2015, Vol. 36
Shanghai University
Article Information
- Wenan JIANG, Guoce ZHANG, Liqun CHEN. 2015.
- Forced response of quadratic nonlinear oscillator: comparison of various approaches
- Appl. Math. Mech. -Engl. Ed., 36(11): 1403-1416
- http://dx.doi.org/10.1007/s10483-015-1991-7
Article History
- Received May 13, 2014;
- Revised Sept. 5, 2014
2. College of Sciences, Shanghai University, Shanghai 200444, China;
3. Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai University, Shanghai 200072, China
Quadratic nonlinearity arises in different models,e.g.,cutting chatter[1],buckled beams[2], pneumatic systems[3],and elastic foundations[4]. A quadratic nonlinear oscillator is a useful test model for approximate analytical approaches. Therefore,the analysis on a quadratic nonlinear oscillator is a fundamental and significant subject.
There are many important investigations on quadratic nonlinear oscillators. Cheung et al.[5] developed a modified Lindstedt-Poincar´e method to calculate the perturbation solution of the free vibration oscillators of quadratic nonlinearity,and demonstrated that the results were in excellent agreement with those obtained by the incremental harmonic balance method. Huseyin and Lin[6] used an intrinsic multiple-scale harmonic balance technique to study the approximate solutions of free quadratic and cubic nonlinear oscillators. Xu and Cheung[7] employed the averaging method based on the generalized harmonic functions to seek the approximate solutions of strongly nonlinear oscillators,and supported the resulting solutions by the numerical integration. Pakdemirli[8] used the multiple-scale method to investigate the continuous free vibration systems with quadratic and cubic nonlinearities. Chen et al. proposed the elliptic perturbation method[9] and the elliptic Lindstedt-Poincar´e method[10] to determine the periodic solution of the free vibration of quadratic nonlinear oscillators,and found that the solution obtained by the elliptic perturbation method was more accurate than that obtained by the Lindstedt-Poincar´e method. Lakrad and Belhaq[11] employed the multiple-scale method with the Jacobian elliptic functions to calculate the solution of nonlinear quadratic and cubic self-excited systems. Mickens[12] used the phase space techniques to compare the free vibrations of three types of quadratic nonlinear oscillators. Cveticanin[13] utilized the Jacobian elliptic function to express the exact analytical solution of a mass-spring oscillator with strong quadratic nonlinearity. Wu and Lim[14] combined the harmonic balance method and the linearization to construct the approximate analytical solutions of a conservative quadratic nonlinear oscillator,and showed that the obtained solutions were in good agreement with the exact solutions. Hu[15, 16, 17] conducted a systematic investigation on the conservative quadratic nonlinear oscillator,expressed the exact solution of the oscillator in the Jacobian elliptic function[15],proposed a modified iteration scheme with a higher accuracy than the first-order harmonic balance method[16],and demonstrated that the harmonic balance method was more accurate than the second approximate solution of the Lindstedt-Poincar´e method[17]. It should be remarked that the harmonic balance method[17] is without the zeroth-order harmonic component.
All the above-mentioned works on quadratic nonlinear oscillators focus on the free vibration without external excitations. There are no investigations on the forced vibration of quadratic nonlinear oscillators,even in the case of weak nonlinearity. To address the lack of research in the aspect,the present work treats the primary resonance of quadratic nonlinear oscillators. Interestingly,various approximate analytical approaches yield different quantitatively and qualitatively results.
The manuscript is organized as follows. Section 2 introduces a quadratic nonlinear oscillator as the truncated model of the snap-through mechanism. Section 3 treats the forced quadratic nonlinear oscillators by use of the Lindstedt-Poincar´e method,the multiple-scale method,the averaging method,and the harmonic balance method. Section 4 compares the analytical outcomes with the numerical calculations. Section 5 ends the paper with concluding remarks.
2 Snap-through mechanism as quadratic nonlinear oscillatorThe research relevant to the snap-through mechanism can be dated back to Thompson and Hunt[18]. Recently,the possibility of the application of the device in harvesting vibration energy was explored[19]. Figure 1 shows the snap-through mechanism under a base excitation. The dynamic equation can be derived from the Newton law as follows:
where m is the mass,x is the displacement of the mass,c is the damping coefficient,ks is the spring stiffness,L is the original length of the spring,l is the distance between the center and the edge of the frame,θis the inclination of the spring with respect to the horizontal,g is the gravitational acceleration,and y is a harmonic base excitation.
