Response of Systems With Damping Materials Modeled Using Fractional Calculus

1995 ◽  
Vol 48 (11S) ◽  
pp. S118-S126 ◽  
Author(s):  
L. Suarez ◽  
A. Shokooh
Keyword(s):  
Fractional Calculus ◽  
Series Solution ◽  
Operator Method ◽  
Damping Force ◽  
Single Degree Of Freedom ◽  
Kelvin Model ◽  
Single Degree ◽  
Damping Materials ◽  
Derivatives Of ◽  
Laplace And Fourier Transform

The mathematical modeling of damping materials based on fractional calculus has been shown to be very effective in representing the frequency dependence of the properties of these materials. In this model, the integer order derivatives in the constitutive equations of the Kelvin model are replaced by derivatives of fractional order. In this paper, we examine the response of a single degree-of-freedom system in which the damping force is proportional to a derivative of order α < 1 of the displacements. Three methods are proposed to obtain the response: the Laplace and Fourier transform methods, and an operator method that results in a series solution. Some interesting features exhibited by the oscillator’s response due to the fractional representation of the damping are unveiled.

A Comparison of the Effects of Nonlinear Damping on the Free Vibration of a Single-Degree-of-Freedom System

2012 ◽  
Vol 134 (2) ◽  
Author(s):  
Bin Tang ◽  
M. J. Brennan
Keyword(s):  
Free Vibration ◽  
Equations Of Motion ◽  
Nonlinear Damping ◽  
Degree Of Freedom ◽  
Damping Force ◽  
Single Degree Of Freedom ◽  
Sdof System ◽  
Single Degree ◽  
Dependent Term ◽  
Quadratic Damping

This article concerns the free vibration of a single-degree-of-freedom (SDOF) system with three types of nonlinear damping. One system considered is where the spring and the damper are connected to the mass so that they are orthogonal, and the vibration is in the direction of the spring. It is shown that, provided the displacement is small, this system behaves in a similar way to the conventional SDOF system with cubic damping, in which the spring and the damper are connected so they act in the same direction. For completeness, these systems are compared with a conventional SDOF system with quadratic damping. By transforming all the equations of motion of the systems so that the damping force is proportional to the product of a displacement dependent term and velocity, then all the systems can be directly compared. It is seen that the system with cubic damping is worse than that with quadratic damping for the attenuation of free vibration.

On Free Vibrations With Combined Viscous and Coulomb Damping

1980 ◽  
Vol 102 (4) ◽  
pp. 283-286 ◽  
Author(s):  
P. H. Markho
Keyword(s):  
Decay Curve ◽  
Closed Form Solution ◽  
Free Vibrations ◽  
Form Solution ◽  
Forced Response ◽  
Experimental Plot ◽  
Damping Force ◽  
Single Degree Of Freedom ◽  
Coulomb Damping ◽  
Single Degree

A closed-form solution of the governing, nonlinear equation for free vibrations of a single-degree-of-freedom system, without stops, under combined viscous and Coulomb damping is first obtained. This is much less involved than forced-response considerations of the same system (with or without stops) the solution of which problem was first obtained by Den Hartog [1]. This note contains the first derivation, as far as the author is aware of, of the equation for the amplitude decay curve (or envelope) for such a system vibrating freely under no-stop conditions. This equation is presented in a form which enables the components of the damping force to be determined from the system’s experimental plot (or record) of displacement versus time.

An improved constant deceleration control method with extended shock velocity range for magnetorheological energy absorber

2021 ◽  
pp. 1045389X2110572
Author(s):  
Zhongqiang Feng ◽  
Dong Yu ◽  
Zhaobo Chen ◽  
Xudong Xing ◽  
Hui Yan
Keyword(s):  
Control Algorithm ◽  
Control Method ◽  
Velocity Range ◽  
Damping Force ◽  
Isolation System ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
Shock Mitigation ◽  
Energy Absorbers ◽  
Feasible Solutions

This paper proposed an extended constant deceleration (ECD) control method that can be used in the shock mitigation system with magnetorheological energy absorbers (MREAs). The ECD control method has three sections: zero controllable force (ZCF) section, constant deceleration (CD) section, and maximum damping force (MDF) section. Under the control of ECD, the system can stop at the end of MREA stroke without exceeding the maximum allowable deceleration. The ECD control algorithm is derived in a single-degree-of-freedom (SDOF) system. The controllable velocity range and the required controllable damping force of ECD control method are also derived, which can provide feasible solutions for the design of shock isolation system with MREAs. The performance of ECD control method is shown by applying to the drop-induced shock mitigation system with different drop velocities, different maximum controllable damping force, and MREA stroke. The results shows that the ECD control method not only has a large controllable velocity range and small controllable damping force requirement, but also can minimize the load transmitted to the system.

Analysis of the Quality Factor and Response Time of Resonant Biosensor

2016 ◽  
Author(s):  
Pezhman A. Hassanpour
Keyword(s):  
Response Time ◽  
Quality Factor ◽  
Governing Equation ◽  
Dynamic Effect ◽  
Equation Of Motion ◽  
Measured Parameter ◽  
Damping Force ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
The Relationship

The relationship between the overall damping and response time of resonant biosensors is investigated in this paper. The governing equation of motion is derived using a single degree-of-freedom model of the resonator considering the dynamic effect of adsorption of the measured parameter. It is shown that the adsorption leads to a damping force on the resonant sensor. If not taken into account, this damping force results in misinter-pretation of the sensor readings.

scholarly journals Vibrations of fractional half- and single-degree of freedom systems

2017 ◽  
Vol 39 (3) ◽  
pp. 259-273
Author(s):  
Valentina Ciaschetti ◽  
Isaac Elishakoff ◽  
Alessandro Marzani
Keyword(s):  
Fractional Derivatives ◽  
Degree Of Freedom ◽  
Benchmark Problems ◽  
Matrix Approach ◽  
Single Degree Of Freedom ◽  
State Variable ◽  
Single Degree ◽  
Rational Order ◽  
Derivatives Of ◽  
Variable Analysis

In this paper we study vibrations of fractional oscillators by two methods: the triangular strip matrix approach, based on the Grunwald-Letnikov discretization of the fractional term, and the state variable analysis, which is suitable for systems with fractional derivatives of rational order. Some examples are solved in order to compare the two approaches and to conduct comparison with benchmark problems.

Natural Vibration with Damping Force Proportional to a Power of the Velocity

1963 ◽  
Vol 67 (630) ◽  
pp. 381-385 ◽  
Author(s):  
A. W. Morley ◽  
W. D. Bryce
Keyword(s):  
Experimental Data ◽  
Natural Vibration ◽  
Dissipated Energy ◽  
Approximate Theory ◽  
Internal Damping ◽  
Damping Force ◽  
Solid Materials ◽  
Single Degree Of Freedom ◽  
Physical Systems ◽  
Single Degree

SummaryThe friction force in many physical systems has been found to depend on a power of the velocity, the magnitude of which varies according to the damping mechanism. This note employs an approximate theory applicable to small damping to examine the free vibration of a single degree of freedom system in which the damping force is proportional to an unspecified power of the velocity.Correlation with experimental data on the internal damping of solid materials shows that this phenomenon also tends to obey a power law although discrepancies exist between the expected and observed behaviours. These arise from the interpretation of the damping coefficient and from the frequency dependence of the dissipated energy.

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