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Table 1 Effect of the parameter m on iterations to converge and the error of close-form solutions of the steady-state PDF of the van der Pol oscillator under Gaussian white noise excitations. function with a relatively large m can help the assumed solution to capture the peculiar shape of the steady-state PDF of strongly nonlinear van der Pol oscillator. For lightly damped van der Pol oscillator, the weighting function with m= 1 is sufficient.

Table 1 Effect of the parameter m on iterations to converge and the error of close-form solutions of the steady-state PDF of the van der Pol oscillator under Gaussian white noise excitations. function with a relatively large m can help the assumed solution to capture the peculiar shape of the steady-state PDF of strongly nonlinear van der Pol oscillator. For lightly damped van der Pol oscillator, the weighting function with m= 1 is sufficient.

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Fig. 1 The simultaneous wind speed-irradiance measurements collected over the total period encountered.
Fig. 1 The simultaneous wind speed-irradiance measurements collected over the total period encountered.
Fig. 2 The thirteen IMF modes and the residual of global horizontal irradiance 7 measurements. Here, we present the EMD analysis for the simultaneous wind speed u and the global horizontal irra- diance J data. Figures 2 and 3 illustrate the corresponding IMF modes C! and C" respectively for the global solar irradiance and the wind speed data. The first mode characterizes the fast fluctuations while the last mode characterize the slowest fluctuations. This indicates that the time scale increases with the mode index i. To highlight this observation quantitatively, we estimate the mean frequency fj, using the following definition [10]:
Fig. 2 The thirteen IMF modes and the residual of global horizontal irradiance 7 measurements. Here, we present the EMD analysis for the simultaneous wind speed u and the global horizontal irra- diance J data. Figures 2 and 3 illustrate the corresponding IMF modes C! and C" respectively for the global solar irradiance and the wind speed data. The first mode characterizes the fast fluctuations while the last mode characterize the slowest fluctuations. This indicates that the time scale increases with the mode index i. To highlight this observation quantitatively, we estimate the mean frequency fj, using the following definition [10]:
Fig. 3 The thirteen IMF modes and the residual of wind speed u measurements
Fig. 3 The thirteen IMF modes and the residual of wind speed u measurements
Fig. 4 Representation of the mean frequency f versus the mode index m, in log-linear plot, for the twe datasets. The fitting slope is 0.28, corresponding to a 1.91-times filter bank.
Fig. 4 Representation of the mean frequency f versus the mode index m, in log-linear plot, for the twe datasets. The fitting slope is 0.28, corresponding to a 1.91-times filter bank.
Table 1 The corresponding mean time period T,, (expressed in days) and the mean frequency fi, (expressed it 1/days) or cycle versus the mode index i: the values obtained being very close for the wind speed u and the global solar irradiance J, here we present the values for u.
Table 1 The corresponding mean time period T,, (expressed in days) and the mean frequency fi, (expressed it 1/days) or cycle versus the mode index i: the values obtained being very close for the wind speed u and the global solar irradiance J, here we present the values for u.
Fig. 5 The spectrum of wind speed u (Q) and global horizontal irradiance J (0) (a) in the fourier space, (b) in the Hilbert space. A Power law behavior is observed for the two datasets.
Fig. 5 The spectrum of wind speed u (Q) and global horizontal irradiance J (0) (a) in the fourier space, (b) in the Hilbert space. A Power law behavior is observed for the two datasets.
Fig. 6 The coherence function A,;(f) of simultaneous wind speed-irradiance measurements. Obviously, both time series are correlated for 12 hour and 24 hour-cycles.
Fig. 6 The coherence function A,;(f) of simultaneous wind speed-irradiance measurements. Obviously, both time series are correlated for 12 hour and 24 hour-cycles.
Fig. 7 The maximum absolute value of cross-correlation function p and TDIC measured versus the mode index i. Contrary to p, TDIC detect strong correlations for all time scales. Here, possible correlations between both datasets are investigated for multiple time scales, providing a time-frequency information. EMD decomposition is applied to the simultaneous wind For that, | speed-irradiance measurements. The results obtained, i.e., the IMF modes, are used for the multi-scale correlation analysis. Figure 7 illustrates the comparison of maximum absolute values obtained by the
Fig. 7 The maximum absolute value of cross-correlation function p and TDIC measured versus the mode index i. Contrary to p, TDIC detect strong correlations for all time scales. Here, possible correlations between both datasets are investigated for multiple time scales, providing a time-frequency information. EMD decomposition is applied to the simultaneous wind For that, | speed-irradiance measurements. The results obtained, i.e., the IMF modes, are used for the multi-scale correlation analysis. Figure 7 illustrates the comparison of maximum absolute values obtained by the
Fig. 8 Illustration of trends, 473-day cycles and 273-day cycles of both time series. For these time scales, It exists a strong positive correlation with a phase difference between the both time series. where the angle braces (.) indicates the statistical average, T is a time lag, 0, and oy are respectively the standard deviations of processes x(t) and y(t). Classical cross-correlation is an average over time defined in the integral sense, while TDIC is defined using instantaneous periods of different scales. As they are not of the same conceptual footing, one would expect them to generate different interpretations on the same time series.
