Construction of approximate analytical solutions to strongly nonlinear damped oscillators

An analytical approximate method for strongly nonlinear damped oscillators is proposed. By introducing phase and amplitude of oscillation as well as a bookkeeping parameter, we rewrite the governing equation into a partial differential equation with solution being a periodic function of the phase. Based on combination of the Newton’s method with the harmonic balance method, the partial differential equation is transformed into a set of linear ordinary differential equations in terms of harmonic coefficients, which can further be converted into systems of linear algebraic equations by using the bookkeeping parameter expansion. Only a few iterations can provide very accurate approximate analytical solutions even if the nonlinearity and damping are significant. The method can be applied to general oscillators with odd nonlinearities as well as even ones even without linear restoring force. Three examples are presented to illustrate the usefulness and effectiveness of the proposed method.     

Non-linear vibrations of suspension bridges with external excitation

Non-linear coupled vertical and torsional vibrations of suspension bridges are investigated. Method of Multiple Scales, a perturbation technique, is applied to the equations to find approximate analytical solutions. The equations are not discretized as usually done, rather the perturbation method is applied directly to the partial differential equations. Free and forced vibrations with damping are investigated in detail. Amplitude and phase modulation equations are obtained. The dependence of non-linear frequency on amplitude is described. Steady-state solutions are analyzed. Frequency-response equation is derived and the jump phenomenon in the frequency-response curves resulting from non-linearity is considered. Effects of initial amplitude and phase values, amplitude of excitation, and damping coefficient on modal amplitudes, are determined.     

Analysis of guided waves with a nonlinear boundary condition caused by internal resonance using the method of multiple scales

We theoretically investigated the cumulative nonlinear guided waves caused by internal resonance, using the method of multiple scales (MMS), which can construct better approximations to the solutions of perturbation problems. In this study, we consider nonlinearity only on the boundary instead of material nonlinearity or geometric nonlinearity. We showed nonlinear effects on the amplitudes of a lower mode and a higher mode depending on the propagation length. Also, we examined effects of wavenumber detuning from a phase matching condition of the two modes. If the wavenumber detuning is exactly equal to zero, the mechanical energy of the lower mode is transferred through nonlinear coupling to the energy of the higher mode, unilaterally. However, if a wavenumber detuning is not equal to zero, amplitude of the two modes change in a cyclic fashion during wave propagation. The amount of this amplitude variation and its cycle length are determined by the eigenfunctions of the two modes, the nonlinear parameter and the wavenumber detuning.     

Analytical study of the nonlinear behavior of a shape memory oscillator: Part II—resonance secondary

In this work, the response of a single-degree-of-freedom shape memory oscillator subjected to the excitation harmonic has been investigated. Equation of motion is formulated assuming a polynomial constitutive model to describe the restitution force of the oscillator. Here the method of multiple scales is used to obtain an approximate solution to the equations of the motion describing the modulation equations of amplitude and phase, and to investigate theoretically its stability. This work is presented in two parts. In Part I of this study we showed the modeling of the problem where the free vibration of the oscillator at low temperature is analyzed, where martensitic phase is stable. Part I also presents the investigation dynamics of the primary resonance of the pseudoelastic oscillator. Part II of the work is focused on the study in the secondary resonance of a pseudoelastic oscillator using the model developed in Part I. The analysis of the system in Part I as well as in Part II is accomplished numerically by means of phase portraits, Lyapunov exponents, power spectrum and Poincare maps. Frequency-response curves are constructed for shape memory oscillators for various excitation levels and detuning parameter. A rich class of solutions and bifurcations, including jump phenomena and saddle-node bifurcations, is found.     

Non-linear oscillator limit cycle analysis using a time transformation approach

A method is presented for the analysis of limit cycle behavior of autonomous non-linear oscillators characterized by second order ordinary differential equations containing a small parameter. The method differs from the classical perturbation methods in that the dependent variable is not expanded in a power series in the small parameter. Rather, a new independent variable is sought such that in its domain the motion is simple harmonic. Use of this time transformation technique to generate limit cycle phase portrait, amplitude and period is presented. We show results of the application of the method to the van der Pol oscillator, to an oscillator with quadratic damping, and to a modified van der Pol oscillator which is statically unstable in the limit of small motion.     

