Computational Approaches to the Min-Max Response of Dynamic Systems With Incompletely Prescribed Input Functions

1967 ◽  
Vol 34 (1) ◽  
pp. 87-90 ◽  
Author(s):  
Eugene Sevin ◽  
Walter Pilkey
Keyword(s):  
Dynamic Systems ◽  
Initial Conditions ◽  
Restoring Force ◽  
Maximum Displacement ◽  
Single Degree Of Freedom ◽  
Load Duration ◽  
Analytical Technique ◽  
Single Degree ◽  
Forcing Function ◽  
Total Impulse

Linear, nonlinear, and dynamic programming formulations are developed for the solution of the min-max response of a single-degree-of-freedom dynamic system with incompletely prescribed input functions. The problem is: Given an oscillator whose equation of motion is mx¨ + g(x, x˙) = f(t), subject to stated initial conditions, and acted upon by a forcing function, f(t), which is nonnegative, and of specified finite duration and total impulse, find the particular forces which produce the least possible maximum displacement of the oscillator, and find this bounding value. Previously, Sevin developed an analytical technique for the solution which is inherently dependent upon a linear undamped form for the restoring force g(x, x˙). In the current work, an alternate statement of the problem is presented which lends itself to tractable computational formulations involving less stringent restrictions on g(x, x˙). Results obtained by dynamic and linear programming for specified forms of g(x, x˙) are given as functions of load duration.

Min-Max Solutions for the Linear Mass-Spring System

1957 ◽  
Vol 24 (1) ◽  
pp. 131-136
Author(s):  
Eugene Sevin
Keyword(s):  
Upper And Lower Bounds ◽  
Constant Force ◽  
Linear Mass ◽  
Maximum Displacement ◽  
Natural Period ◽  
Load Duration ◽  
Forcing Function ◽  
Total Impulse ◽  
Spring System ◽  
Mass Spring

Abstract Absolute upper and lower bounds have been determined for the maximum displacement of an undamped linear mass-spring system acted on by a non-negative forcing function characterized only by total impulse and duration. The upper bound is shown to result from applying the total impulse to the mass as an initial blow. The lower bound is shown to depend upon the ratio of load duration to natural period of the system, and this response results from a forcing function consisting of an initial and final impulse and an intermediate constant force. In the latter case, for sufficiently short durations, the forcing function reduces simply to equal initial and final impulses.

A Computationally Efficient Numerical Algorithm for the Transient Response of High-Speed Elastic Linkages

1979 ◽  
Vol 101 (1) ◽  
pp. 138-148 ◽  
Author(s):  
A. Midha ◽  
A. G. Erdman ◽  
D. A. Frohrib
Keyword(s):  
Transient Response ◽  
Numerical Algorithm ◽  
High Speed ◽  
Relative Size ◽  
Degree Of Freedom ◽  
Time Dependent ◽  
Computationally Efficient ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
Forcing Function

A new numerical algorithm, easily adaptable for computer simulation, is developed to approximate the transient response of a single degree-of-freedom vibrating system; governing differential equation is linear and second order with time-dependent and periodic coefficients. This is accomplished by first solving the classical linear single degree-of-freedom problem with constant coefficients. The system is excited by a periodic forcing function possessing a certain degree of smoothness. The integration terms in the solution are systematically expanded into two groups of terms: one consists of non-integral terms while the other contains only integral terms. The final integral terms are bounded. For certain combinations of frequency and damping, within the sub-resonant frequency range, the relative size of the integral terms are demonstrated to be small. The algebraic expansion (non-integral) terms then approximate the solution. The solution to a single degree-of-freedom system with time-dependent and periodic parameters is made possible by discretizing the forcing period into a number of intervals and assuming the system parameters as constant over each interval. The numerical algorithm is then employed to solve an elastic linkage problem via modal superposition. Convergence of the solution is verified by refining the number of intervals of discretization.

Single Degree-of-Freedom Modeling of the Nonlinear Vibration Response of a Machining Robot

2020 ◽  
Vol 143 (5) ◽  
Author(s):  
Yaser Mohammadi ◽  
Keivan Ahmadi
Keyword(s):  
Control Strategies ◽  
Vibration Response ◽  
Industrial Robots ◽  
Degree Of Freedom ◽  
Restoring Force ◽  
Single Degree Of Freedom ◽  
Machining Forces ◽  
Sdof System ◽  
Single Degree ◽  
Machining Robot

Abstract Highly dynamic machining forces can cause excessive and unstable vibrations when industrial robots are used to perform high-force operations such as milling and drilling. Implementing appropriate optimization and control strategies to suppress vibrations during robotic machining requires accurate models of the robot’s vibration response to the machining forces generated at its tool center point (TCP). The existing models of machining vibrations assume the linearity of the structural dynamics of the robotic arm. This assumption, considering the inherent nonlinearities in the robot’s revolute joints, may cause considerable inaccuracies in predicting the extent and stability of vibrations during the process. In this article, a single degree-of-freedom (SDOF) system with the nonlinear restoring force is used to model the vibration response of a KUKA machining robot at its TCP (i.e., machining tool-tip). The experimental identification of the restoring force shows that its damping and stiffness components can be approximated using cubic models. Subsequently, the higher-order frequency response functions (HFRFs) of the SDOF system are estimated experimentally, and the parameters of the SDOF system are identified by curve fitting the resulting HFRFs. The accuracy of the presented SDOF modeling approach in capturing the nonlinearity of the TCP vibration response is verified experimentally. It is shown that the identified models accurately predict the variation of the receptance of the nonlinear system in the vicinity of well-separated peaks, but nonlinear coupling around closely spaced peaks may cause inaccuracies in the prediction of system dynamics.

