The problem of response prediction is investigated for parametric vibration in terms of a new concept. The response solution is presented in the special form of Fourier series for signal degree freedom of parametric vibration based on modulation feedback. By applying harmonic balance and limitation operation, all coefficients of a harmonic component are fully determined with a set of series. Meanwhile, some important dynamic behaviors are exposed through mathematical deduction, and an instability phenomenon can be discussed through given frequency factors. The investigation result shows that the new approach has an advantage in the complete response expression, and it is very significant for the theoretical research and engineering application concerning parametric vibration.

Dishan Huang

Study of the Response Prediction of a Parametric Vibration Based on Modulation Feedback

A general theory for the free and forced responses of <svg style="vertical-align:-0.09pt" id="M1" height="6.4124999" version="1.1" viewBox="0 0 7.0875001 6.4124999" width="7.0875001"  xmlns="http://www.w3.org/2000/svg"    >
	<g transform="matrix(1.25,0,0,-1.25,0,6.4125)">
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			<text transform="matrix(1,0,0,-1,-71.95,67.01)">
				<tspan style="font-size: 11.21px; " x="0" y="0">𝑛</tspan>
			</text>

		</g>
	</g>
</svg> elastically connected parallel structures is developed. It is shown that if the stiffness operator for an individual structure is self-adjoint with respect to an inner product defined for <svg style="vertical-align:-1.755pt" id="M2" height="14.4625" version="1.1" viewBox="0 0 45.762501 14.4625" width="45.762501"  xmlns="http://www.w3.org/2000/svg"    >
	<g transform="matrix(1.25,0,0,-1.25,0,14.4625)">
		<g transform="translate(72,-60.43)">
			<text transform="matrix(1,0,0,-1,-71.95,62.23)">
				<tspan style="font-size: 11.21px; " x="0" y="0">𝐶</tspan>
			</text>

			<text transform="matrix(1,0,0,-1,-63.53,66.71)">
				<tspan style="font-size: 7.85px; " x="0" y="0">𝑘</tspan>
			</text>

			<text transform="matrix(1,0,0,-1,-58.78,62.23)">
				<tspan style="font-size: 11.21px; " x="0" y="0">[</tspan>
				<tspan style="font-size: 11.21px; " x="3.732264" y="0">0</tspan>
				<tspan style="font-size: 11.21px; " x="9.3362637" y="0">,</tspan>
				<tspan style="font-size: 11.21px; " x="13.998792" y="0">1</tspan>
				<tspan style="font-size: 11.21px; " x="19.602793" y="0">]</tspan>
			</text>

		</g>
	</g>
</svg>, then the stiffness operator for the set of elastically connected structures is self-adjoint with respect to an inner product defined on <svg style="vertical-align:-1.755pt" id="M3" height="14.4625" version="1.1" viewBox="0 0 104.075 14.4625" width="104.075"  xmlns="http://www.w3.org/2000/svg"    >
	<g transform="matrix(1.25,0,0,-1.25,0,14.4625)">
		<g transform="translate(72,-60.43)">
			<text transform="matrix(1,0,0,-1,-71.95,62.23)">
				<tspan style="font-size: 11.21px; " x="0" y="0">𝑈</tspan>
				<tspan style="font-size: 11.21px; " x="11.006256" y="0">=</tspan>
				<tspan style="font-size: 11.21px; " x="21.799561" y="0">𝑅</tspan>
			</text>

			<text transform="matrix(1,0,0,-1,-41.85,66.71)">
				<tspan style="font-size: 7.85px; " x="0" y="0">𝑛</tspan>
			</text>

			<text transform="matrix(1,0,0,-1,-34.96,62.23)">
				<tspan style="font-size: 11.21px; " x="0" y="0">×</tspan>
				<tspan style="font-size: 11.21px; " x="9.6500883" y="0">𝐶</tspan>
			</text>

			<text transform="matrix(1,0,0,-1,-16.88,66.71)">
				<tspan style="font-size: 7.85px; " x="0" y="0">𝑘</tspan>
			</text>

			<text transform="matrix(1,0,0,-1,-12.13,62.23)">
				<tspan style="font-size: 11.21px; " x="0" y="0">[</tspan>
				<tspan style="font-size: 11.21px; " x="3.732264" y="0">0</tspan>
				<tspan style="font-size: 11.21px; " x="9.3362637" y="0">,</tspan>
				<tspan style="font-size: 11.21px; " x="13.998792" y="0">1</tspan>
				<tspan style="font-size: 11.21px; " x="19.602793" y="0">]</tspan>
			</text>

