On the Nonlinear Response of a Relief Valve

1978 ◽  
Vol 100 (4) ◽  
pp. 675-680 ◽  
Author(s):  
M. A. Dokainish ◽  
M. M. Elmadany
Keyword(s):  
Nonlinear Response ◽  
Forced Response ◽  
Distributed Parameter ◽  
Response Curves ◽  
Relief Valve ◽  
Single Degree Of Freedom ◽  
Third Harmonic ◽  
Valve System ◽  
Subharmonic Response ◽  
Subharmonic Vibration

The free and forced response of a relief valve system has been analyzed. The valve system consists of a valve having a mass, resting on a seat having nonlinear spring characteristics, and retained by a helical spring which is considered to be a distributed parameter element. The response curves of the first and third harmonic and one-third subharmonic, the amplitudes of which vary with the frequency of the external force, are investigated and presented graphically. It is found that harmonic, higher-harmonic, and subharmonic response of the system may become unstable within certain frequency ranges. Jump and hysteresis phenomena analogous to that of nonlinear single-degree-of-freedom systems are observed. It is also observed that subharmonic vibration may exist in a number of specified frequency ranges.

Effect of Two-to-One Internal Resonances in a Nonlinearly Restrained Fluid Valve

1999 ◽  
Vol 121 (1) ◽  
pp. 59-63 ◽  
Author(s):  
G. Anlas¸
Keyword(s):  
Nonlinear Response ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Primary Resonance ◽  
Distributed Parameter ◽  
Parameter System ◽  
Response Curves ◽  
Internal Resonances ◽  
Relief Valve ◽  
Steady State Solutions

The effect of two-to-one internal resonances on the nonlinear response of a pressure relief valve is studied. The fluid valve is modeled as a distributed parameter system at one end and nonlinearly restrained at the other. The method of multiple scales is used to solve the system of partial differential equation and boundary conditions. Frequency-response curves are presented for the primary resonance of either mode in the presence of a two-to-one internal resonance. Stability of the steady-state solutions is investigated. Parameters of the system leading to two-to-one internal resonances are tabulated.

High Order Subharmonic Response of High Speed Rotors in Bearing Clearance

1988 ◽  
Vol 110 (1) ◽  
pp. 9-16 ◽  
Author(s):  
F. F. Ehrich
Keyword(s):  
Natural Frequency ◽  
High Speed ◽  
Response Curves ◽  
Whole Number ◽  
Wave Forms ◽  
Vibration Responses ◽  
Subharmonic Response ◽  
Subharmonic Vibration ◽  
Exact Computer ◽  
Very High

Subharmonic vibration refers to the response of a dynamic system to excitation at a whole-number multiple (n) of its natural frequency by vibrating asynchronously at its natural frequency, that is, at (1/n) of the excitation. The phenomenon is generally associated with asymmetry in the stiffness vs. deflection characteristic of the system. It may be characterized as the “bouncing” of the rotor on the surface of the stiff support, energized by every nth unbalance impulse prior to contact. Second, third and fourth order subharmonic vibration responses have previously been observed in high speed rotating machinery with such an asymmetry in the bearing supports. An incident is reported where 8th and 9th order subharmonic vibration responses have been observed in a high speed rotor. A simple but exact computer model of the phenomenon has been evolved based on the numerical integration of a finite difference formulation. Response curves and wave forms of rotor deflection at individual speeds are computed. It is shown that the response is a series of pseudo-critical peaks at whole-number multiples of the rotational speed. Very high orders of subharmonic vibration are found to be possible for systems with low damping and extreme nonlinearity.

A New Method to Find the Forced Response of Nonlinear Systems with Dry Friction

2021 ◽  
Author(s):  
Gregory L. Altamirano ◽  
Meng-Hsuan Tien ◽  
Kiran D'Souza
Keyword(s):  
Dry Friction ◽  
Coulomb Friction ◽  
Nonlinear Response ◽  
Piecewise Linear ◽  
Time Integration ◽  
Nonlinear Behavior ◽  
Forced Response ◽  
New Method ◽  
Analysis Tools ◽  
Compatibility Constraint

Abstract Coulomb friction has an influence on the behavior of numerous mechanical systems. Coulomb friction systems or dry friction systems are nonlinear in nature. This nonlinear behavior requires complex and time demanding analysis tools to capture the dynamics of these systems. Recently, efforts have been made to develop efficient analysis tools able to approximate the forced response of systems with dry friction. The objective of this paper is to introduce a methodology that assists in these efforts. In this method, the piecewise-linear nonlinear response is separated into individual linear responses that are coupled together through compatibility constraint equations. The new method is demonstrated on a number of systems of varying complexity. The results obtained by the new method are validated through the comparison with results obtained by time integration. The computational savings of the new method is also discussed.

scholarly journals Forced Vibration in Cutting Process considering the Nonlinear Curvature and Inertia of a Rotating Composite Cutter Bar

2020 ◽  
Vol 2020 ◽  
pp. 1-19
Author(s):  
Yongsheng Ren ◽  
Donghui Yao
Keyword(s):  
Cutting Force ◽  
Forced Vibration ◽  
Multiple Scales ◽  
Damping Ratio ◽  
Three Dimensional ◽  
Forced Response ◽  
Cutting Process ◽  
Response Curves ◽  
Rotating Speed ◽  
Cutting System

