Evaluation of Parametric Vibration and Stability of Flexible Cam-Follower Systems

1994 ◽  
Vol 116 (1) ◽  
pp. 291-297 ◽  
Author(s):  
A. I. Mahyuddin ◽  
A. Midha ◽  
A. K. Bajaj
Keyword(s):  
Equations Of Motion ◽  
Second Order ◽  
Valve Train ◽  
Time Dependent ◽  
Equivalent Model ◽  
System Response ◽  
Parametric Stability ◽  
Order Linear ◽  
Linear Ordinary Differential Equations ◽  
Cam Follower

A method to study parametric stability of flexible cam-follower systems is developed. This method is applied to an automotive valve train which is modeled as a single-degree-of-freedom vibration system. The inclusion of the transverse and rotational flexibilities of the camshaft results in a system governed by a second-order, linear, ordinary differential equation with time-dependent coefficients. This class of equations, known as Hill’s equations, merits special notice in determination of the system response and stability. The analysis includes development of the equivalent model of the system, derivation of its equation of motion, and a method to evaluate its parametric stability based on Floquet theory. A closed-form numerical algorithm, developed to compute the periodic response of systems governed by second-order, linear, ordinary differential equations of motion with time-dependent coefficients, is utilized. The results of this study are presented in a companion paper in the forms of parametric stability charts and three-dimensional stability and response charts.

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.

Influence of Varying Cam Profile and Follower Motion Event Types on Parametric Vibration and Stability of Flexible Cam-Follower Systems

1994 ◽  
Vol 116 (1) ◽  
pp. 298-305 ◽  
Author(s):  
A. I. Mahyuddin ◽  
A. Midha
Keyword(s):  
Closed Form ◽  
Numerical Algorithm ◽  
Parametric Instability ◽  
Excitation Amplitude ◽  
Companion Paper ◽  
Valve Train ◽  
Positional Error ◽  
Parametric Stability ◽  
Cam Follower ◽  
Cam Profile

A method to study parametric stability of flexible cam-follower systems, based on Floquet theory, as well as a closed-form numerical algorithm to compute periodic response of the system, have been developed in a companion paper. These are applied to an automotive valve train, modeled as a single-degree-of-freedom vibration system. The inclusion of the transverse and rotational flexibilities of the camshaft results in a system that is governed by a linear, second-order, ordinary differential equation with time-dependent coefficients. In this paper, the parametric stability of the system is studied, and the results are presented in the form of parametric stability charts. The regions of instability are plotted on the nondimensionalized frequency and excitation (amplitude) parameter plane. The maximum positional error of the follower motion, analyzed by the closed-form numerical algorithm, enables a novel presentation of three-dimensional stability and response charts. Stability of the system is investigated for three types of follower motion events and four different cam profiles. The effect of damping on parametric instability is also studied. A comparative study of these event and cam profile types reveals some very interesting and hitherto unknown results.

Sign in / Sign up

Export Citation Format

Share Document