In this paper, the influence of physical parameters on the width of saturated control frequency band and the influence of time delay parameters on the stability of saturated control system are studied. The analytical solution of the motion equation of the system when the main resonance and the 1:2 internal resonance occur simultaneously is obtained by multiple scale method, experimentally measured natural frequencies of nonlinear beams. The effects of excitation amplitude, delay feedback coefficients and nonlinear coefficients on saturation control are investigated. The results of the study show that the bandwidth of the saturation control can be increased by increasing the value of the external excitation, the nonlinear coefficients enhance the nonlinear phenomena of the system.
In recent years, the theory and technique of vibration suppression have been studied extensively. In order to suppress the vibration of the main system, various controllers have been developed. The saturation phenomenon means that when the system is stimulated by the main resonance type, the amplitude of the main system response increases with the increase of the amplitude of the external excitation. When the amplitude of the external excitation exceeds a certain critical value, the amplitude of the main system response reaches saturation and no longer increases, then the energy is transferred to the secondary system when the amplitude of the external excitation continues to increase. Saturation phenomenon was first discovered by Nayfeh et al. [
For active control systems, the delay in control processing is caused by transmission delay, measurement of system state, online calculation, data filtering and processing, and the delay caused by the execution of control force. When the control system is applied to a mechanical or structural system, the time delay phenomenon is particularly common, which can limit the performance of the feedback controller. In many systems, the inevitable time delay in the controller will complicate the system behavior and may lead to instability of the dynamic system. Therefore, it is very important to consider the influence of time delay on the system vibration reduction characteristics in nonlinear saturation control vibration reduction systems. This paper mainly studies the influence of physical parameters and time delay parameters on the amplitude and frequency band width of nonlinear beam vibration saturation controller. The structure of this article is as follows. Firstly, the motion equation of the beam is given, and the perturbation analysis and stability analysis of the equilibrium solution are given respectively in Section 2. In Section 3, the natural frequency of the cantilever beam is measured, and in Section 4, the influence of physical parameters and delayed feedback control on the saturation control is studied. In nonlinear saturated control systems, when numerical simulation is carried out, it is found that the nonlinear system is more sensitive to the initial value, and a little change in the initial value leads to system instability, and the study of the time-delay coefficients and the physical parameters of the damping system of the bandwidth is particularly important, and at the same time, it also provides a theoretical basis for engineers and technicians to the real-time operation of the control system.
Warminski et al. [
{ u ¨ ¨ + 2 ζ 1 ω 1 u ˙ + ω 1 2 u + β u 3 − δ ( u u ˙ 2 + u 2 u ¨ ) = f cos ( Ω t ) + γ v 2 ( t − τ 1 ) + g 1 u ˙ ( t − τ 4 ) v ¨ + 2 ζ 2 ω 2 v ˙ + ω 2 2 v = α u ˙ ( t − τ 2 ) v ( t − τ 3 ) + g 2 v ˙ ( t − τ 5 ) (1)
where u, ω 1 and v, ω 2 are the displacements and the natural frequencies of the cantilever beam and controller respectively. ζ 1 and ζ 2 are the damping coefficients of the cantilever beam and controller respectively. β and δ are the system nonlinear coefficients respectively, while γ and α denote the quadratic nonlinear feedback gain coefficients respectively. f and Ω are the external excitation amplitude and frequency respectively. g 1 u ˙ ( t − τ 4 ) and g 2 v ˙ ( t − τ 5 ) are the velocity feedback signals of the cantilever beam and the saturation controller respectively, g 1 and g 2 are the feedback gain coefficients of the cantilever beam and the controller, τ 4 and τ 5 are the feedback time delays. γ v 2 ( t − τ 3 ) is feedback signal of the controller.
In order to analyze the nonlinear saturated system described above, introducing a formal small-scale parameter ε ( 0 < ε < 1 ) to find the solution of steady states of Equation (1). It can be set that: ζ 1 = ε ζ 1 , ζ 2 = ε ζ 2 , g 1 = ε g 1 , g 2 = ε g 2 , β = ε − 1 β , δ = ε − 1 δ , f = ε 2 f . The method of multiple scales is employed to seek second order approximate solutions of (1), Then set the displacements of the cantilever beam and the controller u and v and the time-delay value v τ 1 , u τ 2 , v τ 3 , u τ 4 , v τ 5 are solved as follows:
u ( t , ε ) = ε u 1 ( T 0 , T 1 ) + ε 2 u 2 ( T 0 , T 1 ) + ⋯ (2)
v ( t , ε ) = ε v 1 ( T 0 , T 1 ) + ε 2 v 2 ( T 0 , T 1 ) + ⋯ (3)
v τ 1 ( t , ε ) = ε v 1 τ 1 ( T 0 , T 1 ) + ε 2 v 2 τ 1 ( T 0 , T 1 ) + ⋯ (4)
u τ 2 ( t , ε ) = ε u 1 τ 2 ( T 0 , T 1 ) + ε 2 u 2 τ 2 ( T 0 , T 1 ) + ⋯ (5)
v τ 3 ( t , ε ) = ε v 1 τ 3 ( T 0 , T 1 ) + ε 2 v 2 τ 3 ( T 0 , T 1 ) + ⋯ (6)
u τ 4 ( t , ε ) = ε u 1 τ 4 ( T 0 , T 1 ) + ε 2 u 2 τ 4 ( T 0 , T 1 ) + ⋯ (7)
v τ 5 ( t , ε ) = ε v 1 τ 5 ( T 0 , T 1 ) + ε 2 v 2 τ 5 ( T 0 , T 1 ) + ⋯ (8)
where T 0 = t is the fast-varying time scale, T 1 = ε t is the slow-varying time scale, and T n = ε n t ( n = 0 , 1 , 2 , ⋯ ) is the newly introduced independent variable.
