scholarly journals On the Secondary Resonance of a Spinning Disk Under Space-Fixed Excitations

2004 ◽  
Vol 126 (3) ◽  
pp. 422-429 ◽  
Author(s):  
Jen-San Chen ◽  
Chin-Yi Hua ◽  
Chia-Min Sun
Keyword(s):  
Free Oscillation ◽  
Equations Of Motion ◽  
Excitation Frequency ◽  
Multiple Scale ◽  
Combination Resonance ◽  
Secondary Resonance ◽  
Spinning Disk ◽  
Final Response ◽  
Steady State Solutions ◽  
Special Case

We investigate the possibility of secondary resonance of a spinning disk under space-fixed excitations. Von Karman’s plate model is employed in formulating the equations of motion of the spinning disk. Galerkin’s procedure is used to discretize the equations of motion, and the multiple scale method is used to predict the steady state solutions. Attention is focused on the nonlinear coupling between a pair of forward (with frequency ωmn¯) and backward (with frequency ωmn) traveling waves. It is found that combination resonance may occur when the excitation frequency is close to 2ωmn+ωmn¯,ωmn+2ωmn¯, or 1/2ωmn¯+ωmn. When the combination resonance does occur, the frequencies of the free oscillation components are shifted slightly from the respective natural frequencies ωmn¯ and ωmn. The final response is therefore quasiperiodic. However, in the case when the excitation frequency is close to 1/2ωmn¯−ωmn, no combination resonance is possible. In the case when the excitation frequency is close to 1/3ωmn and 1/2ωmn¯−ωmn simultaneously, internal resonance between the forward and backward modes can occur. The frequencies of the free oscillation components are exactly three times and five times that of the excitation frequency. In this special case both saddle-node and Hopf bifurcations are observed.

scholarly journals On the Internal Resonance of a Spinning Disk Under Space-Fixed Pulsating Edge Loads

2001 ◽  
Vol 68 (6) ◽  
pp. 854-859 ◽  
Author(s):  
Jen-San Chen
Keyword(s):  
Natural Frequency ◽  
Trivial Solution ◽  
Internal Resonance ◽  
Excitation Frequency ◽  
Single Mode ◽  
Multiple Scale ◽  
Hopf Bifurcations ◽  
Solution Path ◽  
Nontrivial Solutions ◽  
Spinning Disk

Internal resonance between a pair of forward and backward modes of a spinning disk under space-fixed pulsating edge loads is investigated by means of multiple scale method. It is found that internal resonance can occur only at certain rotation speeds at which the natural frequency of the forward mode is close to three times the natural frequency of the backward mode and the excitation frequency is close to twice the frequency of the backward mode. For a light damping case the trivial solution can lose stability via both pitchfork as well as Hopf bifurcations when frequency detuning of the edge load is varied. On the other hand, nontrivial solutions experience both saddle-node and Hopf bifurcations. When the damping is increased, the Hopf bifurcations along the trivial solution path disappear. Furthermore, there exists a certain value of damping beyond which no nontrivial solution is possible. Single-mode resonance is also briefly discussed for comparison.

Theoretical modeling on the combination resonance of size-dependent microbeams

2021 ◽  
pp. 107754632110531
Author(s):  
Zhenkun Li ◽  
Qiyou Cheng ◽  
Zhizhuang Feng ◽  
Longtao Xing ◽  
Yuming He
Keyword(s):  
Boundary Conditions ◽  
Equations Of Motion ◽  
Deep Understanding ◽  
Perturbation Approach ◽  
Combination Resonance ◽  
Response Curves ◽  
Size Dependent ◽  
Time Histories ◽  
Different Sources ◽  
Special Case

The combination resonance of size-dependent microbeams is investigated. Two harmonic forces act on the microbeam, and combination resonance is observed while the excitation frequencies differ from the resonant frequency. Microbeams with two different sources of nonlinearities including three kinds of boundary conditions, clamped-free (nonlinearity comes from large curvature and nonlinear inertial), clamped-clamped, and hinged-hinged (nonlinearity originates from mid-plane stretching-bending coupling), are taken into consideration to have a deep understanding of this phenomenon. A traveling load acting on the microbeam is presented as a special case of combination resonance. The modal discretization technique is applied to discretize the equations of motion, and then the Lindstedt–Poincare method, a perturbation approach, is employed to solve the resultant equations. The conditions for combination resonance are presented, and frequency-response curves and time histories at the resonance point are obtained for microbeams of each boundary condition. Results reveal that different sources of nonlinearities result in different performances of combination resonance. The free vibration part constitutes a large percentage of the final response. Furthermore, the situation of coexistence of combination resonance and superharmonic (or subharmonic) resonance is determined. The special case demonstrates a higher amplitude than the common combination resonance for all the boundary conditions. Parametric studies are then carried out to discuss the effects of the length scale parameter, excitation force as well as its position, and damping on the performance of the microbeam.

