Search Results (4)

The harmonic balance method (HBM) is a widely used method for determining the forced response of non-linear systems such as bladed disks. This paper focuses on analyzing the sensitivity of this method to key computational parameters and its robustness. HBM and HBM coupled with pseudo arc length continuation are used in this paper to solve the equation of motion of a test case. The pseudo arc length continuation is necessary because when intermittent contact occurs, natural continuation cannot guarantee solver convergence. Intermittent contact, in addition to turning points, introduces further problems, which are caused by an infinite sequence of decaying, but not zero, Fourier coefficients. This results in the need to oversample the non-linear force time signal to avoid convergence problems. The computational parameters investigated in this paper are the samples per period, which determine the number of points in which the time signal is discretized, and the harmonic truncation order. In addition, the connection of contact parameters, such as friction and contact stiffness, with computational parameters is analyzed. This study shows that the number of time samples per period is the most limiting parameter when intermittent contact occurs; whereas, in the absence of intermittent contact convergence, problems can be avoided with a reasonable number of time points. Poor discretization of the signal leads to a bad computation of Fourier coefficients and thus a lack of convergence. Sensitivity analysis shows that the samples per period depend on the contact parameters, especially normal stiffness. To ensure the solver robustness, it is important to set the computation parameters appropriately to ensure the convergence of the solver while avoiding unnecessary computation effort. Full article

This paper aims to investigate the nonlinear transition to turbulence in generalized 3D Kolmogorov flow. The difference between this and classical Kolmogorov flow is that the forcing term in the x direction $sin\left(y\right)$ is replaced with $sin\left(y\right)cos\left(z\right)$. This drastically complicates the problem. First, a stability analysis is performed by deriving the analog of the Orr–Sommerfeld equation. It is shown that for infinite stretching, the flow is stable, contrary to classical forcing. Next, a neutral curve is constructed, and the stability of the main solution is analyzed. It is shown that for the cubic domain, the main solution is linearly stable, at least for $0<R\le 100$. Next, we turn our attention to the numerical investigation of the solutions in the cubic domain. The main feature of this problem is that it is spatially periodic, allowing one to apply a relatively simple pseudo-spectral numerical method for its investigation. We apply the method of deflation to find distinct solutions in the discrete system and the method of arc length continuation to trace the bifurcation solution branches. Such solutions are called disconnected solutions if these are solutions not connected to the branch of the main solution. We investigate the influence of disconnected solutions on the dynamics of the system. It is demonstrated that when disconnected solutions are formed, the nonlinear transition to turbulence is possible, and dangerous initial conditions are these disconnected solutions. Full article

Significant advances in the field of composite structures continue to be made on a variety of fronts, including theoretical studies based on advances in structural theory kinematics and computer models of structural elements employing advanced theories and unique formulations. Plate vibration is a persistently interesting subject owing to its wider usage as a structural component in the industry. The current study was carried out using the C^o continuous eight-noded quadrilateral shear-flexible element having five nodal degrees of freedom, which is ground on first-order shear deformation theory (FSDT). For small strain and sufficiently large deformation, the geometric nonlinearity is integrated using the Von Kármán assumption. The governing equations in the time domain are solved employing the modified shooting technique along with an arc-length and pseudo-arc-length continuation strategy. This work explored the effect of fiber angle on the steady-state nonlinear forced vibration response. To explain hardening nonlinearity, the strain and stress fluctuation throughout the thickness for a rectangular laminated composite plate is determined. The cyclic fluctuation of the steady-state nonlinear normal stress during a time period at the centre of the top/bottom surfaces is also provided at the forcing frequency ratio of peak amplitude in a nonlinear response. Because of the variation in restoring forces, the frequency spectra for all fiber angle orientations show significantly enhanced harmonic participation in addition to the fundamental harmonic. Full article

The nonlinear steady state large amplitude forced vibration response of a laminated composite annular sector plate is presented. The nonlinear governing equation of motion of the laminated composite annular sector plate has been obtained using kinematics of first-order shear deformation theory (FSDT) and employing Hamilton’s principle. The governing equations of motion have been solved in a time domain using a modified shooting method and arc-length/pseudo-arc length continuation technique. The influence of the boundary condition, sector angle, and annularity ratio on the linear as well as nonlinear steady state forced vibration response has been investigated. The strain/stress variation across the thickness of the annular sector plate is presented to explain the reason for a decrease/increase in hardening nonlinear behaviour. The periodic variation of the non-linear steady state stress has also been obtained to throw light into the factors influencing the unequal stress half cycles and multiple cyclic stress reversals, which is detrimental to the fatigue design of laminated composite annular sectorial plates. The frequency spectra of the steady state stress reveals large even and odd higher harmonic contributions for different cases due to changes in the restoring force dynamics. The modal interaction/exchange during a cycle is demonstrated using a deformed configuration of the laminated annular sector plate. Full article