Multiple-scale analysis

About: Multiple-scale analysis is a research topic. Over the lifetime, 1360 publications have been published within this topic receiving 27530 citations.

Compressible-flow channel flutter

Abstract: The effect of fluid compressibility on the dynamic stability of a two-dimensional flow through a flexible channel is analysed. The compressibility parameter Q is defined as the ratio of a reference elastic wave speed of the wall to the local speed of sound. As the fluid speed increases, the walls become dynamically unstable at the critical fluid speed S0 and start to flutter at critical frequency ω0. The effect of three other dimensionless parameters on the critical condition is also analysed. These are the ratio γ of fluid damping to wall damping, the ratio B of wall bending resistance to elastance, and the ratio μ of wall to fluid mass. Nonlinear analysis using the Poincare–Lindstedt method shows stiffening at supercritical speeds. Further stability analysis using the method of multiple scales shows that the amplitude growth is finite and the nonlinear fluttering state is stable. Both symmetric and antisymmetric modes of oscillation are analysed. A frictionless system is found to be a singular case in the nonlinear theory. The hydraulic approximation employed in the analysis is shown to be a particular limiting form of the corresponding Orr–Sommerfeld system.

Free damped nonlinear vibrations of a viscoelastic plate under two-to-one internal resonance

Abstract: Nonlinear free damped vibrations of a rectangular plate described by three nonlinear differential equations are considered when the plate is being under the conditions of the internal resonance two-to-one. Viscous properties of the system are described by the Riemann-Liouville fractional derivative. The functions of the in-plane and out-of-plane displacements are determined in terms of eigenfunctions of linear vibrations with the further utilization of the method of multiple scales, in so doing the fractional derivative is represented as a fractional power of the differentiation operator. The time-dependence of the amplitudes in the form of incomplete integrals of the first kind is obtained. Using the constructed solutions, the influence of viscosity on the energy exchange mechanism is analyzed which is intrinsic to free vibrations of different structures being under the conditions of the internal resonance. It is shown that each mode is characterized by its damping coefficient which is connected with the natural frequency of this mode by the exponential relationship with a negative fractional exponent.

Simply supported elastic beams under parametric excitation

Abstract: In this paper, the nonlinear characteristics of the parametric resonance of simply supported elastic beams are investigated. Considering a geometrically exact formulation for unsharable and inextensible elastic beams subject to support motions, the integral-partial-differential equation of motion is obtained. The third-order perturbation of the equation of motion is then determined in a form amenable to an asymptotic treatment. The method of multiple scales is used to obtain the equations that describe the modulation of the amplitude and phase of parametric-resonance motions. The stability and bifurcations of the system are investigated considering, in particular, the frequency-response function. Furthermore, experimental results are shown to confirm the theoretically predicted stability and bifurcations.

An analytical study of time-delayed control of friction-induced vibrations in a system with a dynamic friction model

Abstract: We investigate the control of friction-induced vibrations in a system with a dynamic friction model which accounts for hysteresis in the friction characteristics. Linear time-delayed position feedback applied in a direction normal to the contacting surfaces has been employed for the purpose. Analysis shows that the uncontrolled system loses stability via. a subcritical Hopf bifurcation making it prone to large amplitude vibrations near the stability boundary. Our results show that the controller achieves the dual objective of quenching the vibrations as well as changing the nature of the bifurcation from subcritical to supercritical. Consequently, the controlled system is globally stable in the linearly stable region and yields small amplitude vibrations if the stability boundary is crossed due to changes in operating conditions or system parameters. Criticality curve separating regions on the stability surface corresponding to subcritical and supercritical bifurcations is obtained analytically using the method of multiple scales (MMS). We have also identified a set of control parameters for which the system is stable for lower and higher relative velocities but vibrates for the intermediate ones. However, the bifurcation is always supercritical for these parameters resulting in low amplitude vibrations only.

Vibrations of a Stretched Beam with Non-Ideal Boundary Conditions

Abstract: A simply supported Euler-Bernoulli beam with immovable end conditions is considered. The concept of non-ideal boundary conditions is applied to the beam problem. In accordance, the boundaries are assumed to allow small deflections. Approximate analytical solution of the problem is found using the method of multiple scales, a perturbation technique.