Alan G. Haddow

On the response of the non-linear vibration absorber

Abstract The steady state vibrations of a non-linear dynamic vibration absorber are studied using the method of multiple scales, in conjunction with digital simulations. The main results are concerned with certain dynamic instabilities which can occur if the absorber is designed such that the desired operating frequency is approximately the mean of the two linearized natural frequencies of the system. A combination resonance can occur in this case, resulting in large amplitude almost-periodic vibrations. This motion destroys the effectiveness of the absorber and can coexist with the desired low-amplitude periodic response, which leads to initial condition dependent dynamics.

Tautochronic Vibration Absorbers for Rotating Systems

This paper describes an analytical and experimental investigation of the dynamic response and performance of a special type of centrifugal pendulum vibration absorber used for reducing torsional vibrations in rotating systems. This absorber has the property that it behaves linearly out to large amplitudes, and thus experiences no frequency de-tuning. Previous analytical work on such tautochronic absorbers has considered the response, dynamic stability, and performance of single- and multi-absorber systems. In particular, it is known that these absorbers, when perfectly tuned to the order of the applied torque, do not exhibit hysteretic jumps in the response, but multi-absorber systems can experience instabilities that destroy the symmetry of their synchronous response. In this work we extend the theory to include linear de-tuning of the absorbers, which can be used as a design parameter to influence absorber performance, both in terms of rotor vibration reduction and operating range. This paper reviews the basic analysis, which employs scaling and averaging, and extends it to include the detuning. In addition, systematic experiments of systems with one and two absorbers are carried out. The experimental results are unique in that the test facility is capable of varying the excitation order, thereby allowing one to obtain order-response curves that are useful for design purposes. The experimental results are found to be in excellent agreement with the analytical predictions, and these clearly demonstrate the tradeoffs faced when selecting absorber tuning.

Vibration Reduction in a Variable Displacement Engine Using Pendulum Absorbers

This paper describes the design, implementation and testing of crankshaft-mounted pendulum absorbers used for reducing vibrations in a variable displacement engine. The engine can run in V8 and V4 modes, and without absorbers it experiences significant vibration levels, especially in V4 idle. The absorbers are tuned to address the dominant second order vibrations, and are slightly overtuned to account for nonlinear effects. The absorbers were designed to replace the large counterweights at the ends of the crankshaft, and thus serve for both balancing and vibration absorption. The engine was placed in a vehicle and tested for vibration levels at idle under various load conditions, and these results were compared with results obtained from a similar vehicle without absorbers. The tests demonstrate that these absorbers offer an effective means of vibration attenuation in variable displacement engines.

On ‘roller-coaster’ experiments for nonlinear oscillators

We consider the dynamics of ‘roller-coaster’ type experimental models used as analog devices for nonlinear oscillators. It is shown how to chose the shape of the track in order to achieve a desired oscillator equation, in terms of the are length coordinate or its projection onto the horizontal. Explicit calculations are carried out for the linear oscillator, the so-called ‘escape equation’, the two-well Duffing oscillator, and the pendulum.

Experimental Investigation of a System With Multiple Nearly Identical Centrifugal Pendulum Vibration Absorbers

This paper presents experimental results from tests completed on a rotor system fitted with nearly-identical circular path centrifugal pendulum vibration absorbers. A brief review of theoretical background for the absorbers is given along with an overview of the test apparatus. The experimental results for one absorber and for four absorbers are presented and compared with theoretical predictions and expectations. To the best knowledge of the authors, this is the first time that systematic, controlled experiments that monitored both the response of the absorbers and the rotor have been undertaken.Copyright

Properties of Cross-Well Chaos in an Impacting System

In this paper we present results on chaotic motions in a periodically forced impacting system which is analogous to the version of Duffing’s equation with negative linear stiffness. Our focus is on the prediction and manipulation of the cross-well chaos in this system. First, we develop a general method for determining parameter conditions under which homoclinic tangles exist, which is a necessary condition for cross-well chaos to occur. We then show how one may manipulate higher harmonics of the excitation in order to affect the range of excitation amplitudes over which fractal basin boundaries between the two potential wells exist. We also experimentally investigate the threshold for cross-well chaos and compare the results with the theoretical results. Second, we consider the rate at which the system crosses from one potential well to the other during a chaotic motion and relate this to the rate of phase space flux in a Poincare map defined in terms of impact parameters. Results from simulations and experiments are compared with a simple theory based on phase space transport ideas, and a predictive scheme for estimating the rate of crossings under different parameter conditions is presented. The main conclusions of the paper are the following: (1) higher harmonics can be used with some effectiveness to extend the region of deterministic basin boundaries (in terms of the amplitude of excitation) but their effect on steady-state chaos is unreliable; (2) the rate at which the system executes cross-well excursions is related in a direct manner to the rate of phase space flux of the system as measured by the area of a turnstile lobe in the Poincare map. These results indicate some of the ways in which the chaotic properties of this system, and possibly others such as Duffing’s equation, are influenced by various system and input parameters. The main tools of analysis are a special version of Melnikov’s method, adapted for this piecewise-linear system, and ideas of phase space transport.

Modeling and design of materials for controlled wave propagation in plane grid structures

Formulations for the optimal design of plane grids with maximum band gaps are presented. Periodic band-gap structures prevent waves in certain frequency ranges from propagating. Materials or structures with band gaps have many applications, including frequency filters, vibration protection devices and wave guides. Here, a simple model of a periodic plane grid structure is presented and then an optimization problem is formulated where the structure’s band gap above a particular frequency is maximized by the selective addition of non-structural masses. Numerical implementation issues are discussed and examples are presented.Copyright