The Response Frequency Conversion Characteristic of a Nonlinear Curved Panel with a Centre Mass and the Sound Radiations

Abstract

This study investigates the response frequency conversion characteristic of a nonlinear curved panel mounted with a centre mass and the sound radiations. A set of coupled governing differential equations is set up and used to generate the nonlinear vibration responses, which are used to calculate the corresponding radiated sounds. The vibration, sound levels, and the ratio of the antisymmetrical to symmetrical mode responses are plotted against the excitation level and compared with a set of experimental data. The frequency conversion characteristic is investigated from the frequency spectrums of the vibration responses.

1. Introduction

Previous studies of various nonlinear vibration/oscillation systems have been studied over recent decades [1–15]. Some researchers particularly worked on nonlinear curved/buckled beams/plates. Detailed discussions on the quadratic nonlinear responses of a two-degrees-of-freedom beam-mass system can be found in the work of Nayfeh and Mook [16] and Schmidt and Tondl [17]. Besides, some previous studies [18, 19] have shown that the nonlinear antisymmetric mode responses of a curved panel can be excited symmetrically due to the quadratic nonlinearity. It was found that the dominant response frequency was much lower than the excitation frequency. However, there are few research works which focus on the sound radiation of a nonlinear structure. In this study, the response frequency conversion characteristic of nonlinear curved panel vibrations and the sound radiations are further investigated in detail. In order to observe the frequency conversion characteristics, the resonant frequencies of the first symmetrical and antisymmetrical modes of the panel have to be tuned close to each other. According to the experimental experience documented in [20], adding masses on the panel surface makes easier for tuning of the resonant frequencies of a curved panel in a particular range. That is why a centre mass is considered and mounted on the panel in this study.

2. Methodology

Figure 1 shows a simply supported curved panel mounted with a central mass that is subject to a harmonic excitation. It is assumed that the flexural bending along the width is neglected. Thus, the structure is simplified and considered as a beamlike panel. The governing differential equation of a curved structure with a centre mass that is subject to a harmonic uniformly distributed pressure force [19] is given by 𝜌𝐴+𝑀𝑤̈+Ω𝑤̇+𝐸𝐼𝑤^′′′′=𝐸𝐴𝐿𝑤^′′+𝑤^′′𝐿₀𝑤^′𝑤^′+12𝑤^′2𝑑𝑥−𝜅𝜌𝐴+𝑀𝑔sin(𝜔𝑡), (2.1) where 𝑤 is the transverse displacement caused by the panel bending; 𝑤̇ and 𝑤̈ are the first and second derivatives of the transverse displacement with respect to time 𝑡; 𝑤^′,𝑤^′′, and 𝑤^′′′′ are the first, second, and fourth derivatives of the transverse displacement with respect to the spatial variable 𝑥; 𝑤=𝑞₀𝜙₁ is the initial transverse displacement; 𝑞₀ is the transverse displacement at the center; 𝐸 is the Young’s modulus; 𝜌 is the material density; Ω is the damping coefficient; 𝐴 is 𝐵×ℎ = the cross-sectional area; 𝐵 is the width; ℎ is the thickness; 𝐿 is the length; 𝑀 is 𝑚_𝑐𝛿(𝐿/2); 𝑚_𝑐 is the center mass; 𝛿(𝑥) is the Dirac delta function; 𝜔 is the excitation frequency; 𝜅 is the excitation parameter; 𝑔 is gravity is 9.81 ms⁻². The transverse displacement is expressed in terms of the mode shapes and given by 𝑤(𝑥,𝑡)=^𝑁_𝑖=1𝑞_𝑖(𝑡)𝜙_𝑖(𝑥), (2.2) where 𝑞_𝑖 is the modal amplitude of the 𝑖th mode; 𝜙_𝑖 is the 𝑖th sine function mode shape (i.e. sin(𝑖𝜋𝑥/𝐿)); 𝑖 is the mode number; 𝑁 is the number of modes considered (only the first three modes are considered in this study). According to [21], the resonant frequency of a nonlinear isotopic simply supported beam/plate can be accurately obtained using the single-mode approach. Hence, (2.1) can be easily reduced into a Duffing equation which has been solved by various methods (e.g. the variational approach [2], Hamiltonian approach [3, 4], elliptical Integral method [8], the harmonic balance method [8], the numerical integration method [18, 19], etc.). In Appendix, the procedure to obtain the resonant frequency of the Duffing equation is shown.

