Theoretical and Experimental Study of the Nonlinearly Coupled Heave, Pitch, and Roll Motions of a Ship in Longitudinal Waves

14th Biennial Conference on Mechanical Vibration and Noise: Nonlinear Vibrations ◽

10.1115/detc1993-0037 ◽

1993 ◽

Author(s):

I. G. Oh ◽

A. H. Nayfeh ◽

D. T. Mook

Keyword(s):

Large Amplitude ◽

Degrees Of Freedom ◽

Excitation Frequency ◽

Force Response ◽

Governing Equations ◽

Response Curves ◽

Varying Coefficients ◽

Jump Phenomena ◽

Time Varying Coefficients ◽

Three Degrees Of Freedom

Abstract The loss of dynamic stability and the resulting large-amplitude roll of a vessel in a head or following sea were studied theoretically and experimentally. A ship model with three degrees of freedom (roll, pitch, heave) was considered. The governing equations for the heave and pitch modes were linearized and their harmonic solutions were coupled with the nonlinear equation governing roll. The resulting equation, which has time-varying coefficients, was used to predict the response in roll. The principal parametric resonance was considered in which the excitation frequency is twice the natural frequency in roll. Force-response curves were obtained. The existence of jump phenomena and multiple stable solutions for the case of subcritical instability was observed in the experiments and found to be in good qualitative agreement with the results predicted by the theory. The experiments also revealed that the large-amplitude roll is dependent on the location of the model in the standing waves.

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Nonlinear Damping on Large-Amplitude Vibrations of Plates

Volume 4B: Dynamics, Vibration, and Control ◽

10.1115/imece2015-50951 ◽

2015 ◽

Author(s):

Giovanni Ferrari ◽

Marco Amabili ◽

Prabakaran Balasubramanian

Keyword(s):

Large Amplitude ◽

Single Point ◽

Excitation Frequency ◽

Laser Doppler Vibrometer ◽

Testing Procedure ◽

Nonlinear Damping ◽

Harmonic Excitation ◽

Laser Doppler ◽

Rectangular Plates ◽

Response Curves

Large-amplitude (geometrically nonlinear) forced vibrations of completely free sandwich and steel rectangular plates are investigated experimentally. Harmonic excitation is applied by using an electro-dynamic exciter and the plate vibration is measured by using laser Doppler vibrometers. A scanning laser Doppler vibrometer is used for experimental modal analysis since it provides non-contact vibration measurements with very high spatial resolution. The large-amplitude vibration experiments are carried out by using a single point Laser Doppler Vibrometer and a stepped-sine testing procedure. The non-linear frequency response curves are obtained by increasing and decreasing the excitation frequency in very small steps at specific force amplitudes controlled in a closed-loop. The experimental results are compared to numerical simulations obtained by reduced-order models and show very good agreement. The nonlinear damping is experimentally obtained as a function of the vibration amplitude.

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Analysis of a Nonlinear Mechanical System With Three Degrees of Freedom

Journal of Applied Mechanics ◽

10.1115/1.4012109 ◽

1959 ◽

Vol 26 (4) ◽

pp. 546-548

Author(s):

S. A. Hovanessian

Keyword(s):

Steady State ◽

Mechanical System ◽

Degrees Of Freedom ◽

Amplitude Equations ◽

Response Curves ◽

Nonlinear Mechanical ◽

Nonlinear Mechanical System ◽

Nonlinear Mechanical Systems ◽

Steady State Amplitude ◽

Three Degrees Of Freedom

Abstract In recent years the methods of solution of nonlinear vibratory systems with one degree of freedom have been extended to the solution of nonlinear systems with two degrees of freedom. For these systems, the steady-state amplitude equations can be obtained by several methods, such as perturbation and the iteration method introduced by Duffing. In this paper the amplitude-frequency equations for a nonlinear mechanical system with three degrees of freedom are obtained, using Duffing’s method for obtaining the steady-state amplitude equations [1]. The steady-state response curves of two nonlinear mechanical systems, each having three degrees of freedom, are given.

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Tracking Control of Unmanned Surface Vehicles With Abkowitz Steering Model

ASME 2010 Dynamic Systems and Control Conference, Volume 1 ◽

10.1115/dscc2010-4293 ◽

2010 ◽

Author(s):

C. Nataraj ◽

Ramesh Thimmaraya

Keyword(s):