Fig. 8 Illustration of trends, 473-day cycles and 273-day cycles of both time series. For these time scales, It exists a strong positive correlation with a phase difference between the both time series. where the angle braces (.) indicates the statistical average, T is a time lag, 0, and oy are respectively the standard deviations of processes x(t) and y(t). Classical cross-correlation is an average over time defined in the integral sense, while TDIC is defined using instantaneous periods of different scales. As they are not of the same conceptual footing, one would expect them to generate different interpretations on the same time series.
Fig. 9 The TDIC measured for the 273-day cycle, showing a strong positive correlation between both time series (the presence of holes indicates that the R cannot pass the t-test [11]).
Fig. 9 The TDIC measured for the 273-day cycle, showing a strong positive correlation between both time series (the presence of holes indicates that the R cannot pass the t-test [11]).
Fig. 10 The 56-day cycle from EMD. The maximum value of cross-correlation function p is 0.32. Now we consider the correlation between u and J for mean periods T,, < 273 days. Figure 10 illustrates the IMF modes with a mean period of 56 days (approximately two months). One can observe strong positive and weak negative correlations between the wind speed u and the global horizontal irradiance J on some portions. This is confirmed by the corresponding TDIC given in Figure 11. However the classical cross correlation p indicates no correlation with a maximum value equal to 0.32.
Fig. 10 The 56-day cycle from EMD. The maximum value of cross-correlation function p is 0.32. Now we consider the correlation between u and J for mean periods T,, < 273 days. Figure 10 illustrates the IMF modes with a mean period of 56 days (approximately two months). One can observe strong positive and weak negative correlations between the wind speed u and the global horizontal irradiance J on some portions. This is confirmed by the corresponding TDIC given in Figure 11. However the classical cross correlation p indicates no correlation with a maximum value equal to 0.32.
Fig. 11 The TDIC measured for the 56-day cycle, showing local positive and negative correlations locally between both time series. The maximum value of cross-correlation function p is 0.32 (the presence of holes indicates that the R cannot pass the t-test [{11]).
Fig. 11 The TDIC measured for the 56-day cycle, showing local positive and negative correlations locally between both time series. The maximum value of cross-correlation function p is 0.32 (the presence of holes indicates that the R cannot pass the t-test [{11]).
Fig. 12 The TDIC measured for the 14-day cycle, showing strong positive and negative correlations locally between both time series. The maximum value of cross-correlation function p is 0.15.
Fig. 12 The TDIC measured for the 14-day cycle, showing strong positive and negative correlations locally between both time series. The maximum value of cross-correlation function p is 0.15.
Fig. 13 The TDIC measured for the 1.47-day cycle, showing strong and negative correlations locally between both time series. The maximum value of cross-correlation function p is 0.18.
Fig. 13 The TDIC measured for the 1.47-day cycle, showing strong and negative correlations locally between both time series. The maximum value of cross-correlation function p is 0.18.
Fig. 14 The TDIC measured for the 12h cycle, showing strong and negative correlations locally between both time series. The maximum value of cross-correlation function p is 0.21.
Fig. 14 The TDIC measured for the 12h cycle, showing strong and negative correlations locally between both time series. The maximum value of cross-correlation function p is 0.21.
Fig. 15 A zoom of TDIC measured for the 2.2h cycle, showing strong and negative correlations locally between both time series. The maximum value of cross-correlation function p is 0.042 (The presence of hole indicated that the R cannot pass the t-test [11]) .
Fig. 15 A zoom of TDIC measured for the 2.2h cycle, showing strong and negative correlations locally between both time series. The maximum value of cross-correlation function p is 0.042 (The presence of hole indicated that the R cannot pass the t-test [11]) .
Fig. 16 The complementarity, strong negative correlation, considering here the minimal values of TDIC method, between the both times series for time scales for the mean period of modes 2.2h < Ty, < 115.34 days. The elaboration of unit production strategy combining wind and solar energy requires the estimation of the dependence between the wind and solar resources. In [6], the author analyzed the balance between solar and wind resources data in Sweden. This analysis was based on the Pearson correlation coefficient. The results suggested balancing between wind and solar resources due to the negative correlation for monthly time scales. However no correlation was observed for shorter time scales. These methods are based on Fourier framework and provide a global value for the correlation.
Fig. 16 The complementarity, strong negative correlation, considering here the minimal values of TDIC method, between the both times series for time scales for the mean period of modes 2.2h < Ty, < 115.34 days. The elaboration of unit production strategy combining wind and solar energy requires the estimation of the dependence between the wind and solar resources. In [6], the author analyzed the balance between solar and wind resources data in Sweden. This analysis was based on the Pearson correlation coefficient. The results suggested balancing between wind and solar resources due to the negative correlation for monthly time scales. However no correlation was observed for shorter time scales. These methods are based on Fourier framework and provide a global value for the correlation.
