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.

On the Stability Properties of a Periodically Forced, Time-Varying Mass System

2009 ◽  
Author(s):  
W. T. van Horssen ◽  
O. V. Pischanskyy ◽  
J. L. A. Dubbeldam
Keyword(s):  
Periodic Solutions ◽  
Forced Vibrations ◽  
Time Varying ◽  
Mass System ◽  
Single Degree Of Freedom ◽  
Stability Properties ◽  
Single Degree ◽  
Existence Of Periodic Solutions ◽  
The Stability ◽  
Varying Mass

In this paper the forced vibrations of a linear, single degree of freedom oscillator (sdofo) with a time-varying mass will be studied. The forced vibrations are due to small masses which are periodically hitting and leaving the oscillator with different velocities. Since these small masses stay for some time on the oscillator surface the effective mass of the oscillator will periodically vary in time. Not only solutions of the oscillator equation will be constructed, but also the stability properties, and the existence of periodic solutions will be discussed.

A Comprehensive Stability Criterion for Forced Vibrations in Nonlinear Systems

1953 ◽  
Vol 20 (1) ◽  
pp. 9-12
Author(s):  
K. Klotter ◽  
E. Pinney
Keyword(s):  
Differential Equation ◽  
Nonlinear Systems ◽  
Stability Criterion ◽  
Practical Importance ◽  
Nonlinear Function ◽  
Forced Vibrations ◽  
Maximum Value ◽  
The Stability

Abstract This paper deals with the forced vibrations described by the differential equation a q .. + c q + c Φ ( q , q . ) = P cos Ω t wherein Φ denotes a nonlinear function of q and/or q̇. It presents a criterion for determining their stability. It is shown that under very weak restrictions, which equivalently means, for a large variety of cases (including all of practical importance) the stability depends on the sign of ∂q*/∂P (q* denoting the maximum value of q(t) within a period). The motion is stable if this derivative is positive; it is unstable if it is negative.

An Alternative Perturbation Procedure of Multiple Scales for Nonlinear Dynamics Systems

1989 ◽  
Vol 56 (3) ◽  
pp. 667-675 ◽  
Author(s):  
S. L. Lau ◽  
Y. K. Cheung ◽  
Shuhui Chen
Keyword(s):  
Steady State ◽  
Nonlinear Systems ◽  
Degrees Of Freedom ◽  
Multiple Scales ◽  
Steady State Solution ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Almost Periodic ◽  
Perturbation Procedure ◽  
Incremental Harmonic Balance

An alternative perturbation procedure of multiple scales is presented in this paper which is capable of treating various periodic and almost periodic steady-state vibrations including combination resonance of nonlinear systems with multiple degrees-of-freedom. This procedure is a generalization of the Lindstedt-Poincare´ method. To show its essential features a typical example of cubic nonlinear systems, the clamped-hinged beam, is analyzed. The numerical results for the almost periodic-free vibration are surprisingly close to that obtained by the incremental harmonic balance (IHB) method, and the analytical formulae for steady-state solution are, in fact, identical with that of conventional method of multiple time scales. Moreover, detail calculations of this example revealed some interesting behavior of nonlinear responses, which is of significance for general cubic systems.

A Geometrical Approach Assessing Instability Trends for Galloping

1991 ◽  
Vol 58 (3) ◽  
pp. 784-791 ◽  
Author(s):  
P. Yu ◽  
N. Popplewell ◽  
A. H. Shah
Keyword(s):  
Wind Speed ◽  
Parameter Space ◽  
Degrees Of Freedom ◽  
Critical Conditions ◽  
Geometrical Approach ◽  
Two Dimensional ◽  
Electrical Conductor ◽  
Geometrical Method ◽  
Dimensional Parameter ◽  
The Stability

Although the galloping of an iced electrical conductor has been considered by many researchers, no special attention has been given to the galloping’s sensitivity to alternations in the system’s parameters. A geometrical method is presented in this paper to describe these instability trends and to provide compromises for controlling an instability. The conventional but uncontrollable parameter of the wind speed is chosen as the basis for obtaining the critical conditions under which bifurcations occur for a representative two degrees-of-freedom model. Variations in these critical conditions are found in a two-dimensional parameter space in order to determine the trends for the initiation of galloping as well as to evaluate the stability of the ensuring periodic vibrations.

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.

The Steady-State Response of the Double Bilinear Hysteretic Model

1965 ◽  
Vol 32 (4) ◽  
pp. 921-925 ◽  
Author(s):  
W. D. Iwan
Keyword(s):  
Steady State ◽  
Nonlinear Systems ◽  
Steady State Solution ◽  
Degree Of Freedom ◽  
State Solution ◽  
Steady State Response ◽  
Hysteretic Model ◽  
Jump Phenomenon ◽  
State Response ◽  
The Stability

The steady-state response of a one-degree-of-freedom double bilinear hysteretic model is investigated and it is shown that this model gives rise to the jump phenomenon which is associated with certain nonlinear systems. The stability of the steady-state solution is discussed and it is shown that the model predicts an unbounded resonance for finite excitation.

scholarly journals Geometrical Non-Linear Periodic Vibration of Plates

2000 ◽  
Author(s):  
M. Petyt ◽  
P. Ribeiro
Keyword(s):  
Steady State ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Continuation Method ◽  
Harmonic Balance Method ◽  
Internal Resonances ◽  
Resonance Frequencies ◽  
Modal Coupling ◽  
Non Linear ◽  
The Stability

Abstract Periodic, geometrically non-linear free and steady-state forced vibrations of fully clamped plates are investigated. The hierarchical finite element method (HFEM) and the harmonic balance method are used to derive the equations of motion in the frequency domain, which are solved by a continuation method. It is demonstrated that the HFEM requires far fewer degrees of freedom than the h-version of the FEM. Internal resonances due to modal coupling between modes with resonance frequencies related by a rational number, are discovered. In free vibration, internal resonances cause a very significant variation of the mode shape during the period of vibration. A similar behaviour is observed in steady-state forced vibration. The stability of the steady-state solutions is studied by Floquet’s theory and it is shown that stable multi-modal solutions occur.

The Concept of Complex Damping

1952 ◽  
Vol 19 (3) ◽  
pp. 284-286
Author(s):  
N. O. Myklestad
Keyword(s):  
Steady State ◽  
Complex Number ◽  
Forced Vibrations ◽  
Damping Factor ◽  
Viscous Damping ◽  
Mass System ◽  
Free And Forced Vibrations ◽  
Phase Angles ◽  
State Case ◽  
Complex Damping

Abstract In this paper it is shown that if the hysteresis loop for a material has a particular shape the damping can be considered adequately by multiplying the modulus of elasticity of the material by the complex number e2bi where 2b is called the complex damping factor. For small values of b it is shown that both for free and forced vibrations of a simple spring-mass system the motion in the case of complex damping is the same as in the case of viscous damping, with b = c/ccr, except that in the steady-state case the phase angles are slightly different. Also, it is shown how complex damping may be applied to cases of forced vibrations of uniform rods and beams. The greatest advantage of using complex damping, however, is in numerical calculations of forced vibrations of engine crankshafts, airplane wings, and other types of structures; and for such calculations it already has been extensively used.

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