Periodic response of nonlinear systems

A procedure is developed to determine approximate periodic solutions of autonomous and non-autonomous systems. The trignometric collocation method (TCM) is formalized to allow for the analysis of relatively small order systems directly in physical coordinates. The TCM is extended to large order systems by utilizing modal analysis in a component mode synthesis strategy. The procedure was coded and verified by several check cases. Numerical results for two small order mechanical systems and one large order rotor dynamic system are presented. The method allows for the possibility of approximating periodic responses for large order forced and self-excited nonlinear systems

Finite Element Methodology for Nonlinear Free and Harmonic Forced Vibrations of Beam and Plate Structures

The literature and experiments have shown that nonlinear vibrations produce significant effects in structural analysis, especially the frequency-amplitude-force relation and the analysis of strain. An analysis was developed to predict both the frequency-amplitude-force relation and strains of beam and plate structures. Two finite element methods were developed, namely, the iterative single-mode method (method I) and the multiple-mode method (method II). The harmonic force matrix was developed to analyze nonlinear forced vibrations. Nonlinear free vibration was a special case of the general forced vibration by setting the harmonic force matrix equal to zero. The harmonic force matrix represents the external applied force in matrix form, instead of a vector form, so that the analysis of nonlinear force vibrations can be performed as an eigenvalue problem. The study showed that the effect of midplane stretching due to large deflection is to increase the nonlinearity. However, the effects of inplane displacements and inertia (IDI) are to reduce nonlinearity. The concentrated force case yields a more severe response than the uniform distributed force case. For beams and plates with end supports restrained from axial movement (immovable case) only the hardening type nonlinearity is observed. For beams with large slenderness ratio (L/R $\le$ 100) with movable end supports, the increase in nonlinearity due to large deflection is partially compensated by a reduction in nonlinearity due to inplane displacement and inertia. This leads to a negligible hardening type nonlinearity, therefore, the small deflection linear solution can be employed. However, for beams with a small slenderness ratio (L/R = 20) and movable end supports, the softening type nonlinearity is found. The effect of the higher modes is more pronounced for the clamped supported beam than the simply supported one. The beam without inplane displacement and inertia (IDI) yields more effect of the higher modes than the one with inplane displacement and inertia. For beams, method I and method II converge into a true deflection shape, provided the number of modes for method II is high enough. Similarly, both method I and method II yield accurate strains provided the number of modes for method II is high enough

Nonlinear structural vibrations by the linear acceleration method

Numerical integration method for calculating dynamic response of nonlinear elastic structure

Hingeless rotor frequency response with unsteady inflow

Hingeless rotor frequency response calculations are obtained by applying a generalized harmonic balance to the elastic blade flapping equations. Nonuniform, unsteady induced flow effects are included by assuming a simple three-degree-of-freedom description of the rotor wake. Results obtained by using various models of elastic blade bending and induced flow are compared with experimental data obtained from a 7.5-ft diameter wind tunnel model at advance ratios from 0.0 to 0.6. It is shown that the blade elasticity and nonuniform, unsteady induced flow can have a significant effect on the transient response characteristics of rotor systems

Hypersonic panel flutter in a rarefied atmosphere

Panel flutter is a form of dynamic aeroelastic instability resulting from the interaction between motion of an aircraft structural panel and the aerodynamic loads exerted on that panel by air flowing past one of the faces. It differs from lifting surface flutter in the sense that it is not usually catastrophic, the panel's motion being limited by nonlinear membrane stresses produced by the transverse displacement. Above some critical airflow condition, the linear instability grows to a limit cycle . The present investigation studies panel flutter in an aerodynamic regime known as 'free molecule flow', wherein intermolecular collisions can be neglected and loads are caused by interactions between individual molecules and the bounding surface. After collision with the panel, molecules may be reflected specularly or reemitted in diffuse fashion. Two parameters characterize this process: the 'momentum accommodation coefficient', which is the fraction of the specularly reflected molecules; and the ratio between the panel temperature and that of the free airstream. This model is relevant to the case of hypersonic flight vehicles traveling at very high altitudes and especially for panels oriented parallel to the airstream or in the vehicle's lee. Under these conditions the aerodynamic shear stress turns out to be considerably larger than the surface pressures, and shear effects must be included in the model. This is accomplished by means of distributed longitudinal and bending loads. The former can cause the panel to buckle. In the example of a simply-supported panel, it turns out that the second mode of free vibration tends to dominate the flutter solution, which is carried out by a Galerkin analysis. Several parametric studies are presented. They include the effects of (1) temperature ratio; (2) momentum accommodation coefficient; (3) spring parameters, which are associated with how the panel is connected to adjacent structures; (4) a parameter which relates compressive end load to its value which would cause classical column buckling; (5) a parameter proportional to the pressure differential between the front and back faces; and (6) initial curvature. The research is completed by an investigation into the possibility of accounting for molecular collisions, which proves to be infeasible given the speeds of current mainframe supercomputers