conditions we have an opportunity of creating a high-precision inductive (with thin magnetic films) logical nanoelement. The cycling frequency of which is order 150 MHz and has raised capability [1,2]. In this connection the nanodecreasing of logic cell is more important than increasing of frequency [3]. Now the Japanese scientists Imran Mahboob and Hiroshi Yamaguchi from NTT Basic Research Lab. have proposed the logic element constructed by nanobeam with working frequency order 100 MHz [4]. Parametron (Fig.1) contains two ferromagnetic cores and each of them has two coiled reels with 2 Equations describing the dynamic of parametron
conditions we have an opportunity of creating a high-precision inductive (with thin magnetic films) logical nanoelement. The cycling frequency of which is order 150 MHz and has raised capability [1,2]. In this connection the nanodecreasing of logic cell is more important than increasing of frequency [3]. Now the Japanese scientists Imran Mahboob and Hiroshi Yamaguchi from NTT Basic Research Lab. have proposed the logic element constructed by nanobeam with working frequency order 100 MHz [4]. Parametron (Fig.1) contains two ferromagnetic cores and each of them has two coiled reels with 2 Equations describing the dynamic of parametron
Fig. 2 Numerical results with values of parameters: h = 0.35, r=0.07, B =0.25,k=1
Fig. 2 Numerical results with values of parameters: h = 0.35, r=0.07, B =0.25,k=1
Fig. 3 Numerical results with values of parameters: h = 0.5, r= 0.07, = 0.25, k= 1, @ =0.87
Fig. 3 Numerical results with values of parameters: h = 0.5, r= 0.07, = 0.25, k= 1, @ =0.87
Fig. 4 Amplitude-frequency characteristic (AFC) 1—A = 0.35, 2—h=0.4, 3-h=.45, 4-—h=0.5, 5—h=0.55.
Fig. 4 Amplitude-frequency characteristic (AFC) 1—A = 0.35, 2—h=0.4, 3-h=.45, 4-—h=0.5, 5—h=0.55.
All of these curves are calculated for upper branch of AFC with value of h =0.55. At decreasing of frequency from @ = 0.94 until @ =0.75 (point of inflection) the root p; (solid curve) grows from pi = 0.61, retaining with module smaller than 1. In diapason @ < 0.75 the root p; grows and |p,| > 1. At decreasing of frequency in this diapason the root pz (dotted line) decreases from p2 = 0.96 retaining by module smaller then 1. Inside field of lines in Fig.4 corresponds to zone of possible establishment the parametric resonance. And upper border remains stable until a point of inflection. This is a line of amplitude of stable parametric oscillation.
All of these curves are calculated for upper branch of AFC with value of h =0.55. At decreasing of frequency from @ = 0.94 until @ =0.75 (point of inflection) the root p; (solid curve) grows from pi = 0.61, retaining with module smaller than 1. In diapason @ < 0.75 the root p; grows and |p,| > 1. At decreasing of frequency in this diapason the root pz (dotted line) decreases from p2 = 0.96 retaining by module smaller then 1. Inside field of lines in Fig.4 corresponds to zone of possible establishment the parametric resonance. And upper border remains stable until a point of inflection. This is a line of amplitude of stable parametric oscillation.
Fig. 7 Resonance curve at biasing current 1-15 mA, 2-20 mA. This scheme contains two circular toroidal iron (ferrite) circuits. On each of them the number of turns wy = W2 = 200 is coiled. The diameter of conductors is equal 0.08 mm. The first pair of coils with number of turns wz coming from opposite direction is switched to source of pulsatile current. The second pair of coils connected in series with number of turns w, is loaded to capacitor and represents a resonance contour. The values of parameters are C = ImF,, Ry = Ro = 1kQ. The two-radial oscillograph serves for simultaneous observation of exciting signal and signal of origin processes in resonance contour.
Fig. 7 Resonance curve at biasing current 1-15 mA, 2-20 mA. This scheme contains two circular toroidal iron (ferrite) circuits. On each of them the number of turns wy = W2 = 200 is coiled. The diameter of conductors is equal 0.08 mm. The first pair of coils with number of turns wz coming from opposite direction is switched to source of pulsatile current. The second pair of coils connected in series with number of turns w, is loaded to capacitor and represents a resonance contour. The values of parameters are C = ImF,, Ry = Ro = 1kQ. The two-radial oscillograph serves for simultaneous observation of exciting signal and signal of origin processes in resonance contour.
Fig. 6 Circuit for excitement of parametric resonance.
Fig. 6 Circuit for excitement of parametric resonance.
