Parametric Instability of a Periodically Supported Pipe Without and With Vibration Absorbers

1984 ◽  
Vol 51 (1) ◽  
pp. 159-163 ◽  
Author(s):  
A. K. Mallik ◽  
S. B. Kulkarni ◽  
K. S. Ram
Keyword(s):  
Prior Knowledge ◽  
Propagation Constant ◽  
Parametric Instability ◽  
Numerical Results ◽  
Mode Shapes ◽  
Effective Control ◽  
Vibration Absorbers ◽  
Instability Regions

Parametric instability of a periodically supported pipe without and with dynamic absorbers is investigated. A method based on the notion of propagation constant is used. This method requires no prior knowledge of the mode shapes and renders the amount of computation independent of the number of spans. Numerical results are presented for single and two-span pipes. Effective control of the instability regions by means of dynamic absorbers is also demonstrated.

scholarly journals Parametric instability analysis of the top-tensioned riser in consideration of complex pre-stress distribution

2018 ◽  
Vol 10 (1) ◽  
pp. 168781401775389 ◽  
Author(s):  
Leixin Li ◽  
Luyun Chen
Keyword(s):  
Stress Distribution ◽  
Differential Equations ◽  
Parametric Instability ◽  
Beam Theory ◽  
Complex Loading ◽  
Welding Residual Stress ◽  
Parametric Vibration ◽  
Mode Shapes ◽  
Stress Influences ◽  
Euler Bernoulli Beam

The parametric vibration instability of a riser is studied in consideration of a complex pre-stress distribution. Differential equations of the riser are derived according to Euler–Bernoulli beam theory, and a method to solve the differential equations is proposed. With the parametric vibration of a top-tensioned riser as an example, the effects of the amplitude and direction of complex pre-stress on frequency, mode shapes, and instability characteristics are investigated. Results show that welding residual stress influences the dynamic response of the riser structure. A new approach to eliminate the complex loading of the riser is obtained.

scholarly journals Three Dimensional Vibration Analysis of a Class of Traction-Free Solid Elastic Bodies with an Eccentric Cavity

2012 ◽  
Vol 19 (6) ◽  
pp. 1341-1357 ◽  
Author(s):  
Seyyed M. Hasheminejad ◽  
Yaser Mirzaei
Keyword(s):  
Vibration Analysis ◽  
Three Dimensional ◽  
Free Vibration Analysis ◽  
Numerical Results ◽  
Mode Shapes ◽  
Computational Techniques ◽  
Published Data ◽  
Elastic Structures ◽  
Wide Range ◽  
Vibrational Characteristics

A three-dimensional elasticity-based continuum model is developed for describing the free vibrational characteristics of an important class of isotropic, homogeneous, and completely free structural bodies (i.e., finite cylinders, solid spheres, and rectangular parallelepipeds) containing an arbitrarily located simple inhomogeneity in form of a spherical or cylindrical defect. The solution method uses Ritz minimization procedure with triplicate series of orthogonal Chebyshev polynomials as the trial functions to approximate the displacement components in the associated elastic domains, and eventually arrive at the governing eigenvalue equations. An extensive review of the literature spanning over the past three decades is also given herein regarding the free vibration analysis of elastic structures using Ritz approach. Accuracy of the implemented approach is established through proper convergence studies, while the validity of results is demonstrated with the aid of a commercial FEM software, and whenever possible, by comparison with other published data. Numerical results are provided and discussed for the first few clusters of eigen-frequencies corresponding to various mode categories in a wide range of cavity eccentricities. Also, the corresponding 3D mode shapes are graphically illustrated for selected eccentricities. The numerical results disclose the vital influence of inner cavity eccentricity on the vibrational characteristics of the voided elastic structures. In particular, the activation of degenerate frequency splitting and incidence of internal/external mode crossings are confirmed and discussed. Most of the results reported herein are believed to be new to the existing literature and may serve as benchmark data for future developments in computational techniques.

scholarly journals Effect of Location of Delamination on Free Vibration of Cross-Ply Conical Shells