Non-Linear Dynamic Analysis of a Cable-Stayed Beam

2005 ◽  
Author(s):  
Carlos E. N. Mazzilli ◽  
Franz Rena´n Villarroel Rojas
Keyword(s):  
Multiple Scales ◽  
Equations Of Motion ◽  
Multiple Scale ◽  
Local Mode ◽  
Numerical Application ◽  
Global Mode ◽  
Lower Mode ◽  
Reduced Equations ◽  
External Resonance ◽  
Non Linear

The dynamic behaviour of a simple clamped beam suspended at the other end by an inclined cable stay is surveyed in this paper. The sag due to the cable weight, as well as the non-linear coupling between the cable and the beam motions are taken into account. The formulation for in-plane vibration follows closely that of Gattulli et al. [1] and confirms their findings for the overall features of the equations of motion and the system modal properties. A reduced non-linear mathematical model, with two degrees of freedom, is also developed, following again the steps of Gattulli and co-authors [2,3]. Hamilton’s Principle is evoked to allow for the projection of the displacement field of both the beam and the cable onto the space defined by the first two modes, namely a “global” mode (beam and cable) and a “local” mode (cable). The method of multiple scales is then applied to the analysis of the reduced equations of motion, when the system is subjected to the action of a harmonic loading. The steady-state solutions are characterised in the case of internal resonance between the local and the global modes, plus external resonance with respect to either one of the modes considered. A numerical application is presented, for which multiple-scale results are compared with those of numerical integration. A reasonable qualitative and quantitative agreement is seen to happen particularly in the case of external resonance with the higher mode. Discrepancies should obviously be expected due to strong non-linearities present in the reduced equations of motion. That is specially the case for external resonance with the lower mode.

Nonlinear Combination Resonances in Cantilever Composite Plates

1995 ◽  
Author(s):  
Kyoyul Oh ◽  
Ali H. Nayfeh
Keyword(s):  
Natural Frequencies ◽  
Excitation Frequency ◽  
Power Spectra ◽  
Low Frequency ◽  
Composite Plates ◽  
High Amplitude ◽  
Mode Shapes ◽  
Laser Vibrometer ◽  
Combination Resonance ◽  
Response Curves

Abstract We experimentally investigated nonlinear combination resonances in a graphite-epoxy cantilever plate having the configuration (–75/75/75/ – 75/75/ – 75)s. As a first step, we compared the natural frequencies and mode shapes obtained from the finite-element and experimental modal analyses. The largest difference in the obtained frequencies was 2.6%. Then, we transversely excited the plate and obtained force-response and frequency-response curves, which were used to characterize the plate dynamics. We acquired time-domain data for specific input conditions using an A/D card and used them to generate time traces, power spectra, pseudo-state portraits, and Poincaré maps. The data were obtained with an accelerometer monitoring the excitation and a laser vibrometer monitoring the plate response. We observed the external combination resonance Ω≈12(ω2+ω5) and the internal combination resonance Ω≈ω8≈12(ω2+ω13), where the ωi are the natural frequencies of the plate and Ω is the excitation frequency. The results show that a low-amplitude high-frequency excitation can produce a high-amplitude low-frequency motion.

Dynamic Model of a Spur Gear System With Backlash and Friction Consideration

1994 ◽  
Author(s):  
T. K. Shing ◽  
Lung-Wen Tsai ◽  
P. S. Krishnaprasad
Keyword(s):  
Free Oscillation ◽  
Equations Of Motion ◽  
Spur Gear ◽  
Constant Load ◽  
Friction Torque ◽  
Gear System ◽  
Real Time Control ◽  
Dynamical Equations ◽  
Gear Noise ◽  
Time Control

Abstract A new model which accounts for both backlash and friction effects is proposed for the dynamics of a spur gear system. The model estimates average friction torque and uses it to replace the instantaneous friction torque to simplify the dynamical equations of motion. Two simulations, free oscillation and constant load operation, are performed to illustrate the effects of backlash and friction on gear dynamics. The results are compared with that of a previously established model which does not account for the friction. Finally, the effect of adding a damper on the driving shaft is also studied. This model is judged to be more realistic for real time control of electronmechanical systems to reduce gear noise and to achieve high precision.

scholarly journals Experiment and Theory on the Nonlinear Vibration of a Shallow Arch Under Harmonic Excitation at the End

2005 ◽  
Vol 74 (6) ◽  
pp. 1061-1070 ◽  
Author(s):  
Jen-San Chen ◽  
Cheng-Han Yang
Keyword(s):  
Steady State ◽  
Excitation Frequency ◽  
Multiple Scale ◽  
Steady State Response ◽  
Scale Analysis ◽  
Jump Phenomenon ◽  
Shallow Arch ◽  
State Response ◽  
Geometrical Imperfection ◽  
Multiple Scale Analysis

In this paper we study, both theoretically and experimentally, the nonlinear vibration of a shallow arch with one end attached to an electro-mechanical shaker. In the experiment we generate harmonic magnetic force on the central core of the shaker by controlling the electric current flowing into the shaker. The end motion of the arch is in general not harmonic, especially when the amplitude of lateral vibration is large. In the case when the excitation frequency is close to the nth natural frequency of the arch, we found that geometrical imperfection is the key for the nth mode to be excited. Analytical formula relating the amplitude of the steady state response and the geometrical imperfection can be derived via a multiple scale analysis. In the case when the excitation frequency is close to two times of the nth natural frequency two stable steady state responses can exist simultaneously. As a consequence jump phenomenon is observed when the excitation frequency sweeps upward. The effect of geometrical imperfection on the steady state response is minimal in this case. The multiple scale analysis not only predicts the amplitudes and phases of both the stable and unstable solutions, but also predicts analytically the frequency at which jump phenomenon occurs.

