Semilinear Random Vibrations in Discrete Systems

1968 ◽  
Vol 35 (3) ◽  
pp. 560-564 ◽  
Author(s):  
E. T. Foster
Keyword(s):  
Equations Of Motion ◽  
Discrete Systems ◽  
Random Excitation ◽  
Solution Technique ◽  
Response Analysis ◽  
Weakly Nonlinear ◽  
Linear Set ◽  
System Output ◽  
Vibration Problems ◽  
Nonlinear Equations Of Motion

Random vibration problems are discussed for weakly nonlinear, multidegree-of-freedom discrete systems subjected to zero-mean, stationary random excitation. The semilinear solution technique developed involves substituting an optimum linear set of equations of motion for the actual nonlinear equations of motion. Parameters of this optimum linear system are selected on the basis of the system output so that a cyclic solution occurs. The cycles require parameter selection and response analysis until a convergence occurs in the sense that the answers from cycle to cycle are similar.

scholarly journals Stationary rarefaction waves in discrete materials with strain-softening behavior

2017 ◽  
Vol 31 (10) ◽  
pp. 1742005
Author(s):  
Eric B. Herbold
Keyword(s):  
Strain Softening ◽  
Equations Of Motion ◽  
Discrete Systems ◽  
Rarefaction Waves ◽  
Weakly Nonlinear ◽  
Discrete Equations ◽  
Softening Behavior ◽  
Wave Guides ◽  
Traveling Pulse ◽  
Nonlinearity Dispersion

This article details some of the techniques used to derive exact solutions from the discrete equations of motion of strongly and weakly nonlinear discrete systems. The distinction between strongly and weakly nonlinear systems is related to the amplitude of a traveling pulse and the external confining load. Materials with an anomalous strain-softening behavior will be emphasized (i.e., [Formula: see text], [Formula: see text]), though this choice does not preclude applications for strain-hardening systems like those with Hertzian potentials. Discrete materials with tunable acoustic transmission properties and novel impact mitigation capacity have gained interest in recent years due to their practical application across many scientific fields. Wave-guides comprised of discrete materials with a nonlinear interaction potential have been used to investigate the interplay between nonlinearity, dispersion and dissipation.

Nonlinear Free Vibration Analysis of a Fluid-Conveying Microtube

2014 ◽  
Author(s):  
Shamim Mashrouteh ◽  
Mehran Sadri ◽  
Davood Younesian ◽  
Ebrahim Esmailzadeh
Keyword(s):  
Free Vibration ◽  
Natural Frequencies ◽  
Equations Of Motion ◽  
Free Vibration Analysis ◽  
Beam Theory ◽  
Variational Iteration Method ◽  
Solution Technique ◽  
Nonlinear Free Vibration ◽  
Nonlinear Equations Of Motion ◽  
Analytical Expressions

Nonlinear free vibration of a microstructure has been analyzed in this study. A fluid-conveying microtube is mathematically modeled using non-classical beam theory. Partial differential equation of the model is considered in non-dimensional form. Simply-supported boundaries are taken into account and assuming three vibrating modes, an analytical method is employed to obtain the nonlinear equations of motion. Variational iteration method has been utilized as an analytical solution technique. In order to obtain the nonlinear natural frequencies of the system, analytical expressions are found based on this method. A parametric study is also carried out to investigate the effect of different parameters on the vibration characteristics of the microstructure.

scholarly journals Dynamic Analysis of a Pendulum Dynamic Automatic Balancer

2007 ◽  
Vol 14 (2) ◽  
pp. 151-167 ◽  
Author(s):  
Jin-Seung Sohn ◽  
Jin Woo Lee ◽  
Eun-Hyoung Cho ◽  
No-Cheol Park ◽  
Young-Pil Park
Keyword(s):  
Stability Analysis ◽  
Dynamic Stability ◽  
Critical Speed ◽  
Equations Of Motion ◽  
Polar Coordinates ◽  
Response Analysis ◽  
Rotating Speed ◽  
Time Response Analysis ◽  
The Stability ◽  
Nonlinear Equations Of Motion