Fig. 1 Analytical model. Two incompressible immiscible fluids form layers in a rectangular tank and the tank undergoes horizontal excitation as an external disturbance. where, the symbol - denotes the differentiation with respect to time, V= (0/dx,0/dy,0/dz), and re- lations of ¢; = ¢;+%,x, z=Z, and Vy = (%2,0,0) are used. Substituting Eq. (2) into Eq. (1) results in:
Fig. 1 Analytical model. Two incompressible immiscible fluids form layers in a rectangular tank and the tank undergoes horizontal excitation as an external disturbance. where, the symbol - denotes the differentiation with respect to time, V= (0/dx,0/dy,0/dz), and re- lations of ¢; = ¢;+%,x, z=Z, and Vy = (%2,0,0) are used. Substituting Eq. (2) into Eq. (1) results in:
The following relationship for the interfaces is used in the differentiation with respect to the normal 4 -~ « Rn where, 0D; is the closed boundary of the domain D;, ds is the surface element of 0D; and n is the outward pointing normal of the boundary 0D; except on Fy of dD2. The positive direction of n on F, is defined to be the +z side, and n is the inward pointing normal on F; of D2. The last terms in Eqs. (10) and (11) are rewritten as follows by using Green’s first identity again: where, 7) is N; for F = F, and 2 for F = Fy. After substituting Eqs. (12) and (13) into LEE. o;V7 5 o;dxdydz
The following relationship for the interfaces is used in the differentiation with respect to the normal 4 -~ « Rn where, 0D; is the closed boundary of the domain D;, ds is the surface element of 0D; and n is the outward pointing normal of the boundary 0D; except on Fy of dD2. The positive direction of n on F, is defined to be the +z side, and n is the inward pointing normal on F; of D2. The last terms in Eqs. (10) and (11) are rewritten as follows by using Green’s first identity again: where, 7) is N; for F = F, and 2 for F = Fy. After substituting Eqs. (12) and (13) into LEE. o;V7 5 o;dxdydz
of Eqs. (10) and (11), substituting Eqs. (8) to (11) into ] Eq. (7) and using Eq. (14) obtains: By assuming the following Eqs. (18) to (19), the kinematic conditions and dynamic conditions imposed on the interfaces are separated from Eq. (15) to obtain Eqs. (16) and (17) respectively.
of Eqs. (10) and (11), substituting Eqs. (8) to (11) into ] Eq. (7) and using Eq. (14) obtains: By assuming the following Eqs. (18) to (19), the kinematic conditions and dynamic conditions imposed on the interfaces are separated from Eq. (15) to obtain Eqs. (16) and (17) respectively.
* The common damping ratio is applied to all modes. Table 1 Analytical conditions (The conditions correspond to those in the experiments performed by Handa and Tajima [1]).
* The common damping ratio is applied to all modes. Table 1 Analytical conditions (The conditions correspond to those in the experiments performed by Handa and Tajima [1]).
Table 2 Natural frequencies and mode shapes by eigenmode analysis.
Table 2 Natural frequencies and mode shapes by eigenmode analysis.
Fig. 2 Maximum response of interfaces (x = a).
Fig. 2 Maximum response of interfaces (x = a).
Fig. 3. Wave height profile under “peculiar oscillation” [analysis, excitation frequency f = 2.418 Hz (period: T = 1/f = 0.414 sec.) aaa ae “Sa Sa”; T= Tae ea. ~~ -|__- —_:_ ae that of four terms consisting of C22 and D2; are shows j in the upper two time series in Fig. 4 (‘force’). The wave heights on the interface of the two liquids and on the free surface interface, n\(a) and 1)2(a), are also shown in Fig. 4. The components of the relevant four eigenmodes are separated from 711(a) and are also shown in Fig. 4, i-e., 15111(@), Ne121(@), Nsi21(a) and Ns122(a). The frequencies written on the time series in Fig. 4 are the dominant frequencies in FFT analysis. The power spectra of the relevant four eigenmodes by FFT are shown in Fig. 5.
Fig. 3. Wave height profile under “peculiar oscillation” [analysis, excitation frequency f = 2.418 Hz (period: T = 1/f = 0.414 sec.) aaa ae “Sa Sa”; T= Tae ea. ~~ -|__- —_:_ ae that of four terms consisting of C22 and D2; are shows j in the upper two time series in Fig. 4 (‘force’). The wave heights on the interface of the two liquids and on the free surface interface, n\(a) and 1)2(a), are also shown in Fig. 4. The components of the relevant four eigenmodes are separated from 711(a) and are also shown in Fig. 4, i-e., 15111(@), Ne121(@), Nsi21(a) and Ns122(a). The frequencies written on the time series in Fig. 4 are the dominant frequencies in FFT analysis. The power spectra of the relevant four eigenmodes by FFT are shown in Fig. 5.
Fig. 5 Power spectral densities of the relevant four modes, i.e. asym. (1, 1) (Ci1), sym. (2, 1) (Do1), asym. (2, 1) (Cy,) and asym. (2, 2) (Co2). (analysis, o = 0.50, € = 0.02, f =2.418 Hz).
Fig. 5 Power spectral densities of the relevant four modes, i.e. asym. (1, 1) (Ci1), sym. (2, 1) (Do1), asym. (2, 1) (Cy,) and asym. (2, 2) (Co2). (analysis, o = 0.50, € = 0.02, f =2.418 Hz).
Fig. 4 Time series of “peculiar oscillation” (analysis o = 0.50, € = 0.02 f = 2.418 Hz).
Fig. 4 Time series of “peculiar oscillation” (analysis o = 0.50, € = 0.02 f = 2.418 Hz).
Fig. 6 Amplitudes of the asymmetric (1, 1) mode-synchronous component in sums of nonlinear force terms under each excitation frequency (analysis, o = 0.50, ¢ = 0.02). The amplitude is represented by V2 times the standard deviation of the synchronous component.
