Analysis of nonlinear vibration of coupled systems with cubic nonlinearity/Sujungtu sistemu su kubiniu netiesiskumu netiesiniu svyravimu analize.

Bayat, M. ; Pakar, I. ; Shahidi, M. 等

1. Introduction

In the past few decades the motion of multidegree of freedom (multi-DOF) oscillation systems has been widely considered. Moochhala and Raynor [1] proposed an approximate method for the motions of unequal masses connected by (n+1) nonlinear springs and anchored to rigid end walls. Huang [2] studied on the Harmonic oscillations of nonlinear two-degree-of-freedom systems. Gilchrist [3] analyzed the free oscillations of conservative quasilinear systems with two degrees of freedom. Efstathiades [4] developed the work on the existence and characteristic behaviour of combination tones in nonlinear systems with two degrees of freedom. Alexander and Richard [5] considered the resonant dynamics of a two-degree-of-freedom system composed of a linear oscillator weakly coupled to a strongly nonlinear one, with an essential (nonlinearizable) cubic stiffness nonlinearity. Chen [6] used generalized Galerkin's method to nonlinear oscillations of two-degree-of-freedom systems. Ladygina and Manevich [7] investigated the free oscillations of a conservative system with two degrees of freedom having cubic nonlinearities (of symmetric nature) and close natural frequencies by using multiscale method. Cveticanin [8, 9] used a combination of a Jacobi elliptic function and a trigonometric function to obtain an analytical solution for the motion of a two-mass system with two degrees of freedom in which the masses were connected with three springs.

Two degree of freedom (TDOF) systems are very important in physics and engineering and many practical engineering vibration systems such as elastic beams supported by two springs and vibration of a milling machine [10] can be studied by considering them as a TDOF systems. The TDOF oscillation systems consist of two second-order differential equations with cubic nonlinearities. So, a set of differential algebraic equations by introducing new variables was obtained from transforming the equations of motion of a mechanical system which associated with the linear and nonlinear springs. In general, finding an exact analytical solution for nonlinear equations is extremely difficult. Therefore, many analytical and numerical approaches have been investigated. The most useful methods for solving nonlinear equations are perturbation methods. They are not valid for strongly nonlinear equations and there have many shortcomings. Many new techniques have appeared in the open literature to overcome the shortcomings, such as Homotopy perturbation [11], energy balance [12-15], variational approach [16, 17], max-min approach [18], Iteration perturbation method [19] and other analytical and numerical methods [20-32].

In the present paper, we applied He's Max-Min Approach (MMA) and He's Improved Amplitude-Frequency Formulation (IAFF) for nonlinear oscillators which were proposed by J.H. He [26, 30]. Both of them lead us to a very rapid convergence of the solution, and they can be easily extended to other nonlinear oscillations. Comparisons between analytical and exact solutions show that He's MMA and He's IAFF methods can converge to an accurate periodic solution for nonlinear systems.

2. Basic idea of he's max-min approach method

We consider a generalized nonlinear oscillator in the form

u" + uf(u) = 0, u (0) = A, u'(0) = 0 (1)

where f (u) is a nonnegative function of u. According to the idea of the max-min method, we choose a trial function in the form

u (t) = A cos ([omega]t) (2)

where the [omega] unknown frequency to be further is determined. Observe that the square of frequency, [[omega].sup.2], is never less than that in the solution

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3)

of the following linear oscillator

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4)

where [f.sub.min] is the minimum value of the function f (u). In addition, [[omega].sup.2] never exceeds the square of frequency of the solution

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (5)

of the following oscillator

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (6)

where [f.sub.max] is the maximum value of the function f (u) . Hence, it follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (7)

According to He Chentian interpolation [26, 27], we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (8)

or

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (9)

where m and n are weighting factors, k = n/m . So the solution of Eq. (1) can be expressed as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (10)

The value of k can be approximately determined by various approximate methods [26-28]. Among others, hereby we use the residual method [26]. Substituting Eq. (10) into Eq. (1) results in the following residual

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (11)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], if by chance, Eq. (10) is the exact solution, then R (t;k) is vanishing completely. Since our approach is only an approximation to the exact solution, we set

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (12)

where T = 2[pi]/[omega]. Solving the above equation, we can easily obtain.

In the present paper, we consider a general nonlinear oscillator in the form [29]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (13)

Substituting the above equation into Eq. (10), we obtain the approximate solution of Eq. (1).

