Fan/Foundation Interaction—A Simplified Calculation Procedure

1981 ◽  
Vol 103 (4) ◽  
pp. 805-810
Author(s):  
H. M. Chen ◽  
S. B. Malanoski
Keyword(s):  
Dynamic System ◽  
Equations Of Motion ◽  
Rotor Dynamics ◽  
Analysis Procedure ◽  
Initial Assessment ◽  
Design Decision ◽  
Foundation Soil ◽  
Trouble Shooting ◽  
Modes Of Vibration ◽  
Simplified Calculation

This paper presents a simplified analysis procedure to provide initial assessment and guidance on fan rotor dynamics including the foundation interaction. For purposes of early-design decision-making (or trouble-shooting), the interaction of a rotor-bearing dynamic system and a foundation-soil/piling dynamic system is viewed approximately for the vertical, horizontal, and rocking modes of vibration. The equations of motion are written in matrix form and include the pertinent parameters. A numerical example is presented to guide in the interpretation of the analysis; this example considers the unbalance response of the entire system as measured at the bearings.

Concise Equations for Rotor Dynamics Analysis

1995 ◽  
Author(s):  
W. J. Chen
Keyword(s):  
Equations Of Motion ◽  
Rotor Dynamics ◽  
Memory Storage ◽  
The Other ◽  
Dynamics Analysis ◽  
Real Domain ◽  
The Matrix ◽  
System Equations ◽  
Ordering Algorithms ◽  
Matrix Bandwidth

Abstract Concise equations for rotor dynamics analysis are presented. Two coordinate ordering methods are introduced in the element equations of motion. One is in the real domain and the other is in the complex domain. The two proposed ordering algorithms lead to more compact element matrices. A station numbering technique is also proposed for the system equations during the assembly process. This numbering technique can minimize the matrix bandwidth, the memory storage and can increase the computational efficiency.

A Note on Computational Rotor Dynamics

1998 ◽  
Vol 120 (1) ◽  
pp. 228-233 ◽  
Author(s):  
W. J. Chen
Keyword(s):  
Computational Efficiency ◽  
Equations Of Motion ◽  
Rotor Dynamics ◽  
Memory Storage ◽  
Rotor Systems ◽  
Real Domain ◽  
The Matrix ◽  
System Equations ◽  
Ordering Algorithms ◽  
Matrix Bandwidth

Concise equations for improvements in computational efficiency on dynamics of rotor systems are presented. Two coordinate ordering methods are introduced in the element equations of motion. One is in the real domain and the other is in the complex domain. The two coordinate ordering algorithms lead to compact element matrices. A station numbering technique is also proposed for the system equations during the assembly process. The proposed numbering technique can minimize the matrix bandwidth, the memory storage and can increase the computational efficiency. Numerical examples are presented to demonstrate the benefit of the proposed algorithms.

Sliding Mode Control in Ziegler’s Pendulum With Tracking Force: Novel Modeling Considerations

2013 ◽  
Author(s):  
Ehsan Sarshari ◽  
Nastaran Vasegh ◽  
Mehran Khaghani ◽  
Saeid Dousti
Keyword(s):  
Stability Analysis ◽  
Sliding Mode Control ◽  
Dynamic System ◽  
Linear Stability Analysis ◽  
Sliding Mode ◽  
Chaotic Behavior ◽  
Equations Of Motion ◽  
Sliding Mode Controller ◽  
Gravity Effect ◽  
Tracking Force

Ziegler’s pendulum is an appropriate model of a non-conservative dynamic system. By considering gravity effect, new equations of motion are extracted from Newton’s motion laws. The instability of equilibriums is determined by linear stability analysis. Chaotic behavior of the model is shown by numerical simulations. Sliding mode controller is used for eliminating chaos and for stabilizing the equilibriums.