Fast methods to numerically integrate the Reynolds equation for gas fluid films

The alternating direction implicit (ADI) method is adopted, modified, and applied to the Reynolds equation for thin, gas fluid films. An efficient code is developed to predict both the steady-state and dynamic performance of an aerodynamic journal bearing. An alternative approach is shown for hybrid journal gas bearings by using Liebmann's iterative solution (LIS) for elliptic partial differential equations. The results are compared with known design criteria from experimental data. The developed methods show good accuracy and very short computer running time in comparison with methods based on an inverting of a matrix. The computer codes need a small amount of memory and can be run on either personal computers or on mainframe systems

Application of the Finite Element Method to Rotary Wing Aeroelasticity

A finite element method for the spatial discretization of the dynamic equations of equilibrium governing rotary-wing aeroelastic problems is presented. Formulation of the finite element equations is based on weighted Galerkin residuals. This Galerkin finite element method reduces algebraic manipulative labor significantly, when compared to the application of the global Galerkin method in similar problems. The coupled flap-lag aeroelastic stability boundaries of hingeless helicopter rotor blades in hover are calculated. The linearized dynamic equations are reduced to the standard eigenvalue problem from which the aeroelastic stability boundaries are obtained. The convergence properties of the Galerkin finite element method are studied numerically by refining the discretization process. Results indicate that four or five elements suffice to capture the dynamics of the blade with the same accuracy as the global Galerkin method

Nonlinear rotordynamics analysis

Effective analysis tools were developed for predicting the nonlinear rotordynamic behavior of the Space Shuttle Main Engine (SSME) turbopumps under steady and transient operating conditions. Using these methods, preliminary parametric studies were conducted on both generic and actual HPOTP (high pressure oxygen turbopump) models. In particular, a novel modified harmonic balance/alternating Fourier transform (HB/AFT) method was developed and used to conduct a preliminary study of the effects of fluid, bearing and seal forces on the unbalanced response of a multi-disk rotor in the presence of bearing clearances. The method makes it possible to determine periodic, sub-, super-synchronous and chaotic responses of a rotor system. The method also yields information about the stability of the obtained response, thus allowing bifurcation analyses. This provides a more effective capability for predicting the response under transient conditions by searching in proximity of resonance peaks. Preliminary results were also obtained for the nonlinear transient response of an actual HPOTP model using an efficient, newly developed numerical method based on convolution integration. Currently, the HB/AFT is being extended for determining the aperiodic response of nonlinear systems. Initial results show the method to be promising

The effect of general imperfections on the buckling of cylindrical shells

An experimental and theoretical investigation of the effect of general imperfections on the buckling load of a circular cylindrical shell under axial compression was carried out. A non-contact probe has been used to make complete imperfection surveys on electroformed copper shells before and during the loading process up to the buckling load. The data recording process has been fully automated and the data reduction was done on an IBM 7094. Three-dimensional plots were obtained of the measured initial imperfection surfaces and of the growth of these imperfections under increasing axial load. The modal components of the measured imperfection surfaces were also obtained. The theoretical solution located the limit points of the post-buckled states. A simplified imperfection model was used consisting of one axisymmetric and one asymmetric component. For global buckling the correlation between the theoretical buckling loads and the experimental values was found to be good

Component mode synthesis and large deflection vibration of complex structures. Volume 3: Multiple-mode nonlinear free and forced vibrations of beams using finite element method

Multiple-mode nonlinear forced vibration of a beam was analyzed by the finite element method. Inplane (longitudinal) displacement and inertia (IDI) are considered in the formulation. By combining the finite element method and nonlinear theory, more realistic models of structural response are obtained more easily and faster