Spatial Response Concentrations in Extended Structures

1967 ◽  
Vol 89 (4) ◽  
pp. 754-758
Author(s):  
R. H. Lyon
Keyword(s):  
Mode Shapes ◽  
Vibration Modes ◽  
Mean Square ◽  
Spatial Response ◽  
The Central Limit Theorem ◽  
Mean Square Response ◽  
Spatial Coordinates ◽  
The Mean ◽  
Band Limited ◽  
Approach Unity

When an extended structure is excited at frequencies above the resonance of its lowest modes, spatial variations in the mean square response occur since the mode shapes are functions of the spatial coordinates. For excitation consisting of band-limited noise or a pure sinusoid, one may calculate the mean square response relatively easily. The spatial variance of the mean square temporal response can also be found and can be interpreted by application of the “central limit theorem.” Vibration modes generally are coherent to some degree. Rather large variations of response may occur at positions where coherent modes have in-phase antinodes. The probability of the occurrence of such response concentrations is studied in this paper. The probability of the occurrence of a concentration at some position on the plate is found to approach unity for some assumed statistical distribution of the resonating modes. (It is felt that the conclusions are not strongly dependent on these assumptions.)

Random Vibration of Beams

1962 ◽  
Vol 29 (2) ◽  
pp. 267-275 ◽  
Author(s):  
S. H. Crandall ◽  
Asim Yildiz
Keyword(s):  
Timoshenko Beam ◽  
Dynamic Models ◽  
Random Excitation ◽  
Viscous Damping ◽  
Mean Square ◽  
Euler Beam ◽  
Rayleigh Beam ◽  
Mean Square Response ◽  
The Mean ◽  
Band Limited

The calculated response of a uniform beam to stationary random excitation depends greatly on the dynamical model postulated, on the damping mechanism assumed, and on the nature of the random excitation process. To illustrate this, the mean square deflections, slopes, bending moments, and shear forces have been compared for four different dynamical models, with three different damping mechanisms, subjected to a distributed transverse loading process which is uncorrelated spacewise and which is either ideally “white” timewise or band-limited with an upper cut-off frequency. The dynamic models are the Bernoulli-Euler beam, the Timoshenko beam, and two intermediate models, the Rayleigh beam, and a beam which has the shear flexibility of the Timoshenko beam but not the rotatory inertia. The damping mechanisms are transverse viscous damping, rotatory viscous damping, and Voigt viscoelasticity. It is found that many of the mean-square response quantities are finite when the excitation is ideally white (i.e., when the input has infinite mean square); however, some of the responses are unbounded. For these cases the rate of growth of the response as the cut-off frequency of the excitation is increased is obtained.

Wide-Band Random Axisymmetric Vibration of Cylindrical Shells

1979 ◽  
Vol 46 (2) ◽  
pp. 417-422 ◽  
Author(s):  
I. Elishakoff ◽  
A. Th. van Zanten ◽  
S. H. Crandall
Keyword(s):  
Circular Cylindrical Shell ◽  
Time History ◽  
Wide Band ◽  
Axisymmetric Vibration ◽  
Mean Square ◽  
Mean Square Response ◽  
History Of ◽  
Cross Correlations ◽  
The Mean ◽  
Band Limited

Analytical and numerical results are reported for the random vibrations of a uniform circular cylindrical shell excited by a ring load which is uniform around the circumference and random in time. The time history of loading is taken to be a stationary wide-band random process. The shell response is essentially one-dimensional but differs qualitatively and quantitatively from the response distributions for point-excited uniform strings and beams because of the large modal overlaps at the low end of the spectrum of shell natural frequencies. The contributions from the modal cross-correlations (which can usually be neglected for strings and beams) introduce an asymmetry into the distribution of mean-square response and can alter the magnitude of the local response considerably. For example, in a thin shell with a radius-to-length ratio of 0.5 the contribution to the mean-square velocity at the driven section due to the modal cross-correlations can be more than three times that due to the modal autocorrelations when the excitation is a band-limited white noise which includes 81 modes.

Mean-Square Response of a Second-Order System to Nonstationary, Random Excitation

1970 ◽  
Vol 37 (3) ◽  
pp. 612-616 ◽  
Author(s):  
L. L. Bucciarelli ◽  
C. Kuo
Keyword(s):  
Narrow Band ◽  
Random Excitation ◽  
Second Order ◽  
Envelope Function ◽  
Order System ◽  
Mean Square ◽  
Forcing Function ◽  
Mean Square Response ◽  
Noise Function ◽  
The Mean

The mean-square response of a lightly damped, second-order system to a type of non-stationary random excitation is determined. The forcing function on the system is taken in the form of a product of a well-defined, slowly varying envelope function and a noise function. The latter is assumed to be white or correlated as a narrow band process. Taking advantage of the slow variation of the envelope function and the small damping of the system, relatively simple integrals are obtained which approximate the mean-square response. Upper bounds on the mean-square response are also obtained.

