Modal Damping and Frequency Variations in Nonlinearly Coupled Oscillators With Negative Linear Stiffness Components

2018 ◽  
Author(s):  
Fatima K. Alhammadi ◽  
Mohammad A. AL-Shudeifat
Keyword(s):  
Coupled Oscillators ◽  
Nonlinear Dynamical Systems ◽  
Equations Of Motion ◽  
Original System ◽  
Nonlinear Frequency ◽  
Nonlinear Coupling ◽  
Frequency Content ◽  
Damping Coefficients ◽  
Modal Damping ◽  
Coupling Force

A method is applied here to extract the amplitude-dependent modal damping coefficients and frequencies of nonlinearly coupled oscillators with a nonlinear force in which a negative linear stiffness is incorporated. The proposed method can be directly applied into the equations of motion of the original system where the solution is not required to be obtained a priori. The exact nonlinear frequency content in the nonlinear coupling element is employed to obtain an equivalent amplitude-dependent stiffness element using a scaling parameter that preserves the exact frequency content in the original nonlinear element. Therefore, at each amplitude in the nonlinear coupling force, the modal damping coefficients and frequencies are calculated from the eigensolution of the instantaneous amplitude-dependent equivalent system. It is found that the modal damping content is strongly affected by the nonlinear frequency content where the modal damping coefficients become amplitude-dependent quantities. The obtained amplitude-dependent damping coefficients are plotted with respect to the potential energy of the nonlinear coupling force. The method is also applicable with larger degree-of-freedom nonlinear dynamical systems in which negative and non-negative linear stiffness components are incorporated in the nonlinear coupling forces. The amplitude-dependent modal damping matrices of the amplitude-dependent equivalent systems are found to be satisfying all matrix similarity conditions with the linear modal damping matrix of the original system.

Effect of Negative Stiffness Content on the Periodic Motion of Nonlinearly Coupled Oscillators

2021 ◽  
Vol 16 (11) ◽  
Author(s):  
Mohammad A. Al-Shudeifat
Keyword(s):  
Coupled Oscillators ◽  
Dynamical Behavior ◽  
Negative Stiffness ◽  
Nonlinear Frequency ◽  
Nonlinear Stiffness ◽  
Nonlinear Coupling ◽  
Nonlinear Dynamical ◽  
Wide Range ◽  
Coupling Force ◽  
Linear And Nonlinear

Abstract The linear and nonlinear stiffness coupling forces in dynamical oscillators are usually dominated by positive stiffness components. Therefore, plotting the resultant force in y-axis with respect to the change in displacement in x-axis results in an odd symmetry in the first and third quadrants of the xy-plane. However, the appearance of negative stiffness content in coupling elements between dynamical oscillators generates a force that can be dominated by an odd symmetry in the second and fourth quadrants. The underlying nonlinear dynamical behavior of systems employing this kind of force has not been well-studied in the literature. Accordingly, the considered system here is composed of two linear oscillators that are nonlinearly coupled by a force of which the negative stiffness content is dominant. Therefore, the underlying dynamical behavior of the considered system in physical and dimensionless forms is studied on the frequency-energy plots where many backbone curves of periodic solution have been obtained. It is found that within a wide range of nonlinear frequency levels, the nonlinear coupling force is dominated by a strong negative stiffness content at the obtained frequency-energy plots backbones.

Analytical Treatment for Bistable Nonlinearly Coupled Oscillators

2017 ◽  
Author(s):  
Mohammad A. AL-Shudeifat ◽  
Adnan S. Saeed
Keyword(s):  
Exact Solution ◽  
Analytical Solution ◽  
Coupled Oscillators ◽  
Normal Modes ◽  
Nonlinear Coupling ◽  
Analytical Treatment ◽  
Internal Resonances ◽  
Restoring Force ◽  
Forcing Function ◽  
Coupling Force

Here, we introduce an analytical approximation to the exact solution of a bistable nonlinearly coupled oscillators (NLC-LOs) to study the internal resonance at the nonlinear normal modes (NNMs). The considered system is composed of two symmetrical linear oscillators coupled by a bistable nonlinear coupling restoring force. The coupling restoring force includes negative and nonnegative linear and nonlinear stiffness components. The introduced approximate analytical solution for the considered bistable NLC-LOs system is mainly proposed for the cases of which the exact frequency and the exact solution are neither available nor valid. The proposed solution depends on the application of the local equivalent linear stiffness method (LELSM) to linearize the nonlinear coupling force according to the non-linear frequency content in the original system. Accordingly, the bistable nonlinear coupling force in the NLC-LOs is replaced by an equivalent periodic forcing function of which the frequency is equal to that of the original NLC-LOs system. Therefore, the original NLC-LOs system is decoupled into two forced single degree-of-freedom subsystems where the analytical solution can be directly obtained. This obtained analytical solution is found to be highly accurate approximation for the exact solution, especially at internal resonances that occur on some NNMs of the system.

