As can be seen from the definition given by Eq. The attenuation ratio R k , can be expressed as:. The primary resonance response amplitude is determined by the real solution a k 2 of Eq. One or three real solutions can be obtained. Three real solutions exist between two points of vertical tangents saddle-node bifurcation , which are determined by differentiation of Eq.

This leads to the condition:. The critical force amplitude obtained from Eq. The stability of the solutions can be determined by the eigenvalues of the corresponding Jacobian matrix of Eq. The corresponding eigenvalues are the roots of [10]:. As can be seen from Eq. The system will be unstable.

## Dynamics of Controlled Mechanical Systems with Delayed Feedback

The system is also unstable and a Hopf bifurcation may occur. The other eigenvalues is zero when:. From Eq. Based on the analyses mentioned above, the sufficient conditions to guarantee the system stability are:. If there is no real solution to the Eq. The sufficient conditions of guaranteeing the system stability can be written as:. For the fixed time-delay, the range of the feedback gains can be obtained based on the stable conditions of the primary response resonances.

For the given feedback gains of the controllers, the time delay can be designed for the requirement of the suppression of the nonlinear systems.

Substituting Eqs. Substituting 43 into Eq. After substituting them into Eq.

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There are two possibilities: either a trivial solution a k , or non-trivial solutions, which can be acquired by:. The steady-state solutions of subharmonic resonance response are determined by the eigenvalues of the characteristic equation, which are the roots of:. Substituting them into Eq.

For the purpose of comparison, the peak amplitude of the superharmonic resonance without control can be written as:. The attenuation ratio of the peak amplitude of superharmonic resonance response is defined by the ratio of the peak amplitude of superharmonic resonance vibrations of the nonlinear system with and without the controllers. The attenuation ratio R k is [26]:. If there is no real solution for the Eq. We can find that the inequalities satisfy the Eq. We can get:.

For the fixed time-delay, the range of the feedback gains can be obtained based on stable conditions of the superharmonic response resonances. This section illustrates the effect of the feedback gains and time-delays on the non-linear dynamical behavior of the controlled system. The variation of attenuation ratio with time-delays and feedback gains is also discussed.

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The results are shown in a set of figures. The coefficient of elasticity of the spring is The feedback gain of the spring is 5. The coefficient of damping controllers is 0. From Figs.

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For the fixed feedback gains and M 2 k , different control purposes can be easily achieved by the right choice of the time-delays. As can be seen from the figures, under a fixed value of the amplitude of excitation, a small value of the attenuation ratio R k indicates a large reduction in the nonlinear vibrations of the nonlinear primary system.

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The proper selection of the feedback gains and the time-delays can guarantee the stability of the non-linear system. For the fixed feedback gains and M 2 k , the good suppression control for the nonlinear vibration of beams can obtain from the right choice of the time-delays easily. This suggests that saddle node bifurcation and jump phenomenon can be eliminated by certain values of the time-delays.

The bending of the frequency response curves is responsible for a jump phenomenon. Variation of attenuation ratio R 1 with the time-delays. Variation of attenuation ratio R 2 with the time-delays. First mode frequency-response curves of primary resonance for three sets of the time-delays. The curve of the subharmonic resonance with time-delays is shown as Fig. First mode frequency-response curves of subharmonic resonance for three sets of the time-delays. The time-delays can change the range of the occurrence of subharmonic resonance.

It is noted that the regions for the existence of subharmonic responses are different. Thus, the effect of time delays on the system performance has drawn much at tention in the design of robots, active vehicle suspensions, active tendons for tall buildings, as well as the controlled vibro-impact systems.

On the other hand, the properly designed delay control may improve the performance of dynamic sys tems.

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## Advances in Analysis and Control of Time-Delayed Dynamical Systems

For instance, the delayed state feedback has found its applications to the design of dynamic absorbers, the linearization of nonlinear systems, the control of chaotic oscillators, etc. Most controlled mechanical systems with time delays can be modeled as the dynamic systems described by a set of ordinary differential equations with time delays. Recent years have witnessed a rapid development of active control of various mechanical systems. With increasingly strict requirements for control speed and system performance, the unavoidable time delays in both controllers and actuators have become a serious problem.

For instance, all digital controllers, analogue anti- aliasing and reconstruction filters exhibit a certain time delay during operation, and the hydraulic actuators and human being interaction usually show even more significant time delays.

These time delays, albeit very short in most cases, often deteriorate the control performance or even cause the instability of the system, be- cause the actuators may feed energy at the moment when the system does not need it. Thus, the effect of time delays on the system performance has drawn much at- tention in the design of robots, active vehicle suspensions, active tendons for tall buildings, as well as the controlled vibro-impact systems.

On the other hand, the properly designed delay control may improve the performance of dynamic sys- tems.