3. Basic idea of improved amplitude-frequency formulation

We consider a generalized nonlinear oscillator in the form [30]

u" + f (u) = 0, u (0) = A, u'(0) = 0 (14)

We use two following trial functions

[u.sub.1](t) = Acos ([[omega].sub.1]t) (15)

and

[u.sub.2](t) = Acos([[omega].sub.2]t) (16)

The residuals are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (17)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (18)

The original frequency-amplitude formulation

reads [30, 31]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (19)

He used the following formulation [30, 31] and Geng and Cai improved the formulation by choosing another location point [31].

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (20)

This is the improved form by Geng and Cai.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (21)

The point is: cos ([[omega].sub.1] t) = cos ([[omega].sub.2] t) = k.

Substituting the obtained [omega] into u(t) = Acos ([omega] t), we can obtain the constant k in [[omega].sup.2] equation in order to have the frequency without irrelevant parameter.

To improve its accuracy, we can use the following trial function when they are required

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (22)

or

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (23)

But in most cases because of the sufficient accuracy, trial functions are as follow and just the first term

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (24)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (25)

where a and c are unknown constants. In addition we can set cos t = k in [u.sub.1], and cos (at) = k in [u.sub.2].

4. Examples of nonlinear two degree of freedom (TDOF) oscillating systems

In this section, two practical examples of TDOF oscillation systems are illustrated to show the applicability, accuracy and effectiveness of the proposed approach.

4.1. Example 1

A two-mass system connected with linear and nonlinear stiffnesses. Consider the two-mass system model as shown in Fig. 1. The equation of motion is given as [9]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (26)

with initial conditions

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (27)

[FIGURE 1 OMITTED]

Where double dots in Eq. (26) denote double differentiation with respect to time, [k.sub.1] and [k.sub.2] are linear and nonlinear coefficients of the spring stiffness, respectively. Dividing Eq. (26) by mass m yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (28)

Introducing intermediate variables u and v as follows [32]

x: = u (29a)

y - x: = v (29b)

and transforming Eqs. (29.a) and (29.b) yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (30)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (31)

where [alpha] = [k.sub.1]/m and [beta] = [k.sub.2]/m . Eq. (30) is rearranged as follows

u = [alpha] v + [beta][v.sup.3] (32)

Substituting Eq. (32) into Eq. (31) yields

v + 2[alpha]v + 2[beta][v.sup.3] = 0 (33)

with initial conditions

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (34)

4.1.1. Solution using MMA

We can rewrite Eq. (32) in the following form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (35)

We choose a trial-function in the form

v = Acos ([omega]t) (36)

where [omega] the frequency to be is determined the maximum and minimum values of 2a + 2[beta][v.sup.2] will be 2[alpha] + 2[beta][A.sup.2] and 2[alpha] respectively, so we can write

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (37)

According to He Chengtian's inequality [27, 28], we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (38)

where m and n are weighting factors, k = n/m + n. Therefore the frequency can be approximated as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (39)

Its approximate solution reads

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (40)

In view of the approximate solution, Eq. (40), we rewrite Eq. (33) in the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (41)

If by any chance Eq. (30) is the exact solution, then the right side of Eq. (31) vanishes completely. Considering our approach which is just an approximation one, we set

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (42)

where T = 2[pi]/[omega]. Solving the above equation, we can easily obtain

k = 3/4 (43)

Finally the frequency is obtained as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (44)

According to Eqs. (36) and (44), we can obtain the following approximate solution

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (45)

The first-order analytical approximation for u (t) is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (46)

Therefore, the first-order analytical approximate displacements x (t) and y (t) are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (47)

4.1.2. Solution using IAFF

We use trial functions, as follows:

[v.sub.1] (t) = Acos t (48)

and

[v.sub.2] (t)= Acos(2t) (49)

Respectively, the residual equations are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (50)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (51)

Considering cos [t.sub.1] = cos 2[t.sub.2] = k we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (52)

We can rewrite v (t) = A cos ([omega]t) in the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (53)

In view of the approximate solution, we can rewrite the main equation in the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (54)

If by any chance [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] is the exact solution, then the right side of Eq. (54) vanishes completely. Considering our approach which is just an approximation one, we set

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (55)

Considering [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] and substituting to Eq. (55) and solving the integral t, we have

k = 1/2 [square root of 3] (56)

So, substituting Eq. (56) into Eq. (52), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (57)

4.2. Example 2

A two-mass system connected with linear and nonlinear stiffnesses fixed to the body.