2012 ◽  
Vol 19 (4) ◽  
pp. 679-692 ◽  
Author(s):  
Sudip Dey ◽  
Amit Karmakar
Keyword(s):  
Finite Element ◽  
Free Vibration ◽  
Natural Frequencies ◽  
Dynamic Equilibrium ◽  
Twist Angle ◽  
Structural Instability ◽  
Numerical Results ◽  
Mode Shapes ◽  
Iteration Algorithm ◽  
Conical Shells

Location of delamination is a triggering parameter for structural instability of laminated composites. In this paper, a finite element method is employed to determine the effects of location of delamination on free vibration characteristics of graphite-epoxy cross-ply composite pre-twisted shallow conical shells. The generalized dynamic equilibrium equation is derived from Lagrange's equation of motion neglecting Coriolis effect for moderate rotational speeds. The formulation is exercised by using an eight noded isoparametric plate bending element based on Mindlin's theory. Multi-point constraint algorithm is utilized to ensure the compatibility of deformation and equilibrium of resultant forces and moments at the delamination crack front. The standard eigen value problem is solved by applying the QR iteration algorithm. Finite element codes are developed to obtain the numerical results concerning the effects of location of delamination, twist angle and rotational speed on the natural frequencies of cross-ply composite shallow conical shells. The mode shapes are also depicted for a typical laminate configuration. Numerical results obtained from parametric studies of both symmetric and anti-symmetric cross-ply laminates are the first known non-dimensional natural frequencies for the type of analyses carried out here.

Parametric Instability of a Rotating Circular Ring With Moving, Time-Varying Springs

2005 ◽  
Vol 128 (2) ◽  
pp. 231-243 ◽  
Author(s):  
Sripathi Vangipuram Canchi ◽  
Robert G. Parker
Keyword(s):  
Parametric Instability ◽  
Circular Ring ◽  
Time Varying ◽  
Ring Gear ◽  
Geometric Description ◽  
Bending Vibrations ◽  
System Parameters ◽  
Ring Rotation ◽  
Special Cases ◽  
Instability Regions

Parametric instabilities of in-plane bending vibrations of a rotating ring coupled to multiple, discrete, rotating, time-varying stiffness spring-sets of general geometric description are investigated in this work. Instability boundaries are identified analytically using perturbation analysis and given as closed-form expressions in the system parameters. Ring rotation and time-varying stiffness significantly affect instability regions. Different configurations with a rotating and nonrotating ring, and rotating spring-sets are examined. Simple relations governing the occurrence and suppression of instabilities are discussed for special cases with symmetric circumferential spacing of spring-sets. These results are applied to identify possible conditions of ring gear instability in example planetary gears.

scholarly journals The regions of parametric instability of brush-contact device electromagnetic circuit in unstable working conditions

2015 ◽  
Vol 1 ◽  
pp. 84
Author(s):  
O.I. Kozyreva ◽  
I.V. Plokhov ◽  
Y. N. Guraviev ◽  
I. E. Savraev ◽  
A. V. Ilyin
Keyword(s):  
Working Conditions ◽  
Electromagnetic Waves ◽  
Parametric Instability ◽  
Dissipative System ◽  
Degree Of Freedom ◽  
Contact Device ◽  
Instability Regions ◽  
Oscillation Equation ◽  
Selection Of

<p class="R-AbstractKeywords"><span lang="EN-US">The method to determine parametric instability regions of the electromagnetic circuit of the secluded brush contact of brush-contact device with using the oscillation equation  Mathieu II order for a dissipative system with one degree of freedom is developed. Recommendations on the selection of the parameters of the external damping device to eliminate the instability of electromagnetic waves in unstable working conditions brush-contact device are  given.</span></p>

Mesh Stiffness Variation Instabilities in Two-Stage Gear Systems

2001 ◽  
Author(s):  
Jian Lin ◽  
Robert G. Parker
Keyword(s):  
Numerical Solutions ◽  
Parametric Instability ◽  
Operating Conditions ◽  
Mesh Stiffness ◽  
Two Stage ◽  
Instability Regions ◽  
Stiffness Variation ◽  
Parametric Instabilities ◽  
Gear Systems ◽  
Mesh Phasing

