Non-Linear Analysis of Vibrations of Non-Linear System Subjected to Multi-Excitation Forces via a Non-Linear Absorber ()
1. Introduction
Vibrations are the cause of discomfort, disturbance, damage, and sometimes destruction of machines and structures. It must be reduced or controlled or eliminated. One of the most common methods of vibration control is the dynamic absorber. It has the advantages of low cost and simple operation at one modal frequency. In the domain of many mechanical vibration systems the coupled non-linear vibration of such systems can be reduced to non-linear second order differential equations which are solved analytically and numerically. Queini and Nayfeh [1] proposed a non-linear active control law to suppress the vibrations of the first mode of a cantilever beam when subjected to a principal parametric excitation. The method of multiple scales is applied throughout. The analysis revealed that cubic velocity feedback reduced the amplitude of the response. Asfar [2] took material non-linearity into consideration in the analysis of the performance of an elastomeric damper with a spring hardening cubic effects near primary resonance condition applying multiple time scale method. Eissa [3] reported that when using a dynamic absorber, its damping coefficient should be kept minimal for better system performance. Eissa [4] has shown that for controlling the vibration of a system subjected to harmonic excitations, the fundamental or the first harmonic absorber is the most effective one. Eissa and El-Ganaini [5,6] studied the control of both vibration and dynamics chaos of mechanical system having quadratic and cubic non-linearities, subjected to harmonic excitation using multi-absorbers. Kamel and Amer [7] studied the behavior of one-degreeof-freedom system with different quadratic damping and cubic stiffness non-linearities simulating the axial vibration of a cantilever beam under multi-parametric excitation forces. Song et al. [8] investigated the vibration response of the spring mass damper system with a parametrically excited pendulum hinged to the mass using the harmonic balance method. The stability analysis showed that the area of unstable motion of the system obtained from the third order approximation.
Soom [9] and Jordanov [10] studied the optimal parameter design of linear and non-linear dynamic vibration absorbers for damped primary systems. They examined optimization criteria other than the traditional one and obtained small improvements in steady state response by using non-linear springs. However, the presence of the non-linearities introduces dangerous instability, which in some cases may result in amplification rather than reduction of the vibration amplitudes [11,12]. Natsiavas [13] applied the method of averaging to investigate the steady state oscillations and stability of nonlinear dynamic vibration absorbers. He pointed out that proper selection of the system parameters would result insubstantial improvements of non-linear absorbers and avoid dangerous effects that are likely to occur due to the presence of the non-linearities. Zhu et al. [14] studied the non-linear dynamics of a two-degree-of freedom vibrating system having both non-linear damping and nonlinear spring using the averaging method. Results showed that the vibration amplitude can be reduced by properly selecting the values of non-linear damper, non-linear spring stiffness and the range of exciting frequency.
Nayfeh [15] compared application of the method of multiple scales with reconstitution and the generalized method of averaging for determining higher-order approximations of three single-degree-of-freedom systems and a two-degree-of-freedom system. Amer [16] investigated the coupling of two non-linear oscillators of the system and absorber representing ultrasonic cutting process. A threshold value of linear damping has been obtained, where the system vibration can be reduced dramatically. Eissa and Sayed [17-19] and Sayed [20], studied the effects of different active controllers on simple and spring pendulum at the primary resonance via negative velocity feedback or its square or cubic. Hamed et al. [21-23] studied USM model subject to multi-external or both multi-external and multi-parametric and both multi-external and tuned excitation forces. The model consists of multi-degree-of-freedom system consisting of the tool holder and absorbers (tools) simulating ultrasonic machining process. The advantages of using multi-tools are to machine different materials and different shapes at the same time. Sayed and Hamed [24] studied the response of a two-degree-of-freedom system with quadratic coupling under parametric and harmonic excitations. The method of multiple scale perturbation technique is applied to solve the non-linear differential equations and obtain approximate solutions up to and including the second-order approximations. Sayed and Kamel [25,26] investigated the effects of different controllers on the vibrating system and the saturation control of a linear absorber to reduce vibrations due to rotor blade flapping motion. Amer and Sayed [27], studied the response of one-degree-of freedom, non-linear system under multi-parametric and external excitation forces simulating the vibration of the cantilever beam.