Analytical study of the nonlinear behavior of a shape memory oscillator: Part I—primary resonance and free response at low temperatures

In this work, the response of a single-degree-of-freedom shape memory oscillator subjected to the excitation harmonic has been investigated. Equation of motion is formulated assuming a polynomial constitutive model to describe the restitution force of the oscillator. Here the method of multiple scales is used to obtain an approximate solution to the equations of the motion describing the modulation equations of amplitude and phase, and to investigate theoretically its stability. This work is presented in two parts. In Part I of this study we showed the modeling of the problem where the free vibration of the oscillator at low temperature is analyzed, where martensitic phase is stable. Part I also presents the investigation dynamics of the primary resonance of the pseudoelastic oscillator. Part II of the work is focused on the study in the secondary resonance of a pseudoelastic oscillator using the model developed in Part I. The analysis of the system in Part I as well as in Part II is accomplished numerically by means of phase portraits, Lyapunov exponents, power spectrum and Poincare maps. Frequency-response curves are constructed for shape memory oscillators for various excitation levels and detuning parameter. A rich class of solutions and bifurcations, including jump phenomena and saddle-node bifurcations, is found.     

Frequency lock-in during nonlinear vibration of an airfoil coupled with van der Pol Oscillator

The frequency lock-in during the nonlinear vibration of a turbomachinery blade is modeled using a spring-mounted airfoil coupled with a van der Pol Oscillator (VDP) oscillator. The proposed reduced-order model uses the nonlinear VDP oscillator to represent the oscillatory nature of wake dynamics caused by the vortex shedding. The damping term in the VDP oscillator is assumed to be nonlinear. The coupled equations governing the pitch and plunge motion of an airfoil are used to approximate the vibration of a turbomachinery blade. Springs having cubic-order nonlinearity for their stiffnesses are used to mount the airfoil. The unsteady lift acting on the blade is modeled using a self-excited nonlinear wake oscillator. The model for wake dynamics takes into account the influence of blade inertia. The nonlinear coupled three degrees of freedom (dof) aeroelastic system is studied for instability resulting in the frequency lock-in phenomenon. The equations are transformed into non-dimensional form, and then the frequencies of the coupled system are plotted to demonstrate the frequency lock-in. Further, the method of multiple scales is used to derive modulation equations which represent the amplitude and phase of the oscillation. The results obtained using the method of multiple scales are compared with direct numerical solutions to verify the present modeling method. The steady-state amplitudes of the response are plotted against the detuning parameter, which represents the frequency response curve. Further, the sensitivity of non-dimensional parameters such as coupling coefficients, mass ratio, reduced velocity, static unbalance, structural damping coefficient and the ratio of uncoupled pitch and plunge natural frequencies on the frequency response is investigated. The study revealed that parameters such as mass ratio, reduced velocity, structural damping coefficient, and coupling coefficients have a stronger influence in suppressing the amplitude of vibration. Meanwhile, parameters such as the frequency ratio, static unbalance, reduced velocity, and mass ratio significantly affect the range of frequency in which the lock-in phenomenon happens. Further, linear perturbation analysis is done to understand the qualitative effect of the system parameters such as coupling coefficients, mass ratio, frequency ratio, and static unbalance on the range of lock-in.     

Non-linear vibration of variable speed rotating viscoelastic beams

Non-linear vibration of a variable speed rotating beam is analyzed in this paper. The coupled longitudinal and bending vibration of a beam is studied and the governing equations of motion, using Hamilton’s principle, are derived. The solutions of the non-linear partial differential equations of motion are discretized to the time and position functions using the Galerkin method. The multiple scales method is then utilized to obtain the first-order approximate solution. The exact first-order solution is determined for both the stationary and non-stationary rotating speeds. A very close agreement is achieved between the simulation results obtained by the numerical integration method and the first-order exact solution one. The parameter sensitivity study is carried out and the effect of different parameters including the hub radius, structural damping, acceleration, and the deceleration rates on the vibration amplitude is investigated.     

The transition from phase locking to drift in a system of two weakly coupled van der pol oscillators

We investigate the slow flow resulting from the application of the two variable expansion perturbation method to a system of two linearly coupled van der Pol oscillators. The slow flow consists of three non-linear coupled odes on the amplitudes and phase difference of the oscillators. We obtain regions in parameter space which correspond to phase locking, phase entrainment and phase drift of the coupled oscillators. In the slow flow, these states correspond respectively to a stable equilibrium, a stable limit cycle and a stable libration orbit. Phase entrainment, in which the phase difference between the oscillators varies periodically, is seen as an intermediate state between phase locking and phase drift. In the slow flow, the transitions between these states are shown to be associated with Hopf and saddle-connection bifurcations.     

Arbitrary amplitude periodic solutions for parametrically excited systems with time delay

Periodic solutions for parametrically excited system under state feedback control with a time delay are investigated. Using the asymptotic perturbation method, two slow-flow equations for the amplitude and phase of the parametric resonance response are derived. Their fixed points correspond to limit cycles (phase-locked periodic solutions) for the starting system. In the system without control, periodic solutions (if any) exist only for fixed values of amplitude and phase and depend on the system parameters and excitation amplitude. In many cases, the amplitudes of periodic solutions do not correspond to the technical requirements. On the contrary, it is demonstrated that, if the vibration control terms are added, stable periodic solutions with arbitrarily chosen amplitude and phase can be accomplished. Therefore, an effective vibration control is possible if appropriate time delay and feedback gains are chosen.     