Fig. 6 Amplitudes of the asymmetric (1, 1) mode-synchronous component in sums of nonlinear force terms under each excitation frequency (analysis, o = 0.50, ¢ = 0.02). The amplitude is represented by V2 times the standard deviation of the synchronous component.
Fig. 7 Amplitude of resonance component of asymmetric (1, 1) mode (R;) under “peculiar oscillation” (analysis, o = 0.50, € = 0.02).
Fig. 7 Amplitude of resonance component of asymmetric (1, 1) mode (R;) under “peculiar oscillation” (analysis, o = 0.50, € = 0.02).
Fig. 8 Solution of R; by the amplitude equation (o = 0.50, € = 0.02). The steady values for R, of the amplitude equation under each excitation frequency are shown in Fig. 8 along with the original time domain solution. R; by the amplitude equation of Eq. (30) is in generally good agreement with the original time domain solution by | Eqs. (26) and (27) [i-e., Eqs. (24) and (25)]. It should be noticed that effects of cj;~ and cjj; on R, are negligible since the maximum R, changes by +6% in the ‘3-mode-1’ analysis or —0.5% in the ‘3-mode-2’ analysis by omitting cj,- and cys. It also should be noticed that there are nonlinear components that restrict R; and Ro. For example, R; increases by 36% (f = 2.419 Hz) by omitting Rs in the ‘3-mode-1’ analysis, and R; increases by 19% (f = 2.706 Hz) by omitting Ry in the ‘3- mode-2’ analysis. In order to obtain solutions quantitatively close to the original solutions of Eqs. (26) and (27), it is required to incorporate nonlinear components other than nonlinear resonance components of @; and @3.
Fig. 8 Solution of R; by the amplitude equation (o = 0.50, € = 0.02). The steady values for R, of the amplitude equation under each excitation frequency are shown in Fig. 8 along with the original time domain solution. R; by the amplitude equation of Eq. (30) is in generally good agreement with the original time domain solution by | Eqs. (26) and (27) [i-e., Eqs. (24) and (25)]. It should be noticed that effects of cj;~ and cjj; on R, are negligible since the maximum R, changes by +6% in the ‘3-mode-1’ analysis or —0.5% in the ‘3-mode-2’ analysis by omitting cj,- and cys. It also should be noticed that there are nonlinear components that restrict R; and Ro. For example, R; increases by 36% (f = 2.419 Hz) by omitting Rs in the ‘3-mode-1’ analysis, and R; increases by 19% (f = 2.706 Hz) by omitting Ry in the ‘3- mode-2’ analysis. In order to obtain solutions quantitatively close to the original solutions of Eqs. (26) and (27), it is required to incorporate nonlinear components other than nonlinear resonance components of @; and @3.
Fig. 9 Excitation frequency (@) components under “peculiar oscillation” (o = 0.50, ¢ = 0.02).
Fig. 9 Excitation frequency (@) components under “peculiar oscillation” (o = 0.50, ¢ = 0.02).
Fig. 10 Schematic of force flows among three modes in the “peculiar oscillation” “ Based on Eq. (32), the mechanism of the oscillation can be defined as follows. Both nonlinear resonance components R; and R2, form products with Canes, i-e., RoC2nc,s in the equation for R, and é1, and R1Concs in the equation for R2 and 5). This indicates that R; and Ro excite each other through the mediation of C2n¢,5 (excitation frequency component). The schematic of force flows in this relationship is drawn in Fig. 10. This relationship is similar to the equation of the summed and differential harmonic oscillation (combination oscillation) reported by Yamamoto et al. [15]. One different point is that the excitation frequency (@) component (C2n¢,5) is affected by nonlinear components of R; and R> particularly in case of the ‘3-mode-2’ analysis (see Fig. 9). Being based on the force flows, the “peculiar oscillation” is judged to be one of summed and differential harmonic oscillations. The resonance condition in this phenomenon is f = fs,; + fc2; where f is the excitation frequency and the frequencies on the right-hand side are natural frequencies of eigenmodes under nonlinear resonance. In the present case, these are the natural frequency of the asymmetric (1, 1) mode, fs1;, and that of the symmetric (2, 1) mode, fc2}. From the view point of engineering, it is useful that the frequency condition for oscillation can be predicted.
Fig. 10 Schematic of force flows among three modes in the “peculiar oscillation” “ Based on Eq. (32), the mechanism of the oscillation can be defined as follows. Both nonlinear resonance components R; and R2, form products with Canes, i-e., RoC2nc,s in the equation for R, and é1, and R1Concs in the equation for R2 and 5). This indicates that R; and Ro excite each other through the mediation of C2n¢,5 (excitation frequency component). The schematic of force flows in this relationship is drawn in Fig. 10. This relationship is similar to the equation of the summed and differential harmonic oscillation (combination oscillation) reported by Yamamoto et al. [15]. One different point is that the excitation frequency (@) component (C2n¢,5) is affected by nonlinear components of R; and R> particularly in case of the ‘3-mode-2’ analysis (see Fig. 9). Being based on the force flows, the “peculiar oscillation” is judged to be one of summed and differential harmonic oscillations. The resonance condition in this phenomenon is f = fs,; + fc2; where f is the excitation frequency and the frequencies on the right-hand side are natural frequencies of eigenmodes under nonlinear resonance. In the present case, these are the natural frequency of the asymmetric (1, 1) mode, fs1;, and that of the symmetric (2, 1) mode, fc2}. From the view point of engineering, it is useful that the frequency condition for oscillation can be predicted.
