Dynamic Response of Spinning Blades Subjected to Gyroscopic Motion

1994 ◽  
Vol 116 (1) ◽  
pp. 6-15 ◽  
Author(s):  
T. H. Young ◽  
G. T. Liou
Keyword(s):  
Differential Equations ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Parametric Instability ◽  
Spin Axis ◽  
System Response ◽  
First Order ◽  
System Equations ◽  
Element Technique ◽  
Gyroscopic Motion

This paper presents an investigation into the vibration and stability of a blade spinning with respect to a nonfixed axis. Due to the motion of the spin axis, parametric instability of the blade may occur in certain situations. In this work, the discretized equations of motion are first formulated by the finite element technique. Then the system equations are transformed, by a special modal analysis procedure, into independent sets of first-order simultaneous differential equations. Each set of differential equations is solved analytically by the method of multiple scales if the precessional speed of the spin axis is assumed to be small compared to the spin rate of the blade, yielding the system response and the expressions for the boundaries of the unstable regions. Finally, the effects of system parameters on the changes in these boundaries are studied numerically.

Coriolis Effect on the Vibration of a Cantilever Plate With Time-Varying Rotating Speed

1992 ◽  
Vol 114 (2) ◽  
pp. 232-241 ◽  
Author(s):  
T. H. Young ◽  
G. T. Liou
Keyword(s):  
Differential Equations ◽  
Multiple Scales ◽  
Parametric Instability ◽  
System Response ◽  
Coriolis Effect ◽  
Time Varying ◽  
Cantilever Plate ◽  
Rotating Speed ◽  
First Order ◽  
System Equation

A method for investigating the Corilois effect on the vibration of a cantilever plate rotating at a time-varying speed is presented in this paper. Due to this time-dependent speed, parametric instability occurs in the system. Furthermore, owing to the existence of the Coriolis force, the system equation is transformed, by a special modal analysis procedure, into independent sets of first-order simultaneous differential equations. This set of simultaneous differential equations is solved by the method of multiple scales, yielding the system response and the expressions for the boundaries of the unstable regions. Finally, the Coriolis effect on the changes in the boundaries of the unstable regions is investigated numerically.

scholarly journals Nonlinear Response of Cantilever Beams to Combination and Subcombination Resonances

1998 ◽  
Vol 5 (5-6) ◽  
pp. 277-288 ◽  
Author(s):  
Ali H. Nayfeh ◽  
Haider N. Arafat
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Virtual Work ◽  
Equations Of Motion ◽  
Nonlinear Ordinary Differential Equations ◽  
Response Curves ◽  
First Order ◽  
Aspect Ratios

The nonlinear planar response of cantilever metallic beams to combination parametric and external subcombination resonances is investigated, taking into account the effects of cubic geometric and inertia nonlinearities. The beams considered here are assumed to have large length-to-width aspect ratios and thin rectangular cross sections. Hence, the effects of shear deformations and rotatory inertia are neglected. For the case of combination parametric resonance, a two-mode Galerkin discretization along with Hamilton’s extended principle is used to obtain two second-order nonlinear ordinary-differential equations of motion and associated boundary conditions. Then, the method of multiple scales is applied to obtain a set of four first-order nonlinear ordinary-differential equations governing the modulation of the amplitudes and phases of the two excited modes. For the case of subcombination resonance, the method of multiple scales is applied directly to the Lagrangian and virtual-work term. Then using Hamilton’s extended principle, we obtain a set of four first-order nonlinear ordinary-differential equations governing the amplitudes and phases of the two excited modes. In both cases, the modulation equations are used to generate frequency- and force-response curves. We found that the trivial solution exhibits a jump as it undergoes a subcritical pitchfork bifurcation. Similarly, the nontrivial solutions also exhibit jumps as they undergo saddle-node bifurcations.

Dynamic Stability of Axially Moving Plates With Periodically Varying Speeds Under Partially Distributed Edge Tensions

2012 ◽  
Author(s):  
T. H. Young ◽  
S. J. Huang ◽  
A. C. Liu
Keyword(s):  
Dynamic Stability ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Parametric Instability ◽  
Plane Elasticity ◽  
Moving Plate ◽  
System Equations ◽  
Axially Moving ◽  
Axially Moving Web ◽  
Moving Speed