		</g>
	</g>
</svg>. This leads to the definition of energy inner products defined on <svg style="vertical-align:-0.14626pt" id="M4" height="9.4499998" version="1.1" viewBox="0 0 11.5375 9.4499998" width="11.5375"  xmlns="http://www.w3.org/2000/svg"    >
	<g transform="matrix(1.25,0,0,-1.25,0,9.45)">
		<g transform="translate(72,-64.44)">
			<text transform="matrix(1,0,0,-1,-71.95,64.63)">
				<tspan style="font-size: 11.21px; " x="0" y="0">𝑈</tspan>
			</text>

		</g>
	</g>
</svg>. When a normal mode solution is used to develop the free response, it is shown that the natural frequencies are the square roots of the eigenvalues of an operator that is self-adjoint with respect to the energy inner product. The completeness of the eigenvectors in <svg style="vertical-align:-0.18pt" id="M5" height="9.5" version="1.1" viewBox="0 0 16.112499 9.5" width="16.112499"  xmlns="http://www.w3.org/2000/svg"    >
	<g transform="matrix(1.25,0,0,-1.25,0,9.5)">
		<g transform="translate(72,-64.4)">
			<text transform="matrix(1,0,0,-1,-71.95,64.63)">
				<tspan style="font-size: 11.21px; " x="0" y="0">𝑊</tspan>
			</text>

		</g>
	</g>
</svg> is used to develop a forced response. Special cases are considered. When the individual stiffness operators are proportional, the problem for the natural frequencies and mode shapes reduces to a matrix eigenvalue problem, and it is shown that for each spatial mode there is a set of <svg style="vertical-align:-0.09pt" id="M6" height="6.4124999" version="1.1" viewBox="0 0 7.0875001 6.4124999" width="7.0875001"  xmlns="http://www.w3.org/2000/svg"    >
	<g transform="matrix(1.25,0,0,-1.25,0,6.4125)">
		<g transform="translate(72,-66.87)">
			<text transform="matrix(1,0,0,-1,-71.95,67.01)">
				<tspan style="font-size: 11.21px; " x="0" y="0">𝑛</tspan>
			</text>

		</g>
	</g>
</svg> intramodal mode shapes. When the structures are identical, uniform, or nonuniform, the differential equations are uncoupled through diagonalization of a coupling stiffness matrix. The most general case requires an iterative solution.

S. Graham Kelly

Free and Forced Vibrations of Elastically Connected Structures

Theoretical analyses of hydrodynamic fluid film bearings with different bearing profiles rely on solutions of the Reynolds equation. This paper presents an approach used for analysing the so-called pocket bearings formed from a combination of offset circular bearing profiles. The results show that the variation of the dynamic bearing characteristics with different load inclinations for the pocket bearings is less than that for the elliptic bearing counterpart. It is shown that the natural frequencies as well as the critical speeds, and hence the vibrational behaviour, can also be significantly different for an industrial rotor supported by the different bearings.

N. S. Feng

E. J. Hahn

Vibration Characteristics of Hydrodynamic Fluid Film Pocket Journal Bearings

Motive forces by muscles are applied to different parts of the human body in a periodic fashion when walking at a uniform rate. In this study, the whole human body is modeled as a multidegree of freedom (MDOF) system with seven degrees of freedom. In view of the changing contact conditions with the ground due to alternating feet movements, the system under study is considered piecewise time invariant for each half-period when one foot is in contact with the ground. Forces transmitted from the body to the ground while walking at a normal pace are experimentally measured and numerically simulated. Fourth-order Runge-Kutta method is employed to numerically simulate the forces acting on different masses of the body. An optimization problem is formulated with the squared difference between the measured and simulated forces transmitted to the ground as the objective function, and the motive forces on the body masses as the design variables to solve.

Raghdan J. AlKhoury

Suraj Joshi

Rama B. Bhat

Shiping Ma

Identification of Motive Forces on the Whole Body System during Walking

This paper deals with the in-plane vibration of circular annular disks under combinations of different boundary conditions at the inner and outer edges. The in-plane free vibration of an elastic and isotropic disk is studied on the basis of the two-dimensional linear plane stress theory of elasticity. The exact solution of the in-plane equation of equilibrium of annular disk is attainable, in terms of Bessel functions, for uniform boundary conditions. The frequency equations for different modes can be obtained from the general solutions by applying the appropriate boundary conditions at the inner and outer edges. The presented frequency equations provide the frequency parameters for the required number of modes for a wide range of radius ratios and Poisson's ratios of annular disks under clamped, free, or flexible boundary conditions. Simplified forms of frequency equations are presented for solid disks and axisymmetric modes of annular disks. Frequency parameters are computed and compared with those available in literature. The frequency equations can be used as a reference to assess the accuracy of approximate methods.