Abstract Mesh stiffness variation, the change in stiffness of meshing teeth as the number of teeth in contact changes, causes parametric instabilities and severe vibration in gear systems. The operating conditions leading to parametric instability are investigated for two-stage gear chains, including idler gear and countershaft configurations. Interactions between the stiffness variations at the two meshes are examined. Primary, secondary, and combination instabilities are studied. The effects of mesh stiffness parameters, including stiffness variation amplitudes, mesh frequencies, contact ratios, and mesh phasing, on these instabilities are analytically identified. For mesh stiffness variation with rectangular waveforms, simple design formulae are derived to control the instability regions by adjusting the contact ratios and mesh phasing. The analytical results are compared to numerical solutions.

Dynamic Stability of a Translating String With a Sinusoidally Varying Velocity

2011 ◽  
Vol 78 (6) ◽  
Author(s):  
W. D. Zhu ◽  
X. K. Song ◽  
N. A. Zheng
Keyword(s):  
Fixed Point ◽  
Fixed Point Theory ◽  
Parametric Instability ◽  
Point Theory ◽  
Constant Length ◽  
Instability Phenomenon ◽  
Translation Velocity ◽  
Wave Solutions ◽  
Instability Regions ◽  
Phase Functions

A new parametric instability phenomenon characterized by infinitely compressed, shocklike waves with a bounded displacement and an unbounded vibratory energy is discovered in a translating string with a constant length and tension and a sinusoidally varying velocity. A novel method based on the wave solutions and the fixed point theory is developed to analyze the instability phenomenon. The phase functions of the wave solutions corresponding to the phases of the sinusoidal part of the translation velocity, when an infinitesimal wave arrives at the left boundary, are established. The period number of a fixed point of a phase function is defined as the number of times that the corresponding infinitesimal wave propagates between the two boundaries before the phase repeats itself. The instability conditions are determined by identifying the regions in a parameter plane where attracting fixed points of the phase functions exist. The period-1 instability regions are analytically obtained, and the period-i (i>1) instability regions are numerically calculated using bifurcation diagrams. The wave patterns corresponding to different instability regions are determined, and the strength of instability corresponding to different period numbers is analyzed.

PARAMETRIC INSTABILITY OF LAMINATED COMPOSITE CYLINDRICAL PANELS SUBJECTED TO PERIODIC NON-UNIFORM IN-PLANE LOADS

2011 ◽  
Vol 03 (04) ◽  
pp. 845-865 ◽  
Author(s):  
SARAT KUMAR PANDA ◽  
L. S. RAMACHANDRA
Keyword(s):  
Differential Equations ◽  
Dynamic Instability ◽  
Elastic Shell ◽  
Parametric Instability ◽  
Shear Deformation Theory ◽  
Mathieu Equation ◽  
Laminated Composite ◽  
Stress Distributions ◽  
Cylindrical Panels ◽  
Instability Regions

In the present investigation, the dynamic instability regions of shear deformable cross-ply laminated and composite cylindrical panels subjected to periodic nonuniform in-plane loads are reported. Since the applied in-plane load is nonuniform, initially the static part of the nonuniform in-plane loads are applied and the stresses (σx, σy and τxy) within the panel are evaluated by the solution of cylindrical panel membrane problem. Subsequently, superposing the stress distribution due to static and dynamic in-plane loads, the stress distributions within the panel are obtained. Using these stress distributions the governing equations of the problem are derived through Hamilton's variational principle based on higher-order shear deformation theory of elastic shell theory incorporating von Kármán-type nonlinear strain displacement relations. The governing partial differential equations are reduced into a set of ordinary differential equations (Mathieu-type of equations) by employing Galerkin's method. The instability boundaries of Mathieu equation corresponding to periodic solutions of period T and 2T are determined using Fourier series. Effect of various parameters like static and dynamic load factors, aspect ratio, thickness-to-radius ratio, shallowness ratio, linearly varying in-plane load, parabolic in-plane load and various boundary conditions on the instability regions are investigated.

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