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| Fig. 1 Device of mass-spring-damper for snap-through mechanism |
The static equilibrium positions of the system satisfy the equation as follows:
For the excited small-amplitude motion,the snap-through system undergoes a motion around an equilibrium point. In this case,the nonlinear term in (1) can be expanded into the Taylor series at a stable equilibrium (x1,0) solved by (2). Omitting the higher order terms in the resulting expanding expression and shifting the origin of the coordinate by introducing the new variable x ↔ x − x1,one can cast (1) into the dimensionless form as follows:
for small base excitation,where x is the dimensionless generalized coordinate," is a small parameter,c is a damping coefficient,ω is the natural frequency,α is the coefficient of the quadratic nonlinear term,and f and Ω are the amplitude and the frequency of the harmonic force,respectively.
In the following,the primary resonance is treated. In this case,the excitation frequency is very close to the natural frequency. Introduce the detuning parameter σ0 satisfying
3 Analytical approximation of amplitude-frequency relationship 3.1 Lindstedt-Poincar´e method
The Lindstedt-Poincar´e method assumes an approximate expansion of the solution of (3) in the form as follows:
Substituting (4) and (5) into (3) and equating the coefficients of the same order ε yield
where x′ represents the derivative of x with respect to the new time scale τ = Ω t.
The solution of (6) is
with the amplitude a and the initial phase φ. Substituting (8) into (7) and eliminating the secular terms lead to
Eliminating φ from (9),we can obtain the amplitude-frequency response relationship as follows:
which is actually the amplitude-frequency response relationship of a linear oscillator defined by (3) where α = 0. Therefore,up to the first-order approximation,the Lindstedt-Poincar´e method cannot account for the quadratic nonlinearity.
Substituting (8) into (7) and omitting the secular terms yield
According to (5),we can obtain the second-order approximation as follows:
indicating that the amplitude of the second-order approximation comes from the first-order approximation. However,the amplitude-frequency response relationship of the first-order approximation is linear. Consequently,when the excitation amplitude is a first-order small quantity of the bookkeeping parameter,the second-order approximation does not have multivaluedness and bending.
3.2 Multiple-scale methodThe multiple-scale method assumes an approximate expansion of the solution of (3) in the following form:
where
Denote
Then,
Substituting (13) and (14) into (3) and equating the coefficients of ε0 and ε1 yield
The solution of (15) can be expressed as follows:
where A(T1) is an unknown complex function of T1,and c.c. stands for the complex conjugate of the preceding terms. Substituting (17) into (16) and eliminating the secular terms yield
Express the unknown function A in the polar form as follows:
Substituting (19) into (18) and separating the resulting equation into real and imaginary parts, one has
where
For a steady-state response,both a and should be constants. Therefore,from (20),we get
Eliminating γ from (21) yields (10). Thus,the amplitude-frequency response relationship derived from the multiple-scale method is the same as that derived from the Lindstedt-Poincar´e method.
Substituting (19) into (17) and omitting the secular terms yield
Substituting (17) and (22) into (13) yields the second approximate as follows:
Thus,the second approximate derived from the multiple-scale method is the same as that derived from the Lindstedt-Poincar´e method.
3.3 Averaging methodThe averaging method is employed to seek the solution to (3) in the form as follows:
where
The averaging method assumes the solution of (24) in the following form:
Let Ψ= Ωt − φ. Then,(24) and (26) yield
Averaging the right-hand sides of (27) over Ψ from 0 to 2π yields
Substituting (25) into (28) yields
For a steady-state response,both a and ' should be constants. Therefore,
Eliminating φ from (30) yields the amplitude-frequency response relationship as follows:
(10) and (31) indicate that the amplitude-frequency response relationship derived from the averaging method is not identical to that obtained by the Lindstedt-Poincar´e method or the multiple-scale method.
3.4 Harmonic balance methodThe harmonic balance method assumes that the solution of (3) is
where a0,a1,and b1 are the coefficients to be determined.