Expressions of Q-terms for the ‘3-mode-1’ (Cj;, D2; and C;) analysis
Expressions of Q-terms for the ‘3-mode-1’ (Cj;, D2; and C;) analysis
Fig. 1 Curved PZT unimorph beam element. where S? denotes the membrane strain of the neutral surface and represents the changes in the curvatur of neutral surface during deformation.
Fig. 1 Curved PZT unimorph beam element. where S? denotes the membrane strain of the neutral surface and represents the changes in the curvatur of neutral surface during deformation.
where V, and V, are the volume of the piezoelectric material and the volume of the substrate, respec- tively. In this work we assume there is no external force applied to the axial direction, the membrane The total kinetic energy, the total potential energy, and the work due to the external transverse direction force acting on the PZT unimorph strip can be written as,
where V, and V, are the volume of the piezoelectric material and the volume of the substrate, respec- tively. In this work we assume there is no external force applied to the axial direction, the membrane The total kinetic energy, the total potential energy, and the work due to the external transverse direction force acting on the PZT unimorph strip can be written as,
In this section the analytical voltage output based on the mechanical-electrical governing equations of a THUNDER is obtained with the two modes energy harvester circuit and the experimental setup is shown in Fig. 2. Fig. 2 Energy harvesting experiment setup.
In this section the analytical voltage output based on the mechanical-electrical governing equations of a THUNDER is obtained with the two modes energy harvester circuit and the experimental setup is shown in Fig. 2. Fig. 2 Energy harvesting experiment setup.
Fig. 3 Two moded energy harvesting circuit. Fig. 3 is the dialog sketch of the piezoelectric martial and the two modes energy harvester circuit. The left part in the dash line box stands for the mechanical and the electrical representation of the THUNDER structure. The two operation modes 1) full bridge rectifier directly charging mode at low level excitation, and 2) memory stored optimal duty cycle step-down converter mode at high level excitation. To save the energy of the threshold control between the two operation modes, an almost zero energy cost mechanical switch will be designed as a threshold controller. The upper center dash line box of Fig. 3 represents the dialog sketch of the mechanical switch. Considering the output voltage of the piezoelectric material is linearly proportional to the vibration amplitude of the piezoelectric material, the switching between the two energy harvesting modes is determined by the threshold vibration amplitude which is testified by the experiments. When the vibration is small with an amplitude less than the threshold amplitude value the mechanical switch turns on the directly
Fig. 3 Two moded energy harvesting circuit. Fig. 3 is the dialog sketch of the piezoelectric martial and the two modes energy harvester circuit. The left part in the dash line box stands for the mechanical and the electrical representation of the THUNDER structure. The two operation modes 1) full bridge rectifier directly charging mode at low level excitation, and 2) memory stored optimal duty cycle step-down converter mode at high level excitation. To save the energy of the threshold control between the two operation modes, an almost zero energy cost mechanical switch will be designed as a threshold controller. The upper center dash line box of Fig. 3 represents the dialog sketch of the mechanical switch. Considering the output voltage of the piezoelectric material is linearly proportional to the vibration amplitude of the piezoelectric material, the switching between the two energy harvesting modes is determined by the threshold vibration amplitude which is testified by the experiments. When the vibration is small with an amplitude less than the threshold amplitude value the mechanical switch turns on the directly
As shown in Fig. 3, when the vibration amplitude of the piezoelectric material is small, the mechanical switch turns on the full bridge rectifier direct charge mode. The direct charge mode consists of a rectifier bridge, a filtering capacitance, and the external load (a rechargeable battery or a resistor). If the rectifier is connected with a rechargeable battery, the rectified voltage equals the rechargeable batteries voltage. On the other hand if the rectifier is connected with a resistor, the rectified voltage is still constant but its value is determined by the magnitude of the resistor. With a sinusoidal excitation, the displacement of the piezoelectric material and generated voltage by the piezoelectric material are also sinusoidal. Fig. 4 Simulated voltage and current waveforms for the piezoelectric material.
As shown in Fig. 3, when the vibration amplitude of the piezoelectric material is small, the mechanical switch turns on the full bridge rectifier direct charge mode. The direct charge mode consists of a rectifier bridge, a filtering capacitance, and the external load (a rechargeable battery or a resistor). If the rectifier is connected with a rechargeable battery, the rectified voltage equals the rechargeable batteries voltage. On the other hand if the rectifier is connected with a resistor, the rectified voltage is still constant but its value is determined by the magnitude of the resistor. With a sinusoidal excitation, the displacement of the piezoelectric material and generated voltage by the piezoelectric material are also sinusoidal. Fig. 4 Simulated voltage and current waveforms for the piezoelectric material.
Fig. 5 The experiment rectified voltage with full bridge rectifier connected to resistors. The charge lost due to charge and recharge of the internal capacitor of piezoelectric material equals
Fig. 5 The experiment rectified voltage with full bridge rectifier connected to resistors. The charge lost due to charge and recharge of the internal capacitor of piezoelectric material equals
Fig. 6 The experiment rectified voltage with full bridge rectifier connected to resistors.