The automatic dynamic balancer is a device to reduce the vibration from unbalanced mass of rotors. Instead of considering prevailing ball automatic dynamic balancer, pendulum automatic dynamic balancer is analyzed. For the analysis of dynamic stability and behavior, the nonlinear equations of motion for a system are derived with respect to polar coordinates by the Lagrange's equations. The perturbation method is applied to investigate the dynamic behavior of the system around the equilibrium position. Based on the linearized equations, the dynamic stability of the system around the equilibrium positions is investigated by the eigenvalue analysis.The stability analysis provides the design requirements for the pendulum automatic dynamic balancer to achieve a balancing of the system. The efficiency of ball automatic dynamic balancer, for reducing the total vibration of the system, is better than one of pendulum automatic balancer, if the rotating speed is above critical speed. However, pendulum automatic dynamic balancer can achieve balancing even if the rotating speed is below critical speed.The time response analysis demonstrates the stability analysis from computing the radial displacement of the rotating system and the positions of pendulums. Furthermore, in order to confirm the theoretical analysis, various experiments are made on pendulum automatic dynamic balancer.

scholarly journals Effect of Thrust on the Structural Vibrations of a Nonuniform Slender Rocket

2020 ◽  
Vol 25 (2) ◽  
pp. 29
Author(s):  
Desmond Adair ◽  
Aigul Nagimova ◽  
Martin Jaeger
Keyword(s):  
Natural Frequencies ◽  
Equations Of Motion ◽  
Approximate Solutions ◽  
Mode Shapes ◽  
Algebraic Equations ◽  
Structural Vibrations ◽  
Unknown Parameters ◽  
One Dimensional ◽  
Vibration Problems ◽  
Nonuniform Beam

The vibration characteristics of a nonuniform, flexible and free-flying slender rocket experiencing constant thrust is investigated. The rocket is idealized as a classic nonuniform beam with a constant one-dimensional follower force and with free-free boundary conditions. The equations of motion are derived by applying the extended Hamilton’s principle for non-conservative systems. Natural frequencies and associated mode shapes of the rocket are determined using the relatively efficient and accurate Adomian modified decomposition method (AMDM) with the solutions obtained by solving a set of algebraic equations with only three unknown parameters. The method can easily be extended to obtain approximate solutions to vibration problems for any type of nonuniform beam.

On the Parametric Excitation of Pendulum-Type Vibration Absorber

1961 ◽  
Vol 28 (3) ◽  
pp. 330-334 ◽  
Author(s):  
Eugene Sevin
Keyword(s):  
Energy Transfer ◽  
Numerical Solution ◽  
Parametric Excitation ◽  
Equations Of Motion ◽  
Vibration Absorber ◽  
Single Cycle ◽  
Linearized System ◽  
Nonlinear Equations Of Motion ◽  
Beam Oscillation ◽  
Energy Interchange

The free motion of an undamped pendulum-type vibration absorber is studied on the basis of approximate nonlinear equations of motion. It is shown that this type of mechanical system exhibits the phenomenon of auto parametric excitation; a type of “instability” which cannot be accounted for on the basis of the linearized system. Complete energy transfer between modes is shown to occur when the beam frequency is twice the simple pendulum frequency. On the basis of a numerical solution, approximately 150 cycles of the beam oscillation take place during a single cycle of energy interchange.

Less=Moor: A Time-Efficient Computational Tool to Assess the Behavior of Moored Ships in Waves

2017 ◽  
Author(s):  
Yijun Wang ◽  
Alex van Deyzen ◽  
Benno Beimers
Keyword(s):  
Dynamic Response ◽  
Research Effort ◽  
Equations Of Motion ◽  
Line Tension ◽  
Mooring Line ◽  
Induced Response ◽  
Wave Forces ◽  
Computational Tool ◽  
The Time Domain ◽  
Nonlinear Equations Of Motion

In the field of port design there is a need for a reliable but time-efficient method to assess the behavior of moored ships in order to determine if further detailed analysis of the behavior is required. The response of moored ships induced by gusting wind and/or waves is dynamic. Excessive motion response may cause interruption of the (un)loading operation. High line tension may cause lines to snap, introducing dangerous situations. A (detailed) Dynamic Mooring Analysis (DMA), however, is often a time-consuming and expensive exercise, especially when responses in many different environmental conditions need to be assessed. Royal HaskoningDHV has developed a time-efficient computational tool in-house to assess the wave (sea or swell) induced dynamic response of ships moored to exposed berths. The mooring line characteristics are linearized and the equations of motion are solved in the frequency domain with both the 1st and 2nd wave forces taken into account. This tool has been termed Less=Moor. The accuracy and reliability of the computational tool has been illustrated by comparing motions and mooring line forces to results obtained with software that solves the nonlinear equations of motion in the time domain (aNySIM). The calculated response of a Floating Storage and Regasification Unit (FSRU) moored to dolphins located offshore has been presented. The results show a good comparison. The computational tool can therefore be used to indicate whether the wave induced response of ships moored at exposed berths proves to be critical. The next step is to make this tool suitable to assess the dynamic response of moored ships with large wind areas, e.g. container ships, cruise vessels, RoRo or car carriers, to gusting wind. In addition, assessment of ship responses in a complicated wave field (e.g. with reflected infra-gravity waves) also requires more research effort.