Consider a two-mass system connected with linear and nonlinear springs and fixed to a body at two ends as shown in Fig. 2.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (58)

with initial conditions

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (59)

[FIGURE 2 OMITTED]

Where double dots in Eq. (58) denote double differentiation with respect to time t, [k.sub.1] and [k.sub.2] are linear and nonlinear coefficients of the spring stiffness and [k.sub.3] is the nonlinear coefficient of the spring stiffness. Dividing Eq. (58) by mass m yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (60)

Like in Example 1, transforming the above equations using intermediate variables in Eqs. (29.a) and (29.b) yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (61)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (62)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. Eq. (61) is rearranged as follows

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (63)

Substituting Eq. (61) into Eq. (62) yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (64)

with initial conditions

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (65)

4.2.1. Solution using MMA

We can re-write Eq. (64) in the following form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (66)

We choose a trial-function in the form

v = Acos ([omega]t) (67)

where [omega] the frequency to be is determined the maximum and minimum values of [alpha] + 2[beta] + 2[xi][v.sup.2] will be [alpha] + 2[beta] + 2[xi][A.sup.2] and [alpha] + 2[beta] respectively, so we can write

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (68)

According to He Chengtian's inequality [27, 28], we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (69)

where m and n are weighting factors, k = n/m + n . Therefore the frequency can be approximated as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (70)

Its approximate solution reads

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (71)

In view of the approximate solution, Eq. (71), we re-write Eq. (64) in the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (72)

If by any chance Eq. (71) is the exact solution, then the right side of Eq. (72) vanishes completely. Considering our approach which is just an approximation one, we set

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (73)

where T = 2[pi]/[omega]. Solving the above equation, we can easily obtain

k = 3/4 (74)

Finally the frequency is obtained as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (75)

According to Eqs. (75) and (67), we can obtain the following approximate solution

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (76)

The first-order analytical approximation for u(t) is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (77)

Therefore, the first-order analytical approximate displacements x(t) and y (t) are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (78)

4. 2.2. Solution using IAFF

We use trial functions, as follows

[v.sub.1] (t)= Acos t (79)

and

[v.sub.2] (t) = Acos(2t) (80)

Respectively, the residual equations are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (81)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (82)

Considering cos [t.sub.1] = cos 2[t.sub.2] = k we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (83)

We can rewrite v(t)= A cos ([omega]t) in the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (84)

In view of the approximate solution, we can rewrite the main equation in the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (85)

If by any chance Eq. (84) is the exact solution, then the right side of Eq. (85) vanishes completely. Considering our approach which is just an approximation one, we set

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (86)

Considering the term [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] and substituting the term to Eq. (86) and solving the integral term, we have

k = 1/2 [square root of 3] (87)

So, substituting Eq. (87) into Eq. (86), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (88)

5. Discussion of the examples

Comparisons with published data and exact solutions [8, 9] are presented and tabulated to illustrate and verify the accuracy of the MMA and IAFF .The first-order approximate solutions is of a high accuracy and the percentage error improves significantly from lower order to higher order analytical approximations for different parameters and initial amplitudes. Hence, it is concluded that excellent agreement with the exact so.

Tables 1 and 2; give the comparison of obtained results with the exact solutions [8, 9] for different m, [k.sub.1], [k.sub.2], [k.sub.3] and initial conditions. It can be observed from Tables 1 and 2 that there are an excellent agreement between the results obtained from the MMA and IAFF method and exact one [8, 9]. The maximum relative error between the MMA and IAFF results and exact results is 2.220415%.

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

[FIGURE 6 OMITTED]

[FIGURE 7 OMITTED]

[FIGURE 8 OMITTED]

A comparison of the time history oscillatory displacement response for the two masses with exact solutions are presented in Figs. 3-6 for example 1 and Figs. 7-10 for example 2. From the Figs. 3 and 7, motions of the systems are periodic motions and the amplitude of vibrations is function of the initial conditions. As shown in Figs. 3-10, it is apparent that the MMA and IAFF have an excellent agreement with the numerical solution using the exact solution. These expressions are valid for a wide range of vibration amplitudes and initial conditions. The proposed methods are quickly convergent and can also be readily generalized to two-degree-of-freedom oscillation systems with quadratic nonlinearity by combining the transformation technique. The accuracy of the results shows that the MMA and IAFF can be potentially used for the analysis of strongly nonlinear vibration problems with high accuracy.

