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.

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.

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.

Powered-Flight Trajectories of Rockets Under Constant Transverse Thrust

1963 ◽  
Vol 30 (4) ◽  
pp. 559-561 ◽  
Author(s):  
Chong-Hung Zee
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Series Expansion ◽  
Practical Problem ◽  
Nonlinear Differential Equations ◽  
Equations Of Motion ◽  
Variable Time ◽  
Truncation Errors ◽  
Independent Variable ◽  
Special Case

The second-order nonlinear differential equations of motion in the case of a rocket in drag-free powered-flight under constant transverse (normal to the focal radius) thrust are solved by series expansion developed to the seventh power of the independent variable “time.” The coefficients of the powers of time are in terms of given boundary conditions. The truncation errors of the series are estimated, hence the accuracy for any practical problem based on the analysis presented in this paper could be well established. The case of constant transverse thrust acceleration, which may be conceived as a special case of the present analysis, is also solved.

On nonlinear perturbation analysis of a structure carrying a circular cylindrical liquid tank under horizontal excitation

2018 ◽  
Vol 25 (5) ◽  
pp. 1058-1079 ◽  
Author(s):  
N. K. A. Attari ◽  
F. R. Rofooei ◽  
Z. Waezi
Keyword(s):  
Differential Equations ◽  
Perturbation Analysis ◽  
Multiple Scales ◽  
Nonlinear Differential Equations ◽  
Equations Of Motion ◽  
Structural Response ◽  
Single Mode ◽  
Excitation Amplitude ◽  
Combined System ◽  
The Stability

The lateral response of a single degree of freedom structural system containing a rigid circular cylindrical liquid tank under harmonic and earthquake excitations at a 1:2 autoparametric resonance case is considered. The governing nonlinear differential equations of motion for the combined system are solved by means of a multiple scales method considering the first three liquid sloshing modes (1,1), (0,1), and (2,1), under horizontal excitation. The fixed points of the gyroscopic type of governing differential equations are determined and their stability is investigated employing the perturbation method. The obtained results reveal an increase in the stability region for a single-mode response with respect to the excitation amplitude. The saturation phenomenon is observed in the decoupled modes of the system; however, the structural mode and the first anti-symmetric mode of liquid are a combination of the saturated mode and another mode whose scale factor is crucial for the structural response. The results of perturbation analysis are in good agreement with results obtained from numerical methods.

scholarly journals Nonlinear Differential Equations Modeling the Antarctic Circumpolar Current

2021 ◽  
Vol 23 (4) ◽  
Author(s):  
Jifeng Chu ◽  
Kateryna Marynets
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Topological Degree ◽  
Antarctic Circumpolar Current ◽  
Fixed Point Theorems ◽  
Nonlinear Differential Equations ◽  
Existence Results ◽  
Circumpolar Current ◽  
The Antarctic

AbstractThe aim of this paper is to study one class of nonlinear differential equations, which model the Antarctic circumpolar current. We prove the existence results for such equations related to the geophysical relevant boundary conditions. First, based on the weighted eigenvalues and the theory of topological degree, we study the semilinear case. Secondly, the existence results for the sublinear and superlinear cases are proved by fixed point theorems.

Approximate Formulas for Cylindrical Shell Free Vibration Based on Vlasov’s and Enhanced Vlasov’s Semi-Momentless Theory

2018 ◽  
Author(s):  
Igor Orynyak ◽  
Yaroslav Dubyk
Keyword(s):  
Cylindrical Shell ◽  
Boundary Conditions ◽  
Circular Cylindrical Shell ◽  
Equations Of Motion ◽  
Middle Surface ◽  
Axial Direction ◽  
Mode Shapes ◽  
Natural Mode ◽  
Transverse Deflection ◽  
Approximate Formulas

Simple approximate formulas for the natural frequencies of circular cylindrical shells are presented for modes in which transverse deflection dominates. Based on the Donnell-Mushtari thin shell theory the equations of motion of the circular cylindrical shell are introduced, using Vlasov assumptions and Fourier series for the circumferential direction, an exact solution in the axial direction is obtained. To improve the results assumptions of Vlasov’s semimomentless theory are enhanced, i.e. we have used only the hypothesis of middle surface inextensibility to obtain a solution in axial direction. Nonlinear characteristic equations and natural mode shapes, are derived for all type of boundary conditions. Good agreement with experimental data and FEM is shown and advantage over the existing formulas for a variety of boundary conditions is presented.