Fig. 6 The experiment rectified voltage with full bridge rectifier connected to resistors.
Fig. 7 The experiment output power with full bridge rectifier connected to resistors. Combining Eqs. (11), (13), and (14), obtain the rectified voltage and output power,
Fig. 7 The experiment output power with full bridge rectifier connected to resistors. Combining Eqs. (11), (13), and (14), obtain the rectified voltage and output power,
Fig. 8 The experiment output power with full bridge rectifier connected to resistors. The natural frequency @, of the piezoelectric structure is defined by Eqs. (9) and (10). When the excitation frequency closer to the natural frequency of the THUNDER structures the rectified output voltage is higher for the same load resistor. On the other hand as the excitation frequency moves away from the natural frequency of the THUNDER the output voltage reduces with the same load resistor. Fig. 6 illustrated t he variation trend of the rectified voltage as the resistance load increases. Fig. 6 verifies Eq. (19), | Eq. (19) demonstrates rectified voltage is monotonically increase function of the resistance and the rectified voltage is limited by
Fig. 8 The experiment output power with full bridge rectifier connected to resistors. The natural frequency @, of the piezoelectric structure is defined by Eqs. (9) and (10). When the excitation frequency closer to the natural frequency of the THUNDER structures the rectified output voltage is higher for the same load resistor. On the other hand as the excitation frequency moves away from the natural frequency of the THUNDER the output voltage reduces with the same load resistor. Fig. 6 illustrated t he variation trend of the rectified voltage as the resistance load increases. Fig. 6 verifies Eq. (19), | Eq. (19) demonstrates rectified voltage is monotonically increase function of the resistance and the rectified voltage is limited by
Fig. 9 The theoretical comparison between direct charge power and the available maximum power
Fig. 9 The theoretical comparison between direct charge power and the available maximum power
Fig. 10 Current and switch states of the step down converter: (a) CCM (b) DCM.
Fig. 10 Current and switch states of the step down converter: (a) CCM (b) DCM.
The inductor current reaches it maximum value after the time reaches DT,,
The inductor current reaches it maximum value after the time reaches DT,,
Fig. 11 Theoretical Optimum Duty Cycle versus the normalized excitation frequency of the CCM converter. where 6 is the time period that the transistor is off, from Eq. (32) 6 can be expressed as,
Fig. 11 Theoretical Optimum Duty Cycle versus the normalized excitation frequency of the CCM converter. where 6 is the time period that the transistor is off, from Eq. (32) 6 can be expressed as,
Fig. 12 Theoretical Optimum Duty Cycle versus the normalized excitation frequency of the DCM converter. From Eq. (40), we can see that the optimum duty cycle of the DCM converter is a function of inductance, switching frequency, the piezoelectric element?s capacitance, and the vibration amplitude of the piezoelectric device. The optimum switching frequency is calculated based on minimum power lost analysis of the circuit. There are into three major categories of power losses: 1) conduction power losses which describe the Ohmic power dissipated in the parasitic series resistances and diodes present in the power-conducting switches, inductors, and capacitors; 2) switching power losses, i.e. the energy needed to charge and discharge gate-drive capacitors, the energy lost due to the voltage-current overlaps across the switches, and all other energy lost due to the switching actions of the converter; 3) quiescent reducing switch frequency f, will cause degrade of the energy east energy losses and achieves the highest energy harvesting e! harvesting efficiency. In there is an optimum relationship between switch frequency f; and the output current t converter?s inductor with 10 millihenry inductance, based on our energy lost analysis switch frequency f, we set equals 1 khz. power losses which refer to the steady-state current required by the controller to function and operate at the prescribed switching frequency. While reducing switch frequency f, in the DCM converter decreases quiescent and switching losses, it also increases ac conduction losses because of the inductor peak current increase. When the switching frequency is reduced compared to the CC harvesting efficiency will first increase but there exist and optimum point beyond this , the energy point further other word, hat produces ficiency. In our work, we choose the the optimum Similar to the CCM case, the optimum duty cycle versus normalize excitation frequency of the DCM converter is given in Fig. 12. Compared to Fig. 11, with the same normalized excitation frequency, the optimum duty cycle of the DCM converter is much smaller than that of the CCM converter.
