On the definition of the natural frequency of oscillations in nonlinear large rotation problems

doi:10.1016/j.jsv.2010.02.015

Journal of Sound and Vibration

Volume 329, Issue 15, 19 July 2010, Pages 3171-3181

https://doi.org/10.1016/j.jsv.2010.02.015 Get rights and content

Abstract

Computational multibody system algorithms allow for performing eigenvalue analysis at different time points during the simulation to study the system stability. The nonlinear equations of motion are linearized at these time points, and the resulting linear equations are used to determine the eigenvalues and eigenvectors of the system. In the case of linear systems, the system eigenvalues remain the same under a constant coordinate transformation; and zero eigenvalues are always associated with rigid body modes, while nonzero eigenvalues are associated with non-rigid body motion. These results, however, cannot in general be applied to nonlinear multibody systems as demonstrated in this paper. Different sets of large rotation parameters lead to different forms of the nonlinear and linearized equations of motion, making it necessary to have a correct interpretation of the obtained eigenvalue solution. As shown in this investigation, the frequencies associated with different sets of orientation parameters can differ significantly, and rigid body motion can be associated with non-zero oscillation frequencies, depending on the coordinates used. In order to demonstrate this fact, the multibody system motion equations associated with the system degrees of freedom are presented and linearized. The resulting linear equations are used to define an eigevalue problem using the state space representation in order to account for general damping that characterizes multibody system applications. In order to demonstrate the significant differences between the eigenvalue solutions associated with two different sets of orientation parameters, a simple rotating disk example is considered in this study. The equations of motion of this simple example are formulated using Euler angles, Euler parameters and Rodriguez parameters. The results presented in this study demonstrate that the frequencies obtained using computational multibody system algorithms should not in general be interpreted as the system natural frequencies, but as the frequencies of the oscillations of the coordinates used to describe the motion of the system.

Introduction

In the linear theory of vibration, the system stability is examined using the eigenvalues that remain constant with time since the mass, damping and stiffness matrices are assumed to be constant [1], [2], [3]. Negative real parts of the eigenvalues are associated with stable modes, positive real parts are associated with unstable modes, and zero real parts are associated with modes that exhibit sustained oscillations. One mode with an eigenvalue that has a positive real part is sufficient to render the system unstable. In the case of a general damping matrix, a state space representation is often used to solve for the system eigenvalues and eigenvectors. In the case of linear systems, one can show that a constant coordinate transformation does not lead to a change in the eigenvalues, and as a consequence, conclusions on the stability and nature of oscillations obtained using one set of coordinates apply also when other sets of coordinates are used. Furthermore, in the case of linear systems, zero eigenvalues are always associated with rigid body modes; and nonzero eigenvalues are associated with non-rigid body motion.

In the case of nonlinear systems where the mass, damping, and/or the stiffness matrices do not remain constant; the eigenvalues and eigenvectors become configuration dependent and vary with time. The stability of nonlinear systems is often examined by linearizing the governing dynamic equations of motion at different system configurations. The resulting linearized equations are used to formulate a linear problem that can be solved for the eigenvalues and eigenvectors. The eigenvalues can be used to examine the system stability at the configurations at which the nonlinear equations are linearized. Frequencies of oscillations as well as damping ratios can be extracted from the solution of the eigenvalue problem in a straight forward manner. Unlike linear systems, as will be demonstrated in this paper, the eigenvalue solution depends on the set of coordinates used. Furthermore, rigid body motion can lead to non-zero eigenvalues, depending on the set of coordinates used. For this reason, the definition of the natural frequencies and interpretation of the eigenvalue solution of nonlinear systems is not as simple as in the case of linear systems. This issue is of particular significance in the study of the highly nonlinear multibody system applications.

The dynamics of multibody systems is governed by a system of differential and algebraic equations (DAEs). The differential equations represent the equations of motion of the system, while the nonlinear algebraic equations represent the kinematic constraints imposed on the motion of the system. General multibody system algorithms implemented in general purpose computer programs are designed to satisfy the constraint equations at the position, velocity, and acceleration levels. In order to solve for the eigenvalues and eigenvectors of the multibody system at different simulation time points, the constraint forces can be eliminated by writing the system accelerations in terms of the independent accelerations using a velocity transformation [4], [5], [6], [7], [8]. By eliminating the constraint forces, one obtains a minimum set of equations of motion associated with the system degrees of freedom. These equations can be linearized at different configurations in order to obtain a system of linear equations that can be used to formulate the eigenvalue problem. In order to account for the general damping matrix that characterizes most multibody system applications, the eigenvalue problem is formulated in multibody system algorithms using the state space approach.

In the multibody system applications, as previously mentioned, it is important to have a correct interpretation of the results of the eigenvalue solution. Different multibody system formulations employ different sets of orientation parameters. Some formulations employ Euler angles to describe the orientation of the body reference in space. In order to avoid the singularity problems associated with the use of the three Euler angles, some other multibody system formulations employ the four Euler parameters to describe the orientation of the body reference. The four Euler parameters are related by one nonlinear kinematic constraint equation that must be adjoined to the system equations of motion as an algebraic constraint equation. This constraint equation must be satisfied at the position, velocity and acceleration levels.

