The Acceleration of a Single Degree of Freedom System Through its Resonant Frequency

It is Sometimes required to find the maximum amplitudes of vibration attained and the speeds at which they occur when a machine is run through its critical speed with different accelerations. The solution of this problem for single degree of freedom systems has been obtained by Lewis and by Ellington and McCallion for mechanical vibrations and by Hok for the equivalent electrical case. These solutions require higher mathematics (contour integration, Fresnel's integrals or series solutions leading to Bessel functions). The purpose of this note is to show how, by using simple integration only, an alternative method of solution can be obtained for both zero and small values of damping.

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Deceleration of an Unbalanced Rotor Through a Critical Speed

Journal of Engineering for Industry ◽

10.1115/1.3610113 ◽

1967 ◽

Vol 89 (4) ◽

pp. 582-585

Author(s):

W. K. Bodger

Keyword(s):

Critical Speed ◽

Degree Of Freedom ◽

Bending Stress ◽

Rotor Speed ◽

Single Degree Of Freedom ◽

Closed Solution ◽

Rotor Shaft ◽

Single Degree ◽

Unbalanced Rotor ◽

Maximum Increment

The problem of a single-degree-of-freedom rotor decelerating slowly through its critica speed is solved by an energy approach; a closed solution is obtained. A small discontinuous downward jump of rotor speed across the critical speed is shown to be required, either with or without damping in the system. The maximum increment of deflection, hence bending stress, in the rotor shaft is shown to be small, provided the rotor is carefully balanced and/or the system is sufficiently damped.

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Effect of Third Harmonic Vibratory Force on a Linear Spring-Mass System

Journal of the Royal Aeronautical Society ◽

10.1017/s0001924000061674 ◽

1963 ◽

Vol 67 (636) ◽

pp. 799-803

Author(s):

C. L. Kirk

Keyword(s):

Resonant Frequency ◽

Elastic System ◽

Rate Of Change ◽

Degree Of Freedom ◽

The Other ◽

Mass System ◽

Single Degree Of Freedom ◽

Third Harmonic ◽

Single Degree ◽

Linear Spring

SummaryThe response of an elastic system having a single degree of freedom, to a vibratory force whose waveform can be varied, is examined. The variable waveform is produced by a system of two pairs of unbalanced rotors in which one pair rotates at three times the speed of the other pair. The waveform depends on the frequency of excitation, the phasing of the rotors and the ratio of their amounts of unbalance. If the rotors are run at a speed at which the faster pair rotates above resonance while the slower pair rotates below resonance, a frequency is found at which the rate of change of amplitude with respect to frequency is zero. At this point, however, the waveform is quite sensitive to small changes in the frequency of excitation. If the rotor speeds cannot be maintained constant, and if stable vibration waveforms are required, it is necessary to run the slowest rotor well above the resonant frequency where both the amplitude and waveform will be virtually independent of frequency.

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On the Critical Speed of Continuous Shaft-Disk Systems

Journal of Vibration and Acoustics ◽

10.1115/1.3269154 ◽

1984 ◽

Vol 106 (1) ◽

pp. 59-61 ◽

Cited By ~ 19

Author(s):

H. Nevzat O¨zgu¨ven

Keyword(s):

Critical Speed ◽

Degree Of Freedom ◽

Mass Ratio ◽

Disk System ◽

Single Degree Of Freedom ◽

Single Degree

The critical speed of a shaft-disk system can be approximately determined from a single degree-of-freedom model. The errors in the critical speed predictions obtained from such a model are investigated. The percentage errors are plotted against disk to shaft mass ratio for different bearings and various disk locations.

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Sistem Respon Satu Derajat Kebebasan terhadap Beban Harmonik pada Struktur Portal 2D

Jurnal Penelitian Enjiniring ◽

10.25042/jpe.112019.07 ◽

2019 ◽

Vol 23 (2) ◽

pp. 136-140

Author(s):

Muhammad Zubair Muis Alie ◽

Indah Melati Suci ◽

Astika Rajmi ◽

Andi Muhammad Alfian Arafat

Keyword(s):

Finite Element Method ◽

Finite Element ◽

Structural Dynamics ◽

Cyclic Load ◽

Degree Of Freedom ◽

Harmonic Loading ◽

Fe Method ◽

Mechanical Vibrations ◽

Single Degree Of Freedom ◽

Single Degree

Response of One Degree of Freedom System to Harmonic Loading on the structure idealized as single degree of freedom systems excited harmonically, that is structure subjected to force or displacement where the magnitude may be represented by a sine or cosine function of time. This type of excitation results one of the most important motions in the study of mechanical vibrations as well as in applications to structural dynamics. Structure is very often subjected to the dynamic action such cyclic load acting and resulting response due to the the unavoidable load eccentricity. The objective of the present study is to analyze the response of one-degree of freedom system to the portal 2D. The structure is modelled and analyzed using finite element method. The result obtained by FE method is joint displacement of the structure.

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Some Extensions to an Algorithm for the Identification of Single-Degree-of-Freedom Machines

Journal of Mechanisms Transmissions and Automation in Design ◽

10.1115/1.3258601 ◽

1984 ◽

Vol 106 (4) ◽

pp. 498-502

Author(s):

J. J. Nitao ◽

J. L. Wiederrich

Keyword(s):

Sufficient Conditions ◽

Operating Conditions ◽

Degree Of Freedom ◽

Operating Characteristics ◽

Deterministic Method ◽

Single Degree Of Freedom ◽

Single Degree ◽

Periodic Data ◽

Method Of Solution ◽

Necessary And Sufficient

In a previous work the second author presented a theory for the identification of a single-degree-of-freedom machine from its forced periodic operating characteristics. In this work that concept is extended by presenting two new methods of solution which, unlike the previous work, do not use Fourier series. The first method is a deterministic method of solution which is not restricted to using periodic data. This deterministic method is used to establish necessary and sufficient conditions for the existence of a unique solution. In particular, it is shown that three observations of the instantaneous drive shaft speed and torque characteristics under different operating conditions are necessary and that the observed speeds must satisfy two conditions. These conditions assure that the observations are independent of each other and that the observations uniquely define three periodic functions which characterize a machine belonging to the specified class of machines. The second method of solution is a least squares formulation which is derived from the first method; however, the derivation of this second method requires that the data be periodic.

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