scholarly journals Repeated alternating loading of a elastoplastic three-layer plate in a temperature field

2021 ◽  
Vol 21 (1) ◽  
pp. 60-75
Author(s):  
Eduard I. Starovoitov ◽  
◽  
Denis V. Leonenko ◽  
Keyword(s):  
Temperature Field ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Nonlinear Differential Equations ◽  
Boundary Value ◽  
Plastic Region ◽  
Local Load ◽  
Natural State ◽  
Elastic Plastic

Axisymmetric deformation of a three-layer circular plate under repeated alternating loading from the plastic region by a local load is considered. To describe kinematics of asymmetrical on the thickness of the plate pack is adopted the hypothesis of a broken line. In a thin elastic-plastic load-bearing layers are used the hypothesis of Kirchhoff. A non-linearly elastic relatively thick filler is incompressible in thickness. It is taken to be a hypothesis of Tymoshenko regarding the straightness and the incompressibility of the deformed normals with linear approximation of the displacements through the thickness layer. The work of the filler in the tangential direction is taken into account. The physical relations of stress-strain relations correspond to the theory of small elastic-plastic deformations. The effect of heat flow is taken into account. The temperature field in the plate was calculated by the formula obtained by averaging the thermophysical parameters over the thickness of the package. The system of differential equations of equilibrium under loading of the plate from the natural state is obtained by the Lagrange variational method. Boundary conditions on the plate contour are formulated. The solution of the corresponding boundary value problem is reduced to finding the three desired functions: deflection, shear and radial displacement of the shear surface of the filler. A non-uniform system of ordinary nonlinear differential equations is written for these functions. Its analytical iterative solution is obtained in Bessel functions by the method of elastic solutions of Ilyushin. In case of repeated alternating loading of the plate, the solution of the boundary value problem is constructed using the theory of variable loading of Moskvitin. In this case, the hypothesis of similarity of plasticity functions at each loading step is used. Their analytical form is taken independent of the point of unloading. However, the material constants included in the approximation formulas will be different. The cyclic hardening of the material of the bearing layers is taken into account. The parametric analysis of the obtained solutions under different boundary conditions in the case of a local load distributed in a circle is carried out. The influence of temperature and nonlinearity of layer materials on the displacements in the plate is numerically investigated.

Effects of Tuned Mass Damper on Fixed-Fixed 3D Nonlinear String Resting on Nonlinear Elastic Foundation

2017 ◽  
Vol 17 (04) ◽  
pp. 1750047 ◽  
Author(s):  
Yi-Ren Wang ◽  
Li-Ping Wu
Keyword(s):  
Elastic Foundation ◽  
Internal Resonance ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Maximum Amplitude ◽  
Tuned Mass Damper ◽  
Mode Shapes ◽  
Nonlinear Elastic Foundation ◽  
Mass Damper ◽  
Nonlinear Elastic

This paper studies the vibration of a nonlinear 3D-string fixed at both ends and supported by a nonlinear elastic foundation. Newton’s second law is adopted to derive the equations of motion for the string resting on an elastic foundation. Then, the method of multiple scales (MOMS) is employed for the analysis of the nonlinear system. It was found that 1:3 internal resonance exists in the first and fourth modes of the string when the wave speed in the transverse direction is [Formula: see text] and the elasticity coefficient of the foundation is [Formula: see text]. Fixed point plots are used to obtain the frequency responses of the various modes and to identify internal resonance through observation of the amplitudes and mode shapes. To prevent internal resonance and reduce vibration, a tuned mass damper (TMD) is applied to the string. The effects of various TMD masses, locations, damper coefficients ([Formula: see text]), and spring constants ([Formula: see text]) on overall damping were analyzed. The 3D plots of the maximum amplitude (3D POMAs) and 3D maximum amplitude contour plots (3D MACPs) are generated for the various modes to illustrate the amplitudes of the string, while identifying the optimal TMD parameters for vibration reduction. The results were verified numerically. It was concluded that better damping effects can be achieved using a TMD mass ratio [Formula: see text]–0.5 located near the middle of the string. Furthermore, for damper coefficient [Formula: see text], the use of spring constant [Formula: see text]–13 can improve the overall damping.

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