The response of single degree of freedom systems with quadratic and cubic non-linearities to a subharmonic excitation

The response of single degree of freedom systems with quadratic and cubic nonlinearities to a subharmonic excitation is investigated. The method of multiple scales is used to derive two first order ordinary differential equations that govern the evolution of the amplitude and phase of the subharmonic. These equations are used to obtain the steady state solutions and their stability. The results identify two critical values ζ₁ and ζ₂, where ζ₂>ζ₁, for the excitation amplitude f. The value ζ₂ is the threshold for the stability of the trivial solution. When f>ζ₂, subharmonic oscillations of finite amplitude are always excited. When f<ζ₁, subharmonic oscillations cannot be excited. But when ζ₁<f<ζ₂, subharmonic oscillations may or may not be excited, depending on the initial conditions. Also, the method of harmonic balance is applied to a special case of the problem considered. It is shown that, although the method seems straightforward, it can lead to erroneous results if extreme care is not taken in the ordering of the different terms.     

The response of two-degree-of-freedom systems with quadratic non-linearities to a parametric excitation

The method of multiple scales is used to analyze the response of two-degree-of-freedom systems with quadratic non-linearities to a parametric harmonic excitation having the frequency Ω. Four ordinary differential equations are derived to describe the modulation of the amplitudes and the phases when ω₂ ≈ 2ω₁ and either $\text{Ω ≈ 2ω}{}_1$ or $\text{Ω ≈ 2ω}{}_2$, where ω₁ and ω₂ are the linear undamped natural frequencies of the system. Two critical values ζ₁ and ζ₂ of the amplitude F of the excitation are identified in the analysis. When F >ζ₂, the amplitude of the directly excited mode grows exponentially with time according to the linear analysis, whereas the amplitudes of both modes achieve steady state constant values, irrespective of the initial amplitudes, according to the non-linear analysis. When F < ζ₁, the motion decays to zero according to both the linear and non-linear analyses. When ζ₁ ? F ? ζ₂, the motion decays to zero according to the linear analysis, whereas it achieves a periodic steady state or decays to zero depending on the initial amplitudes according to the non-linear analysis. This is an example of subcritical instability. When $\text{Ω ≈ 2ω}{}_2$, the steady state value of the higher mode, which is directly excited, is a constant that is independent of the excitation of amplitude F, whereas the amplitude of the lower mode, which is indirectly excited through internal resonance, grows with the excitation amplitude F. This is another example of saturation.     

The response of multidegree-of-freedom systems with quadratic non-linearities to a harmonic parametric resonance

An analysis is presented of the response of multidegree-of-freedom systems with quadratic non-linearities to a harmonic parametric excitation in the presence of an internal resonance of the combination type ω₃ ≈ ω₂ + ω₁, where the ω_n are the linear natural frequencies of the systems. In the case of a fundamental resonance of the third mode (i.e., $\text{Ω ≈ω}{}_3$, where Ω is the frequency of the excitation), one can identify two critical values _{ζ1 and ζ2, where ζ2 ? ζ1}, of the amplitude F of the excitation. The value F = ζ₂ corresponds to the transition from stable to unstable solutions. When F < ζ₁, the motion decays to zero according to both linear and non-linear theories. When F >ζ₂, the motion grows exponentially with time according to the linear theory but the non-linearity limits the motion to a finite amplitude steady state. The amplitude of the third mode, which is directly excited, is independent of F, whereas the amplitudes of the first and second modes, which are indirectly excited through the internal resonance, are functions of F. When ζ₁ ? F ? ζ₂, the motion decays or achieves a finite amplitude steady state depending on the initial conditions according to the non-linear theory, whereas it decays to zero according to the linear theory. This is an example of subcritical instability. In the case of a fundamental resonance of either the first or second mode, the trivial response is the only possible steady state. When F ? ζ₂, the motion decays to zero according to both linear and non-linear theories. When F >ζ₂, the motion grows exponentially with time according to the linear theory but it is aperiodic according to the non-linear theory. Experiments are being planned to check these theoretical results.     

The response of two-degree-of-freedom systems with quadratic and cubic non-linearities to multifrequency parametric excitations

The response of two d.o.f. systems with quadratic and cubic non-linearities to multi-frequency parametric excitations is determined by using the method of multiple scales. Four first-order ordinary differential equations are derived to describe the modulation of the amplitudes and the phases when principal parametric resonances of both modes and combination resonances of the additive and difference type occur simultaneously. In all cases the steady state solutions and their stability are determined. Numerical results depicting the various resonances are presented.     

