• Journal of Advanced Marine Engineering and Technology
• /
• v.10 no.4
• /
• pp.31-40
• /
• 1986

With the tendency toward high speed and high pressure in centrifugal pumps, the problem of sub-synchronous vibration has arisen, caused by the hydraulic forces of the working fluid, such as wearring, balance piston, impeller, etc.. These forces can drastically alter the rotor critical speeds and stability characteristics, and can be acted significant destabilizing forces. For preventing such self-excited vibration, the desing of the rotor system needs, which would secure the stability of the machine. In this paper, a procedure is presented for dynamic modeling of rotor-bearing-seal-impeller systems which consist of rigid disks, distributed parameter finite rotor elements and discrete bearings, seals and impellers. A finite element model including the effects of rotatory inertia and gyroscopic moments is developed using the consistent matrix approach. The technique of dynamic matrix reduction is applied to the shaft matrices to reduce them to a set of matrices of dynamic of significantly fewer degrees of freedom. The representation of bearing, seal and impeller elements is in term of linearized stiffness and damping matrices by reasonably small perturbations from equilibrium. The stability behavior of a typical double suction centrifugal pump is presented. Results show the influence of clearance and flow conditions on running speeds and stability characteristics.

• Transactions of the Korean Society of Mechanical Engineers A
• /
• v.29 no.12 s.243
• /
• pp.1574-1585
• /
• 2005

In this paper, a spectral element model is developed for the uniform straight pipelines conveying internal unsteady fluid. Four coupled pipe-dynamics equations are derived first by using the Hamilton's principle and the principles of fluid mechanics. The transverse displacement, the axial displacement, the fluid pressure and the fluid velocity are all considered as the dependent variables. The coupled pipe-dynamics equations are then linearized about the steady state values of the fluid pressure and velocity. As the final step, the spectral element model represented by the exact dynamic stiffness matrix, which is often called spectral element matrix, is formulated by using the frequency-domain solutions of the linearized pipe-dynamics equations. The FFT-based spectral dynamic analyses are conducted to evaluate the accuracy of the present spectral element model and also to investigate the structural dynamic characteristics and the internal fluid transients of an example pipeline system.

• Journal of KSNVE
• /
• v.10 no.1
• /
• pp.97-106
• /
• 2000

The dynamic behavior of crankshaft-bearing system in scroll compressor has been investigated using the combined methodologies of finite elements and transfer matrices. The finite element formulation is proposed including the field element for a shaft section and the point element at balancer weight locations, bearing locations, etc., whereas the conventional method is used with the elements. The Houbolt method is used to consider the time march for the integration of the system equations. The linear stiffness and damping coefficients are calculated for a finite cylindrical fluid-film bearing by solving the Reynolds equation, using finite difference method. The orbital response of crankshaft supported on the linear bearing model is obtained, considering balancer weights of motor rotor. And, the steady state displacement of crankshaft are compared with a variation in balancer weight. The loci of crankshaft at bearing locations are composed of the synchronous whirl component and the non-synchronous whirl component.

• Journal of the Korean Society for Railway
• /
• v.6 no.2
• /
• pp.129-134
• /
• 2003

This paper investigates the dynamic characteristics of a beam axially moving over multiple elastic supports. The spectral element matrix is derived first for the axially moving beam element and then it is used to formulate the spectral element matrix for the moving beam element with an interim elastic support. The moving speed dependance of the eigenvalues is numerically investigated by varying the applied axial tension and the stiffness of the elastic supports. Numerical results show that the fundamental eigenvalue vanishes first at the critical moving speed to generate the static instability.

• Advances in nano research
• /
• v.14 no.2
• /
• pp.127-139
• /
• 2023

This paper studies the free vibration behavior of bi-dimensional functionally graded (BFG) nanobeams subjected to arbitrary boundary conditions. According to Eringen's nonlocal theory and Hamilton's principle, the underlying equations of motion have been obtained for BFG nanobeams. Moreover, the variable substitution method is utilized to establish the structure's state-space differential equations, followed by forming the dynamic stiffness matrix based on state-space differential equations. In order to compute the natural frequencies, the current study utilizes the Wittrick-Williams algorithm as a solution technique. Moreover, the nonlinear vibration frequencies calculated by employing the proposed method are compared to the frequencies obtained in previous studies to evaluate the proposed method's performance. Some illustrative numerical examples are also given in order to study the impacts of the nonlocal parameters, material property gradient indices, nanobeam length, and boundary conditions on the BFG nanobeam's frequency. It is found that reducing the nonlocal parameter will usually result in increased vibration frequencies.

