scholarly journals Nonlinear vibration of a lumped system with springs-in-series

Meccanica ◽  
2020 ◽  
Author(s):  
Jan Awrejcewicz ◽  
Roman Starosta ◽  
Grażyna Sypniewska-Kamińska
Keyword(s):  
Multiple Scales ◽  
Approximate Solutions ◽  
Nonlinear Damping ◽  
Harmonic Excitation ◽  
Algebraic Equations ◽  
Lumped Mass ◽  
Nonlinear Springs ◽  
Differential Algebraic System ◽  
In Series ◽  
The Mathematical Model

AbstractThe paper deals with the dynamics of a lumped mass mechanical system containing two nonlinear springs connected in series. The external harmonic excitation, linear and nonlinear damping are included into considerations. The mathematical model contains both differential and algebraic equations, so it belongs to the class of dynamical systems governed by the differential–algebraic system of equations (DAEs). An approximate analytical approach is used to solve the initial value problem for the DAEs. We adopt the multiple scales method (MSM) that allows one to obtain the sufficiently correct approximate solutions both far from the resonance and at the resonance conditions. The steady and non-steady resonant vibrations are analyzed by employing the modulation equations of the amplitudes and phases which are yielded by the MSM procedure.

scholarly journals Approximate Solutions of Time Fractional Diffusion Wave Models

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 923 ◽  
Author(s):  
Abdul Ghafoor ◽  
Sirajul Haq ◽  
Manzoor Hussain ◽  
Poom Kumam ◽  
Muhammad Asif Jan
Keyword(s):  
Finite Difference ◽  
Wave Equations ◽  
Quadrature Rule ◽  
Approximate Solutions ◽  
Fractional Diffusion ◽  
Haar Wavelets ◽  
Test Problems ◽  
Algebraic Equations ◽  
Diffusion Wave ◽  
Wave Models

In this paper, a wavelet based collocation method is formulated for an approximate solution of (1 + 1)- and (1 + 2)-dimensional time fractional diffusion wave equations. The main objective of this study is to combine the finite difference method with Haar wavelets. One and two dimensional Haar wavelets are used for the discretization of a spatial operator while time fractional derivative is approximated using second order finite difference and quadrature rule. The scheme has an excellent feature that converts a time fractional partial differential equation to a system of algebraic equations which can be solved easily. The suggested technique is applied to solve some test problems. The obtained results have been compared with existing results in the literature. Also, the accuracy of the scheme has been checked by computing L 2 and L ∞ error norms. Computations validate that the proposed method produces good results, which are comparable with exact solutions and those presented before.

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.

Feedback control of a nonlinear aeroelastic system with non-semi-simple eigenvalues at the critical point of Hopf bifurcation

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Licai Wang ◽  
Yudong Chen ◽  
Chunyan Pei ◽  
Lina Liu ◽  
Suhuan Chen
Keyword(s):  
Hopf Bifurcation ◽  
Feedback Control ◽  
Critical Point ◽  
Control Parameter ◽  
Eigenvalue Problems ◽  
Multiple Scales ◽  
Approximate Solutions ◽  
State Matrix ◽  
Aeroelastic System ◽  
Center Manifold Reduction

Abstract The feedback control of Hopf bifurcation of nonlinear aeroelastic systems with asymmetric aerodynamic lift force and nonlinear elastic forces of the airfoil is discussed. For the Hopf bifurcation analysis, the eigenvalue problems of the state matrix and its adjoint matrix are defined. The Puiseux expansion is used to discuss the variations of the non-semi-simple eigenvalues, as the control parameter passes through the critical value to avoid the difficulty for computing the derivatives of the non-semi-simple eigenvalues with respect to the control parameter. The method of multiple scales and center-manifold reduction are used to deal with the feedback control design of a nonlinear system with non-semi-simple eigenvalues at the critical point of the Hopf bifurcation. The first order approximate solutions are developed, which include gain vector and input. The presented methods are based on the Jordan form which is the simplest one. Finally, an example of an airfoil model is given to show the feasibility and for verification of the present method.

scholarly journals New efficiency algorithm for solving fractional order differential-algebraic system

2016 ◽  
Vol 5 (3) ◽  
pp. 152
Author(s):  
Sameer Hasan ◽  
Eman Namah
Keyword(s):  
Exact Solution ◽  
Fractional Order ◽  
Lagrange Multiplier ◽  
Iteration Method ◽  
Algebraic System ◽  
Analytic Solution ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Differential Algebraic System ◽  
Fractional Differential

This work provided the evolution of the algorithm for analytic solution of system of fractional differential-algebraic equations (FDAEs).The algorithm referred to good effective method for combination the Laplace Iteration method with general Lagrange multiplier (LLIM). Through this method we have reached excellent results in comparison with exact solution as we illustrated in our examples.

Free and Forced Vibrations of the Strongly Nonlinear Cubic-Quintic Duffing Oscillators

2017 ◽  
Vol 72 (1) ◽  
pp. 59-69 ◽  
Author(s):  
M.M. Fatih Karahan ◽  
Mehmet Pakdemirli
Keyword(s):  
Numerical Simulations ◽  
Numerical Solutions ◽  
Multiple Scales ◽  
Approximate Solutions ◽  
Response Curves ◽  
Strongly Nonlinear ◽  
Free And Forced Vibrations ◽  
Approximate Analytical Solutions ◽  
Strongly Nonlinear Oscillators ◽  
Good Agreement

AbstractStrongly nonlinear cubic-quintic Duffing oscillatoris considered. Approximate solutions are derived using the multiple scales Lindstedt Poincare method (MSLP), a relatively new method developed for strongly nonlinear oscillators. The free undamped oscillator is considered first. Approximate analytical solutions of the MSLP are contrasted with the classical multiple scales (MS) method and numerical simulations. It is found that contrary to the classical MS method, the MSLP can provide acceptable solutions for the case of strong nonlinearities. Next, the forced and damped case is treated. Frequency response curves of both the MS and MSLP methods are obtained and contrasted with the numerical solutions. The MSLP method and numerical simulations are in good agreement while there are discrepancies between the MS and numerical solutions.