A note on isoenergetic stability

In this note we consider certain two-degree-of-freedom Hamiltonian systems which may be regarded as perturbations of integrable systems governed by a real parameter ε. We wish to study the stability, at fixed energy, of certain periodic solutions. Two constants are defined, computable in terms of the original Hamiltonian function and the energy. The main theorem then states that if these constants are not zero, the periodic solutions are isoenergetically stable for sufficiently small ε. The proof is an application of the Twist Theorem of Kolmogorov-Arnol'd-Moser. By way of illustration, we apply the theorem to a mechanical system consisting of coupled non-linear oscillators. The periodic solutions are the “normal modes” and ε governs the non-linearity of the system. One obtains stability criteria for arbitrary energies and small ε, or, alternatively, for arbitrary ε and small energies.     

Usefulness of passive non-linear energy sinks in controlling galloping vibrations

The suppression of vibration amplitudes of an elastically-mounted square prism subjected to galloping oscillations by using a non-linear energy sink is investigated. The non-linear energy sink consists of a secondary system with linear damping and non-linear stiffness. A representative model that couples the transverse displacement of the square prism and the non-linear energy sink is constructed. A linear analysis is performed to determine the impacts of the non-linear energy sink parameters (mass, damping, and stiffness) on the coupled frequency and onset speed of galloping. It is demonstrated that increasing the damping of the non-linear energy sink can result in a significant increase in the onset speed of galloping. Then, the normal form of the Hopf bifurcation is derived to identify the type of instability and to determine the effects of the non-linear energy sink stiffness on the performance of the aeroelastic system near the bifurcation. The results show that the non-linear energy sink can be efficiently implemented to significantly reduce the galloping amplitude of the square prism. It is also shown that the multiple stable responses of the coupled aeroelastic system are obtained as well as the periodic responses, which are dependent on the considered non-linear energy sink parameters.     

Bifurcation Analysis of Parametrically Excited Rayleigh–Liénard Oscillators

A parametrically excited Rayleigh–Liénard oscillator is investigatedby an asymptotic perturbation method based on Fourier expansion and timerescaling. Two coupled equations for the amplitude and the phase ofsolutions are derived and the stability of steady-state periodic solutionsas well as parametric excitation-response and frequency-response curvesare determined. Comparison with the parametrically excited Liénardoscillator is performed and analytic approximate solutions are checkedusing numerical integration. Dulac's criterion, thePoincaré–Bendixson theorem, and energy considerations are used in order to study the existence and characteristics of limit cycles of the twocoupled equations. A limit cycle corresponds to a modulated motion forthe Rayleigh–Liénard oscillator. Modulated motion can be also obtainedfor very low values of the parametric excitation, and in this case, anapproximate analytic solution is easily constructed. If the parametricexcitation is increased, an infinite-period bifurcation is observed because the modulation period lengthens and becomes infinite, while themodulation amplitude remains finite and suddenly the attractor settlesdown into a periodic motion. Floquet's theory is used to evaluatethe stability of the periodic solutions, and in certain cases,symmetry-breaking bifurcations are predicted. Numerical simulationsconfirm this scenario and detect chaos and unbounded motions in theinstability regions of the periodic solutions.     

Nonlinear response of an initially buckled beam with 1:1 internal resonance to sinusoidal excitation

The nonlinear response of an initially buckled beam in the neighborhood of 1:1 internal resonance is investigated analytically, numerically, and experimentally. The method of multiple time scales is applied to derive the equations in amplitudes and phase angles. Within a small range of the internal detuning parameter, the first mode; which is externally excited, is found to transfer energy to the second mode. Outside this region, the response is governed by a unimodal response of the first mode. Stability boundaries of the unimodal response are determined in terms of the excitation level, and internal and external detuning parameters. Boundaries separating unimodal from mixed mode responses are obtained in terms of the excitation and internal detuning parameters. Stationary and non-stationary solutions are found to coexist in the case of mixed mode response. For the case of non-stationary response, the modulation of the amplitude depends on the integration increment such that the motion can be periodically or chaotically modulated for a choice of different integration increments. The results obtained by multiple time scales are qualitatively compared with those obtained by numerical simulation of the original equations of motion and by experimental measurements. Both numerical integration and experimental results reveal the occurrence of multifurcation, escaping from one well to the other in an irregular manner. and chaotic motion.     

