Additive Resonances Research Articles

Numerical simulation of stability and responses of dynamic systems under parametric excitation

The stability and responses of dynamic systems under parametric excitation are often encountered in many fields of science and engineering, such as slender columns and thin plates under axial loadings. This paper proposes a novel numerical simulation method to simultaneously construct stability diagrams and predict the responses of multiple degree-of-freedom (DOF) dynamic systems under arbitrary parametric loadings. The method divides an arbitrary load into discrete segments to approximate the variable excitation function by a step function and then accumulates the system responses of each segment using a matrix method. Dynamic stability and response analysis of undamped and damped 1DOF, 2DOF, 3DOF, and multiple DOF systems are conducted, and numerical examples are compared to conventional methods to show the accuracy and advantage of the proposed numerical simulation method. The instability diagrams are also substantiated by vibration response curves obtained from the same method. The method is applied to calibrate the validity and ranges of applicability of the Hill infinite determinants in the Floquet theory of 1DOF Mathieu-Hill equations. The second order approximation of the first instability region from the Hill determinant is acceptable. However, the fourth order approximation of the second instability region must be used to obtain a relatively accurate result. For 2DOF, 3DOF, and multi-parameter linear periodic systems, the method can simultaneously render four types of instability regions in a single procedure: no resonances, simple parametric resonances, combination additive resonances, and combination differential resonances. These results could provide a better understanding of stability and responses of dynamic systems under parametric excitation.

Selectively Enhancing Solar Scattering for Direct Radiative Cooling through Control of Polymer Nanofiber Morphology.

Radiative cooling can alleviate urban heat island effects and passively improve personal thermal comfort. Among many emerging approaches, infrared (IR) transparent films and fabrics are promising because they can allow objects to directly radiate heat through bands of atmospheric transparency while blocking solar heating. However, achieving high solar reflectance while maintaining IR transmittance using scalable nanostructured materials requires control over the shape and size distribution of the nanoscale building blocks. Here, we investigate the scattering and transmission properties of electrospun polyacrylonitrile (PAN) nanofibers that feature spherical, ellipsoidal, and cylindrical morphologies. We find that nanofibers that have ellipsoidal beads exhibit the most efficient solar scattering, mainly due to the additive dielectric resonances of the ellipsoidal and cylindrical geometries, as confirmed through electromagnetic simulations. This favorable scattering decreases the amount of material needed to reach above 95% solar reflectance, which, in turn, enables high infrared transmittance (>70%) despite PAN's intrinsic IR absorption. We further show that these PAN nanofibers (nanoPAN) can enable cooling of surfaces with relatively low solar reflectance, which is demonstrated by covering a reference blackbody surface with beaded nanoPAN. During peak solar hours, this configuration lowers the temperature of the black surface by approximately 50 °C and is able to achieve as low as 3 °C below the ambient air temperature. More broadly, our demonstration using PAN, which is not as IR transparent as more commonly used polyethylene, provides a method for utilizing lower purity materials in radiative cooling.

Parametric vibrations and instabilities of an elliptical gear pair

Parametric vibrations and instabilities can be induced in an elliptical gear pair. This study examined the effects of basic parameters on vibration instability. Geometric parameters and time-variant mesh stiffness are calculated. A torsional dynamic model is established by introducing the lumped parameter assumption. Based on the model, the parametric vibrations caused by the excitations of eccentricity and time-variant mesh stiffness are investigated, and the additive and difference combination resonances are identified. Comparison between the elliptical and circular gears implies that the vibration amplitude at mesh frequency of the elliptical gears is lower than that of the regular circular gears. The vibrations caused by load torque are also examined. The results show that the vibrations with different frequencies are coupled with each other. The inherent reason behind the coupling is the interaction between the eccentricity, time-variant stiffness, and load.