Approximate Solutions to a Cantilever Beam Using Optimal Homotopy Asymptotic Method

2013 ◽  
Vol 430 ◽  
pp. 22-26 ◽  
Author(s):  
Vasile Marinca ◽  
Nicolae Herisanu ◽  
Traian Marinca
Keyword(s):  
Small Parameter ◽  
Cantilever Beam ◽  
Asymptotic Method ◽  
Approximate Solutions ◽  
Lumped Mass ◽  
Engineering Problems ◽  
Approximate Frequency ◽  
Optimal Homotopy Asymptotic Method ◽  
Analytical Expressions

The response of a cantilever beam with a lumped mass attached to its free end subject to harmonical excitation at the base is investigated by means of the Optimal Homotopy Asymptotic Method (OHAM). Approximate accurate analytical expressions for the solutions and for approximate frequency are determined. This method does not require any small parameter in the equation. The obtained results prove that our method is very accurate, effective and simple for investigation of such engineering problems.

The Application of the Ritz Averaging Method to Determining the Response of Systems with Time Varying Stiffness to Harmonic Excitation

1980 ◽  
Vol 102 (2) ◽  
pp. 384-390 ◽  
Author(s):  
M. Benton ◽  
A. Seireg
Keyword(s):  
Linear System ◽  
Averaging Method ◽  
Approximate Solutions ◽  
Harmonic Excitation ◽  
Time Varying ◽  
Parametric Vibrations ◽  
Nonlinear Characteristics ◽  
Closed Form Solutions ◽  
Mathieu’S Equation ◽  
Stiffness Variation

Parametric vibrations occur in many mechanical systems such as gears where the stiffness variation and external excitations generally occur at integer multiples of the rotational speed. This paper describes a procedure based on the Ritz Averaging Method for developing closed form solutions for the response of such systems to harmonic excitations. Although the method is illustrated in the paper by the case of a linear system with harmonic stiffness fluctuation (defined by Mathieu’s equation) it can be readily applied to determine approximate solutions for systems with nonlinear characteristics and any periodic variations of parameters.

Numerical Simulation of Subsea Cluster Manifold Installation by the Sheave Method

2018 ◽  
Author(s):  
Yu Zhao ◽  
Yingying Wang ◽  
Liwei Li ◽  
Chao Yang ◽  
Yang Du ◽  
...  
Keyword(s):  
Numerical Simulation ◽  
Mathematical Model ◽  
South China Sea ◽  
Deep Sea ◽  
High Reliability ◽  
Two Dimensional ◽  
Complex Environment ◽  
Lumped Mass ◽  
Lumped Mass Method ◽  
The Mathematical Model

The sheave installation method (SIM) is an effective and non-conventional method to solve the installation of subsea equipment in deep water (>1000m), which has been developed to deploy the 175t Roncador Manifold I into 1,885 meters water depth in 2002. With the weight increment of subsea cluster manifold, how to solve its installation with the high reliability in the deep sea is still a great challenge. In this paper, the installation of the 300t subsea cluster manifold using the SIM is studied in the two-dimensional coordinate system. The mathematical model is established and the lumped mass method is used to calculate the hydrodynamic forces of the wireropes. Taking into account the complex environment loads, the numerical simulation of the lowering process is carried out by OrcaFlex. The displacement and vibration of the subsea cluster manifold in the z-axis direction and the effective tension at the top of the wireropes can be gotten, which can provide guidance for the installation of the cluster manifold in the South China Sea.

scholarly journals THE VALUATION OF SELF-FUNDING INSTALMENT WARRANTS

2017 ◽  
Vol 20 (04) ◽  
pp. 1750025
Author(s):  
J. N. DEWYNNE ◽  
N. EL-HASSAN
Keyword(s):  
Monte Carlo ◽  
Finite Difference ◽  
Multiple Scales ◽  
Correction Term ◽  
Approximate Solutions ◽  
Fair Value ◽  
Small Time ◽  
Point Of View ◽  
First Order Correction ◽  
Short Time

We present two models for the fair value of a self-funding instalment warrant. In both models we assume the underlying stock process follows a geometric Brownian motion. In the first model, we assume that the underlying stock pays a continuous dividend yield and in the second we assume that it pays a series of discrete dividend yields. We show that both models admit similarity reductions and use these to obtain simple finite-difference and Monte Carlo solutions. We use the method of multiple scales to connect these two models and establish the first-order correction term to be applied to the first model in order to obtain the second, thereby establishing that the former model is justified when many dividends are paid during the life of the warrant. Further, we show that the functional form of this correction may be expressed in terms of the hedging parameters for the first model and is, from this point of view, independent of the particular payoff in the first model. In two appendices we present approximate solutions for the first model which are valid in the small volatility and the short time-to-expiry limits, respectively, by using singular perturbation techniques. The small volatility solutions are used to check our finite-difference solutions and the small time-to-expiry solutions are used as a means of systematically smoothing the payoffs so we may use pathwise sensitivities for our Monte Carlo methods.

Sign in / Sign up

Export Citation Format

Share Document