Natural Forcing Functions in Nonlinear Systems

1958 ◽  
Vol 25 (3) ◽  
pp. 352-356
Author(s):  
T. J. Harvey
Keyword(s):  
Frequency Response ◽  
Maximum Amplitude ◽  
Second Order ◽  
Point Of View ◽  
Order System ◽  
Nonlinear Frequency ◽  
Restoring Force ◽  
Forcing Function ◽  
Special Cases ◽  
Forcing Functions

Abstract The response of nonlinear, second-order systems is examined from a new point of view which greatly simplifies presentation of the usual frequency-response diagrams. The use of “natural” forcing functions results in a general equation relating the maximum amplitude of the applied force to the maximum amplitude of the restoring force. The relationship is found to be a function of the ratio of the period of free oscillation to the period of the forcing function. The results apply for any second-order system without damping and with a nonlinear (or linear) restoring force. The special cases of a linear system and of Duffing’s equation are considered to illustrate similarities as well as differences between treatment of linear and nonlinear frequency-response problems.

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.

Identifying parametric variation in second-order system from frequency response measurement

2016 ◽  
Vol 24 (5) ◽  
pp. 879-891 ◽  
Author(s):  
A Hegde ◽  
J Tang
Keyword(s):  
Frequency Response ◽  
Parametric Identification ◽  
Second Order ◽  
Order System ◽  
Model Parameters ◽  
Order Model ◽  
Response Measurement ◽  
Electrical Systems ◽  
Angle Measurements ◽  
Second Order Model

Fundamentally, second-order model is the foundation of describing the dynamic characteristics of many mechanical and electrical systems. This paper investigates a parametric identification scheme for single degree-of-freedom second-order model in which the model parameters are subject to normal variation. By utilizing frequency response magnitude and phase angle measurements, we construct a linear-in-the-parameters model and build a related maximum likelihood estimator for both parametric means as well as variances. The validity of the approach is demonstrated through a collection of case analyses, and the results show considerable levels of accuracy in the presence of sufficient data.

scholarly journals Application of Butterworth high pass filter as an approximation of Wood Anderson seismometer frequency response to earthquake signal recording

ACTA IMEKO ◽  
2020 ◽  
Vol 9 (5) ◽  
pp. 379
Author(s):  
Hamidatul Husna Matondang ◽  
Endra Joelianto ◽  
Sri Widiyantoro
Keyword(s):  
Frequency Response ◽  
Signal To Noise Ratio ◽  
Cutoff Frequency ◽  
Maximum Amplitude ◽  
Second Order ◽  
Pass Filter ◽  
High Pass Filter ◽  
Local Earthquake ◽  
Seismic Magnitude ◽  
High Pass

The method for generating maximum amplitude and signal to noise ratio values by using second order high pass Butterworth filter on local seismic magnitude scale calculations is proposed. The test data are signals from local earthquake that have been occurred in Sunda Strait on April 8th 2012. Based on the experimental results, a 8 Hz cutoff frequency and a gain of 2200 of second order Butterworth high pass filter as an approach to simulating the frequency response of Wood Anderson seismometer can provide maximum amplitude value, SNR, and the magnitude better than simulated Wood Anderson frequency response.

Frequency response of human soleus muscle

1976 ◽  
Vol 39 (4) ◽  
pp. 788-793 ◽  
Author(s):  
P. Bawa ◽  
R. B. Stein
Keyword(s):  
Frequency Response ◽  
Natural Frequency ◽  
Soleus Muscle ◽  
Damping Ratio ◽  
Low Frequency ◽  
Second Order ◽  
Order System ◽  
Pass Filter ◽  
Stimulus Pulse ◽  
Low Pass Filter

1. The properties of human soleus muscle were studied by systems analysis. Single stimulus pulses and random stimulus pulse trains were applied to a branch of the nerve to soleus muscle and the resultant tension fluctuations were recorded. 2. The frequency-response function between stimulus pulses and tension conforms to that of a second-order, low-pass filter. The parameters of the second-order system, low frequency gain, natural frequency, and damping ratio, varied systematically with the angle of the ankle. As the ankle was flexed (the length of the muscle was increased), the low frequency gain increased, the natural frequency decreased, and the damping ratio was unaffected or increased slightly. 3. These results are discussed in relation to the twitch responses of human soleus muscles and the responses previously observed in cat muscles.