Periodic and Chaotic Motion of a Time-Delayed Nonlinear System Under Two Coexisting Families of Additive Resonances

A time-delayed quadratic nonlinear mechanical system can exhibit two coexisting stable bifurcating solutions (SBSs) after two-to-one resonant Hopf bifurcations occur in the corresponding autonomous time-delayed system. One SBS is of small-amplitude and has the Hopf bifurcation frequencies (HBFs), while the other is of large-amplitude and contains the shifted Hopf bifurcation frequencies (the shifted HBFs). When the forcing frequency is tuned to be the sum of two HBFs or the sum of two shifted HBFs, two families of additive resonances can be induced in the forced response. The forced response under the additive resonance related to the HBFs can demonstrate periodic, quasi-periodic and chaotic motion. On the contrary, the forced response under the additive resonance associated with the shifted HBFs may exhibit period-three periodic motion and quasi-periodic motion. Bifurcation diagrams, time trajectories, frequency spectra, phase portraits and Poincaré sections are presented to show periodic, quasi-periodic, and chaotic motion of the time-delayed nonlinear system under the two families of additive resonances.

Multifrequency excitation of a clamped-clamped microbeam: Analytical and experimental investigation.

Using partial electrodes and a multifrequency electrical source, we present a large-bandwidth, large-amplitude clamped–clamped microbeam resonator excited near the higher order modes of vibration. We analytically and experimentally investigate the nonlinear dynamics of the microbeam under a two-source harmonic excitation. The first-frequency source is swept around the first three modes of vibration, whereas the second source frequency remains fixed. New additive and subtractive resonances are demonstrated. We illustrated that by properly tuning the frequency and amplitude of the excitation force, the frequency bandwidth of the resonator is controlled. The microbeam is fabricated using polyimide as a structural layer coated with nickel from the top and chromium and gold layers from the bottom. Using the Galerkin method, a reduced order model is derived to simulate the static and dynamic response of the device. A good agreement between the theoretical and experimental data are reported.

Nonlinear dynamic response of a fractionally damped cylindrical shell with a three-to-one internal resonance

Nonlinear vibrations of a cylindrical shell embedded in a fractional derivative viscoelastic medium and subjected to the different conditions of the internal resonance are investigated. The viscous properties of the surrounding medium are described by Riemann–Liouville fractional derivative. The displacement functions are determined in terms of eigenfunctions of linear vibrations of a free-simply supported shell. A novel procedure resulting in decoupling linear parts of equations is proposed with the further utilization of the method of multiple scales for solving nonlinear governing equations of motion by expanding the amplitude functions into power series in terms of the small parameter and different time scales. It is shown that the phenomenon of internal resonance can be very critical, since in a circular cylindrical shell the one-to-one, two-to-one, and three-to-one internal resonances, as well as additive and difference combinational resonances are always present. The three-to-one internal resonance is analyzed in detail. Since the internal resonances belong to the resonances of the constructive type, i.e., all of them depend on the geometrical dimensions of the shell under consideration and its mechanical characteristics, that is why such resonances could not be ignored and eliminated for a particularly designed shell. It is shown that the energy exchange could occur between two or three subsystems at a time: normal vibrations of the shell, its torsional vibrations and shear vibrations along the shell axis. Such an energy exchange, if it takes place for a rather long time, could result in crack formation in the shell, and finally to its failure. The energy exchange is illustrated pictorially by the phase portraits, wherein the phase trajectories of the phase fluid motion are depicted.

Additive resonances of a controlled van der Pol–Duffing oscillator

The trivial equilibrium of a controlled van der Pol–Duffing oscillator with nonlinear feedback control may lose its stability via a non-resonant interaction of two Hopf bifurcations when two critical time delays corresponding to two Hopf bifurcations have the same value. Such an interaction results in a non-resonant bifurcation of co-dimension two. In the vicinity of the non-resonant Hopf bifurcations, the presence of a periodic excitation in the controlled oscillator can induce three types of additive resonances in the forced response, when the frequency of the external excitation and the frequencies of the two Hopf bifurcations satisfy a certain relationship.With the aid of centre manifold theorem and the method of multiple scales, three types of additive resonance responses of the controlled system are investigated by studying the possible solutions and their stability of the four-dimensional ordinary differential equations on the centre manifold. The amplitudes of the free-oscillation terms are found to admit three solutions; two non-trivial solutions and the trivial solution. Of two non-trivial solutions, one is stable and the trivial solution is unstable. A stable non-trivial solution corresponds to a quasi-periodic motion of the original system. It is also found that the frequency-response curves for three cases of additive resonances are an isolated closed curve. It is shown that the forced response of the oscillator may exhibit quasi-periodic motions on a three-dimensional torus consisting of three frequencies; the frequencies of two bifurcating solutions and the frequency of the excitation. Illustrative examples are given to show the quasi-periodic motions.