Substituting (32) into (3) and equalizing the coefficients of the constant terms and the first harmonic components yield
Decoupling a0 from a1 and b1 in (33) yields
Once a0 has been determined,a1 and b1 can be solved as follows:
Then,the amplitude-frequency response relationship can be calculated by
3.5 Three-time scale method
We seek a solution of (3) in the three time scales form
where
Denote
Then,
Use the three-time scale method to rescale the damping coefficient,the amplitude of excitation, and the detuning parameter,respectively,i.e.,
Then,we can rewrite the equation of the system as follows:
Substituting (37),(38),and (39) into (40) and equating the coefficients of the same power " yield
The solution of (41) can be expressed as follows:
Substituting (44) into (42) and eliminating the secular terms yield
With this result,the solution of (42) can be written as follows:
Substituting (44) and (46) into (43) and eliminating the secular terms yield
Express the unknown function A in the polar form as follows:
Substituting (48) into (47) and separating the resulting equation into real and imaginary parts yield
where
For a steady-state response,both a and should be constants. Therefore,(49) leads to
Eliminating γ from (50) yields the amplitude-frequency response relationship derived from the three-time scale method as follows:
where the characteristic of nonlinearity exists.
The amplitude of (50) satisfies the following condition:
When the amplitude of the steady-state response is the maximum,the detuning parameter is
It should be noticed that the detuning is negative for any quadratic nonlinear coefficients. Therefore,the quadratic nonlinear system results in the softening type nonlinear characteristic.
Substituting (48) into (44) and (46),according to (37),we can obtain the second-order approximate as follows:
Using the same scale,the second-order approximation derived from the Lindstedt-Poincar´e method is the same as that derived from the three-time scale method.
Although the form of (54) is the same as (12) and (23),the amplitude of (54) comes from the amplitude-frequency response relationship (51) which is nonlinear. Consequently,the secondorder approximation of the three-time scale method has multivaluedness and bending. Therefore, the three-time scale method can predict the steady-state response of the quadratic nonlinear system.
3.6 Primary resonance under strong harmonic excitationNow,consider a quadratic nonlinearity oscillator subject to a strong external harmonic excitation,i.e.,
In the primary resonance,it is difficult to directly apply the Lindstedt-Poincar´e method and the multiple-scale method,because the secular term occurs in the zeroth-order approximation. The averaging method still works. The similar procedure in Subsection 3.4 leads to the amplitude-frequency response relationship for (55) as follows:
The harmonic balance method is also valid. The similar procedure in Subsection 3.4 leads to the amplitude-frequency response relationship (36). Therefore,we can obtain a0 by
and a1 and b1 by
4 Comparisons of numerical results
To examine the analytical results in the preceding section,the fourth-order Runge-Kutta algorithm is employed to integrate (3) and (55). The parameters are chosen as follows: ε = 0.01, ω = 1,c = 2,and α = 5.
4.1 Primary resonance under weak harmonic excitationFigure 2 shows the amplitude-frequency response curves of (3) with (4) for two excitation amplitudes. In Fig. 2,the hollow dot lines,the solid (stable) lines,and the solid lines with solid dots stand for the outcomes of the numerical integration,the harmonic balance method, and the multiple-scale method,respectively. The outcome of the Lindstedt-Poincar´e method is identical to that of the multiple-scale method,and the outcome of the averaging method is so close that there are no differences among the results in the figure. For rather small excitation amplitude shown in Fig. 2(a),all curves are qualitatively the same in the sense that the amplitude-frequency response relationships are all single-valued,while the response curve obtained by the harmonic balance method is quantitatively more close to the numerical result than to that obtained by the multiple-scale method. For relatively large excitation amplitude shown in Fig. 2(b),the outcome of the numerical integration agrees qualitatively with that of the harmonic balance method. Both the curves bend to the left,while the cubic nonlinearity with the positive coefficient bends the curve to the right. In each of the curve, there is multi-valuedness,resulting in jumping. In spite of the qualitative agreement,there are certain quantitative differences.
