An externally excited Duffing oscillator under feedback control is discussed and analyzed under the worst resonance case. Multiple time scales method is applied for this system to find analytic solution with the existence and nonexistence of the time delay on control loop. An appropriate stability analysis is also performed and appropriate choices for the feedback gains and the time delay are found in order to reduce the amplitude peak. Different response curves are involved to show and compare controller effects. In addition, analytic solutions are compared with numerical approximation solutions using Rung-Kutta method of fourth order.

The existence of vibrations and dynamic chaos in machinery and structural systems is an unavoidable phenomenon, which ultimately can lead to machine failures and dangerous accidents. Different reasons lead such systems for nonlinear vibrations, as nonlinear properties of materials, geometric nonlinearities, and nonlinear excitation forces. Much time, money and efforts are spent for minimizing vibrations and oscillations for longer time life of these systems and prevent them from failures or damaging.

In the last decades resonantly forced systems with control has been investigated in various engineering fields. For control speed and system performance, the time-delay in the controllers and the actuators has become an urgent problem for increasing strict requirements. For instance, a nonlinear delayed dynamic system of one dimension which may exhibit chaotic behavior was studied in [

Wang et al. [

Kruthika et al. [

To research the dynamics of the diffusive system with a cubic non-linearity time-delayed category of the damping Duffing equation, a coupling between the multiple scales approach and the homotopy disturbance was used in Yusry [

In this paper, we consider the effect of time delay in a Duffing oscillator under a parametric excitation force. The system amplitude-phase modulating equations are extracted by applying the multiple scales perturbation technique. The frequency-response curves to the system are obtained with and without delay time in control feedback. The effects of the coupling parameters, absorber, linear damping coefficient, and excitation amplitude on the frequency-response curve are explored. Numerical confirmations for the all acquired results are performed, time-histories are conducted before and after control effect. Finally, important notes are included for the optimal working conditions.

As a consequence of the motion of a body subjected to a nonlinear spring force, linear sticky damping and intermittent force, the Duffing oscillator occurs. Oscillations of mechanical systems under the action of a periodic parametric force can be revealed using Duffing oscillator. The mathematical model for the Duffing oscillator under a parametric excitation force is given in Wenlin et al. [

The terms associated with the system in

where the time derivative will take the values:

and

Now we will study system behavior in both cases with and without delay time in control system.

Applying

And the homogenous solution of (4) is given by:

where

Let

Expanding

Substituting from (21) into (20) and using only the first term approximation so:

Introducing (6), and (10) into (5), yields:

Now we will study the system worst operating modes due to resonance case, this worst case is the sub-harmonic resonance case:

where

Converting the function

where

where

For obtaining the steady state solution for amplitude and phase putting

where

To discuss the stability behavior of these solutions, linearizing

where

Accordingly, the stability of the steady-state solution varies depending on analyzing and solving characteristic equation, the Jacobian matrix’s eigenvalues that can be obtained by:

or

where

The state’s solution is asymptotically stable, as seen by the Routh-Huriwitz criterion, if and only if all real parts for roots of

Applying the active velocity and displacement control without time delay, i.e.,

In this section we illustrate the behavior of the system amplitude and phase with and without resonance case. We will show a comparison between active and time delay control and the effect of some system parameters on its amplitude.

In this Sub-section we compare the analytic solution of Eqs. (24), and (25) induced by (MTSM) and numerical solution of

Let us consider the parameters given in Sub-section 3.1 unless otherwise specified. In this sub-section we show the change of amplitude range with varying of the system parameters. We observe that the value of

In this research, we use multiple time scales method to find analytic solution and discuss the worst resonance case of an externally excited Duffing oscillator under feedback control. We studied the existence and nonexistence of the time delay effect on the system amplitude in case of the worst resonance cases that were principal parametric resonance. An appropriate stability analysis has been also performed and appropriate choices for the feedback gains and the time delay have been found in order to reduce the amplitude peak. In addition, analytic solutions were compared with numerical approximation solutions using Rung-Kutta method, these comparisons give a good agreement between analytic and numerical solutions.