Dean T. Mook

Nonlinear Coupling of Pitch and Roll Modes in Ship Motions

An analysis is presented for the nonlinear coupling of the pitch (heave) and roll modes of ship motions in regular seas when their frequencies are in the ratio of two to one. When the frequency of encounter (excitation frequency) is near the pitch frequency, the pitch mode is excited if the encountered wave amplitude (excitation amplitude) is small. As the excitation amplitude increases, the amplitude of the pitch mode increases until it reaches a critical small value. As the excitation amplitude increases further, the pitch amplitude does not change from the critical value (i.e., the pitch mode is saturated), and all of the extra energy is transferred to the roll mode. Thus, for large excitation amplitudes, the amplitude of the roll mode is very much larger than that of the pitch mode. When the excitation frequency is near the roll frequency, there is no saturation phenomenon and at close to perfect resonance, there is no steady state response in some cases. The present results indicate that large roll amplitudes are likely in this case also.

Theoretical and experimental study of modal interaction in a two-degree-of-freedom structure

Abstract Fpr a two-degree-of-freedom structure, an experimental and theoretical investigation has been made of the primary resonances of the system, which occur when the frequency of excitation is near one of the natural frequencies, ω1 and ω2. The additional constraint of ω 2 2 ω 1 has been included and its profound influence on the response studied. The investigation has revealed some characteristics of the response that are the result of both non-linearity and more than one degree of freedom, the principal one of these being saturation. As far as is known this is the first time that saturation has beeb observed and studied in detail in a physical structure.

An overview on non-ideal vibrations

We analyze the dynamical coupling between energy sources and structural response that must not be ignored in real engineering problems, since real motors have limited output power.

A refined nonlinear model of composite plates with integrated piezoelectric actuators and sensors

Abstract A fully nonlinear theory for the dynamics and active control of elastic laminated plates with integrated piezoelectric actuators and sensors undergoing large-rotation and small-strain vibrations is presented. The theory fully accounts for geometric nonlinearities (large rotations and displacements) by using local stress and strain measures and an exact coordinate transformation. Moreover, the model accounts for continuity of interlaminar shear stresses, extensionality, orthotropic properties of piezoelectric actuators, dependence of piezoelectric strain constants on induced strains, and arbitrary orientations of the integrated actuators and sensors. Extension and shearing forces and bending and twisting moments are introduced onto the plate along the boundaries of the piezoelectric actuators. Five nonlinear partial differential equations describing the extension-extension-bending-shear-shear vibrations of laminated plates are obtained, which display linear elastic and nonlinear geometric couplings among all motions. Piezoelectric actuator-induced warping is also addressed, and comparisons with other simplified models and nonlinear theories are made.

On methods for continuous systems with quadratic and cubic nonlinearities

Methods for determining the response of continuous systems with quadratic and cubic nonlinearities are discussed. We show by means of a simple example that perturbation and computational methods based on first discretizing the systems may lead to erroncous results whereas perturbation methods that attack directly the nonlinear partial-differential equations and boundary conditions avoid the pitfalls associated with the analysis of the discretized systems. We describe a perturbation technique that applies either the method of multiple scales or the method of averaging to the Lagrangian of the system rather than the partial-differential equations and boundary conditions.

Analytical study of the subsonic wing-rock phenomenon for slender delta wings

An analytic expression describing the aerodynamic roll moment has been obtained from the numerical simulation of wing rock. This expression is used in the equation governing the rolling motion of a delta wing around its midspan axis. The result is used to construct phase planes, which reveal the general global nature of wing rock—stable limit cycles, unstable foci, saddle points, and domains of initial conditions leading to oscillatory motion and divergence. An asymptotic approximation to the solution of the governing equation is obtained; this result provides expressions for the amplitudes and frequencies of limit cycles. The present analysis provides a penetrating global view of the wing-rock phenomenon.

