A theoretical and experimental investigation of a three-degree-of-freedom structure

The response of a structure to a simple-harmonic excitation is investigated theoretically and experimentally. The structure consists of two light-weight beams arranged in a T-shape turned on its side. Relatively heavy and concentrated weights are placed at the upper and lower free ends and at the point where the two beams are joined. The base of the T is clamped to the head of a shaker. Because the masses of the concentrated weights are much larger than the masses of the beams, the first three natural frequencies are far below the fourth; consequently, for relatively low frequencies of the excitation, the structure has, for all practical purposes, only three degrees of freedom. The lengths and weights are chosen so that the third natural frequency is approximately equal to the sum of the two lower natural frequencies, an arrangement that produces an autoparametric (also called an internal) resonance. A linear analysis is performed to predict the natural frequencies and to aid in the design of the experiment; the predictions and observations are in close agreement. Then a nonlinear analysis of the response to a prescribed transverse motion at the base of the T is performed. The method of multiple scales is used to obtain six first-order differential equations describing the modulations of the amplitudes and phases of the three interacting modes when the frequency of the excitation is near the third natural frequency. Some of the predicted phenomena include periodic, two-period quasiperiodic, and phase-locked (also called synchronized) motions; coexistence of multiple stable motions and the attendant jumps; and saturation. All the predictions are confirmed in the experiments, and some phenomena that are not yet explained by theory are observed.     

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.     

Non-linear resonances in the forced responses of plates,part II: Asymmetric responses of circular plates

The dynamic analogue of the von Karman equations is used to study the forced response, including asymmetric vibrations and traveling waves, of a clamped circular plate subjected to harmonic excitations when the frequency of excitation is near one of the natural frequencies. The method of multiple scales, a perturbation technique, is used to solve the non-linear governing equations. The approach presented provides a great deal of insight into the nature of the non-linear forced resonant response. It is shown that in the absence of internal resonance (i.e., a combination of commensurable natural frequencies) or when the frequency of excitation is near one of the lower frequencies involved in the internal resonance, the steady state response can only have the form of a standing wave. However, when the frequency of excitation is near the highest frequency involved in the internal resonance it is possible for a traveling wave component of the highest mode to appear in the steady state response.     

Vibrations of nearly annular and circular plates

A general procedure is presented to determine the natural frequencies and mode shapes of nearly annular and circular plates. A straightforward perturbation procedure is used with a transfer of the boundary conditions to obtain solutions given by simple expressions. As well as yielding good accuracy quantitatively, the method accurately predicts qualitative behavior. Numerical results are presented for clamped elliptical and square plates.     

Non-linear vibrations of parametrically excited cantilever beams subjected to non-linear delayed-feedback control

Non-linear feedback control provides an effective methodology for vibration mitigation in non-linear dynamic systems. However, within digital circuits, actuation mechanisms, filters, and controller processing time, intrinsic time-delays unavoidably bring an unacceptable and possibly detrimental delay period between the controller input and real-time system actuation. If not well-studied, these inherent and compounding delays may inadvertently channel energy into or out of a system at incorrect time intervals, producing instabilities and rendering controllers’ performance ineffective. In this work, we present a comprehensive investigation of the effect of time delays on the non-linear control of parametrically excited cantilever beams. More specifically, we examine three non-linear cubic delayed-feedback control methodologies: position, velocity, and acceleration delayed feedback. Utilizing the method of multiple scales, we derive the modulation equations that govern the non-linear dynamics of the beam. These equations are then utilized to investigate the effect of time delays on the stability, amplitude, and frequency–response behavior. We show that, when manifested in the feedback, even the minute amount of delays can completely alter the behavior and stability of the parametrically excited beam, leading to unexpected behavior and responses that could puzzle researchers if not well-understood and documented.     

Two-to-one internal resonance in microscanners

To realize large scanning angles, torsional microscanners are normally excited at their natural frequencies. Usually, a bias DC voltage is also applied to scan around a desired nonzero tilt angle. As a result, a deep understanding of the mirror’s response to a DC-shifted primary resonance excitation is imperative. Along these lines, we use the method of multiple scales to obtain a second-order nonlinear approximate analytical solution of the mirror steady-state response. We show that the response of the mirror exhibits a softening-type behavior that increases as the magnitude of the DC component increases. For a given mirror, we can also identify a DC voltage range wherein the mirror exhibits a two-to-one internal resonance between the first two modes; that is, ω ₂≈2ω ₁. To analyze the mirror behavior within that range, we first treat the case where the excitation frequency is near the first-mode frequency; that is, Ω≈ω ₁. Then we treat the case where the excitation frequency is near the second-mode frequency; that is, Ω≈ω ₂. We analyze the stability of the response and compare the analytical results to numerical solutions obtained via long-time integration of the equations of motion. We show that, due to the internal resonance, the mirror exhibits complex dynamic behavior characterized by aperiodic responses to primary resonance excitations. This behavior results in undesirable oscillations that are detrimental to the mirror performance, namely bringing the target point in and out of focus and resulting in distorted images.