Multivalued Response Research Articles

Effect of variation of input banwidth on the response of hardening Duffing oscillators

The response of hardening Duffing oscillators to coloured noise excitation is considered. It is shown that for certain combinations of excitation intensity and bandwidth the system realises multi-valued response states. Theoretical predictions of bounds in the parameter space which define regions of multiple statistical moments are supported by numerical simulation results. The effect of the occurrence and persistence of multi-level responses on the probability distribution of the peaks is also considered.

Exact Solutions for the Free Vibration in an Asymmetrical Duffing Equation.

An algorithm is proposed to find exact solutions for the free vibration in an asymmetrical Duffing oscillator. The system is composed of a spring function of the Duffing type (with linear and cubic terms with respect to displacement) accompanied by a constant force. It is reduced from the most comprehensive 3rd order polynomial having arbitrary terms from constant to the 3rd power. By employing a bilinear transformation, the system is successfully converted into a regular Duffing equation whose exact solutions already exist and are abstracted here. The whole procedure to obtain the present solution is summarized in a flow chart. Numerical values of conversion constants, skeleton curves and waveforms are computed and illustrated for the typical cases of asymmetrical (hard, soft and snap-through) spring systems. It is demonstrated that the skeleton curves and waveforms are asymmetrical and that, some of them exhibit multivalued responses for a given frequency.

Nonlinear random coupled motions of structural elements with quadratic nonlinearities

The second-order closure method is used to analyze the nonlinear response of two-degree-of-freedom systems with quadratic nonlinearities. The excitation is assumed to be the sum of a deterministic harmonic component and a random component. The case of primary resonance of the second mode in the presence of a two-to-one internal (autoparametric) resonance is investigated. The method of multiple scales is used to obtain four first-order ordinary-differential equations that describe the modulation of the amplitudes and phases of the two modes. Applying the second-order closure method to the modulation equations, we determine the stationary mean and mean-square responses. For the case of a narrow-band random excitation, the results show that the presence of the nonlinearity causes multi-valued mean-square responses. The multi-valuedness is responsible for a jump phenomenon. Contrary to the results of the linear analysis, the nonlinear analysis reveals that the directly excited second mode takes a small amount of the input energy (saturates) and spills over the rest of the input energy into the first mode, which is indirectly excited through the autoparametric resonance.

Non-linear response of a fluid valve

The method of multiple scales is used to analyse the non-linear response of a relief valve under combined static and dynamic loadings. First-order ordinary-differential equations are derived for the evolution of the amplitudes and phases for the following resonances: primary resonance; subharmonic resonance of order one-third; superharmonic resonance of order-three; subharmonic resonance of order one-half; superharmonic resonance of order-two; and combination resonances of both the additive and the difference types. The evolution equations are used to determine the steady-state motions and representative frequency-response curves are presented for each resonance. The stability of the steady-stae solutions is investigated. Moreover, unlike linear systems, multivalued responses are possible for non-linear systems, and hence the initial conditions determine the response. The multivaluedness of the solution may cause jumps, and hence may cause unsatisfactory performance of the relief valve.

Continuation methods in semiconductor device simulation

Predictor-corrector continuation methods for characterizing the voltage-current ( V, I) behavior of semiconductor devices are presented. Numerical simulations of some complex CMOS structures demonstrate the efficacy of continuation methods; in particular, it is possible to accurately determine the limit points of certain ( V, I) curves, corresponding to latchup triggering and holding points. Continuation techniques, coupled with grid adaption, provide substantial improvement in computational efficiency over previous approaches and are well suited to deal with multivalued current responses.

Random superharmonic response of a Duffing oscillator

The third order superharmonic response of a Duffing oscillator to narrow band random excitation is analyzed. The analysis shows the effect of excitation bandwidth on the response and stability of the non-linear oscillator. In particular the analysis shows that multivalued superharmonic response can occur only when the excitation bandwidth is small. Stability of the response is examined by constructing a mean square phase plane from a time dependent Fokker-Planck equation and also by perturbing the stationary solutions.