Nonlinear response of a parametrically excited buckled beam

A nonlinear analysis of the response of a simply-supported buckled beam to a harmonic axial load is presented. The method of multiple scales is used to determine to second order the amplitude- and phase-modulation equations. Floquet theory is used to analyze the stability of periodic responses. The perturbation results are verified by integrating the governing equation using both digital and analog computers. For small excitation amplitudes, the analytical results are in good agreement with the numerical solutions. The large-amplitude responses are investigated by using a digital computer and are compared with those obtained via an analog-computer simulation. The complicated dynamic behaviors that were found include period-multiplying and period-demultiplying bifurcations, period-three and period-six motions, jump phenomena, and chaos. In some cases, multiple periodic attractors coexist, and a chaotic attractor coexists with a periodic attractor. Phase portraits, spectra of the responses, and a bifurcation set of the many solutions are presented.

Numerical Model of Unsteady Subsonic Aeroelastic Behavior

A method for predicting unsteady, subsonic aeroelastic responses was developed. The technique accounts for aerodynamic nonlinearities associated with angles of attack, vortex-dominated flow, static deformations, and unsteady behavior. The fluid and the wing together are treated as a single dynamical system, and the equations of motion for the structure and flow field are integrated simultaneously and interactively in the time domain. The method employs an iterative scheme based on a predictor-corrector technique. The aerodynamic loads are computed by the general unsteady vortex-lattice method and are determined simultaneously with the motion of the wing. Because the unsteady vortex-lattice method predicts the wake as part of the solution, the history of the motion is taken into account; hysteresis is predicted. Two models are used to demonstrate the technique: a rigid wing on an elastic support experiencing plunge and pitch about the elastic axis, and an elastic wing rigidly supported at the root chord experiencing spanwise bending and twisting. The method can be readily extended to account for structural nonlinearities and/or substitute aerodynamic load models. The time domain solution coupled with the unsteady vortex-lattice method provides the capability of graphically depicting wing and wake motion.

A numerical-perturbation method for the nonlinear analysis of structural vibrations

A numerical-perturbation method is proposed for the determination of the nonlinear forced response of structural elements. Purely analytical techniques are capable of determining the response of structural elements having simple geometries and simple variations in thickness and properties, but they are not applicable to elements with complicated structure and boundaries. Numerical techniques are effective in determining the linear response of complicated structures, but they are not optimal for determining the nonlinear response of even simple elements when modal interactions take place due to the complicated nature of the response. Therefore, the optimum is a combined numerical and perturbation technique. The present technique is applied to beams with varying cross sections. ~ 4Y large-amplitude deflection of a beam or a plate which is restrained at its ends or along its edges results in some midplane stretching/One must account for this stretching with nonlinear strain-displacement relationships. The nonlinear equations of motion describing this situation were the basis of a number of earlier investigations and are the basis for the present paper as well. The purpose of the present paper is to present a new scheme for determining the response to a harmonic excitation. Emphasis is placed on the case when the frequency of the excitation is near a natural frequency. A convenient way to attack this nonlinear problem involves representing the deflection curve or surface with an expansion in terms of the linear, free-oscillation modes. The deflection is then determined in two steps. First, the damping, the forcing, and the nonlinear terms are deleted and the linear modes (eigenfunctions) and natural frequencies (eigenvalues) are determined. Second, the time-dependent coefficients in the expansion are obtained from a set of coupled, nonlinear, ordinary, second-order differential equations, the linear modes being used to determine the coefficients in these equations. (The procedure is described in detail in Sec. II.) Generally, one cannot obtain the linear modes analytically for structural elements having complicated boundaries and composition, and one cannot easily determine the character of the timedependent coefficients through numerical integration of the set of nonlinear equations. (The results obtained in the present numerical example are typical of the complicated manner in which the steady-state amplitudes of the various modes making up the response can vary with the amplitude and the frequency of the excitation.) Consequently, an optimal procedure involves a numerical method to determine the linear, free-oscillation modes and an analytical method to determine the time-dependent coefficients. The present procedure combines either a finiteelement or a finite-difference method with the method of multiple scales (see, for example, Ref. 1). The following brief review mentions representative examples of the work that was and is

Nonlinear analysis of the forced response of structural elements

A general procedure is presented for the nonlinear analysis of the forced response of structural elements to harmonic excitations. Internal resonances (i.e., modal interactions) are taken into account. All excitations are considered, with special consideration given to resonant excitations. The general procedure is applied to clamped‐hinged beams. The results reveal that exciting a higher mode may lead to a larger response in a lower interacting mode, contrary to the results of linear analyses.