In the following, the system is analyzed in the presence of a 1:1 primary resonance and a 1:2 internal resonance, whose coefficients are required to be equal to zero, and convert small-divisors into secular terms by introducing the detuning parameters σ s and σ c according to:
Ω = ω 1 + ε σ s , 2 ω 2 = ω 1 + ε σ c (9)
we can obtain the solvability conditions:
− A 2 2 γ e i σ c T 1 − 2 i τ 1 ω 2 + 3 A 1 2 β A 1 ¯ + 2 A 1 2 δ A 1 ¯ ω 1 2 + 2 i A 1 ζ 1 ω 1 2 + 2 i A ′ 1 ω 1 − 1 2 f e i σ s T 1 + A 1 g 1 i ω 1 e − i τ 4 ω 1 = 0 (10)
α i ω 1 A 1 A 2 ¯ e i ( τ 3 ω 2 − τ 2 ω 1 − T 1 σ c ) − i ω 2 ( 2 ζ 2 ω 2 A 2 + 2 A ′ 2 ) − A 2 g 2 i ω 2 e − i τ 5 ω 2 = 0 (11)
Introducing polar notation A 1 = 1 2 a 1 ( T 1 ) e i θ 1 ( T 1 ) and A 2 = 1 2 a 2 ( T 1 ) e i θ 2 ( T 1 ) into
Equation (10) and Equation (11), then separating the real and imaginary parts leads to the following equation:
∀ a ∈ B (12)
θ ′ 1 a 1 = − a 2 2 γ 4 ω 1 [ cos ( 2 τ 1 ω 2 + ϕ 2 ) ] − f cos ( ϕ 1 ) 2 ω 1 + 1 2 a 1 g 1 sin ( τ 4 ω 1 ) + 3 a 1 3 β 8 ω 1 + 1 4 a 1 3 δ ω 1 (13)
a ′ 2 = α ω 1 a 1 a 2 4 ω 2 [ cos ( ϕ 2 + τ 3 ω 2 − τ 2 ω 1 ) ] − a 2 [ ζ 2 ω 2 − 1 2 g 2 cos ( τ 5 ω 2 ) ] (14)
θ ′ 2 a 2 = α ω 1 a 1 a 2 4 ω 2 [ sin ( ϕ 2 + τ 3 ω 2 − τ 2 ω 1 ) ] − 1 2 a 2 g 2 sin ( τ 5 ω 2 ) (15)
where ϕ 1 = σ s T 1 − θ 1 , ϕ 2 = θ 1 − 2 θ 2 − σ c T 1 .
Assuming that the amplitudes of both the cantilever beam and the controller are positive, there exist two equilibrium solutions: a single-modal solution and a coupled-modal solution, and the two modal solutions are as follows:
where a 2 = 0 , obtain single-mode solution:
a 1 2 ( ( 3 β 8 ω 1 + 1 4 δ ω 1 ) a 1 2 + 1 2 g 1 cos ( τ 4 ω 1 ) − σ s ) 2 + a 1 2 [ ζ 1 ω 1 + 1 2 g 1 sin ( τ 4 ω 1 ) ] 2 = ( f 2 ω 1 ) 2 (16)
where a 2 ≠ 0 , obtain coupled-mode solution:
a 1 = 4 ω 2 α ω 1 ( ζ 2 ω 2 − 1 2 g 2 cos ( τ 5 ω 2 ) ) 2 + ( σ s − σ c 2 + 1 2 g 2 sin ( τ 5 ω 2 ) ) 2 (17)
To get the steady-state solutions of Equations (12)-(15) and to determine the stability of the equilibrium solutions, the Cartesian coordinates is introduced:
p ′ 1 = − γ ( p 2 2 − q 2 2 ) sin ( 2 τ 1 ω 2 ) 4 ω 1 − γ p 2 q 2 cos ( 2 τ 1 ω 2 ) 2 ω 1 − ζ 1 ω 1 p 1 − q 1 σ s + 1 2 g 1 ( q 1 sin ( τ 4 ω 1 ) − p 1 cos ( τ 4 ω 1 ) ) + q 1 ( p 1 2 + q 1 2 ) ( 3 β 8 ω 1 + 1 4 δ ω 1 ) (18)
q ′ 1 = γ ( p 2 2 − q 2 2 ) cos ( 2 τ 1 ω 2 ) 4 ω 1 − γ p 2 q 2 sin ( 2 τ 1 ω 2 ) 2 ω 1 + f 2 ω 1 + p 1 σ s − ζ 1 ω 1 q 1 − 1 2 g 1 ( p 1 sin ( τ 4 ω 1 ) + q 1 cos ( τ 4 ω 1 ) ) − p 1 ( p 1 2 + q 1 2 ) ( 3 β 8 ω 1 + 1 4 δ ω 1 ) (19)
p ′ 2 = α ω 1 ( p 1 p 2 + q 1 q 2 ) cos ( τ 3 ω 2 − τ 2 ω 1 ) 4 ω 2 − q 2 ( σ s − σ c ) 2 − α ω 1 ( p 1 q 2 − p 2 q 1 ) sin ( τ 3 ω 2 − τ 2 ω 1 ) 4 ω 2 − ζ 2 ω 2 p 2 + 1 2 g 2 [ p 2 cos ( τ 5 ω 2 ) − q 2 sin ( τ 5 ω 2 ) ] (20)
q ′ 2 = α ω 1 ( p 2 q 1 − p 1 q 2 ) cos ( τ 3 ω 2 − τ 2 ω 1 ) 4 ω 2 + p 2 ( σ s − σ c ) 2 − α ω 1 ( p 1 p 2 + q 1 q 2 ) sin ( τ 3 ω 2 − τ 2 ω 1 ) 4 ω 2 − ζ 2 ω 2 q 2 + 1 2 g 2 [ p 2 sin ( τ 5 ω 2 ) + q 2 cos ( τ 5 ω 2 ) ] (21)
where p 1 = a 1 cos ( ϕ 1 ) , q 1 = a 1 sin ( ϕ 1 ) , p 2 = a 2 cos ( ϕ 1 + ϕ 2 2 ) , q 2 = a 2 sin ( ϕ 1 + ϕ 2 2 ) .
The cantilever beam selected for the experiment is a carbon fiber beam, for which modal experiments are conducted to measure the natural frequency and vibration mode of the cantilever beam by force measurement method.