 |
| Fig. 2 Amplitude-frequency response curves for weak excitations |
To highlight the quantitative differences among different approaches,the phase portraits of the steady-state responses are illustrated in Fig. 3 for a few sets of parameters. In Fig. 3,the solid dot lines,the solid lines,the dash lines,and the dash dot lines represent the results of the numerical integrations,the harmonic balance method,the multiple-scale method (also the Lindstedt-Poincar´e method),and the averaging method,respectively. In all cases,the results of the method of harmonic balance are the best approximations supported by the numerical results.
 |
| Fig. 3 Phase portraits of steady-state responses to weak excitations for different parameters |
Figure 4 depicts the amplitude-frequency response curves of the second-order approximation derived from the two-time scale method. The outcome of the Lindstedt-Poincar´e method under the same scale is identical to those of the two-time scale method. In Fig. 4,the solid triangles are the numerical results based on the original (1),the hollow dot lines are the numerical results based on the truncated (3),and the solid and dashed lines stand for the results of the second approximate and the first approximate of the two-time scale method,respectively. The numerical solutions of (1) and (3) are demonstrated to be in good agreement. It is evident that the second-order approximation derived from the two-time scale method cannot predict any bending in the response curves.
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| Fig. 4 Amplitude-frequency response curves under weak excitations |
Figure 5 shows the amplitude-frequency response curves of the second-order approximation derived from the three-time scale method,the two-time scale method,the harmonic balance method,and the numerical results,respectively. The results indicate that the second-order approximation of the three-time scale method bends to the left,agreeing qualitatively with the numerical observations. The simulations and theoretical analyses reveal that the harmonic balance method can outperform the averaging method,the Lindstedt-Poincar´e method,and the multiple-scale method for the quadratic nonlinear system under weak external excitations.
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| Fig. 5 Amplitude-frequency response curves under weak excitations |
Figure 6 depicts the amplitude-frequency response curves of (55) with (4) determined by the numerical integration (hollow dot lines),the harmonic balance method (solid lines for stable portions and dashed lines for unstable portions),and the averaging method (solid lines with solid dots). The averaging method cannot reveal the effect of quadratic nonlinearity,since the resulting response curves are without bending. The harmonic balance method predicts bending of the response curves to the left,and the prediction is supported by the numerical results. For the quadratic nonlinear term with negative coefficients,the only change made by the transform x ↔ -x is the sign of the forcing term in the resulting governing equation. According to (36),(57),and (58),the forcing term sign change does alter the amplitudefrequency response relationship. Contrast to the cubic nonlinearity,the quadratic nonlinearity, regardless its coefficient sign,bends the response curves to the left instead of the right.
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| Fig. 6 Frequency response curve under different excitations |
For a few sets of parameters,Fig. 7 plots the phase portraits of the steady-state responses determined by the numerical integration (solid dot lines),the harmonic balance method (solid lines),and the averaging method (dashed lines). The quantitative differences between
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| Fig. 7 Phase portraits of steady-state responses under strong excitations with different parameters |
The present work analyzes the steady-state response in the primary resonance of a quadraticly nonlinear oscillator subject to weak external excitations and strong external excitations. The analytical approximations of the amplitude-frequency response relationships are derived from the Lindstedt-Poincar´e method,the multiple-scale method ,the averaging method,and the harmonic balance method. The analytical results are compared with the numerical integrations. The investigation yields the following conclusions:
(i) The harmonic balancemethod reveals that the quadratic nonlinearity bends the amplitudefrequency response curves to the left.
(ii) The left-bending predicted by the harmonic balance method is supported by the numerical integrations.
(iii) Up to the first-order approximation,the Lindstedt-Poincar´e method,the multiple-scale method,and the averaging method cannot account for the quadratic nonlinearity,and thus cannot predict any bending in the amplitude-frequency response curves.
(iv) It is demonstrated that there exists a zeroth-order harmonic component in the application of the harmonic balance method to the forced quadratic nonlinearity oscillator.
(v) Up to the second-order approximation,the two-time scale method cannot predict any bending in the response curves,while the three-time scale method can predict any bending.
(vi) Simulations and theoretical analyses reveal that the harmonic balance method can outperform the averaging method,the Lindstedt-Poincar´e method,and the multiple-scale method for the quadratic nonlinear system under weak external excitations.
(vii) It is also demonstrated that the quadraticly nonlinear system indicates a softening type nonlinearity for any quadratic nonlinear coefficients.
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