Tracking Control ◽

Control Algorithm ◽

Degrees Of Freedom ◽

Closed Loop ◽

Linear Time ◽

Time Varying ◽

Unmanned Surface Vehicles ◽

Varying Coefficients ◽

Ship Steering ◽

Time Varying Coefficients

This paper is concerned with the tracking control of unmanned surface vehicles. Steering dynamics is modeled using nonlinear equations with three degrees of freedom following Abkowitz. Tracking control of this nonlinear system leads to the need to derive a control algorithm for linear error equations which have time-varying coefficients. Next, a control algorithm has been derived for this set of linear time-varying equations. Lyapunov transformations have been applied to transform the error equation into a canonical form. A desired closed-loop PD-spectrum and the desired right PD-modal matrix have been chosen and the resulting Sylvester equation has been solved to obtain a matrix of time-varying controller gains. This leads to the closed loop equations for controlling the ship steering of an unmanned ship. The controller algorithm is applied to the motion control of ships with parametric values from published reports. Several tracking trajectories have been generated with and without obstacles, and time-varying control has been investigated and presented. The control algorithm is shown to be quite effective for tracking of unmanned surface vehicles. Stability conditions are derived to ensure convergence. Present work in experimental verification is outlined.

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Induced Vibration of Suspension Bridges Traversed by Moving Vehicles

10.1115/imece2001/de-23218 ◽

2001 ◽

Author(s):

Ebrahim Esmailzadeh ◽

Nader Jalili

Keyword(s):

Degrees Of Freedom ◽

Beam Theory ◽

Bending Moment ◽

Suspension Bridge ◽

Suspension Bridges ◽

Vehicle Speed ◽

Moving Vehicle ◽

Varying Coefficients ◽

Moving Vehicles ◽

Time Varying Coefficients

Abstract An investigation into the dynamics of vehicle-structure interaction of a suspension bridge traversed by a moving vehicle is presented. The vehicle including the occupants is modeled as a half-car model with six degrees-of-freedom, and the bridge is assumed to obey the Euler-Bernoulli beam theory. Due to the continuously moving location of the loads on the bridge, the governing differential equations will have time-varying coefficients and hence, become rather complicated. The relationship between the bridge vibration characteristics and the vehicle speed is rendered, which yields into a search for a particular speed that determines the maximum values of dynamic deflection and the bending moment of the bridge. Results at different vehicle speeds demonstrate that the maximum dynamic deflection occurs at the vicinity of the bridge mid-span (±3%), while the maximum bending moment is found at ±20% of the mid-span. It is shown that one can find a critical speed at which the maximum values of bridge dynamic deflection and bending moment attain their global maxima.

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Dynamic interaction of vehicles moving on uniform bridges

Proceedings of the Institution of Mechanical Engineers Part K Journal of Multi-body Dynamics ◽

10.1243/146441902320992437 ◽

2002 ◽

Vol 216 (4) ◽

pp. 343-350 ◽

Cited By ~ 3

Author(s):

N Jalili ◽

E Esmailzadeh

Keyword(s):

Degrees Of Freedom ◽

Practical Importance ◽

Beam Theory ◽

Bending Moment ◽

Dynamic Interaction ◽

Suspension Bridges ◽

Vehicle Speed ◽

Varying Coefficients ◽

Moving Vehicles ◽

Time Varying Coefficients

The dynamic interaction problem of moving vehicles on uniform suspension bridges is studied. The resulting variable moving loads acting on the bridge are of great practical importance to both bridge and automotive engineers. The vehicle, including the occupants, is modelled as a planar half-car with six degrees of freedom, and the bridge is assumed to obey the Euler-Bernoulli beam theory with arbitrary conventional boundary conditions. However, the numerical simulations presented here are for the case of a vehicle travelling at a constant speed on a bridge with simply supported end conditions. Owing to the continuously moving location of the loads on the bridge, the governing differential equations will have time-varying coefficients and hence become rather complicated. The relationship between the bridge vibration characteristics and the vehicle speed is established, resulting in a search for a particular speed that determines the maximum values of dynamic deflection and the bending moment of the bridge.

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Generalized Frequency Response Functions for Systems with Time-Varying Coefficients

Proceedings of the Institution of Mechanical Engineers Part C Journal of Mechanical Engineering Science ◽

10.1243/pime_proc_1994_208_112_02 ◽

1994 ◽

Vol 208 (3) ◽

pp. 145-153 ◽

Cited By ~ 2

Author(s):

M R Belmont

Keyword(s):

Frequency Response ◽

Time Domain ◽

Illustrative Case ◽

Time Varying ◽

Response Functions ◽

Governing Equations ◽

Varying Coefficients ◽

Wide Range ◽

Time Varying Coefficients ◽

The Time Domain

An extension of the concept of frequency response is introduced which can be applied to systems described by differential equations whose coefficients vary periodically or almost periodically with time. Such systems are not accessible to traditional frequency response function methods because while the governing equations may be linear in the time domain they are non-linear in frequency. The basic theory of the technique is introduced and results are obtained for a wide range of systems. Time domain solutions are also deduced to complement the spectral development. Numerical results are calculated for an illustrative case that deals with a photochemical problem driven by a solar daylight cycle.

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