The purpose of this investigation is to demonstrate the significant difference between the eigenvalue solutions obtained when different sets of rotation parameters are used to describe the orientation of the body reference in space. In particular, the most widely used Euler angles and Euler parameters are employed in this investigation. Euler parameters are bounded, and therefore, simple free rotations that represent rigid body modes do not lead to zero frequency modes as in the case of Euler angles that can increase linearly if the system is torque free. For this reason, many of the concepts and conclusions drawn from the analysis of linear systems cannot be generalized and used in the case of nonlinear multibody systems. This problem is particularly important when comparing the vibration and stability results obtained using two different multibody system codes that employ the same reference frames but use different sets of parameters to define the orientations of these frames. The two computer codes can yield the same dynamics results and define the correct state of the system. The eigenvalue solutions obtained using the two codes, on the other hand, may look significantly different despite the fact that both solutions are correct and are associated with a correct system configuration. This paper addresses this important issue and explains the source of the differences between two eigenvalue solutions obtained using two different sets of orientation parameters that describe the motion of the same frame of reference. The paper also shows that the eigenvalues associated with rigid body modes can depend on the system initial conditions when a set of orientation parameters is used. For this reason, one should be careful in interpreting these eigenvalues as the system natural frequencies.

Section snippets

Background

In the case of linear vibration, the equations of motion of a mechanical system can be written in the following form [1], [2]: $M_{q} \ddot{q} + K_{q} q = Q_{q}$

In this equation, q is the vector of system coordinates; M_q and K_q are, respectively, the constant symmetric system mass and stiffness matrices associated with the coordinates q; and Q_q is the vector of generalized forces associated with q. The vector of generalized forces Q_q is assumed to be independent of the coordinates and velocities, and therefore, such a

Nonlinear large rotation problems

Most general multibody system computer codes allow for using a systematic procedure to linearize the highly nonlinear constrained differential equations of motion about nominal configurations at different time points specified by the user of the code. Quite often the eigenvalue results are used to study the system stability. Multibody system algorithms are designed to solve differential and algebraic equations (DAEs). The differential equations represent the equations of motion, while the

Large rotations

Different sets of rotation parameters can be used to define the orientation of a rigid frame of reference in space. Among these sets are the three Euler angles and the four Euler parameters. Euler parameters are often used to avoid the singularity problem associated with the use of three parameter representations. The four Euler parameters, however, are related by one algebraic equation that must be introduced to the dynamic formulation as a kinematic constraint equation. This equation must be

Large rotation rigid body modes

In the case of nonlinear large rotation problems, rigid body motion is not always associated with zero frequency. The natural frequency of oscillations depends on the set of orientation parameters used. For this reason, it is important to recognize that the natural frequencies should not be interpreted as the system natural frequencies, but as the natural frequencies of oscillations of the coordinates used to describe the motion of the system. Different coordinates lead to different forms of

Rodriguez parameters

Another example of a set of orientation coordinates that can lead to eigenvalue results different from the results obtained using Euler angles and Euler parameters is the set of Rodriguez parameters. The three Rodriguez parameters, which are not bounded, are defined in terms of the angle of rotation θ and the components of the unit vector $v = [v_{1} v_{2} v_{3}]^{T}$ as [5], [8] $γ_{1} = v_{1} \tan \frac{θ}{2}, γ_{2} = v_{2} \tan \frac{θ}{2}, γ_{3} = v_{3} \tan \frac{θ}{2}$ It is clear from this definition that when θ=π, singularities are encountered when Rodriguez parameters

Numerical results

The general procedure described in Section 3 is implemented in the general purpose multibody system computer code SAMS/2000 [7] which is used in this investigation to obtain the numerical results presented in this section for a simple rotating system. Euler parameters are used to describe the orientation of the rotating body, and the kinematic constraints imposed on the motion of the system are introduced using nonlinear algebraic equations that are satisfied at the position, velocity, and

Summary and conclusions

Computational multibody system algorithms allow for the linearization of the nonlinear system equations of motion at different time points that correspond to different system configurations. The resulting linear equations are used to formulate an eigenvalue problem that can be solved for the eigenvalues and eigenvectors. The eigen solution is often used to shed light on the system stability at different configurations and time points [10], [11], [12]. Different multibody system algorithms,

Acknowledgements

This work was supported by the US Army Research Office, Research Triangle Park, NC; and by the Federal Railroad Administration, Washington, DC. The author would also like to thank Mr. Bassam Hussein of University of Illinois at Chicago for his help in providing some of the formulas used in Section 6 of the paper.

References (12)

R.W. Clough et al.
Dynamics of Structures
(1975)
L. Meirovitch
Elements of Vibration Analysis
(1986)
W.T. Thomson
Theory of Vibration with Applications
(1988)
T.R. Kane et al.
Dynamics: Theory and Applications
(1985)
R.E. Roberson et al.
Dynamics of Multibody Systems
(1988)
R.L. Huston
Multibody Dynamics
(1990)

There are more references available in the full text version of this article.

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On the definition of the natural frequency of oscillations in nonlinear large rotation problems

Abstract

Introduction

Section snippets

Background

Nonlinear large rotation problems

Large rotations

Large rotation rigid body modes

Rodriguez parameters

Numerical results

Summary and conclusions

Acknowledgements

Dynamics of Structures

Elements of Vibration Analysis

Theory of Vibration with Applications

Dynamics: Theory and Applications

Dynamics of Multibody Systems

Multibody Dynamics