S. Bashmal

R. Bhat

S. Rakheja

Frequency Equations for the In-Plane Vibration of Circular Annular Disks

The frequency distribution of a short interval period, the SIP distribution, obtained
from the vibration of a structure that is excited by the force of non-stationary vibration is available for the robust estimation of the dynamic property of the structure. This paper shows experiments of the health monitoring of a model structure using the SIP distribution. Comparisons of SIP distributions with average DFT spectra are also shown.

Yoshimutsu Hirata

Mikio Tohyama

Mitsuo Matsumoto

Satoru Gotoh

Health Monitoring for a Structure Using Its Nonstationary Vibration

The thickness noise predicted by the Ffowcs Williams and Hawkings
(FW&#x26;H) equation depends on the normal velocity <svg style="vertical-align:-3.20526pt;width:15.23125px;" id="M1" height="12.3975" version="1.1" viewBox="0 0 15.23125 12.3975" width="15.23125"  xmlns="http://www.w3.org/2000/svg">
	<g transform="matrix(1.25,0,0,-1.25,0,12.3975)">
		<text transform="matrix(1,0,0,-1,0.498,3.694)">
			<tspan style="font-size: 12.50px; " x="0" y="0">𝑣</tspan>
		</text>

		<text transform="matrix(1,0,0,-1,6.825,0.568)">
			<tspan style="font-size: 8.75px; " x="0" y="0">𝑛</tspan>
		</text>

	</g>
</svg> which is very sensitive
to the meshing size. Isom showed that in far field a monopolar source is
equivalent to a dipolar source induced by a uniform distribution of the
load on the entire moving surface. The main objective of this paper is
to determine a specific expression of Isom's thickness noise in time and
frequency domains for subsonic fans. The scope of the proposed expression
of Isom's thickness noise is to define a benchmark test of consistency for
thickness and loading noise codes in both time and frequency domains for
subsonic fans when using the free field solution of FW&#x26;H's equation.

Sofiane Khelladi

Farid Bakir

A Consistency Test of Thickness and Loading Noise Codes Using Ffowcs Williams and Hawkings Equation

Active noise control systems of simple ducts are investigated. In particular, open-loop characteristics
and closed-loop performances corresponding to various structures of control sources are compared based on both mathematical models and experimental results. In addition to the standard single loudspeaker and the Swinbanks&#39; source, we propose and examine a single loudspeaker with a rear sound interference
as a novel structure of control source, where the rear sound radiated from the loudspeaker is interfered with the front sound in order to reduce the net upstream sound directly radiated from the control source. The comparisons of the control structures are performed as follows. First, the open-loop transfer
function is derived based on the standard wave equation, where a generalized control structure unifying the three structures mentioned above is considered. Secondly, by a comparison of the open-loop transfer functions from the first principle modeling and frequency response experiments, it is shown that a certain
phase-lag is imposed by the Swinbanks&#39; source and the rear sound interference. Thirdly, effects on control performances of control source structures are examined by control experiments with robust controllers.

Yasuhide Kobayashi

Hisaya Fujioka

Naoki Jinbo

A Control Source Structure of Single Loudspeaker and Rear Sound Interference for Inexpensive Active Noise Control

An approach is presented for analyzing the transient elastodynamic problem of a plate under an impact loading. The plate is considered to be in the form of a long strip under plane strain conditions. The loading is taken as a concentrated line force applied normal to the plate surface. It is assumed that this line force is suddenly applied and maintained thereafter (i.e., it is a Heaviside step function of time). Inertia effects are taken into consideration and the problem is treated exactly within the framework of elastodynamic theory. The approach is based on multiple Laplace transforms and on certain asymptotic arguments. In particular, the one-sided Laplace transform is applied to suppress time dependence and the two-sided Laplace transform to suppress the dependence upon a spatial variable (along the extent of the infinite strip). Exact inversions are then followed by invoking the asymptotic Tauber theorem and the Cagniard-deHoop technique. Various extensions of this basic analysis are also discussed.

Penelope Michalopoulou

George A. Papadopoulos

An Asymptotic Approach for the Elastodynamic Problem of a Plate under Impact Loading

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<p><em>Advances in Acoustics and Vibration</em> acts as a forum for the dissemination of innovative and original research and development work in the area of acoustics and vibration.</p>


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Advances in Acoustics and Vibration acts as a forum for the dissemination of innovative and original research and development work in the area of acoustics and vibration.

Advances in Acoustics and Vibration