Forced vibration of the cutting system with a three-dimensional composite cutter bar is investigated. The composite cutter bar is simplified as a rotating cantilever shaft which is subjected to a cutting force including regenerative delay effects and harmonic exciting items. The nonlinear curvature and inertia of the cutter bar are taken into account based on inextensible assumption. The effects of the moment of inertia, gyroscopic effect, and internal and external damping are also considered, but shear deformation is neglected. Equation of motion is derived based on Hamiltonʼs extended principle and discretized by the Galerkin method. The analytical solutions of the steady-state response of the cutting system are constructed by using the method of multiple scales. The response of the cutting system is studied for primary and superharmonic resonances. The effects of length-to-diameter ratio, damping ratio, cutting force coefficients, ply angle, rotating speed, and internal and external damping are investigated. The results show that nonlinear curvature and inertia imposed a significant effect on the dynamic behavior of the cutting process. The equivalent nonlinearity of the cutting system shows hard spring characteristics. Multiple solutions and jumping phenomenon of typical Duffing system are found in forced response curves.

A Probabilistic Analysis of Single-Degree-of-Freedom Blade Vibration

1993 ◽  
Author(s):  
Kenan Y. Sanliturk ◽  
Mehmet Imregun ◽  
David J. Ewins
Keyword(s):  
Natural Frequencies ◽  
Probabilistic Approach ◽  
Substantial Reduction ◽  
Forced Response ◽  
Degree Of Freedom ◽  
Cumulative Probability ◽  
Blade Vibration ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
Stiffness And Damping

The effects of random stiffness and damping variations on damped natural frequencies and response levels of turbomachinery blades are investigated by employing probabilistic approach using a single-degree-of-freedom (SDOF) model. An important feature of this study is the determination of the cumulative probability distributions for damped natural frequencies and receptance frequency response functions without having to compute their probability density distributions since it is shown that those of stiffness and damping can be used directly. The advantage of this approach is not only in the simplicity of problem formulation but also in the substantial reduction of computational requirements. Furthermore, results suggest that both stiffness and damping properties should be considered as random parameters in statistical analyses of forced response.

Forced Response of a SDOF Friction Oscillator Colliding With a Hysteretic Obstacle

2001 ◽  
Author(s):  
Ugo Andreaus ◽  
Paolo Casini
Keyword(s):  
Constant Velocity ◽  
Driving Force ◽  
Practical Interest ◽  
Forced Response ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
Moving Base ◽  
Friction Oscillator ◽  
Sample Application ◽  
Made In

Abstract The forced dynamics of non-smooth oscillators have not yet been sufficiently investigated, when damping is simultaneously due to friction and impact. Because of the theoretical and practical interest of this type of systems, an effort is made in this paper to lighten the behaviour of a single-degree-of-freedom oscillator colliding with a hysteretic obstacle and excited by an harmonic driving force and by a moving base with constant velocity. A friction-contact model has been proposed which allows simulating an exponential velocity-dependent friction law, and a deformable (hysteretic) obstacle. This model has been numerically tested via a sample application.

Forced Response of Continuous System With Hysteresis Loop Characteristics: System With Quadrilateral Hysteresis Loop Characteristics

2005 ◽  
Author(s):  
Shigeru Aoki ◽  
Takeshi Watanabe
Keyword(s):  
Approximate Solution ◽  
Steady State ◽  
Energy Loss ◽  
Hysteresis Loop ◽  
Nonlinear Response ◽  
Continuous System ◽  
Forced Response ◽  
Steady State Response ◽  
State Response ◽  
Discontinuous Line

This paper deals with steady-state response of a continuous system with collision characteristics. Considering the energy loss in a collision, an analytical method of approximate solution for the continuous system with symmetrical hysteresis loop characteristics is presented. The resonance curves of nonlinear response obtained from approximate solution are shown as discontinuous line, and are discussed the phenomenon.

Modelling of a cantilevered energy harvester with partial piezoelectric coverage and shunted to practical interface circuits

2019 ◽  
Vol 30 (13) ◽  
pp. 1896-1912 ◽  
Author(s):  
Guobiao Hu ◽  
Lihua Tang ◽  
Junrui Liang ◽  
Raj Das
Keyword(s):  
Energy Harvester ◽  
Degree Of Freedom ◽  
Distributed Parameter ◽  
Interface Circuits ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
Distributed Parameter Model ◽  
Modelling Method ◽  
Equivalent Impedance ◽  
Standard Energy

This article presents a modelling methodology for a cantilevered energy harvester with partial piezoelectric coverage and shunted to practical power conditioning interface circuits. First, the distributed parameter model of the partially covered piezoelectric energy harvester is developed and the associated analytical solution is derived. Subsequently, the single-degree-of-freedom representation model is developed and the explicit expressions of equivalent lumped parameters are derived by taking the static deflection as the approximated fundamental vibration mode. Based on the comparison with the single-mode expression of the distributed parameter model, a correction factor is proposed to improve the accuracy of the single-degree-of-freedom model. The results of both the distributed parameter and the corrected single-degree-of-freedom models are compared. The accuracy of the corrected single-degree-of-freedom representation model is verified against the analytical and the finite element models. Finally, practical interface circuits including the standard energy harvesting circuit and the parallel synchronized switch harvesting on inductor circuit are considered. A modified equivalent impedance modelling method is proposed for the analysis of the standard energy harvesting and parallel synchronized switch harvesting on inductor circuits. The results of the modified equivalent impedance modelling method are verified against the existing method in the literature.

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