Application of Lifting-Surface Theory to the Prediction of Hydroelastic Response of Hydrofoil Boats

1968 ◽  
Vol 12 (04) ◽  
pp. 286-301
Author(s):  
C. J. Henry
Keyword(s):  
Equations Of Motion ◽  
Three Dimensional ◽  
Numerical Procedure ◽  
Mode Shapes ◽  
Operator Technique ◽  
Elastic Mode ◽  
Surface Theory ◽  
Lifting Surface ◽  
Modes Of Vibration ◽  
Elastic Modes

In this report a theoretical procedure is developed for the prediction of the dynamic response elastic or rigid body, of a hydrofoil-supported vehicle in the flying condition— to any prescribed transient or periodic disturbance. The procedure also yields the stability indices of the response, so that dynamic instabilities such as flutter can also be predicted. The unsteady hydrodynamic forces are introduced in the equations of motion for the elastic vehicle in terms of the indicia I pressure-response functions, which are de rived herein from lifting-surface theory. Thus, the predicted vehicle-response includes the effects of three-dimensional unsteady flow conditions at specified forward speed. The natural frequencies and elastic modes of vibration of the vehicle and foil system in the absence of hydrodynamic effects are presumed known. A numerical procedure is presented for the solution of the downwash integral equations relating the unknown indicial pressure distributions to the specified elastic-mode shapes. The procedure is based on use of the generalized-lift-operator technique together with the collocation method.

Conditions for the planes of symmetry of an elastically supported rigid body

2009 ◽  
Vol 223 (8) ◽  
pp. 1755-1766 ◽  
Author(s):  
S J Jang ◽  
Y J Choi
Keyword(s):  
Rigid Body ◽  
Eigenvalue Problems ◽  
Mass Matrix ◽  
Equations Of Motion ◽  
Mass Matrices ◽  
Out Of Plane ◽  
Modes Of Vibration ◽  
Compliance Matrices ◽  
Special Points ◽  
Elastically Supported

Introducing the planes of symmetry into an oscillating rigid body suspended by springs simplifies the complexity of the equations of motion and decouples the modes of vibration into in-plane and out-of-plane modes. There have been some research results from the investigation into the conditions for planes of symmetry in which prior conditions for the simplification of the equations of motion are required. In this article, the conditions for the planes of symmetry that do not need prior conditions for simplification are presented. The conditions are derived from direct expansions of eigenvalue problems for stiffness and mass matrices that are expressed in terms of in-plane and out-of-plane modes and the orthogonality condition with respect to the mass matrix. Two special points, the planar couple point and the perpendicular translation point are identified, where the expressions for stiffness and compliance matrices can be greatly simplified. The simplified expressions are utilized to obtain the analytical expressions for the axes of vibration of a vibration system with planes of symmetry.

An Improved Transfer Matrix-Direct Integration Method for Rotor Dynamics

1986 ◽  
Vol 108 (2) ◽  
pp. 182-188 ◽  
Author(s):  
Jialiu Gu
Keyword(s):  
Transfer Matrix ◽  
Integration Method ◽  
Equations Of Motion ◽  
Rotor Dynamics ◽  
Direct Integration ◽  
Rotating System ◽  
Direct Integration Method ◽  
Mass Moment ◽  
Mass Moment Of Inertia ◽  
Matrix Iteration

A transfer matrix-direct integration combined method is proposed, which employs the transfer matrix method to derive the equations of motion of a “characteristic disk,” and uses the direct integration method to determine the critical speeds, modes and unbalance response of a rotor-bearing system, and to analyze its stability. Despite the complexity of the system, the number of governing equations is not greater than eight. For a single-spool rotating system, the number of equations is only four. A transfer matrix for a uniform shaft is derived to consider its distributed mass, moment of inertia and the effect of shearing force. An impedance matrix iteration method is proposed to consider the effect of a complicated bearing-supporting system on the rotor dynamics. Two examples are given, and the results agree satisfactorily with the experiments.

Unbalance Response of Rotors Supported on Hydrodynamic Bearings Placed Close to Nodal Points of Excited Vibration Modes

2002 ◽  
Author(s):  
Demetrio C. Zachariadis
Keyword(s):  
Tilt Angle ◽  
Rotor Dynamics ◽  
Journal Bearings ◽  
Vibration Modes ◽  
Time Varying ◽  
Rotor Systems ◽  
Unbalance Response ◽  
Nodal Points ◽  
Excited Vibration ◽  
Modes Of Vibration

The traditional 8-coefficient bearing model, used in linear rotor dynamics, is shown here to be inadequate for the unbalance response calculation of rotor systems supported on hydrodynamic journal bearings placed close to nodal points of excited modes of vibration. In such situations, one cannot neglect the time varying tilt angle between journals and bearings, whose consideration leads to the adoption of a 32-coefficient bearing model. Numerical results indicate that the differences between vibration amplitudes calculated using both bearing models can be greater than 100%, while discrepancies in the predicted stability thresholds are small. The conclusions of the study are coherent with previously published theoretical and experimental results.