Random superharmonic resonance

The third‐order superharmonic mean‐square response of a Duffing oscillator to narrow‐band random excitation is analyzed. The analysis uses harmonic balance to separate the primary and superharmonic responses and statistical linearization to obtain a differential equation for the autocorrelation of the superharmonic response. The main emphasis of the analysis is to demonstrate the effect of excitation bandwidth on the response and stability of the oscillator. The analysis shows that a tripling of both excitation frequency and bandwidth occur and that occurrence of multivalued superharmonic response and jump phenomena depends upon the bandwidth of excitation. Stability of the superharmonic response is considered by generating a mean‐square phase plane which shows that multiple values represented by two stable sinks and an unstable saddle point occur when the excitation bandwidth is small. The phase plane reduces to a single sink when the excitation bandwidth is increased. Stability is also examined by pertu...

Isolas in four-wave mixing optical bistability

Optical bistability based on the cross-interaction of two counterpropagating plane waves in a Kerr-like medium is considered. It is shown that the input-output characteristics of such a system have not only multivalued hysteretic response (which was demonstrated by Winful and Marburger in 1980), but also multiple isolated branches (so called isolas) with nonhysteretic jumps from these branches to the main curve (but not vice versa). The isolas were found both in the case in which the problem can be solved analytically (particularly for electrostriction mechanism of optical nonlinearity) and in the general case of arbitrary light-induced anisotropy.

Response characteristics of knollenberg light-scattering aerosol counters

Response calculations are presented for four commercially available “Knollenberg” light-scattering aerosol counters: the classical scattering aerosol spectrometer probe (CSASP), the active scattering aerosol spectrometer probe (ASASP), the forward scattering spectrometer probe (FSSP) and the axially scattering spectrometer probe (ASSP). The results show sensitivity of the response to aerosol refractive index for values of indexes characteristic of atmospheric aerosols, and for a particular index, multivalued response for particles in the 0.5–4 μm radius range. The response calculations have been validated for two of these instruments (the CSASP and ASASP) by measurement of monodisperse spherical particles. The size resolution of these two instruments is significantly less than advertised by the manufacturer and measurement of irregular particles causes additional loss of resolution.

Response Calculations for Light-Scattering Aerosol Particle Counters

Response calculations for five commercial light-scattering aerosol particle counters have been carried out that take into account the emissive power of the light source, the spectral sensitivity of the phototube, and the specific geometrical factors for each instrument. Earlier calculations had been published for two of these instruments, but these had not considered each of these factors appropriately. The results indicate a strong dependence of response upon both the real and imaginary part of the refractive index and, for a given refractive index a multivalued response in the submicrometer range for three of the five instruments.

Response of Hard-Spring Oscillator to Narrow-Band Excitation

The response of an oscillator with a nonlinear stiffness of the hard-spring type to narrow band Gaussian random noise is analyzed theoretically by using the method of quasi linearization. The particular point of interest is the possibility of an occurrence of multiple valued response or “jumps” such as one observes when the exciting force is sinusoidal. It is argued that while it does not seem possible to have multivalued response with wide-band (white noise) excitation, the response to narrow band excitation might exhibit such behavior owing to the temporal correlation of the source with the response. Numerical computations of the response curves do suggest multivalued response. In experiments with a nonlinear oscillator, multivalued response was observed, but agreement between theory and experiment may only be claimed to be qualitative.

Stability of forced oscillations in nonlinear feedback systems

It has been known for a considerable time that non-linear systems may exhibit multivalued response under external periodic excitation. The stability of these forced oscillations has been extensively investigated for second-order systems and a general criterion for their stability was found. Stability of higher order systems was investigated by means of the incremental describing function. This function must be calculated for every case of interest. In this paper the general criterion formerly derived for second-order systems is extended to higher order systems. Thus it is not necessary to make a special stability investigation in every case.