Figure 1 

Side view of a curved panel mounted with a mass.

The residual can be found by substituting (2.2) into (2.1) and then multiplied by 𝜙_𝑖, integrated with respect to the beam length, and set it to zero. A set of weighted residual differential equations is generated (see (2.3)-(2.4)). These differential equations can be solved using the Runge-Kutta numerical integration to obtain the modal responses Δ=𝜌𝐴+𝑀^𝑁_𝑖=1𝑞̈_𝑖𝜙_𝑖+𝜉𝜔_𝑖^𝑁_𝑖=1𝑞̇_𝑖𝜙_𝑖+𝐸𝐼^𝑁_𝑖=1𝑞_𝑖𝜙_𝑖^′′′′−𝐸𝐴𝐿^𝑁_𝑖=1𝑞_𝑖𝜙_𝑖^′′+𝑞₀𝜙₁^′′×^𝑁_𝑖=1𝑞₀𝑞_𝑖^𝐿₀𝜙^′₁𝜙^′_𝑖1𝑑𝑥+2^𝑁𝑁_𝑖=1𝑗=1𝑞_𝑖𝑞_𝑗^𝐿₀𝜙^′_𝑖𝜙^′_𝑗𝑑𝑥+𝐹,_𝑁𝑖=1𝑞̈_𝑖𝜌𝐴𝛼^0,0_𝑖,𝑚+𝑚_𝑐𝛽_𝑖,𝑚+^𝑁_𝑖=1𝑞̇_𝑖𝜉𝜔_𝑖𝛼^0,0_𝑖,𝑚+𝐸𝐼^𝑁_𝑖=1𝑞_𝑖𝛼^4,0_𝑖,𝑚−𝐸𝐴𝐿×+^𝑁_𝑖=1𝛼^1,1_1,𝑖𝛼^2,0_1,𝑚𝑞²₀𝑞_𝑖+^𝑁𝑁_𝑖=1𝑗=1𝛼^1,1_1,𝑖𝛼^2,0_𝑗,𝑚+12𝛼^1,1_𝑖,𝑗𝛼^2,0_1,𝑚𝑞₀𝑞_𝑖𝑞_𝑗+12^𝑁𝑁_𝑖=1^𝑁_𝑗=1𝑘=1𝛼^1,1_𝑖,𝑗𝛼^2,0_𝑘,𝑚𝑞_𝑖𝑞_𝑗𝑞_𝑘+𝐹_𝑚=0, (2.3)(2.4) where 𝜙^′_𝑖,𝜙_𝑖^′′, and 𝜙_𝑖^′′′′ are the first, second, and fourth derivatives of the 𝑖th mode shape, respectively, and 𝑖,𝑗,𝑚,𝑘 are the mode numbers. 𝜉𝜔_𝑖=Ω, where 𝜉 is the modal damping coefficient (),𝐹𝐹=𝜅𝜌𝐴+𝑀𝑔sin𝜔𝑡_𝑚=^𝐿₀𝐹𝜙_𝑚𝑑𝑥,𝛽_𝑖,𝑚=𝜙_𝑖𝐿2𝜙_𝑚𝐿2,𝛼^0,0_𝑖,𝑚=^𝐿₀𝜙_𝑖𝜙_𝑚𝛼𝑑𝑥,_2,0𝑖,𝑚=^𝐿₀𝜙_𝑖^′′𝜙_𝑚𝛼𝑑𝑥,_4,0𝑖,𝑚=^𝐿₀𝜙_𝑖^′′′′𝜙_𝑚𝑑𝑥. (2.5a)(2.5b)(2.5c)(2.5d)(2.5e)Then, the sound power radiated from the 𝑖th mode can be calculated using the radiation efficiency formulas which have been adopted in [22, 23]. The following two formulas are the simplified version in [22] for quick calculation: 𝜎_𝑖=32𝐾²𝐵𝐿𝑖²𝜋⁵𝐾1−²𝐵𝐿8121−(𝑖𝜋)²𝐿𝐵+81−𝜋²𝐵𝐿𝜎,forsymmetricmodes,_𝑖=8𝐾⁴𝐵𝐿3𝑖²𝜋⁵𝐾1−²𝐵𝐿8201−(𝑖𝜋)²𝐿𝐵+1−24𝜋²𝐵𝐿,forantisymmetricmodes, (2.6a)(2.6b)where 𝐾=𝜔/𝐶_air, 𝐶_air = sound speed in air.