Performance Characteristics of a Smart Flexible Beam With Distributed ACLD Elements

2002 ◽  
Author(s):  
L. C. Hau ◽  
Eric H. K. Fung
Keyword(s):  
Finite Element ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Viscoelastic Model ◽  
Element Model ◽  
Flexible Beam ◽  
Constrained Layer Damping ◽  
Optimal Output ◽  
Modes Of Vibration ◽  
Original Size

The finite element method, in conjunction with the Golla-Hughes-McTavish (GHM) viscoelastic model, is employed to model a clamped-free beam partially treated with active constrained layer damping (ACLD) elements. The governing equations of motion are converted to a state-space form for control system design. Prior to this, since the resultant finite element model has too many degrees of freedom due to the addition of dissipative coordinates, a model reduction is performed to revert the system back to its original size. Finally, optimal output feedback gains are designed based on the reduced models. Numerical simulations are performed to study the effect of different element configurations, with various spacing and locations, on the vibration control performance of a “smart” flexible ACLD treated beam. Results are presented for the damping ratios of the first two modes of vibration. It is found that improvement on the second mode damping can be achieved by splitting a single ACLD element into two and placing them at appropriate positions of the beam.

The Use of Normal Modes in Problems of Forced Vibration and Impact

1957 ◽  
Vol 61 (560) ◽  
pp. 552-559
Author(s):  
R. P. N. Jones
Keyword(s):  
Forced Vibration ◽  
Virtual Work ◽  
Equations Of Motion ◽  
Normal Modes ◽  
Very High Frequency ◽  
Linear Elastic ◽  
Modes Of Vibration ◽  
Fundamental Equations ◽  
Very High ◽  
Simple Exposition

SummaryA simple exposition, using d'Alembert's principle and methods of virtual work, is given of the properties and applications of the normal modes of vibration of a linear elastic system. The use of the normal modes in problems of free and forced vibration and dynamic loading is discussed with the aid of simple examples, and it is shown that by these methods dynamical problems for any linear system may be solved without the use of the fundamental equations of motion, provided the natural frequencies and modes of the system are known. In most problems the solutions converge rapidly, so that only the first few modes of vibration need be considered, and in these cases the solution may be modified to give further improvement in convergence. Unsatisfactory convergence may be obtained, however, in problems where there is an exciting force of very high frequency, or an impact of short duration. An approximate allowance may be made for damping, provided this is small.

Dynamic analysis of a microelectromechanical systems resonant gyroscope excited using combination parametric resonance

2006 ◽  
Vol 220 (9) ◽  
pp. 1463-1479 ◽  
Author(s):  
B J Gallacher ◽  
J S Burdess
Keyword(s):  
Perturbation Analysis ◽  
Parametric Excitation ◽  
Microelectromechanical Systems ◽  
Equations Of Motion ◽  
Multiple Time ◽  
Excitation Scheme ◽  
Electrostatically Actuated ◽  
Modes Of Vibration ◽  
The Stability ◽  
Electrostatic Coupling

This paper investigates the application of parametric excitation to a resonant microelectromechanical systems (MEMS) gyroscope. The modal equations of motion of an electrostatically actuated ring are derived and shown to be coupled via the electrostatic stiffness. Such electrostatic coupling between in-plane modes of vibration permits parametric instabilities that may be exploited in a novel excitation scheme. A multiple time scale perturbation method is used to analyse the response of the ring gyroscope to the combination parametric excitations with the principal objective of separating the drive and response frequencies of the ring gyroscope. As pairs of flexural modes of the perfect ring are degenerate, the combination excitation between distinct modes demand the ring to be analysed as a four degree of freedom system. Slight mis-tuning between the otherwise degenerate modes is incorporated in the perturbation analysis. The results of the perturbation analysis are subsequently used to determine the stability boundaries for a typical ring gyroscope when excited using a sum combination resonance between the flexural modes of order 2 and 5. In this case, the ratio of the drive and response frequencies is approximately 10:1. Drive and sense configurations that enable effective parametric excitation of a desired mode are investigated. Simulation of the oscillator scheme is achieved using MATLAB Simulink and this validates the perturbation analysis. Agreement between the models within 10 per cent is demonstrated.

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