Nonlinear Oscillators with a Power-Form Restoring Force: Non-Isochronous and Isochronous Case

2013 ◽  
Vol 430 ◽  
pp. 14-21
Author(s):  
Ivana Kovacic
Keyword(s):  
Kinetic Energy ◽  
Constant Coefficient ◽  
Nonlinear Term ◽  
Nonlinear Oscillators ◽  
Equation Of Motion ◽  
Constant Amplitude ◽  
Restoring Force ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
First Case

This work is concerned with single-degree-of-freedom conservative nonlinear oscillators that have a fixed restoring force, which comprises a linear term and an odd-powered nonlinear term with an arbitrary magnitude of the coefficient of nonlinearity. There are two cases of interest: i) non-isochronous, when the system has an amplitude-dependent frequency and ii) isochronous, when the frequency of oscillations is constant (amplitude-independent). The first case is associated with the constant coefficient of the kinetic energy, while the frequency-amplitude relationship and the solution for motion need to be found. To that end, the equation of motion is solved by introducing a new small expansion parameter and by adjusting the Lindstedt-Poincaré method. In the second case, the condition for the frequency of oscillations to be constant is derived in terms of the expression for the position-dependent coefficient of the kinetic energy. The corresponding solution for isochronous oscillations is obtained. Numerical verifications of the analytical results are also presented.

The Steady State Response of Systems With Hardening Hysteresis

1978 ◽  
Vol 100 (1) ◽  
pp. 193-198 ◽  
Author(s):  
R. K. Miller
Keyword(s):  
Steady State ◽  
Degree Of Freedom ◽  
General Nature ◽  
Model Parameters ◽  
Steady State Response ◽  
Single Degree Of Freedom ◽  
Analytical Technique ◽  
State Response ◽  
Single Degree ◽  
Critical Model

A physical model for hardening hysteresis is presented. An approximate analytical technique is used to determine the steady-state response of a single-degree-of-freedom system and a multi-degree-of-freedom system incorporating this model. Certain critical model parameters which determine the general nature of the responses are identified.

Forced Vibration of a Base-Excited Single-Degree-of-Freedom System With Coulomb Friction

1984 ◽  
Vol 106 (4) ◽  
pp. 280-285 ◽  
Author(s):  
Etsuo Marui ◽  
Shinobu Kato
Keyword(s):  
Friction Force ◽  
Coulomb Friction ◽  
Damping Ratio ◽  
Force Frequency ◽  
Vibratory System ◽  
Single Degree Of Freedom ◽  
Analytical Technique ◽  
Single Degree ◽  
Dimensional Parameters ◽  
Coulomb Friction Force

Using the “stopping region of motion” concept, a brief analytical technique is worked out for the behavior of the linear forced vibratory system under the influence of a Coulomb friction force. The following points are clarified by the above technique: 1. The behavior of the system is completely determined by the three non-dimensional parameters of nondimensional friction force, frequency ratio and damping ratio. 2. The vibratory system undergoes a periodic vibration with stopping periods when the mass cannot move. These stopping periods increase at lower exciting frequencies, owing to Coulomb friction. 3. The relation between the kind of motion occurring in the system and the above three parameters can be obtained theoretically and verified experimentally.

Variational Solution for an Initial Conditions Problem

1997 ◽  
Vol 50 (11S) ◽  
pp. S50-S55 ◽  
Author(s):  
C. P. Filipich ◽  
M. B. Rosales
Keyword(s):  
Finite Number ◽  
Classical Solution ◽  
Time Domain ◽  
Initial Conditions ◽  
Boundary Value ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
Impulse Load ◽  
Initial Conditions Problem ◽  
The Time Domain

An initial conditions problem is addressed by first transforming it into a boundary value one. An appropriate functional and an extremizing sequence are proposed. The methodology has been previously named WEM (Whole Element Method) by the authors. In the present paper, this name is justified since division of the time domain is avoided even when loads with finite number of impulses of arbitrary duration are involved. The method is theoretically founded by theorems and corolaries. Their statements are included in the work. A numerical example of an undamped single degree of freedom (SDOF) system subjected to a rectangular impulse load is carried out. Comparison is made with the well-known classical solution.

Single Degree of Freedom Forced Vibration System

2021 ◽  
pp. 41-79
Keyword(s):  
Forced Vibration ◽  
Influence Factor ◽  
Initial Conditions ◽  
Degree Of Freedom ◽  
Transient Solution ◽  
Vibration System ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
Damped System ◽  
Arbitrary Loading

This chapter concerns the study of forced vibration of a single degree of freedom system, treating undamped and damped system under harmonic, periodic, and arbitrary loading with different cases and examples. Passing by all components of the general solution of an undamped forced system, which are a transient solution, depends only on initial conditions, transient solution due to the load at the end the stationary solution. In this chapter, a study of the dynamic influence factor depends on the ration between load frequency and structure one is presented.

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