Simulation of Engine Vibration on Nonlinear Hydraulic Engine Mounts

2005 ◽  
Author(s):  
A. R. Ohadi ◽  
G. Maghsoodi
Keyword(s):  
Equations Of Motion ◽  
Low Frequency ◽  
Governing Equations ◽  
Engine Mounts ◽  
Engine Vibration ◽  
Engine Mount ◽  
Rotational Motions ◽  
The Time Domain ◽  
Nonlinear Equations Of Motion ◽  
Four Cylinders

In this paper, vibration behavior of engine on nonlinear hydraulic engine mount including inertia track and decoupler is studied. In this regard, after introducing the nonlinear factors of this mount (i.e. inertia and decoupler resistances in turbulent region), the vibration governing equations of engine on one hydraulic engine mount are solved and the effect of nonlinearity is investigated. In order to have a comparison between rubber and hydraulic engine mounts, a 6 degree of freedom four cylinders V-shaped engine under inertia and balancing masses forces and torques is considered. By solving the time domain nonlinear equations of motion of engine on three inclined mounts, translational and rotational motions of engines body are obtained for different engine speeds. Transmitted base forces are also determined for both types of engine mount. Comparison of rubber and hydraulic mounts indicates the efficiency of hydraulic one in low frequency region.

Towards a Technique for Nonlinear Modal Analysis

2012 ◽  
Author(s):  
Simon A. Neild ◽  
Andrea Cammarano ◽  
David J. Wagg
Keyword(s):  
Normal Form ◽  
Modal Analysis ◽  
Equations Of Motion ◽  
Transformation Process ◽  
Second Order ◽  
Identity Transformation ◽  
Modal Testing ◽  
System Response ◽  
Weakly Nonlinear ◽  
Nonlinear Modal Analysis

In this paper we discuss a theoretical technique for decomposing multi-degree-of-freedom weakly nonlinear systems into a simpler form — an approach which has parallels with the well know method for linear modal analysis. The key outcome is that the system resonances, both linear and nonlinear are revealed by the transformation process. For each resonance, parameters can be obtained which characterise the backbone curves, and higher harmonic components of the response. The underlying mathematical technique is based on a near identity normal form transformation. This is an established technique for analysing weakly nonlinear vibrating systems, but in this approach we use a variation of the method for systems of equations written in second-order form. This is a much more natural approach for structural dynamics where the governing equations of motion are written in this form as standard practice. In fact the first step in the method is to carry out a linear modal transformation using linear modes as would typically done for a linear system. The near identity transform is then applied as a second step in the process and one which identifies the nonlinear resonances in the system being considered. For an example system with cubic nonlinearities, we show how the resulting transformed equations can be used to obtain a time independent representation of the system response. We will discuss how the analysis can be carried out with applied forcing, and how the approximations about response frequencies, made during the near-identity transformation, affect the accuracy of the technique. In fact we show that the second-order normal form approach can actually improve the predictions of sub- and super-harmonic responses. Finally we comment on how this theoretical technique could be used as part of a modal testing approach in future work.

A Method for Use of Cyclic Symmetry Properties in Analysis of Nonlinear Multiharmonic Vibrations of Bladed Disks

2004 ◽  
Vol 126 (1) ◽  
pp. 175-183 ◽  
Author(s):  
E. P. Petrov
Keyword(s):  
High Efficiency ◽  
Equations Of Motion ◽  
Linear Part ◽  
Forced Response ◽  
Mode Shapes ◽  
Solution Technique ◽  
Disk Model ◽  
Sector Model ◽  
Bladed Disk ◽  
Symmetry Properties

An effective method for analysis of periodic forced response of nonlinear cyclically symmetric structures has been developed. The method allows multiharmonic forced response to be calculated for a whole bladed disk using a periodic sector model without any loss of accuracy in calculations and modeling. A rigorous proof of the validity of the reduction of the whole nonlinear structure to a sector is provided. Types of bladed disk forcing for which the method may be applied are formulated. A multiharmonic formulation and a solution technique for equations of motion have been derived for two cases of description for a linear part of the bladed disk model: (i) using sector finite element matrices and (ii) using sector mode shapes and frequencies. Calculations validating the developed method and a numerical investigation of a realistic high-pressure turbine bladed disk with shrouds have demonstrated the high efficiency of the method.

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