Forced Vibration in Nonlinear Systems With Various Combinations of Linear Springs

1936 ◽  
Vol 3 (4) ◽  
pp. A127-A130
Author(s):  
J. P. Den Hartog ◽  
R. M. Heiles
Keyword(s):  
Steady State ◽  
Nonlinear Systems ◽  
Forced Vibration ◽  
Mass System ◽  
Special Cases ◽  
Single Mass ◽  
Set Up ◽  
Force Displacement ◽  
Spring Constants ◽  
Single Set

Abstract This paper deals with a single mass system containing a combination of linear springs having a force-displacement characteristic as shown in Fig. 1b. By varying the ratio k1/k2 of the spring constants, various modifications occur, e.g., for k1/k2 = 0 there is only one set of springs, and the system has a clearance x0 on each side of the equilibrium position; for k1/k2 = ∞ there is again a single set of springs which is now given an initial set-up force F. The steady-state motion of these systems under the influence of a harmonic external force is wanted. Detailed exact solutions have been published for the cases k1/k2 = ∞ and k1/k2 = 0. In this paper these two special cases have been brought down to a simpler form, and two additional cases (k1/k2 = 2 and k1/k2 = 0.5) have been calculated. All these results, together with those of the classical linear case k1/k2 = 1 are shown in Figs. 5 to 9, inclusive, plotted in two different dimensionless forms.

On Steady-State Harmonic Vibrations of Nonlinear Systems With Many Degrees of Freedom

1966 ◽  
Vol 33 (2) ◽  
pp. 406-412 ◽  
Author(s):  
W. M. Kinney ◽  
R. M. Rosenberg
Keyword(s):  
Steady State ◽  
Nonlinear Systems ◽  
Degrees Of Freedom ◽  
Forced Vibrations ◽  
Geometrical Method ◽  
Mass System ◽  
Harmonic Vibrations ◽  
Forcing Functions ◽  
Exciting Forces ◽  
The Stability

A nonlinear spring-mass system with many degrees of freedom, and subjected to periodic exciting forces, is examined. The class of admissible systems and forcing functions is defined, and a geometrical method is described for deducing the steady-state forced vibrations having a period equal to that of the forcing functions. The methods used combine the geometrical methods developed earlier in the problem of normal mode vibrations and Rauscher’s method. The stability of these steady-state forced vibrations is examined by Hsu’s method. The results are applied to an example of a system having two degrees of freedom.

scholarly journals Stability analysis in terms of two measures for dynamic systems on time scales

1993 ◽  
Vol 6 (4) ◽  
pp. 325-344 ◽  
Author(s):  
Billûr Kaymakçalan
Keyword(s):  
Stability Analysis ◽  
Time Scales ◽  
Dynamic Systems ◽  
Lyapunov’S Second Method ◽  
Two Measures ◽  
Set Up ◽  
Stability Results ◽  
Stability Concepts ◽  
Single Set

Using the theory of Lyapunov's second method developed earlier for time scales, we extend our stability results to two measures which give rise to unification of several stability concepts in a single set up.

Causal inference via string diagram surgery

2021 ◽  
pp. 1-22
Author(s):  
Bart Jacobs ◽  
Aleks Kissinger ◽  
Fabio Zanasi
Keyword(s):  
Causal Inference ◽  
Probabilistic Reasoning ◽  
Causal Effect ◽  
Sufficient Conditions ◽  
Causal Effects ◽  
Stochastic Matrices ◽  
Counterfactual Reasoning ◽  
Special Cases ◽  
String Diagram ◽  
Set Up

Abstract Extracting causal relationships from observed correlations is a growing area in probabilistic reasoning, originating with the seminal work of Pearl and others from the early 1990s. This paper develops a new, categorically oriented view based on a clear distinction between syntax (string diagrams) and semantics (stochastic matrices), connected via interpretations as structure-preserving functors. A key notion in the identification of causal effects is that of an intervention, whereby a variable is forcefully set to a particular value independent of any prior propensities. We represent the effect of such an intervention as an endo-functor which performs ‘string diagram surgery’ within the syntactic category of string diagrams. This diagram surgery in turn yields a new, interventional distribution via the interpretation functor. While in general there is no way to compute interventional distributions purely from observed data, we show that this is possible in certain special cases using a calculational tool called comb disintegration. We demonstrate the use of this technique on two well-known toy examples: one where we predict the causal effect of smoking on cancer in the presence of a confounding common cause and where we show that this technique provides simple sufficient conditions for computing interventions which apply to a wide variety of situations considered in the causal inference literature; the other one is an illustration of counterfactual reasoning where the same interventional techniques are used, but now in a ‘twinned’ set-up, with two version of the world – one factual and one counterfactual – joined together via exogenous variables that capture the uncertainties at hand.