Analysis of free non-linear vibrations of a viscoelastic plate under the conditions of different internal resonances

Non-linear free damped vibrations of a rectangular plate described by three non-linear differential equations are considered when the plate is being under the conditions of the internal resonance one-to-one, and the internal additive or difference combinational resonances. Viscous properties of the system are described by the Riemann-Liouville fractional derivative of the order smaller than unit. The functions of the in-plane and out-of-plane displacements are determined in terms of eigenfunctions of linear vibrations with the further utilization of the method of multiple scales, in so doing the amplitude functions are expanded into power series in terms of the small parameter and depend on different time scales, but the fractional derivative is represented as a fractional power of the differentiation operator. It is assumed that the order of the damping coefficient depends on the character of the vibratory process and takes on the magnitude of the amplitudes’ order. The time-dependence of the amplitudes in the form of incomplete integrals of the first kind is obtained. Using the constructed solutions, the influence of viscosity on the energy exchange mechanism is analyzed which is intrinsic to free vibrations of different structures being under the conditions of the internal resonance. It is shown that each mode is characterized by its damping coefficient which is connected with the natural frequency of this mode by the exponential relationship with a negative fractional exponent. It is shown that viscosity may have a twofold effect on the system: a destabilizing influence producing unsteady energy exchange, and a stabilizing influence resulting in damping of the energy exchange mechanism.

On the interaction between forced and combination resonances

The purpose of this paper is to present a general theory for analysis of the interaction between forced and additive combination resonances arising from the in-plane dynamic excitation of an imperfect rectangular plate. The dynamic Non Kármán non-linear equations are adopted. The stress function and lateral deflection are expanded into two series with time and space dependent coefficients. The system of differential equations furnished by the Galerkin procedure is analyzed by the first order generalized asymptotic method. The numerical results show that interaction between forced and combination resonances depend on the distance between frequencies and the degree of overlapping between the instability zones and can manifest itself in different ways.

Linear gain and phase in seismic neurons

In nearly quiescent seismic environments, many bullfrog (Rana catesbeiana) saccular afferent axons exhibit ongoing activity, typically ranging from 10 to 80 spikes/s. Low‐level vibration of the whole animal [see H. Koyama et al., Brain Res. 250, 168–172 (1982) for methods] produces linear modulation of this activity, allowing one to define small‐signal linear transfer ratios and to observe tuning curves for such axons. Each tuning curve to date reflects multiple resonances, sometimes coalescing with no intervening antiresonances, sometimes separated by simple antiresonances. The latter typically are much sharper than the resonances on either side, indicating that they are not simply concomitants of additive parallel resonances. Phase lag typically increases by two cycles or more with increasing frequency through the tuning range (roughly 10 to 200 Hz), with abrupt half‐cycle declines through the antiresonances. Resonant and antiresonant frequencies vary from axon to axon, usually lying between 20 and 120 Hz. The amplitudes of peak transfer ratios vary approximately from 50 to 200 spikes/s per cm/s2. At 100 Hz, where it occasionally is found, the latter corresponds to approximately one spike/s response for whole‐animal displacement of 10 pm. [Supported by NSF Grant BNS‐8005834.]

Stationary and non-stationary responses of a reinforced cylindrical shell near parametric resonance

An investigation of nonstationary and stationary responses was conducted for a reinforced circular cylindrical shell subjected to a time varying longitudinal load. Linearly varying external excitation frequencies were used when studying the nonstationary responses (passage through resonance). Checkerboard radial modes were employed in this study and the effect of shell imperfections in harmony with the radial modes were studied. Existence of both parametric and combination additive resonances have been established.