This paper investigates the dynamic stability of an axially moving web which translates with periodically varying speeds and is subjected to partially distributed tensions on two opposite edges. The web is modeled as a rectangular plate simply supported at two opposite edges where the tension is applied, and free at the other two edges. The plate is assumed to possess internal damping, which obeys the Kelvin-Voigt model. The moving speed of the plate is expressed as the sum of a constant speed and a periodical perturbation with a zero mean. Due to the periodically varying speed of the moving plate, terms with time-dependent coefficients appear in the equations of motion, which may bring about parametric instability under certain situations. First, the in-plane stresses of the plate due to the partially distributed edge tensions is determined exactly by the theory of plane elasticity. Then, the dependence on the spatial coordinates in the equations of motion is eliminated by the Galerkin method, which results in a set of discretized system equations in time. Finally, the method of multiple scales is utilized to solve this set of system equations analytically if the periodical perturbation of the moving speed is much smaller as compared with the average speed of the plate, from which the stability boundaries of the moving plate are obtained. Numerical results reveal that only combination resonances of the sum-type appear between modes having the same symmetry class in the transverse direction. Unstable regions of main resonances are generally larger than those of sum-type resonances.

Spatial Transient Analysis of Inertia-Variant Flexible Mechanical Systems

1984 ◽  
Vol 106 (2) ◽  
pp. 172-178 ◽  
Author(s):  
A. A. Shabana ◽  
R. A. Wehage
Keyword(s):  
Differential Equations ◽  
Transient Analysis ◽  
Large Scale ◽  
Equations Of Motion ◽  
Integration Algorithm ◽  
System Equations ◽  
Element Technique ◽  
Coordinate Partitioning ◽  
Mode Technique ◽  
Generalized Coordinate

An analytical method for transient dynamic simulation of large-scale inertia-variant spatial mechanical and structural systems is presented. Multibody systems consisting of interconnected rigid and flexible substructures which may undergo large angular rotations are analyzed. A finite element technique is used to characterize the elastic properties of deformable substructures. A component mode technique is then employed to eliminate insignificant substructure modes. Nonlinear holonomic constraint equations are used to define joints between different substructures. The system equations of motion are written in terms of a mixed set of modal and physical coordinates. A generalized coordinate partitioning technique is then employed to eliminate redundant differential equations. An implicit-explicit numerical integration algorithm solves the remaining set of differential equations and the approximate physical system state is recovered. The transient analysis of a spatial vehicle with flexible chassis is presented to demonstrate the method.

Vibration Response and Parametric Instability in Beams With Combined Quadratic and Cubic Material Nonlinearities

1995 ◽  
Author(s):  
David Chelidze ◽  
Kambiz Farhang ◽  
Tyler J. Selstad
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Multiple Scales ◽  
Nonlinear Differential Equations ◽  
Equations Of Motion ◽  
Parametric Instability ◽  
Response Amplitude ◽  
Nonlinear Material ◽  
Mode Shapes ◽  
Cross Sectional

Abstract Parametric stability in beams with combined quadratic and cubic material nonlinearities is examined. A general mathematical model is developed for parametrically excited beams accounting for their nonlinear material characteristic. Second- and forth-order nonlinear differential equations are found to govern the axial and transverse motions, respectively. Expansions for displacements are assumed in terms of the linear undamped free-oscillation modes. Boundary conditions are applied to the expansions for displacements to determine the mode shapes. Multiplying the equations of motion by the corresponding shape functions, accounting for their orthogonal properties, and integrating over the beam length, a set of coupled nonlinear differential equations in the time-dependent modal coefficients is obtained. Utilizing the method of multiple scales, frequency response as well as response versus excitation amplitude are obtained for two beams of different cross sectional areas. Results are presented for three boundary conditions. It is found that, qualitatively, the response is similar for all the boundary conditions. Quantitative comparison of the cases considered indicate that the highest response amplitude occurs for the cantilever beam with the end mass. The bifurcation points for simply supported beam occur at lower excitation parameter value. It is apparent that more slender columns have larger response amplitude.

Model Based Temperature Control in RTP Yielding ±0.1 °C accuracy on A 1000 °C, 2 second, 100 °C/s Spike Anneal

1998 ◽  
Vol 525 ◽  
Author(s):  
Peter Vandenabeele ◽  
Wayne Renken
Keyword(s):  
Differential Equations ◽  
Temperature Control ◽  
Control Method ◽  
Nonlinear Differential Equations ◽  
System Response ◽  
Model Based Control ◽  
First Order ◽  
Model Based ◽  
Measured System ◽  
Spike Anneal

ABSTRACTA Model Based Control method is presented for accurate control of RTP systems. The model uses 4 states: lamp filament temperature, wafer temperature, quartz temperature and TC temperature. A set of 4 first order, nonlinear differential equations describes the model. Feedback is achieved by updating the model, based on a comparison between actual (measured) system response and modeled system response.