The experimental setup, which was the same as that in [19], contained a curved steel panel with a thickness of 0.5 mm, an arc length of 310 mm, and a breadth of 350 mm (see Figure 2). It was mounted with a centre mass of 20 g. The first symmetrical and antisymmetrical mode resonant frequencies are 𝜔_sym=101 and 𝜔_anti=50 Hz, respectively (i.e. 𝜔_sym/𝜔_anti≈2). Note that the structure was not perfectly simply supported. The curved panel was sinusoidally excited by a shaker table. The modal vibrations of the structure and the base frame were measured using accelerometers placed at 𝑥=𝐿/4, 𝐿/2, and 3/4𝐿, while another accelerometer was used to monitor the dynamic response of the shaker table.

Figure 2 

Experimental setup.

3. Results and Discussions

In Figure 3(a), the relative total vibration levels are plotted against the excitation parameter, 𝜅. The dimensions of the curved panel are the same as those in the experimental setup. The modal damping coefficient is equal to 𝜉=0.02. The centre mass is also equal to 20 g. The ratios of the resonant frequencies of the first symmetrical and antisymmetrical modes are tuned to two by adjusting the curvature or the initial centre deflection. The vibration level in the case of (ℎ=0.7 mm at 𝜅=0.01) is normalized as zero. The total vibration levels are monotonically increasing for all cases. The excitation frequency is equal to the resonant frequency of the first mode. Generally, the thinner the panel is, the higher the vibration level is. Figure 3(b) shows the frequency spectrum of the point A marked in Figure 3(a). It can be seen that the response frequency of the dominant mode, which is the first antisymmetrical mode, is much lower than the excitation frequency (only one-half!). This phenomenon is described as “high-frequency excitation input-low-frequency response output.” The response frequencies of the first and second symmetrical modes are the same as the excitation frequency, but the modal vibration amplitudes of the first and second symmetrical modes are much lower than that of the first antisymmetrical mode (even though the excitation and the structure are symmetrical). In Figure 3(c), the relative total radiated sound levels of the three cases are plotted against the normalized excitation. The sound level of the solid line is exponentially deceasing and almost constant for 𝜅>0.1. The sound level of the dotted line is exponentially deceasing for 𝜅<0.1, bottom around 𝜅=0.1, and slowly increasing for 𝜅>0.1. Similarly, the sound level of the dash line is exponentially deceasing for 𝜅<0.04, bottom around 𝜅=0.04, and slowly increasing for 𝜅>0.04. It is found that although the sound level of the solid line is the highest for 𝜅<0.1, the corresponding vibration level is always the lowest; on the contrary, the sound level of the dash line is highest for 𝜅<0.07, the corresponding vibration level is always the highest. It is because the radiation efficiency of the first antisymmetric mode is much lower than the others, and thus the overall sound radiation is smaller.

(a)

(b)

(c)

Figure 3 

(a) Relative total vibration level (𝜔sym/𝜔anti-sym=2, 𝜔=𝜔sym), (b) relative modal vibration levels of point A in (a), (c) relative total sound level (𝜔sym/𝜔anti=2, 𝜔=𝜔sym).

Figure 4(a) shows other 3 cases in which the ratios of the resonant frequencies are not tuned to two. No centre mass is considered in these 3 cases. The vibration level of the thinnest beam is monotonically deceasing. For the other two cases in Figure 4(a), the vibration levels increase from 𝜅=0.01 to 0.02, peak around 𝜅=0.02 to 0.03, and decease 𝜅=0.03 to 0.2. Figure 4(b) shows the frequency spectrum of the point B marked in Figure 4(a). It can be seen that the first antisymmetrical mode is not as dominant as the one in Figure 3(b). It is because the ratio of the resonant frequencies is not tuned to two. The vibration levels of the first symmetrical and antisymmetrical modes are almost equal and much higher than that of the second symmetrical mode. In Figure 4(c), the relative total radiated sound levels of the three cases in Figure 4(a) are plotted against the normalized excitation. The sound level of the dash line is exponentially deceasing. The sound levels of the solid and dotted lines are increasing for 𝜅<0.03, peak around 𝜅=0.03, and decreasing for 𝜅>0.03. The sound level of the solid line is lower for 𝜅<0.03 and higher for 𝜅>0.03 than that of dotted lines, respectively. It is found that although the sound level of the dash line is lowest for 𝜅>0.015, the corresponding vibration level is always the highest in Figure 4(a); on the contrary, the sound levels of the dash and dotted lines are higher for 𝜅>0.015, the corresponding vibration levels always lower than that of the solid line. Besides, the vibration level of each case in Figure 4(a) is always higher than that of the case with the same thickness in Figure 3(a). It is because the resonant frequency ratios are not tuned to two. The smaller energy transfer to antisymmetrical modes results into the higher overall vibration.

