are observable
in the power spectrum despite excitations by an axial force and rotor
unbalance. Obtained results are consistent with existing literature and
could be adopted into rotor condition monitoring strategies.
Abstract: In this paper, an analytical approach is used to study the coupled lateral-torsional vibrations of laminated composite beam. It is known that in such structures due to the fibers orientation in various layers, any lateral displacement will produce a twisting moment. This phenomenon is modeled by the bending-twisting material coupling rigidity and its main feature is the coupling of lateral and torsional vibrations. In addition to the material coupling, the effects of shear deformation and rotary inertia are taken into account in the definition of the potential and kinetic energies. Then, the governing differential equations are derived using the Hamilton-s principle and the mathematical model matches the Timoshenko beam model when neglecting the effect of bending-twisting rigidity. The equations of motion which form a system of three coupled PDEs are solved analytically to study the free vibrations of the beam in lateral and rotational modes due to the bending, as well as the torsional mode caused by twisting. The analytic solution is carried out in three steps: 1) assuming synchronous motion for the kinematic variables which are the lateral, rotational and torsional displacements, 2) solving the ensuing eigenvalue problem which contains three coupled second order ODEs and 3) imposing different boundary conditions related to combinations of simply, clamped and free end conditions. The resulting natural frequencies and mode shapes are compared with similar results in the literature and good agreement is achieved.
Abstract: In this paper the Differential Quadrature Method (DQM) is employed to study the coupled lateral-torsional free vibration behavior of the laminated composite beams. In such structures due to the fiber orientations in various layers, the lateral displacement leads to a twisting moment. The coupling of lateral and torsional vibrations is modeled by the bending-twisting material coupling rigidity. In the present study, in addition to the material coupling, the effects of shear deformation and rotary inertia are taken into account in the definition of the potential and kinetic energies of the beam. The governing differential equations of motion which form a system of three coupled PDEs are solved numerically using DQ procedure under different boundary conditions consist of the combinations of simply, clamped, free and other end conditions. The resulting natural frequencies and mode shapes for cantilever beam are compared with similar results in the literature and good agreement is achieved.
Abstract: In this article an isotropic linear elastic half-space with
a cylindrical cavity of finite length is considered to be under the
effect of a ring shape time-harmonic torsion force applied at an
arbitrary depth on the surface of the cavity. The equation of
equilibrium has been written in a cylindrical coordinate system. By
means of Fourier cosine integral transform, the non-zero
displacement component is obtained in the transformed domain. With
the aid of the inversion theorem of the Fourier cosine integral
transform, the displacement is obtained in the real domain. With the
aid of boundary conditions, the involved boundary value problem for
the fundamental solution is reduced to a generalized Cauchy singular
integral equation. Integral representation of the stress and
displacement are obtained, and it is shown that their degenerated
form to the static problem coincides with existing solutions in the
literature.
Abstract: In this study, rotating flexible shaft-disk system
having flexible beams is considered as a dynamic system. After
neglecting nonlinear terms, torsional vibration of the shaft-disk
system and lateral and longitudinal vibration of the flexible beam are
still coupled through the motor speed. The system has three natural
frequencies; the flexible shaft-disk system torsional natural
frequency, the flexible beam lateral and longitudinal natural
frequencies. Eigenvalue calculations show that while the shaft speed
changes, torsional natural frequency of the shaft-disk system and the
beam longitudinal natural frequency are not changing but the beam
lateral natural frequency changes. Beam lateral natural frequency
stays the same as the nonrotating beam lateral natural frequency ωb
until the motor speed ωm is equal to ωb. After then ωb increases and
remains equal to the motor speed ωm until the motor speed is equal to
the shaft-disk system natural frequency ωT. Then the beam lateral
natural frequency ωb becomes equal to the natural frequency ωT and
stays same while the motor speed ωm is increased. Modal amplitudes
and phase angles of the vibrations are also plotted against the motor
speed ωm.