The general flow of this experiment is as follows: establish a cantilever beam model in the modal module, stimulate the cantilever beam with a force hammer, collect the response signals from the acceleration sensor and the force hammer.
In order to solve for the effective saturation control damping bandwidth, and in order to further analyze the effect of the parameters on reducing the amplitude and bandwidth of the main system, it is discussed in the following two cases.
Case 1: σ c = 0 , ω 1 = 16.44 Hz, ω 2 = ω 1 / 2 = 8.22 Hz.
Case 2: σ s = 0 , ω 1 = 16.44 Hz, ω 2 = ω 1 / 2 = 8.22 Hz.
The equilibrium solutions can be solved by setting p ′ 1 = q ′ 1 = p ′ 2 = q ′ 2 = 0 , the amplitude equations of the cantilever beam and the controller can be written as:
a i = p i 2 + q i 2 ( i = 1 , 2 ) (22)
The effect of external excitation on saturation control is discussed below in two cases, from
From
| Physical parameters | α | β | γ |
|---|---|---|---|
| numerical value | 0.8 | 12.21 | 0.01 |
| Physical parameters | δ | ζ 1 | ζ 2 |
| numerical value | 1 | 0.01 | 0.0001 |
| Physical parameters | ω 1 | ω 2 | f |
| numerical value | 16.44 | 8.22 | 0.03 |
| Physical parameters | g 1 | g 2 | τ 1 |
| numerical value | 0 | 0 | 0 |
| Physical parameters | τ 2 | τ 3 | |
| numerical value | 0 | 0 |
From
In this paper, the effects of physical and control parameters on saturation control are investigated. The approximate solution of the vibrating system when the main resonance and the 1:2 internal resonance occur simultaneously is obtained by using the multiscale method. Through the above analysis, the following main conclusions are drawn.
The bandwidth of the saturation control can be increased by increasing the value of the external excitation, the nonlinearity of the beam is enhanced by increasing the cubic nonlinearity factor β . As the feedback gain coefficient g 1 decreases, the saturation control interval increases. Case 1 and Case 2 are the amplitude-frequency response curves of two different frequency bases; when Case 1, the variation curve of the main system amplitude in the saturation control interval is not exactly in the form of an even function, i.e., it is not exactly axisymmetric with respect to the outer tuning parameter σ s = 0 ; whereas, when Case 4, the variation curve of the main system amplitude in the saturation control interval is exactly axisymmetric with respect to the inner tuning parameter σ c = 0 , and the amplitude of the main system outside the saturation control interval is unchanged.
This work is supported by the National Nature and Science Foundation of China under Grant No. 12072140, Natural Science Foundation of Jiangxi Province of China under Grant No. 20202ACBL201003.
The authors declare no conflicts of interest regarding the publication of this paper.
Li, Q.Q., Zhao, Y.Y., Zhou, J.C., Wang, W.K. and Dai, H. (2023) Effects of Physical Parameters and Time-Delay Coefficients on the Amplitude and Frequency Bandwidth of Saturation Controller for a Beam Vibration. Open Journal of Applied Sciences, 13, 2186-2197. https://doi.org/10.4236/ojapps.2023.1311170