• Journal of the Korean Society of Manufacturing Process Engineers
• /
• v.18 no.2
• /
• pp.58-64
• /
• 2019

This paper describes the analysis of dynamic characteristics and prediction of the stiffness for the joint between structural members. In the process of deriving the governing equations, the stiffness values responsible for the moment and shear force were modelled by using linear and torsional springs in the middle of a clamped-clamped beam. The sensitivities of the natural frequency and modal assurance criterion were investigated as a function of the dimensionless linear and torsional spring stiffness. The reliability of the predictions for the linear and torsional stiffness values was verified by the inverse computations of the stiffness matrix. The predictive and exact theoretical stiffness values were compared for the stiffness element in the finite element formulation, and their results show an excellent correlation. It is strongly anticipated that although the proposed methodology is currently limited to the analytical utilization, it will provide a useful tool to estimate unknown joint stiffness values based on the experimental natural frequency and mode shape.

• Transactions of the Korean Society for Noise and Vibration Engineering
• /
• v.20 no.10
• /
• pp.923-930
• /
• 2010

Finite element models of dynamic systems can be updated in two stages. In the first stage, mass and stiffness matrices are updated neglecting damping, and in the second stage, damping matrices are estimated with the mass and stiffness matrices fixed. Three methods to estimate damping matrices for this purpose are proposed in this paper. The methods include one for proportional damping systems and two for non-proportional damping systems. Method 1 utilizes orthogonality of normal modes and estimates damping matrices using the modal parameters extracted from the measured responses. Method 2 estimates damping matrices from impedance matrices which are the inverse of FRF matrices. Method 3 estimates damping using the equation which relates a damping matrix to the difference between the analytical and measured FRFs. The characteristics of the three methods are investigated by applying them to simulated discrete system data and experimental cantilever beam data.

• Journal of KSNVE
• /
• v.5 no.4
• /
• pp.565-575
• /
• 1995

Gearboxes are often used in the petrochemical and electrical power plants to transmit mechanical power between two branches of a machinery train rotating at different speeds. When the gearboxes are connected with rotors supported by journal bearings, bearing loads vary in magnitude and direction with rotor speed and torque transmitted by the gearboxes. In this study, dynamic characteristics of the system which consists of gearboxes and a rotor supported by journal bearings are investigated analytically and experimentally by employing the polynomial transfer matrix method and modal analysis under different speeds and torque levels. Journal bearing loads due to the transmitted torque are claculated analytically and the stiffness and damping coefficient of the journal bearings are obtained using finite element method. Comparison of the analytical and experimental results shows that the cross coupled stiffness coefficients increase with increasing rotor speed, while the cross coupled damping coefficients decrease. This generates the oil whirl instability in the journal bearings. As the transmitted torque level goes up, the stiffness coefficients of the journal bearing and the first horizontal natural frequency increase. High levels of the transmitted torque produce high bearing stiffness since the contact loads of the mating gear teeth increase. The logarithmic decrement, which is a stability indicator, is shown to decrease with increasing speed and decreasing torque. Thus, at the low torque level, the system become unstable even at the low shaft speed.

• Journal of the Korean Society for Precision Engineering
• /
• v.32 no.2
• /
• pp.117-125
• /
• 2015

This paper presents a dynamic modeling method for the indeterminate spindle-bearing system supported by multiple bearings of different types. A spindle-bearing system supported by ball and cylindrical roller bearings is considered. The de Mul's bearing model is extended for calculating ball and cylindrical roller bearing stiffness matrices with inclusion of centrifugal force and gyroscopic moment. The dependence between spindle shaft reaction forces and bearing stiffness is effectively resolved using an iterative approach. The spindle rotor dynamics is established with the Timoshenko beam theory based finite elements. The spindle reaction forces, bearings stiffness and spindle natural frequencies are obtained with taking into account spindle radial load, ball bearing axial preload and rotational speed effects. The developed method is verified by comparing the simulation results with those from a commercial program.

• Proceedings of the Korean Society for Noise and Vibration Engineering Conference
• /
• 2005.05a
• /
• pp.221-226
• /
• 2005

In this paper, a dynamic model for the rotor shaft-coupling-blade system is developed. The blades are attached to a disk and driven by an electric motor shaft which is flexible in torsion. We assumed that the shaft torsional flexibility is lumped in the flexible coupling which is usually adopted in rotor systems. The Lagrangian approach with the small deformation theory for both blade-bending and shaft-torsional deformations is employed for developing the equation of the motion. The assumed modes method is used for estimating the blade transverse deflection. The numerical results highlight the effects of both structural damping of the system and the torsional stiffness of the flexible coupling to the dynamic response of the blade. The results showed strong coupling between the blade bending and shaft torsional vibrations in the form of inertial nonlinearif, stiffness hardening and softening.