TOPOLOGICAL DEFECTS AND THE TRIAL ORBIT METHOD

2002 ◽  
Vol 17 (29) ◽  
pp. 1945-1953 ◽  
Author(s):  
D. BAZEIA ◽  
W. FREIRE ◽  
L. LOSANO ◽  
R. F. RIBEIRO
Keyword(s):  
Differential Equations ◽  
Equations Of Motion ◽  
Topological Defects ◽  
Scalar Fields ◽  
Explicit Solutions ◽  
Orbit Method ◽  
First Order ◽  
Real Scalar ◽  
Second Order Equations ◽  
First Order Differential Equations

We deal with the presence of topological defects in models for two real scalar fields. We comment on defects hosting topological defects and search for explicit defect solutions using the trial orbit method. As we know, under certain circumstances the second-order equations of motion can be solved by solutions of first-order differential equations. In this case we show that the trial orbit method can be used very efficiently to obtain explicit solutions.

The Influence of Nonlinear Boundary Conditions on the Nonplanar Autoparametric Responses of an Inextensible Beam

2001 ◽  
Author(s):  
Haider N. Arafat ◽  
Ali H. Nayfeh
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Nonlinear Boundary ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Primary Resonance ◽  
Nonlinear Behavior ◽  
Nonlinear Boundary Conditions ◽  
Nonlinear Springs ◽  
Out Of Plane

Abstract The nonplanar responses of a beam clamped at one end and restrained by nonlinear springs at the other end is investigated under a primary resonance base excitation. The beam’s geometry and the springs’ linear stiffnesses are such that the system possesses a one-to-one autoparametric resonance between the nth in-plane and out-of-plane modes. The beam is modeled using Euler-Bernoulli theory and includes cubic geometric and inertia nonlinearities. The objective is to assess the influence of the nonlinear boundary conditions on the beam’s oscillations. To this end, the method of multiple scales is directly applied to the integral-partial-differential equations of motion and associated boundary conditions. The result is a set of four nonlinear ordinary-differential equations that govern the slow dynamics of the system. Solutions of these modulation equations are then used to characterize the system’s nonlinear behavior.

Internal Resonance and Nonlinear Vibrations of Eccentric Rotating Composite Laminated Circular Cylindrical Shell

2019 ◽  
Author(s):  
Tao Liu ◽  
Wei Zhang ◽  
Yan Zheng ◽  
Yufei Zhang
Keyword(s):  
Cylindrical Shell ◽  
Differential Equations ◽  
Internal Resonance ◽  
Nonlinear Vibrations ◽  
Multiple Scales ◽  
Circular Cylindrical Shell ◽  
Equations Of Motion ◽  
Nonlinear Partial Differential Equations ◽  
Shear Deformation Theory ◽  
Internal Resonances

Abstract This paper is focused on the internal resonances and nonlinear vibrations of an eccentric rotating composite laminated circular cylindrical shell subjected to the lateral excitation and the parametric excitation. Based on Love thin shear deformation theory, the nonlinear partial differential equations of motion for the eccentric rotating composite laminated circular cylindrical shell are established by Hamilton’s principle, which are derived into a set of coupled nonlinear ordinary differential equations by the Galerkin discretization. The excitation conditions of the internal resonance is found through the Campbell diagram, and the effects of eccentricity ratio and geometric papameters on the internal resonance of the eccentric rotating system are studied. Then, the method of multiple scales is employed to obtain the four-dimensional nonlinear averaged equations in the case of 1:2 internal resonance and principal parametric resonance-1/2 subharmonic resonance. Finally, we study the nonlinear vibrations of the eccentric rotating composite laminated circular cylindrical shell systems.

Dynamics of a Basketball Rolling Around the Rim

2005 ◽  
Vol 128 (2) ◽  
pp. 359-364
Author(s):  
C. Q. Liu ◽  
Fang Li ◽  
R. L. Huston
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Initial Conditions ◽  
Equations Of Motion ◽  
Dynamical Equations ◽  
First Order ◽  
Coaching Strategies ◽  
Ball Motion

Governing dynamical equations of motion for a basketball rolling on the rim of a basket are developed and presented. These equations form a system of five first-order, ordinary differential equations. Given suitable initial conditions, these equations are readily integrated numerically. The results of these integrations predict the success (into the basket) or failure (off the outside of the rim) of the basketball shot. A series of examples are presented. The examples show that minor changes in the initial conditions can produce major changes in the subsequent ball motion. Shooting and coaching strategies are recommended.

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