Amplitude-Dependent Stiffness Method for Studying Frequency and Damping Variations in Nonlinear Dynamical Systems

2017 ◽  
Author(s):  
Mohammad A. AL-Shudeifat
Keyword(s):  
Dynamical System ◽  
Nonlinear Dynamical Systems ◽  
Energy Content ◽  
Equations Of Motion ◽  
A Priori ◽  
Nonlinear Dynamical System ◽  
Nonlinear Coupling ◽  
Nonlinear Dynamical ◽  
Coupling Stiffness ◽  
Nonlinear Energy

A method is introduced here for extracting the fundamental backbone branches of the frequency energy plot in which the obtained nonlinear frequencies of the nonlinear dynamical system are plotted with respect to the nonlinear energy content. The proposed method is directly applied to the equations of motion where the solution is not required to be known a priori. The method is based on linearizing the nonlinear coupling force where a scaled amplitude-dependent coupling stiffness force is obtained to replace the original nonlinear coupling stiffness force. Accordingly, the backbone branches in the frequency-nonlinear-energy plot are extracted from the eigensolution of the mass-normalized amplitude-dependent global stiffness matrix of the nonlinear dynamical system. Moreover, the variations in the damping content under the effect of the nonlinear coupling stiffness are also studied. Interesting behavior of damping content under the effect of the amplitude-dependent stiffness has been observed and discussed.

Some Effects of Piston Friction and Crank or Gudgeon Pin Offset on Crankshaft Torsional Vibration

2010 ◽  
Vol 54 (01) ◽  
pp. 41-52
Author(s):  
A. L. Guzzomi ◽  
D. C. Hesterman ◽  
B. J. Stone
Keyword(s):  
Natural Frequencies ◽  
Nonlinear Frequency ◽  
Significant Finding ◽  
Nonlinear Coupling ◽  
Frequency Content ◽  
Secondary Resonance ◽  
Single Cylinder Engine ◽  
Coupling Behavior ◽  
Frequency Coupling ◽  
Resonance Detection

The varying inertia associated with reciprocating mechanisms leads to nonlinear frequency coupling between rotational speed and an engine system's average torsional natural frequencies. This coupling can cause secondary resonance problems. Recent work by the authors has shown that piston-to-cylinder friction and gudgeon pin or crank offset can modify coupling behavior. These effects can be demonstrated by analysis of an engine's receptance function and through time simulations. This paper presents the derivation of a single-cylinder engine receptance in the presence of piston-to-cylinder friction. Simulations are then used to investigate some of the effects of piston-to-cylinder friction, offset, and excitation phase on the frequency content of the crankshaft velocity. Simulations indicate that nonlinear coupling is affected by these variables, which has implications for secondary resonance detection and prevention. The most significant finding is that stronger coupling behavior can occur when piston-to-cylinder lubrication breaks down.

Analytical Solution of Two Coupled Oscillators With a Nonlinear Coupling Resorting Force

2014 ◽  
Author(s):  
Mohammad A. Al-Shudeifat ◽  
Thomas D. Burton
Keyword(s):  
Analytical Solution ◽  
Nonlinear Dynamical System ◽  
Degree Of Freedom ◽  
Nonlinear Frequency ◽  
Nonlinear Coupling ◽  
Restoring Force ◽  
Nonlinear Dynamical ◽  
Forcing Function ◽  
Local Equivalent ◽  
Coupling Force

An approach for accurate analytical solution of a two degree-of-freedom nonlinear dynamical system coupled with a strongly nonlinear restoring force is presented here. The approach is based on the application of the local equivalent linear stiffness method (LELSM) to linearize the nonlinear coupling stiffness in the system based on the nonlinear frequency calculation. Consequently, the system can be decoupled into two forced single degree-of-freedom subsystems by replacing the nonlinear coupling force with a forcing function where the solution can be analytically obtained. Different combinations of the positive and negative linear and cubic stiffness components are considered in the nonlinear coupling force. For all considered stiffness combinations, the obtained analytical solution strongly agrees with the numerical simulation of the system. In addition, the internal resonance is found not to significantly affect the accuracy of the analytical solution.