Analytical Solution of Two Coupled Oscillators With a Nonlinear Coupling Resorting Force

2014 ◽  
Author(s):  
Mohammad A. Al-Shudeifat ◽  
Thomas D. Burton
Keyword(s):  
Analytical Solution ◽  
Nonlinear Dynamical System ◽  
Degree Of Freedom ◽  
Nonlinear Frequency ◽  
Nonlinear Coupling ◽  
Restoring Force ◽  
Nonlinear Dynamical ◽  
Forcing Function ◽  
Local Equivalent ◽  
Coupling Force

An approach for accurate analytical solution of a two degree-of-freedom nonlinear dynamical system coupled with a strongly nonlinear restoring force is presented here. The approach is based on the application of the local equivalent linear stiffness method (LELSM) to linearize the nonlinear coupling stiffness in the system based on the nonlinear frequency calculation. Consequently, the system can be decoupled into two forced single degree-of-freedom subsystems by replacing the nonlinear coupling force with a forcing function where the solution can be analytically obtained. Different combinations of the positive and negative linear and cubic stiffness components are considered in the nonlinear coupling force. For all considered stiffness combinations, the obtained analytical solution strongly agrees with the numerical simulation of the system. In addition, the internal resonance is found not to significantly affect the accuracy of the analytical solution.

Bubble Dynamics and Cavitation Inception Theory

1988 ◽  
Vol 32 (03) ◽  
pp. 155-167
Author(s):  
Blaine R. Parkin ◽  
Brian B. Baker
Keyword(s):  
Initial Conditions ◽  
Second Order ◽  
Order System ◽  
Refined Theory ◽  
Zeroth Order ◽  
Order Problem ◽  
Cavitation Inception ◽  
First Order ◽  
Order Solution ◽  
Forcing Function

In order to provide some theoretical background and to motivate the more refined theory introduced herein, some encouraging known theoretical results on bubble-ring cavitation inception are reviewed. This review is followed by the development of the theory of bubble-ring cavitation cutoff. Its outcome, when compared with experiment, shows the need for a more refined inception theory. The above comparison and the basic ideas behind the cutoff theory's formulation suggest a possible approach for a refinement based on a multiple scales expansion. This seems reasonable because the forcing function pulse in "laboratory time" f, varies slowly compared with the characteristic "bubble time,", which characterizes the response time of a typical microscopic cavitation nucleus. The ratio of these two times gives a small parameter, , appearing in the forcing function, with the result that this problem involves only a soft excitation. Expanding the forced Rayleigh-Plesset equation and its initial conditions to the second order in c, the zeroth-order problem is found to be the well-known autonomous nonlinear equation with nonhomogeneous initial conditions, giving free oscillations of a typical nucleus. The first-order system is a nonautonomous linear system with homogeneous initial conditions which governs the forced bubble growth. The second-order system consists of a linear autonomous differential equation and homogeneous initial conditions. It is needed to establish integrability conditions for the first-order solution. The first-order solution is left for future research and the zeroth-order problem is analyzed in the phase plane. Then a novel approximate integration, = t(u), is given in terms of elliptic integrals and functions. It was not possible to invert this solution and so the inverse u = u() is found numerically. These data are then used to find an analytical approximation for use in future first-order calculations.

A nonlinear model for indirect combustion noise through a compact nozzle

2013 ◽  
Vol 733 ◽  
pp. 268-301 ◽  
Author(s):  
Maxime Huet ◽  
Alexis Giauque
Keyword(s):  
Nonlinear Response ◽  
Regime Change ◽  
Low Frequency ◽  
Maximum Amplitude ◽  
Second Order ◽  
Combustion Noise ◽  
Order System ◽  
Order Model ◽  
Describing Functions ◽  
Low Amplitude

AbstractThe present paper deals with the generation of sound by the passage of acoustic or entropy perturbations through a nozzle in the nonlinear regime and in the low-frequency limit. The analytical model of Marble and Candel for compact nozzles (J. Sound Vib., vol. 55, 1977, pp. 225–243), initially developed for excitations in the linear regime, is rederived and extended to the nonlinear domain. Full nonlinear and second-order models are written for both subcritical and supercritical nozzles in the absence of shock and a detailed methodology is provided for the resolution of the second-order system. The accuracy of the second-order model is assessed for entropy forcings. It is shown to be accurate for all waves, with the exception of the upstream generated wave for subcritical diverging geometries where higher-order nonlinear contributions cannot be neglected. In the context of indirect combustion noise, the phenomenon of regime change of the nozzle due to an incoming entropy fluctuation is also addressed. Regime change is related to a Mach number modification induced by temperature and velocity fluctuations. In the present study, it translates into a limitation of the maximum amplitude of the incoming entropy forcing. Such limitations are to be considered for subcritical nozzles with significant inlet or outlet Mach numbers, where the flow transition is observed even for very low-amplitude entropy excitations. With the constraint of those limitations, the analytical extended nozzle describing functions representing the full nonlinear response for indirect combustion noise are validated through detailed comparisons with numerical simulations.

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