Response regimes of a linear oscillator with a nonlinear energy sink involving an active damper with delay

Abstract

In this paper, we consider a linear oscillator subject to periodic excitation coupled to a nonlinear energy sink with piecewise-quadratic damping characteristics. The damping model includes a time delay to take into account the time needed to change the damping characteristics. In the framework of singular perturbation theory, an analytical/numerical procedure combining complexification, averaging, and multiple scale approaches is used to predict period and quasiperiodic response regimes. The analytical predictions agree satisfactorily with numerical results. It is shown that introducing a relatively small time delay causes significant modifications in the system regimes including suppression of quasiperiodic response regimes and reduction of the response level.

Appendix: 1 D mapping procedure for relaxation oscillations (SMR)

This 1D mapping procedure was developed from [24, 26] to investigate the annihilation of the SMR (or relaxation oscillation).

This procedure is limited to CM shapes as given in Fig. 3. They correspond to the following assumptions: the CM admits two (at $N_2^1$ and $N_2^2$ or at $N_2^1$ and 1) or four (at $N_2^1$, 1, $N_2^2$ and $N_2^3$) local extrema. The first local maximum is located at $N_2^{1}$ and it is differentiable local maximum. We also assume that the excitation amplitude A satisfies condition (76) at $N_2^{1}$ that is to say two singular equilibrium points exist on the fold line associated with $N_2^{1}$. They are characterized by the angular coordinates $\varDelta _2^{1-}$ and $\varDelta _2^{1+}$ given by Eq. (75). In the sequel, we will assume that $\varDelta _2^{1-} < \varDelta _2^{1+}$.

Existence of singular equilibrium points is a necessary condition but not sufficient to ensure the existence of a SMR also named relaxation oscillation. A relaxation oscillation regime corresponds to a motion that is a succession of relaxation cycles. A relaxation cycle alternates between epochs of fast-slow-fast-slow motions [as shown in Fig. 11 curves (1) to (4)]. The epochs of fast motions correspond to jumps from a point on a fold line to a point on the CM (see curves (1) and (3) Fig. 11). The epochs of slow motions correspond to a motion on the CM with an endpoint on a fold line (see curves (2) and (4) Fig. 11).

The objective of the procedure is to verify numerically that, $\forall \;\varDelta _2^{\text{ init }} \in [\varDelta _2^{1-},\varDelta _2^{1+}]$, the end point $(N_2^{\text{ end }},\varDelta _2^{\text{ end }})$ of the cycle starting at $(N_2^{1},\varDelta _2^{\text{ init }})$ satisfies the two conditions: (i) $N_2^{\text{ end }} = N_2^{1}$ and (ii) $\varDelta _2^{\text{ end }}\in [\varDelta _2^{1-},\varDelta _2^{1+}]$.

Fig. 11

Geometry of the CM with a cycle of relaxation oscillation

A cycle corresponds to the following four steps:

• Step 1: (fast motion, curve (1) Fig. 11)

  This step corresponds to a jump from $(N_2^{1},\varDelta _2^{\text{ init }})$ to a point $(N_2^{1(1)},\varDelta _2^{\text{ init }(1)})$ on the CM assuming Equation (47) preserved.

  $(N_2^{1(1)},\varDelta _2^{\text{ init }(1)})$ is obtained solving

  $$\begin{aligned} H(N_2^{1(1)}) = H(N_2^{1}) \end{aligned}$$

  (83)

  with respect to $N_2^{1(1)}$ and using Equation (52) to deduce $\varDelta _2^{\text{ init }(1)}$.

• Step 2: (slow motion, curve (2) Fig. 11)

  This step corresponds to a slow motion in the CM from $(N_2^{1(1)},\varDelta _2^{\text{ init }(1)})$ to $(N_2^{1(2)},\varDelta _2^{\text{ init }(2)})$ where $N_2^{1(2)}=N_2^{2}$ or $N_2^{\text{ sw }}$ or $N_2^{3}$ depending the assumption on the local minima.

  $(N_2^{1(2)},\varDelta _2^{\text{ init }(2)})$ is obtained solving the differential equations (64, 65) with the initial condition $(N_2^{1(1)},\varDelta _2^{\text{ init }(1)})$ up to the time $t^{(2)}$ where $N_2(t^{(2)}) = N_2^{2}$ or $N_2^{\text{ sw }}$ or $N_2^{3}$ as $N_2^{1(2)} = N_2(t^{(2)})$ and $\varDelta _2^{\text{ init }(2)} = \varDelta _2(t^{(2)})$.

• Step 3: (fast motion, curve (3) Fig. 11)

  This step corresponds to a jump from $(N_2^{1(2)},\varDelta _2^{\text{ init }(2)})$ to a point $(N_2^{1(3)},\varDelta _2^{\text{ init }(3)})$ on the CM assuming Equation (47) preserved.

  $(N_2^{1(3)},\varDelta _2^{\text{ init }(3)})$ is obtained solving

  $$\begin{aligned} H(N_2^{1(3)}) = H(N_2^{1(2)}) \end{aligned}$$

  (84)

  with respect to $N_2^{1(3)}$ and using Equation (52) to deduce $\varDelta _2^{\text{ init }(3)}$.

  Note that due to the geometry of the CM (see Fig. 3), $N_2^{1(3)}$ is always smaller than $N_2^{1}$ and Step 4 is valid.

• Step 4: (slow motion, curve (4) Fig. 11)

  This step corresponds to a slow motion in the CM from $(N_2^{1(3)},\varDelta _2^{\text{ init }(3)})$ to $(N_2^{\text{ end }},\varDelta _2^{\text{ end }})$ where $N_2^{\text{ end }}=N_2^{1}$.

  $(N_2^{\text{ end }},\varDelta _2^{\text{ end }})$ is obtained solving the differential equations (64, 65) with the initial condition $(N_2^{1(3)},\varDelta _2^{\text{ init }(3)})$ up to the time $t^{\text{ end }}$ where $N_2(t^{\text{ end }}) = N_2^{1}$ as $N_2^{\text{ end }} = N_2(t^{\text{ end }})$ and $\varDelta _2^{\text{ init }} = \varDelta _2(t^{\text{ end }})$.

In practice, cycles are computed using N points linearly spaced between $\varDelta _2^{1-}$ and $\varDelta _2^{1+}$ as starting points. If, for all the points, the cycles can be built and at the end of step 4 the points return inside the interval $[\varDelta _2^{1-},\varDelta _2^{1+}]$, the relaxation oscillation (SMR) is stable. On the other hand, the relaxation oscillation (SMR) do not appear.

Numerical results presented in this paper were obtained using $N=100$.