[FIGURE 9 OMITTED]

[FIGURE 10 OMITTED]

6. Conclusion

Two powerful explicit analytical approaches have been developed for a set of second-order coupled differential equations with cubic nonlinearities that govern the nonlinear free vibration of conservative two degree of freedom systems. The solutions have been achieved using the MMA and IAFF. Excellent agreement between approximate frequencies and the exact one are demonstrated and discussed. The methods which are proved to be powerful mathematical tools for studying of nonlinear oscillators. According to the results, the precision and convergence rate of the solutions increase using MMA and IAFF. In conclusion, two practical examples of two-mass systems with free and fixed ends and with linear and nonlinear stiffnesses have been presented and discussed. The first-order approximate solutions are of a high accuracy. Of course, the accuracy can be improved upon using a higher order approximate solution. The result shows that the proposed method for solving TDOF system problems gives results that are highly consistent with published data and exact solutions. The MMA and IAFFare two well-established methods for the analysis of nonlinear systems and could be easily extended to any nonlinear equations. The achieved results indicated that MMA and IAFF are extremely simple, easy, powerful, and triggers good accuracy.

Received December 21, 2011

Accepted November 10, 2011

References

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M. Bayat *, I. Pakar **, M. Shahidi ***

* Department of Civil Engineering, Torbateheydarieh Branch, Islamic Azad University, Torbateheydarieh, Iran E-mail: [email protected]

** Department of Civil Engineering, Torbateheydarieh Branch, Islamic Azad University, Torbateheydarieh, Iran E-mail: [email protected]

*** Department of Civil Engineering, Torbateheydarieh Branch, Islamic Azad University, Torbateheydarieh, Iran E-mail: [email protected]

Table 1
Comparison of frequency corresponding to various parameters of system

Constant parameters

m     [k.sub.1]   [k.sub.2]   [k.sub.3]   [X.sub.0]   [Y.sub.0]

1     0.5         0.5         1           5           5
1     1           1           5           1           1
5     2           0.5         5           10          10
10    5           5           10          20          20
20    40          50          20          10          10
50    100         50          -10         20          20
100   400         100         50          -50         -50
200   100         50          100         300         300
500   500         1000        400         600         600

Constant    Analytical               Exact
parameter   solution                 solution
m           [[omega].sub.MMA=IAFF]   [[omega].sub.Exact] [9]

1           3.605551                 3.539243
1           5.09902                  5.005246
5           4.421538                 4.333499
10          8.717798                 8.533586
20          19.46792                 19.05429
50          36.79674                 36.00234
100         122.5071                 119.8489
200         183.7145                 179.7239
500         244.9571                 239.6368

Constat
parameters   Relative
             error %
m            ([[omega].sub.MMA=IAFF]-
             [[omega].SUB.Ex])/
             [[omega].sub.Ex]

1            1.873506
1            1.873506
5            2.031592
10           2.158667
20           2.170820
50           2.206522
100          2.217969
200          2.220354
500          2.220179

Table 2
Comparison of frequency corresponding to various parameters of system

Constant parameters

m     [k.sub.1]   [k.sub.2]  [k.sub.3]   [X.sub.0]   [Y.sub.0]

1        0.5        0.5         0.5         1           5
1        1          1           2           5           1
5        2          0.5         5           5          10
10       5          5           10          10         20
20       40          50         50          20         10
50       100         50         100         -10        20
100      400        100         200         50         -50
200      100         50         400         200        300
500      500        1000        500         400        600

Constant           Analytical                  Exact
parameters          solution                  solution
m            [[omega].sub.MMA=IAFF]    [[omega].sub.Exact] [7]

1                   3.674235                  3.611743
1                   7.141428                  7.004694
5                    6.17252                  6.042804
10                  12.30853                  12.04665
20                  19.54482                  19.13632
50                  52.00000                  50.87391
100                 173.2224                  169.4611
200                  244.951                  239.6302
500                 346.4174                  338.8929

Constant             Relative
parameters            error %
m               ([[omega].sub.MMA=IAFF] -
             [[omega].sub.Ex])/[[omega].sub.Ex]

1                    1.730234
1                    1.952045
5                    2.146618
10                   2.173874
20                   2.134672
50                   2.213492
100                  2.219547
200                  2.220415
500                  2.220297