scholarly journals On the Existence of Normal Modes of Damped Discrete-Continuous Systems

1998 ◽  
Vol 65 (4) ◽  
pp. 980-989 ◽  
Author(s):  
H. T. Banks ◽  
Zheng-Hua Luo ◽  
L. A. Bergman ◽  
D. J. Inman
Keyword(s):  
Hilbert Space ◽  
Arbitrary Dimension ◽  
Normal Modes ◽  
Continuous Systems ◽  
Motion Equations ◽  
Elastic Supports ◽  
Sesquilinear Form ◽  
Abstract Evolution Equation ◽  
Concentrated Masses ◽  
Form Approach

In this paper we investigate a class of combined discrete-continuous mechanical systems consisting of a continuous elastic structure and a finite number of concentrated masses, elastic supports, and linear oscillators of arbitrary dimension. After the motion equations for such combined systems are derived, they are formulated as an abstract evolution equation on an appropriately defined Hilbert space. Our main objective is to ascertain conditions under which the combined systems have classical normal modes. Using the sesquilinear form approach, we show that unless some matching conditions are satisfied, the combined systems cannot have normal modes even if Kelvin-Voigt damping is considered.

scholarly journals On Dirichlet's boundary value problem for some formally hypoelliptic differntial operators

1988 ◽  
Vol 44 (3) ◽  
pp. 324-349 ◽  
Author(s):  
Niels Jacob
Keyword(s):  
Hilbert Space ◽  
Dirichlet Problem ◽  
Differential Operator ◽  
Boundary Value Problem ◽  
Differential Operators ◽  
Boundary Value ◽  
Divergence Form ◽  
Alternative Theorem ◽  
Sesquilinear Form ◽  
Homogeneous Dirichlet Problem

AbstractFor a class of formally hypoelliptic differential operators in divergence form we prove a generalized Gårding inequality. Using this inequality and further properties of the sesquilinear form generated by the differential operator a generalized homogeneous Dirichlet problem is treated in a suitable Hilbert space. In particular Fredholm's alternative theorem is proved to be valid.

Nonlinear Normal Modes of a Continuous System With Quadratic Nonlinearities

1995 ◽  
Vol 117 (2) ◽  
pp. 199-205 ◽  
Author(s):  
A. H. Nayfeh ◽  
S. A. Nayfeh
Keyword(s):  
Boundary Conditions ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Normal Modes ◽  
Continuous System ◽  
Nonlinear Normal Modes ◽  
Continuous Systems ◽  
Quadratic Nonlinearities ◽  
Nonlinear Normal ◽  
Quadratic And Cubic Nonlinearities

We use two approaches to determine the nonlinear modes and natural frequencies of a simply supported Euler-Bernoulli beam resting on an elastic foundation with distributed quadratic and cubic nonlinearities. In the first approach, we use the method of multiple scales to treat the governing partial-differential equation and boundary conditions directly. In the second approach, we use a Galerkin procedure to discretize the system and then determine the normal modes from the discretized equations by using the method of multiple scales and the invariant manifold approach. Whereas one- and two-mode discretizations produce erroneous results for continuous systems with quadratic and cubic nonlinearities, all methods, in the present case, produce the same results because the discretization is carried out by using a complete set of basis functions that satisfy the boundary conditions.

scholarly journals Frames of subspaces in Hilbert spaces with W-metrics

2015 ◽  
Vol 23 (2) ◽  
pp. 5-22
Author(s):  
Primitivo Acosta-Humánez ◽  
Kevin Esmeral ◽  
Osmin Ferrer
Keyword(s):  
Hilbert Space ◽  
Lebesgue Measure ◽  
Hilbert Spaces ◽  
Sesquilinear Form

Abstract In this paper we start considering a sesquilinear form 〈W·,·〉 defined over a Hilbert space (ℌ,〈·,·〉) where W is bounded (W* = W ∈ Ɓ(ℌ)) and ker W = {0}. We study the dynamic of frame of subspaces over the completion of (ℌ, 〈W·,·〉) which is denoted by ℌW and is called Hilbert space with W-metric or simply W-space. The sense of dynamics studied here refers to the behavior of frame of subspaces comparing ℌW with ℌ as well ℌ with ℌW. Furthermore, we show that for any Hilbert space with W-metric ℌW, being 0 an element of the spectrum of W (0 ∈σ(W)), has a decomposition ℌW = ⊕n∈ℕ∪{∞}ℌWψn where ℌWψn ≃ L2(σ(W), χdμn(χ)) for all n ∈ ℕ ∪ {∞}, L2 denotes a Hilbert space square integrable and μ a Lebesgue measure. Finally, the case when W is unbounded also considered.

Structural Dynamics with Parameter Uncertainties

1987 ◽  
Vol 40 (3) ◽  
pp. 309-328 ◽  
Author(s):  
R. A. Ibrahim
Keyword(s):  
Structural Dynamics ◽  
Structural Parameters ◽  
Normal Modes ◽  
Forced Response ◽  
Experimental Investigations ◽  
Continuous Systems ◽  
Parameter Uncertainties ◽  
Response Characteristics ◽  
Random Eigenvalues ◽  
The Impact

The treatment of structural parameters as random variables has been the subject of structural dynamicists and designers for many years. Several problems have been involved during the last few decades and resulted in new theorems and interesting phenomena. This paper reviews a number of topics pertaining to structural dynamics with parameter uncertainties. These include direct problems such as random eigenvalues and random responses of discrete and continuous systems. The impact of these problems on related areas of interest such as sensitivity of structural performance to parameter variations, design optimization, and reliability analysis is also addressed. The paper includes the results of experimental investigations, the phenomenon of normal modes localization, and the effect of mistuning of turbomachinery blades on their flutter and forced response characteristics.

Vibrations

2017 ◽  
pp. 44-93
Keyword(s):  
Degrees Of Freedom ◽  
Normal Modes ◽  
Mode Shapes ◽  
Continuous Systems ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
Mdof Systems ◽  
Parametric Coupling ◽  
Bending Modes ◽  
Stability Concepts

Vibration concepts are reviewed. Single degree-of-freedom vibration (SDOF) are analyzed. Subsequently, the analysis is extended to two degrees-of-freedom (2DOF) systems and coupling in a 2DOF system. The analysis of parametric coupling is introduced. Two sections on energy flow and the modeling of damping follow. Normal modes and mode shapes for systems with multiple degrees-of-freedom (MDOF) will then be considered. By generalizing MDOF systems to continuous systems, we can analyze bending modes in plates. Experimental modal analysis is introduced to prepare the reader for later application of this technique to full-scale operational gates. The second section of this chapter reviews fundamental concepts of fluid-structure systems with resonance. The chapter concludes with a short discussion of stability concepts.

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