Two-degree-of-freedom systems with quadratic non-linearities subjected to parametric and self excitation

The method of multiple scales is used to study the response of two-degree-of-freedom systems with quadratic non-linearities under the simultaneous effects of a harmonic parametric excitation and self excitation. The principal parametric resonance of the first mode and a three-to-one internal resonance is considered, followed by the case of internal and parametric resonance of the second mode. In both cases the stability of the system is also studied. Amplitude and frequency response curves are presented for both cases. The character of stability and the mode in which the system loses its stability is also discussed.     

The response of two-degree-of-freedom systems with quadratic non-linearities to a combination parametric resonance

The response of two-degree-of-freedom systems with quadratic non-linearities to a combination parametric resonance in the presence of two-to-one internal resonances is investigated. The method of multiple scales is used to construct a first order uniform expansion yielding four first order non-linear ordinary differential equations governing the modulation of the amplitudes and the phases of the two modes. Steady state responses and their stability are computed for selected values of the system parameters. The effects of detuning the internal resonance, detuning the parametric resonance, the phase and magnitude of the second mode parametric excitation, and the initial conditions are investigated. The first order perturbation solution predicts qualitatively the trivial and non-trivial stable steady state solutions and illustrates both the quenching and saturation phenomena. In addition to the steady state solutions, other periodic solutions are predicted by the perturbation amplitude and phase modulation equations. These equations predict a transition from constant steady state non-trivial responses to limit cycle responses (Hopf bifurcation). Some limit cycles are also shown to experience period doubling bifurcations. The perturbation solutions are verified by numerically integrating the governing differential equations.     

Combination tones in the response of single degree of freedom systems with quadratic and cubic non-linearities

The method of multiple scales is used to analyze the response of a single-degree-of-freedom system to either the combination resonance of the additive type $\text{Ω}{}_2\text{+ Ω}{}_1\text{≈ ω}{}_0$ or the combination resonance of the difference type $\text{Ω}{}_2\text{? Ω}{}_1\text{≈ ω}{}_0$, where $\text{Ω}{}_1$ and $\text{Ω}{}_2$ are the frequencies of the excitation and ω₀ is the linear undamped natural frequency of the system. To the second approximation, the combination resonance of the additive type has three effects on the steady state response. First, it produces terms having the frequencies $\text{Ω}{}_1\text{, Ω}{}_2$ and $\text{Ω}{}_2\text{+ Ω}{}_1$ at first order and terms having the frequencies 0, $\text{2Ω}{}_1\text{, 2Ω}{}_2\text{, Ω}{}_2\text{? Ω}{}_1$, $\text{2(Ω}{}_2\text{+ Ω}{}_1\text{), Ω}{}_2\text{+ 2Ω}{}_1$ and $\text{2Ω}{}_2\text{+ Ω}{}_1$ at second order. Second, it produces a shift in the natural frequency of the system. Third, it produces a virtual primary-resonant excitation having the frequency $\text{Ω}{}_2\text{+ Ω}{}_1\text{≈ ω}{}_0$ that makes the component having the frequency $\text{Ω}{}_2\text{+ Ω}{}_1$ be of first rather than second order. Similar effects are produced by a combination resonance of the difference type or a superharmonic resonance of order two.     

Fermions with cubic and quartic spectrum

We study exotic fermions with spectrum E ² ∝ p ^2N . Such spectrum emerges in the vicinity of the Fermi point with multiple topological charge N, if special symmetry is obeyed. When this symmetry is violated, the multiple Fermi point typically splits into N elementary Fermi points, i.e., Dirac points with N = 1 and spectrum E ² ∝ p ².     

Normal modes of a continuous system with quadratic and cubic non-linearities

A power series solution is presented for the free vibrations of simply supported beams resting on elastic foundation having quadratic and cubic non-linearities. The time-dependence is assumed harmonic and the problem is posed as a non-linear eigenvalue problem. The spatial variable is transformed into an independent variable that satisfies the boundary conditions. This permits a power series expansion of the beam motion in terms of the new variable. A recurrence relation is obtained from the governing equation and used in conjunction with the Rayleigh energy principle to compute the natural frequencies. The results show that, for a first order approximation, only the lower frequencies and first mode shape are significantly affected by the cubic non-linearity.     

Integrable potentials with cubic and quartic invariants

The necessary general form for a two-dimensional integrable potential to admit an invariant of the type: I=(xp_y-yp_x)ⁿ+ lower order terms, is obtained and an explicit construction is carried out at n = 3,4. A generalization is made to other similar types of the invariant.     