A Very Simple Method to Calculate the (Positive) Largest Lyapunov Exponent Using Interval Extensions

2016 ◽  
Vol 26 (13) ◽  
pp. 1650226 ◽  
Author(s):  
Eduardo M. A. M. Mendes ◽  
Erivelton G. Nepomuceno
Keyword(s):  
Lyapunov Exponent ◽  
Lower Bound ◽  
Equations Of Motion ◽  
Original System ◽  
Glass System ◽  
Largest Lyapunov Exponent ◽  
Simple Method ◽  
Interval Extensions

In this letter, a very simple method to calculate the positive Largest Lyapunov Exponent (LLE) based on the concept of interval extensions and using the original equations of motion is presented. The exponent is estimated from the slope of the line derived from the lower bound error when considering two interval extensions of the original system. It is shown that the algorithm is robust, fast and easy to implement and can be considered as alternative to other algorithms available in the literature. The method has been successfully tested in five well-known systems: Logistic, Hénon, Lorenz and Rössler equations and the Mackey–Glass system.

On the Frequency Response of a Spinning Disk With Large Deformations

2010 ◽  
Author(s):  
Ramin M. H. Khorasany ◽  
Stanley G. Hutton
Keyword(s):  
Frequency Response ◽  
Plate Theory ◽  
Equations Of Motion ◽  
Rotating Disk ◽  
Nonlinear Frequency ◽  
Response Characteristics ◽  
Geometrical Nonlinear ◽  
Oscillation Frequencies ◽  
Multi Mode ◽  
Different Levels

In this paper, the effect of geometrical nonlinear terms, caused by a space fixed point force, on the frequencies of oscillations of a rotating disk with clamped-free boundary conditions is investigated. The nonlinear geometrical equations of motion are based on Von Karman plate theory. Using the eigenfunctions of a stationary disk as approximating functions in Galerkin’s method, the equations of motion are transformed into a set of coupled nonlinear Ordinary Differential Equations (ODEs). These equations are then used to find the equilibrium positions of the disk at different discrete blade speeds. At any given speed, the governing equations are linearized about the equilibrium solution of the disk under the application of a space fixed external force. These linearized equations are then used to find the oscillation frequencies of the disk considering the effect of large deformation. Using multi mode approximation and different levels of nonlinearity, the frequency response of the disk considering the effect of geometrical nonlinear terms are studied. It is found that at the linear critical speed, the nonlinear frequency of the corresponding mode is not zero. Results are presented that illustrate the effect of the magnitude of disk displacement upon the frequency response characteristics. It is also found that for each mode, including the effect of the geometrical nonlinear terms due to the applied load causes a separation in the frequency responses of its backward and forward traveling waves when the disk is stationary. This effect is similar to the effect of a space fixed constraint in the linear problem. In order to verify the numerical results, experiments are conducted and the results are presented.

FREQUENCY SYNCHRONIZATION IN NETWORKS OF COUPLED OSCILLATORS, A MONOTONE DYNAMICAL SYSTEMS APPROACH

2009 ◽  
Vol 19 (12) ◽  
pp. 4107-4116 ◽  
Author(s):  
WEN-XIN QIN
Keyword(s):  
Dynamical Systems ◽  
Coupling Strength ◽  
Coupled Oscillators ◽  
Systems Approach ◽  
System Size ◽  
Coupling Scheme ◽  
Frequency Synchronization ◽  
Nonlinear Coupling ◽  
New Approach ◽  
Monotone Dynamical Systems

We propose a new approach to investigate the frequency synchronization in networks of coupled oscillators. By making use of the theory of monotone dynamical systems, we show that frequency synchronization occurs in networks of coupled oscillators, provided the coupling scheme is symmetric, connected, and strongly cooperative. Our criterion is independent of the system size, the coupling strength and the details of the connections, and applies also to nonlinear coupling schemes.

Modal Loss Factor Predication in Flexible Mechanism Systems With Constrain Viscoelastic Layers

1998 ◽  
Author(s):  
Zhang Xianmin ◽  
Liu Jike
Keyword(s):  
Equations Of Motion ◽  
Sandwich Beam ◽  
Viscoelastic Layer ◽  
Treatment Technique ◽  
Constrained Layer Damping ◽  
Modal Strain Energy ◽  
Frame Element ◽  
Modal Damping ◽  
Flexible Mechanism ◽  
Viscoelastic Layers

Abstract Control of dynamic vibration is critical to the operational success of many flexible mechanism systems. This paper addresses the problem of vibration control of such mechanisms through passive damping, using constrained layer damping treatment technique. A new type of shape function for three layer frame element containing a viscoelastic layer is developed. The equations of motion of the damped flexible mechanism are derived. Modal loss factors of this kind mechanisms are predicated from undamped normal mode by means of the modal strain energy method. Comparisons between the results obtained by this paper and the results obtained by exact solution of the governing equations for a well known sandwich beam demonstrate that the method presented in this paper is correct and reliable. Application of this method in predication of modal damping ratios for damped mechanisms is discussed. It is believed that the method of this paper hold the greatest potential for optimal design of damped flexible mechanism systems.

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