(a)

(b)

(c)

Figure 4 

(a) Relative total vibration level (𝜔sym/𝜔anti-sym=1.85, 𝜔=𝜔sym), (b) relative modal vibration levels of point B in (a), (c) relative total sound level (𝜔sym/𝜔anti=1.85, 𝜔=𝜔sym).

Figure 5 shows the ratio of the antisymmetrical mode to symmetrical mode vibration levels for various resonant frequency ratios. It can be seen that if the resonant frequency ratio is closer to two, the antisymmetrical mode vibration level is higher. When the resonant frequency ratio is equal or close to two, the antisymmetrical mode vibration level is increasing against the excitation level for 𝜅<0.02, peaks around 𝜅=0.02, and decreasing for 𝜅>0.02. When the resonant frequency ratio is far from two, the antisymmetrical mode vibration level is monotonically increasing against the excitation level. The experiment data was obtained using the curved panel with a resonant frequency ratio of close to two. It is shown that that the experimental antisymmetrical mode vibration level is monotonically increasing against the excitation level; it only agrees well with the theoretical case in which the resonant ratio is 1.66. As the panel was not perfectly simply supported and symmetrical, that would result into a smaller energy transfer to the antisymmetrical mode. Thus the experiment data look likes the case of the resonant frequency ratio not tuned to two.

Figure 5 

Ratio of the antisymmetrical to symmetrical responses, (dB scale).

4. Conclusions

The nonlinear curved panels mounted a centre mass and the sound radiations have been studied. The results indicate that if the resonant frequency ratio is more far from two and the excitation and the structure are symmetrical, a higher excitation level is required to induce the antisymmetrical mode vibration. In the frequency spectrums, it can be seen in some cases studied that the response frequency of the dominant mode, which is the first antisymmetrical mode, is much lower than the excitation frequency. This can be considered as “high-frequency excitation input-low-frequency response output.” Besides, as the radiation efficiency of the antisymmetric mode is much lower than the others, the overall sound radiation is thus much smaller.