Nonlinear vibration of a curved beam under uniform base harmonic excitation with quadratic and cubic nonlinearities

This paper presents nonlinear vibration analysis of a curved beam subject to uniform base harmonic excitation with both quadratic and cubic nonlinearities. The Galerkin method is employed to discretize the governing equations. A high-dimensional model that can take nonlinear model coupling into account is derived, and the incremental harmonic balance (IHB) method is employed to obtain the steady-state response of the curved beam. The cases investigated include softening stiffness, hardening stiffness and modal energy transfer. The stability of the periodic solutions for given parameters is determined by the multi-variable Floquet theory using Hsu's method. Particular attention is paid to the anti-symmetric response with and without excitation, as the excitation frequency is close to the first and third natural frequencies of the system. The results obtained with the IHB method compare very well with those obtained via numerical integration.     

The critical excitation and response of simple dynamic systems

A method for defining the critical excitations and responses of dynamic systems is examined. The critical excitations are those functions which maximize some response norm with respect to the constraints placed on the admissible excitations. A class of critical responses for linear, elastoplastic and hysteretic single degree of freedoms systems is studied, showing the frequency and amplitude relations for these solutions. For linear systems it is shown that the critical excitations producing either a maximum displacement response or maximum energy input are harmonic and derivable from the harmonically excited response functions for the same linear system. The critical excitations for elastoplastic systems, however, are not harmonic and at low frequencies the response is significantly larger than the harmonically excited response. The critical response solutions require higher multiple frequency components to exist. Both periodic and inelastic offset types of critical response are examined for a hysteretic, elastoplastic system and the response characteristics for these solutions are discussed.     

The covariance response of linear systems to locally stationary random excitation

A simple method is given for calculating the covariance response of linear, time-invariant systems to random excitation processes which are locally stationary, or approximately so. As an illustration, the method is used to estimate the response of an idealized model of a ten-storey building to non-stationary ground acceleration; the accuracy of the estimated response is assessed by a comparison with the results of a less approximate, but lengthier, general calculation method, previously published.     

An analytical model of two-dimensional impact/sliding response to harmonic excitation

An analytical method for the generation of periodic solutions for impact/sliding response to harmonic excitation of a two-dimensional linear oscillator is outlined and applied to generate a simple symmetric solution. The method yields impact reaction forces and sliding distances and hence enables wear rate calculations to be performed.     

The effect of rotation on the steady state response of a spring-mass system under harmonic excitation

A two degree-of-freedom system, consisting of a point mass which is constrained to move in one plane, is considered. The motion is controlled by linear springs and viscous damping. A constant amplitude harmonic force is applied along one axis in the plane, which is rotating at a constant angular velocity about an axis perpendicular to the plane. Due to the rotation, oscillation takes place in the direction perpendicular to, as well as along, the axis of excitation.The amplitude and phase of the steady state vibrations are derived as a function of the excitation frequency and the rate of turn. For rates of turn very much less than the system natural frequencies, this theory covers the principles of vibratory rate sensors such as the tuning fork; however, the emphasis here is on the performance of the system when the angular velocity is of the same order as the natural frequencies of the system.     

Oscillations of a bose-einstein condensate rotating in a harmonic plus quartic trap

We study the normal modes of a two-dimensional rotating Bose-Einstein condensate confined in a quadratic plus quartic trap. Hydrodynamic theory and sum rules are used to derive analytical predictions for the collective frequencies in the limit of high angular velocities Omega where the vortex lattice produced by the rotation exhibits an annular structure. We predict a class of excitations with frequency sqrt[6]Omega in the rotating frame, irrespective of the mode multipolarity m, as well as a class of low energy modes with frequency proportional to |m|/Omega. The predictions are in good agreement with results of numerical simulations based on the 2D Gross-Pitaevskii equation. The same analysis is also carried out at even higher angular velocities, where the system enters the giant vortex regime.     

Stationary response of combined linear dynamic systems to stationary excitation

The stationary response of a broad class of combined linear systems to stationary random excitation is determined by the normal mode method. The systems are characterized by a viscously damped simple beam (or string, membrane, thin plate or shell, etc.) connected at discrete points to a multiplicity of viscously damped linear oscillators and/or masses. The solution of the free vibration problem by way of Green functions and the deterministic forced vibration problem by modal analysis for both proportional and non-proportional damping is reviewed. The orthogonality relation for the natural modes of vibration is used to derive a unique relationship between the cross-spectral density functions of the applied forces and the cross-spectral density functions of the generalized forces. Finally, the response spectral density functions and the mean square responses of the beam and oscillators are derived in closed form, exact for the proportionally damped system and approximate for the non-proportionally damped system.