Parametric Excitation for Two Internally Resonant van der Pol Oscillators

The response of a system of two nonlinearly coupled van der Poloscillators to a principal parametric excitation in the presence ofone-to-one internal resonance is investigated. The asymptoticperturbation method is applied to derive the slow flow equationsgoverning the modulation of the amplitudes and the phases of the twooscillators. These equations are used to determine steady-stateresponses, corresponding to a periodic motion for the starting system(synchronisation), and parametric excitation-response andfrequency-response curves. Energy considerations are used to studyexistence and characteristics of limit cycles of the slow flowequations. A limit cycle corresponds to a two-period amplitude- andphase-modulated motion for the van der Pol oscillators. Two-periodmodulated motion is also possible for very low values of the parametricexcitation and an approximate analytic solution is constructed for thiscase. If the parametric excitation increases, the oscillation period ofthe modulations becomes infinite and an infinite-period bifurcationsoccur. Analytical results are checked with numerical simulations.     

Accurate Higher-Order Analytical Approximate Solutions to Large-Amplitude Oscillating Systems with a General Non-Rational Restoring Force

A new approximate analytical approach for accurate higher-order nonlinear solutions of oscillations with large amplitude is presented in this paper. The oscillatory system is subjected to a non-rational restoring force. This approach is built upon linearization of the governing dynamic equation associated with the method of harmonic balance. Unlike the classical harmonic balance method, simple linear algebraic equations instead of nonlinear algebraic equations are obtained upon linearization prior to harmonic balancing. This approach also explores large parameter regions beyond the classical perturbation methods which in principle are confined to problems with small parameters. It has significant contribution as there exist many nonlinear problems without small parameters. Through some examples in this paper, we establish the general approximate analytical formulas for the exact period and periodic solution which are valid for small as well as large amplitudes of oscillation.     

Oscillators with a quasi-constant restoring force: approximations for motion

In this work approximate solutions to conservative single-degree of freedom oscillators with a restoring force close to the one with a constant magnitude are derived. Approximate solutions are assumed as a truncated Fourier series and harmonic balancing is applied. In addition, the assumption that the response of the oscillators considered is close to the response of the antisymmetric oscillator is introduced. It is suggested in a novel way how to modify the differential equation of motion with the assumed solution so as to derive explicit expressions for the frequency and the amplitudes of harmonics in the first, second and third approximation are presented. The comparison of the results obtained with numerical solutions as well as with some existing approximate analytical results from the literature is also carried out, showing excellent accuracy.     

Non-linear analysis of creeping flow on the inclined permeable substrate plane subjected to an electric field

The effect of an externally applied electric field on the stability of a thin fluid film over an inclined porous plane is analyzed using linear and non-linear stability analysis in the long wave limit. The principle aim of this study is to illustrate the influence of electric field on the non-linear stability of a thin liquid layer flow down incline substrate when the plane is porous. The driving force for the instability under an electric field is an electrostatic force exerted on the free charges accumulated at the dividing interface. The coupled non-linear evolution equations for the local film thickness and the interfacial charge for two-dimensional disturbances are derived to analyze the effect of long-wave instabilities. The method of multiple scales is applied to obtain approximate solutions and analyze the stability criteria. Numerical simulations of this system of non-linear evolution equations are performed. It is found that the permeability parameter as well as the inclination of the plane plays a destabilizing role in the stability criteria, while the damping influence is observed for increasing of the electrical conductivity in both linear and non-linear behavior.     

On the encounter of an acoustic shear pulse with a phase boundary in an elastic material: reflection and transmission behavior

The fully dynamical motion of a phase boundary is considered for a specific class of elastic materials whose stress-strain relation in simple shear is nonmonotone. It is shown that a preexisting stationary phase boundary in a prestressed layer composed of such a material can be set in motion by a finite amplitude shear pulse. An infinity of solutions is possible according to the present theory, each of which is characterized by different reflected and transmitted waves at the phase boundary. A global analysis gives exact bounds on the size of the solution family for different shear pulse amplitudes. For certain ranges of shear pulse amplitudes a completely reflecting solution will exist, while for an in general different range of shear pulse amplitudes a completely transmitting solution will exist. The properties of these different solutions are examined. In particular, it is observed that the ringing of a shear pulse between the external boundaries and the internal phase boundary gives rise to periodic phase boundary motion for both the case of a completely reflecting phase boundary and a completely transmitting phase boundary.     

Synthesis of oscillators for exact desired periodic solutions having a finite number of harmonics

The synthesis of autonomous oscillators with exact desired periodic steady-state solution is described in this contribution. The vector field of the oscillator differential equation is built up with a conservative and a dissipative part. Both parts are synthesized using an algebraic function describing the desired limit cycle. The desired periodic motion is restricted by a finite numbers of harmonics, whereby the amplitude and the phase shift of every harmonic can be freely chosen, depending on the specific application. Afterwards the synthesis of a periodically driven oscillator with an exact desired periodic response is described. For this purpose, the differential equation of the autonomous oscillator is extended by a state-dependent compensation term that equals the excitation at the steady-state solution. Here the freely definable amplitudes and phase angles of the oscillator motion are restricted by the existence and stability conditions for synchronization.