Appendix

Harmonic Balance Method
The harmonic balance method was employed for solving the Duffing equation, which represents the large amplitude free vibration of a beam in a previous study, and the result agreed well with the elliptic solution in [24]. Therefore, the method is selected here. As aforementioned, the single-mode approach is employed. Then, (2.3) can be reduced to the following form: 𝑞̈_𝑛+𝜀₁𝑞_𝑛+𝜀₂𝑞²_𝑛+𝜀₃𝑞³_𝑛=0, (A.1) where 𝜀₁, 𝜀₂, and 𝜀₃ are the linear, quadratic, cubic nonlinear modal coefficients which depend on the mass, initial deflection, linear stiffness, nonlinear stiffnesses, and modal contribution factors; 𝑛 is the mode number.
The solution of a nonlinear system is assumed to be of the form of a Fourier series: 𝑞_𝑛=𝐶₀+𝐶₁cos𝜔_𝑛𝑡+𝐶₂cos2𝜔_𝑛𝑡+⋯,𝐶=𝐶₀+𝐶₁+𝐶₂+⋯, (A.2)(A.3) where 𝐶₀,𝐶₁,𝐶₂… and so forth are the amplitudes of the harmonic components; 𝐶 is the initial amplitude; 𝜔_𝑛 is the resonant frequency.
For example, one constant and two harmonic terms are considered in (A.2). The Fourier expansion of the quadratic and cubic terms of the output 𝑞(𝑡) in (A.1) can be expressed, when retaining 2 harmonic components, as 𝑞²_𝑛=𝐶₀+𝐶₁cos𝜔_𝑛𝑡+𝐶₂cos2𝜔_𝑛𝑡𝑞³_𝑛=𝐶₀+𝐶₁cos𝜔_𝑛𝑡+𝐶₂cos2𝜔_𝑛𝑡, (A.4) where 𝐶₀=𝜔_𝑛2𝜋^{2𝜋/𝜔𝑛}₀𝐶₀+𝐶₁cos𝜔_𝑛𝑡+𝐶₂cos2𝜔_𝑛𝑡²𝑑𝑡,𝐶₁=𝜔_𝑛2𝜋^{2𝜋/𝜔𝑛}₀𝐶₀+𝐶₁cos𝜔_𝑛𝑡+𝐶₂cos2𝜔_𝑛𝑡²cos𝜔_𝑛𝑡𝑑𝑡,𝐶₂=𝜔_𝑛2𝜋^{2𝜋/𝜔𝑛}₀𝐶₀+𝐶₁cos𝜔_𝑛𝑡+𝐶₂cos2𝜔_𝑛𝑡²cos2𝜔_𝑛𝑡𝑑𝑡,𝐶₀=𝜔_𝑛2𝜋^{2𝜋/𝜔𝑛}₀𝐶₀+𝐶₁cos𝜔_𝑛𝑡+𝐶₂cos2𝜔_𝑛𝑡³𝑑𝑡,𝐶₁=𝜔_𝑛2𝜋^{2𝜋/𝜔𝑛}₀𝐶₀+𝐶₁cos𝜔_𝑛𝑡+𝐶₂cos2𝜔_𝑛𝑡³cos𝜔_𝑛𝑡𝑑𝑡,𝐶₂=𝜔_𝑛2𝜋^{2𝜋/𝜔𝑛}₀𝐶₀+𝐶₁cos𝜔_𝑛𝑡+𝐶₂cos2𝜔_𝑛𝑡³cos2𝜔_𝑛𝑡𝑑𝑡. (A.5) Substituting (A.2) into (A.1) and equating coefficients associated with each harmonic component yields 3 equations as follows: 𝜀₁𝐶₀+𝜀₂𝐶₀+𝜀₃𝐶₀=0,−𝜔²_𝑛𝐶₁+𝜀₁𝐶₁+𝜀₂𝐶₁+𝜀₃𝐶₁=0,−𝜔²_𝑛𝐶₂+𝜀₁𝐶₂+𝜀₂𝐶₂+𝜀₃𝐶₂=0. (A.6) There are four equations (A.3), (A.6) and four unknowns 𝜔_𝑛,𝐶₀,𝐶₁,𝐶₂. Hence the resonant frequency 𝜔_𝑛, can be found.

Hamiltonian Approach
If the initial centre deflection, 𝑞_𝑜=0 in (2.3) (i.e., it is a flat panel), (A.1) can be rewritten in the following form: 𝑞̈_𝑛+𝜀₁𝑞_𝑛+𝜀₃𝑞³_𝑛=0. (A.7) The Hamiltonian approach is now employed here for solving the Duffing equation in (A.1) to obtain the resonant frequency.
The Hamiltonian in (A.7) can be easily obtained, which reads 1𝐻=2𝑞̈²_𝑛+12𝜀₁𝑞²_𝑛+14𝜀₃𝑞⁴_𝑛=0. (A.8) Integrating (A.8) with respect to 𝑡 from 0 to 𝑇/4 yields 𝐻𝑞_𝑛=₀^𝑇/412𝑞̈²_𝑛+12𝜀₁𝑞²_𝑛+14𝜀₃𝑞⁴_𝑛𝑑𝑡. (A.9) Consider that the solution can be expressed as 𝑞_𝑛=𝐶cos𝜔_𝑛𝑡 and substitute it to (A.9) as follows: 𝐻𝑞_𝑛=𝜋8𝐶²𝜔_𝑛+𝜋8𝜔_𝑛𝐶²𝜀₁+38𝜀₃𝐶². (A.10) According to [3, 4], setting 𝜕𝜕𝜕𝐶𝐻(𝑢)𝜕1/𝜔_𝑛=0. (A.11) By solving (A.11), the only unknown 𝜔_𝑛 is given by 𝜔_𝑛=𝜀₁+34𝜀₃𝐶². (A.12)

Acknowledgments

The work described in this paper was fully supported by a grant from the Research Grants Council of the Hong Kong, China [Project no. 9041496 (CityU 116209)]. The author would like to express his sincere gratitude and appreciation to Dr. WY Poon, Dr. CF Ng, and Ms. CK Hui for experimental hardware tuning and data acquisition.