A General Approach for the Coordination of Maintenance Frequencies in Cases with a Single Set-Up

2000 ◽  
pp. 245-282 ◽  
Author(s):  
R. Dekker ◽  
R. E. Wildeman ◽  
J. B. G. Frenk ◽  
R. Van Egmond
Keyword(s):  
Set Up ◽  
Single Set

scholarly journals Connected oscillations in a nonlinearly coupled system

2003 ◽  
Vol 25 (3) ◽  
pp. 129-136
Author(s):  
Nguyen Van Dao
Keyword(s):  
Nonlinear Systems ◽  
Coupled System ◽  
External Excitation ◽  
Plane System ◽  
Restoring Forces ◽  
Single Mass

A single-mass plane system subjected to symmetric restoring forces can strongly oscillate in the direction which is free from the external excitation. The raising oscillation is called the connected one. This phenomenon exists only in nonlinear systems under certain resonance conditions and was first investigated by Kononenko V. 0.

scholarly journals The truncated Hyper-Poisson queues: Hk/Ma,b/C/N with balking, reneging and general bulk service rule

2008 ◽  
Vol 18 (1) ◽  
pp. 23-36 ◽  
Author(s):  
A.I. Shawky ◽  
M.S. El-Paoumy
Keyword(s):  
Steady State ◽  
Analytical Solution ◽  
Explicit Form ◽  
Recurrence Relations ◽  
Expected Number ◽  
Special Cases ◽  
Queue Discipline ◽  
Bulk Service ◽  
General Bulk Service Rule ◽  
Steady State Probabilities

The aim of this paper is to derive the analytical solution of the queue: Hk/Ma,b/C/N with balking and reneging in which (I) units arrive according to a hyper-Poisson distribution with k independent branches, (II) the queue discipline is FIFO; and (III) the units are served in batches according to a general bulk service rule. The steady-state probabilities, recurrence relations connecting various probabilities introduced are found and the expected number of units in the queue is derived in an explicit form. Also, some special cases are obtained. .

Steady-State Performance of Squeeze Film Damper Supported Flexible Rotors

1977 ◽  
Vol 99 (4) ◽  
pp. 552-558 ◽  
Author(s):  
M. D. Rabinowitz ◽  
E. J. Hahn
Keyword(s):  
Steady State ◽  
Critical Speed ◽  
Rolling Element Bearing ◽  
Squeeze Film ◽  
Response Curves ◽  
Flexible Rotors ◽  
Steady State Operation ◽  
Bearing Life ◽  
Rolling Element ◽  
Single Mass

The synchronous steady-state operation of a centrally preloaded single mass flexible rotor supported in squeeze film bearing dampers is examined theoretically. Assuming the short bearing approximation and symmetric motions, frequency response curves are presented exhibiting the effect of relevant system parameters on rotor excursion amplitudes and unbalance transmissibilities for both pressurized and unpressurized lubricant supply. Hence, the influence of rotor flexibility, rotor mass distribution, rotor speed, bearing dimensions, lubricant viscosity, support flexibility can be readily determined, allowing for optimal rotor bearing system design. It is shown that with pressurized bearing mounts, the possibility of undesirable operation modes is eliminated and a smooth passage through the first pin-pin critical speed of the rotor is feasible, while absence of pressurization significantly limits the maximum safe unbalance in the vicinity of this critical speed. Significant decrease in transmissibility and rotor excursion amplitudes over those obtainable with rigid mounts are shown to be a practical possibility, with consequent decrease in the vibration level of the rotor mounts and prolongation of rolling element bearing life, while maintaining acceptable rotor vibration amplitudes. A design example is included to illustrate the use of the data.

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