2. Mathematical Analysis
Using a non-linear absorber connected to the main system, a model of a two degree-of-freedom oscillator under consideration can be written in the forms:
(1)
(2)
where x1, x2 are the displacement of the main system and absorber, w1, w2 are the natural frequencies, Wj (j = 1, 2) forcing frequencies, e small perturbation parameter, fj the forcing amplitudes, ai, bj non-linear parameters (i = 1, ···, 5), zs damping coefficient (s = 1, ···, 4). A first-order uniform solution of equations (1) and (2) is sought using the method of multiple scales [28] in the form:
(3)
(4)
where T0 = t is the fast time scale, which is associated with changes occurring at the frequencies wj and Wj and T1 = et is the slow time scale, which is associated with modulations in the amplitudes and phases resulting from the non-linearities and parametric resonance. In term of T0 and T1 the time derivatives became
(5)
where Dn differential operators; Dn = ¶/¶Tn (n = 0,1). Substituting equations (3) and (4) into equations (1) and (2) and equating the coefficients of same power of e in both sides, we obtain:
(6)
(7)
(8)
(9)
The general solution of equations (6) and (7) can be expressed in the form
(10)
(11)
where 
and A1, A2 are unknown functions in T1, which can be determined from eliminating the secular terms at the next approximation, and cc stands for the conjugate of the preceding terms. The particular solutions of equations (8) and (9) after eliminateing the secular terms are given by:
(12)
(13)
where Ui, i = 1 to 4, Hi, i = 1 to 18, are complex functions in T1. The general solution of x1 and x2 up to the second-order approximation is given by
From the above derived solutions, the reported resonance cases are
1) Primary resonance: W1 @ w1, W1 @ w2
2) Sub-harmonic resonance: W2 @ 2w1
3) Internal resonance: w1 @ nw2, w2 @ nw1, n = 1, 2, 3
4) Combined resonance: W2 @ ± w1 ± w2
5) Simultaneous resonance: Any combination of the above resonance cases is considered as simultaneous resonance.
3. Stability of the System
The stability of the system is investigated at one of the worst resonance cases (confirmed numerically), which is the simultaneous primary, sub-harmonic and internal resonance where W1 @ w1, W2 @ 2w1 and w2 @ w1 Using the detuning parameters s1, s2 and s3 such that:
(14)
Substituting equation (14) into equations (8) and (9) and eliminating the secular terms leads to the solvability conditions as
(15)
(16)
To analyze the solutions of Equations (15) and (16), we express A1 and A2 in the polar form
(17)
where an and mn are real of T1. Substituting Equation (17) into Equations (15) and (16) and separating real and imaginary part yields,
(18)
(19)
(20)
(21)
where
(22)
For the steady state solution
. Then from equation (22) yields
(23)
Then it follows from equations (18)-(21) that the steady state solutions are given by
(24)
(25)
(26)
(27)
Solving the resulting algebraic equations yields two possibilities for the fixed points for each case.
Case (1): The controller is deactivated (a1 ≠ 0, a2 = 0), the frequency response equation can be obtained in the form (after eliminating all parameters corresponding to the controller).
(28)
Case (2): The controller is activated (a1 ≠ 0, a2 ≠ 0), the resulting two equations are obtained
(29)
(30)
4. Results and Discussion
The differential equation (1) governing the oscillating main system is solved numerically (applied Runge-Kutta 4th order method) after eliminating all parameters corresponding to the controller.
Table 1 summaries the worst resonance cases with and without absorber.
Figure 1 shows the steady state amplitude and phase plane of the main system at the primary resonance where W1 @ w1. In this figure, we observe that the steady state amplitude is about four of the maximum excitation force amplitude F1. The system is stable with multi-limit cycle and the response of the system is tuned with some chaos.
Figure 2 shows the steady state amplitude and phase plane of the main system at the sub-harmonic resonance case where W2 @ 2w1. We note that the steady state amplitude is about 3 of the maximum excitation force amplitude F1. Also the chaotic wave motion will increase.
The steady state amplitude and phase plane of the main system at simultaneous primary and principle parametric resonance respectively where W1 @ w1, W2 @ 2w1 is shown in Figure 3. In this figure, the steady state amplitude is about 5 of the maximum excitation force amplitude F1 respectively. The system is stable with fine
Table 1. Summary of worst resonance cases.
Figure 1. System behavior without absorber at W1 @ w1 and W2 is a way from w1. ξ1 = 0.05, α1 = 0.03, α2 = 0.04, F2 = 0.3.
Figure 2. System behavior without absorber at W2 @ 2w1 and W1 is a way from w1.
limit cycle, denoting that the system is free from dynamic chaos.
4.1. Effect of the Controller
Figure 4 shows that the steady state amplitude of the system with absorber at the simultaneous resonance W1 @ w1, W2 @ 2w1 and w2 @ w1. It can be seen for the main system that the steady state amplitude is 0.33%, but the steady state amplitude of the controller is about 42% of excitation amplitude F1. This means that the effectiveness of the absorber Ea (Ea = the steady state amplitude of the main system without absorber/the steady state amplitude of main system with absorber) is about 1500. Also, the oscillations of the system and absorber have multi-limit cycle and limit cycle respectively.