Dynamics of the Elastica With End Mass and Follower Loading

1990 ◽  
Vol 57 (1) ◽  
pp. 203-208 ◽  
Author(s):  
J. M. Snyder ◽  
J. F. Wilson
Keyword(s):  
Internal Pressure ◽  
Large Deformations ◽  
Equations Of Motion ◽  
Dynamic Responses ◽  
Cantilever Beams ◽  
Static Responses ◽  
Tube Type ◽  
Payload Mass ◽  
Nonlinear Equations Of Motion ◽  
Special Case

Orthotropic, polymeric tubes subjected to internal pressure may undergo large deformations while maintaining linear moment-curvature behavior. Such tubes are modeled herein as inertialess, elastic cantilever beams (the elastica) with a payload mass at the tip and with internal pressure as the eccentric tip follower loading that drives the configurations through large deformations. From the nonlinear equations of motion, dynamic beam trajectories are calculated over a range of system parameters for the special case of a point mass at the tip and a terminated ramp pressure loading. The dynamic responses, which are unique because the loading history and the range of motion are fully defined, are presented in nondimensional form and are compared to static responses presented in a companion study. These results are applicable to the dynamic design of high flexure, tube-type, robotic manipulator arms.

Simultaneous Resonance and Anti-Resonance in Dynamical Systems Under Asynchronous Parametric Excitation

2020 ◽  
Vol 15 (9) ◽  
Author(s):  
Artem Karev ◽  
Peter Hagedorn
Keyword(s):  
Dynamical Systems ◽  
Parametric Excitation ◽  
Analytical Method ◽  
Normal Forms ◽  
High Sensitivity ◽  
Combination Resonance ◽  
Concurrent Study ◽  
The Stability ◽  
Analytical Expressions ◽  
Special Case

Abstract Since the discovery of parametric anti-resonance, parametric excitation has also become more prominent for its stabilizing properties. While resonance and anti-resonance are mostly studied individually, there are systems where both effects appear simultaneously at each combination resonance frequency. With a steep transition between them and a high sensitivity of their relative positions, there is a need for a concurrent study of resonance and anti-resonance. The semi-analytical method of normal forms is used to derive approximate analytical expressions describing the magnitude of the stability impact as well as the precise locations of stabilized and destabilized areas. The results reveal that the separate appearance of resonance and anti-resonance is only a special case occurring for synchronous parametric excitation. In particular, in circulatory systems the simultaneous appearance is expected to be much more common.

Electrostatically Actuated Coupled MEMS Resonators

2011 ◽  
Author(s):  
Dumitru I. Caruntu ◽  
Kyle N. Taylor
Keyword(s):  
Nonlinear Response ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
System Structure ◽  
Lagrange Equations ◽  
Electrostatically Actuated ◽  
Beam System ◽  
Mems Resonators ◽  
The Mathematical Model ◽  
Steady State Solutions

This paper deals with the nonlinear response of an electrostatically actuated cantilever beam system composed of two micro beam resonators near natural frequency. The mathematical model of the system is obtained using Lagrange equations. The equations of motion are nondimensionalized and then the method of multiple scales is used to find steady state solutions. Both AC and DC actuation voltages of the first beam are considered, while the influence on the system of DC on the second beam is explored. Graphical representations of the influence of the detuning parameters are provided for a typical micro beam system structure.

Use of Mass as a Perturbation Parameter in Vibrations

1967 ◽  
Vol 89 (4) ◽  
pp. 639-644 ◽  
Author(s):  
R. Chicurel ◽  
J. Counts
Keyword(s):  
Perturbation Technique ◽  
Excitation Frequency ◽  
Perturbation Parameter ◽  
Harmonic Excitation ◽  
Power Series Solution ◽  
Continuous Systems ◽  
Linear Vibration ◽  
Denominator Polynomial ◽  
Vibration Problems ◽  
Special Case

Linear vibration problems involving harmonic excitation of discrete and continuous systems are solved by using the classical perturbation technique. The perturbation parameter is proportional to a mass and the square of the excitation frequency. The power series solution for the displacement of some point in the system is converted to the quotient of two polynomials by the use of continued fractions. The eigenvalues (natural frequencies) of the problem are calculated by finding the roots of the denominator polynomial. The situation wherein a quantity which cannot vanish at any frequency can be found is treated as a special case.

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