Stationary random vibration of a viscoelastic Timoshenko cantilever beam under diverse random processes

2019 ◽  
Vol 234 (4) ◽  
pp. 849-861
Author(s):  
Qingzhao Zhou ◽  
David He ◽  
Yaping Zhao
Keyword(s):  
White Noise ◽  
Cantilever Beam ◽  
Time Correlation ◽  
Random Excitation ◽  
Transverse Displacement ◽  
Mean Square ◽  
Concentrated Force ◽  
The Mean ◽  
Definite Integration ◽  
Band Limited

In this paper, the stochastic properties of a uniform Timoshenko cantilever beam are investigated systematically. Based on the external viscous damping and Kelvin–Voigt viscoelastic damping, the partial differential equations of the Timoshenko beam subjected to random excitation are derived. The applied load is the concentrated force, and the excitation related to includes the ideal white noise, the band-limited white noise, and the exponential noise. Expressions are obtained for the space–time correlation functions and the space–frequency power spectral density functions of the transverse displacement response. The evident improvement is that the infinite integral and the definite integration in the mean square responses are worked out by means of the residue integral method and the integration by partial fraction, and the exact solutions of the mean square response are obtained in the form of an infinite series finally. This improvement provides a basis for both the mode truncation and the modal cross-spectral densities whether which can be ignored. Providing the numerical example, the numerical results obtained show the effectiveness of the theoretical analysis.

Stochastic Response and Stability Analysis of Two-Point Mooring System

2011 ◽  
Author(s):  
A. K. Banik ◽  
T. K. Datta
Keyword(s):  
Stability Analysis ◽  
Probability Density Function ◽  
Probability Density ◽  
Mooring System ◽  
Mean Square ◽  
Stochastic Response ◽  
Sea State ◽  
Mean Square Response ◽  
Linear Damping ◽  
The Mean

The stochastic response and stability of a two-point mooring system are investigated for random sea state represented by the P-M sea spectrum. The two point mooring system is modeled as a SDOF system having only stiffness nonlinearity; drag nonlinearity is represented by an equivalent linear damping. Since no parametric excitation exists and only the linear damping is assumed to be present in the system, only a local stability analysis is sufficient for the system. This is performed using a perturbation technique and the Infante’s method. The analysis requires the mean square response of the system, which may be obtained in various ways. In the present study, the method using van-der-Pol transformation and F-P-K equation is used to obtain the probability density function of the response under the random wave forces. From the moment of the probability density function, the mean square response is obtained. Stability of the system is represented by an inequality condition expressed as a function of some important parameters. A two point mooring system is analysed as an illustrative example for a water depth of 141.5 m and a sea state represented by PM spectrum with 16 m significant height. It is shown that for certain combinations of parameter values, stability of two point mooring system may not be achieved.

Transient Response of a Shear Beam to Correlated Random Boundary Excitation

1977 ◽  
Vol 44 (3) ◽  
pp. 487-491 ◽  
Author(s):  
S. F. Masri ◽  
F. Udwadia
Keyword(s):  
Time Delays ◽  
Spectral Characteristics ◽  
Mean Square Displacement ◽  
Dimensionless Time ◽  
Mean Square ◽  
Random Boundary ◽  
Mean Square Response ◽  
Shear Beam ◽  
The Mean ◽  
Support Motion

The transient mean-square displacement, slope, and relative motion of a viscously damped shear beam subjected to correlated random boundary excitation is presented. The effects of various system parameters including the spectral characteristics of the excitation, the delay time between the beam support motion, and the beam damping have been investigated. Marked amplifications in the mean-square response are shown to occur for certain dimensionless time delays.

Mean-Square Response of Thermoviscoelastic Medium to Nonstationary Random Excitation

1976 ◽  
Vol 43 (1) ◽  
pp. 150-158 ◽  
Author(s):  
W. Mosberg ◽  
M. Yildiz
Keyword(s):  
Random Excitation ◽  
Irreversible Thermodynamics ◽  
Envelope Function ◽  
Statistical Characteristics ◽  
Mean Square ◽  
Statistical Property ◽  
Step Functions ◽  
Mean Square Response ◽  
The Mean ◽  
Thermoviscoelastic Medium

The mean-square wave response of a lightly damped thermoviscoelastic medium to a special type of nonstationary random excitation is determined. The excitation function on the thermoviscoelastic medium is taken in the form of a product of a well-defined, slowly varying envelope function, and a part which prescribes the statistical characteristics of the excitation. Both the unit step and rectangular step functions are used for the envelope function, and both white noise and noise with an exponentially decaying harmonic correlation function are used to prescribe the statistical property of the excitation. By taking into consideration the slow variation envelope function and the wave characteristics of the lightly damped thermoviscoelastic medium, the mean-square response (as a function of temperature, excitation, and damping parameters with the aid of reversible and irreversible thermodynamics) is evaluated.

Random Response Relationships to Transfer Function and State-Space Properties

2007 ◽  
Vol 129 (5) ◽  
pp. 672-677
Author(s):  
Robin C. Redfield
Keyword(s):  
Transfer Function ◽  
State Space ◽  
Dynamic Systems ◽  
System Response ◽  
Mean Square ◽  
System Matrix ◽  
Matrix Transfer Function ◽  
Mean Square Response ◽  
The Mean ◽  
Insight Into

Output variables of dynamic systems subject to random inputs are often quantified by mean-square calculations. Computationally for linear systems, these typically involve integration of the output spectral density over frequency. Numerically, this is a straightforward task and, analytically, methods exist to find mean-square values as functions of transfer function (frequency response) coefficients. These formulations offer analytical relationships between system parameters and mean-square response. This paper develops further analytical relationships in calculating mean-square values as functions of transfer function and state-space properties. Specifically, mean-square response is formulated from (i) system pole-zero locations, (ii) as a spectral decomposition, and (iii) in terms of a system matrix transfer function. Direct, closed-form relationships between response and these properties are afforded. These new analytical representations of the mean-square calculation can provide significant insight into dynamic system response and optimal design/tuning of dynamic systems.

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