Fig. 12 Theoretical Optimum Duty Cycle versus the normalized excitation frequency of the DCM converter. From Eq. (40), we can see that the optimum duty cycle of the DCM converter is a function of inductance, switching frequency, the piezoelectric element?s capacitance, and the vibration amplitude of the piezoelectric device. The optimum switching frequency is calculated based on minimum power lost analysis of the circuit. There are into three major categories of power losses: 1) conduction power losses which describe the Ohmic power dissipated in the parasitic series resistances and diodes present in the power-conducting switches, inductors, and capacitors; 2) switching power losses, i.e. the energy needed to charge and discharge gate-drive capacitors, the energy lost due to the voltage-current overlaps across the switches, and all other energy lost due to the switching actions of the converter; 3) quiescent reducing switch frequency f, will cause degrade of the energy east energy losses and achieves the highest energy harvesting e! harvesting efficiency. In there is an optimum relationship between switch frequency f; and the output current t converter?s inductor with 10 millihenry inductance, based on our energy lost analysis switch frequency f, we set equals 1 khz. power losses which refer to the steady-state current required by the controller to function and operate at the prescribed switching frequency. While reducing switch frequency f, in the DCM converter decreases quiescent and switching losses, it also increases ac conduction losses because of the inductor peak current increase. When the switching frequency is reduced compared to the CC harvesting efficiency will first increase but there exist and optimum point beyond this , the energy point further other word, hat produces ficiency. In our work, we choose the the optimum Similar to the CCM case, the optimum duty cycle versus normalize excitation frequency of the DCM converter is given in Fig. 12. Compared to Fig. 11, with the same normalized excitation frequency, the optimum duty cycle of the DCM converter is much smaller than that of the CCM converter.
Fig. 13 Power stored by a 3.6 volt battery obtained from experiments with different kinds of charging modes Fig. 13 is the comparison of the power obtained from experiment for different kinds of charging modes: 1) the direct charging mode; 2) the step down CCM mode; and 3) the step down DCM mode. Again our experiment the rechargeable battery voltage is 3.6 volt.
Fig. 13 Power stored by a 3.6 volt battery obtained from experiments with different kinds of charging modes Fig. 13 is the comparison of the power obtained from experiment for different kinds of charging modes: 1) the direct charging mode; 2) the step down CCM mode; and 3) the step down DCM mode. Again our experiment the rechargeable battery voltage is 3.6 volt.
Fig. 1 Case-I (a) x; and y, with respect to time;(b) x2 and yz with respect to time ;(c) x3 and y3 with respect to time.
Fig. 1 Case-I (a) x; and y, with respect to time;(b) x2 and yz with respect to time ;(c) x3 and y3 with respect to time.
Fig. 2 Case-II (a) x; and y, with respect to time; ;(b) x2 and y2 with respect to time ;(c) x3 and y3 with respect to time
Fig. 2 Case-II (a) x; and y, with respect to time; ;(b) x2 and y2 with respect to time ;(c) x3 and y3 with respect to time
Fig. 3 Case-III (a) x; and y; with respect to time; ;(b) x2 and yz with respect to time; ;(c) x3 and y3 with respect to time.
Fig. 3 Case-III (a) x; and y; with respect to time; ;(b) x2 and yz with respect to time; ;(c) x3 and y3 with respect to time.
Fig. 5 Case-V (a) x1; and y, with respect to time; (b) x2 and y2 with respect to time; (c) x3 and y3 witk respect to time
Fig. 5 Case-V (a) x1; and y, with respect to time; (b) x2 and y2 with respect to time; (c) x3 and y3 witk respect to time
Fig. 6 Time evolution of the synchronization error e
Fig. 6 Time evolution of the synchronization error e
Analogously, proceeding for the remaining auxiliary functions, we get
Analogously, proceeding for the remaining auxiliary functions, we get
Applying integration by parts for the second term gives
Applying integration by parts for the second term gives
Fig. 1 Numerical Hopf bifurcation curves for € = 0.5, @ = 0.05 and y= 1 for Eq.(2) obtained by using DDE-BIFTOOL . 2 Derivation of slow flow
Fig. 1 Numerical Hopf bifurcation curves for € = 0.5, @ = 0.05 and y= 1 for Eq.(2) obtained by using DDE-BIFTOOL . 2 Derivation of slow flow
Fig. 2 Numerical Hopf bifurcation curves (blue/solid) and analytical Hopf condition Eq.(22) (black /dashdot) for €=0.5, @=0.05 and y=1 for Eq.(2). Since this condition is based on the bold step of replacing the delay quantities in the slow flow by their undelayed counterparts, the question arises as to the correctness of such a procedure and the validity of Eiq.(22). See Fig.2 where Eq.(22) is plotted along with the numerically-obtained conditions for a Hopf.
Fig. 2 Numerical Hopf bifurcation curves (blue/solid) and analytical Hopf condition Eq.(22) (black /dashdot) for €=0.5, @=0.05 and y=1 for Eq.(2). Since this condition is based on the bold step of replacing the delay quantities in the slow flow by their undelayed counterparts, the question arises as to the correctness of such a procedure and the validity of Eiq.(22). See Fig.2 where Eq.(22) is plotted along with the numerically-obtained conditions for a Hopf.
Fig. 3 Critical delay vs. the feedback magnitude k for ¢ = 0.25 (left). Critical delay vs € for k =2 (right). Red/solid curves: Eqs.(38)-(39). Blue dots: numerical roots of Eqs.(28)-(29). These results are for Eq.(2) with parameter of = 0.05. Eq.(33) appears to be the front end of a geometric series. Assuming the series (33) actually is a reometric series, we can sum it:
Fig. 3 Critical delay vs. the feedback magnitude k for ¢ = 0.25 (left). Critical delay vs € for k =2 (right). Red/solid curves: Eqs.(38)-(39). Blue dots: numerical roots of Eqs.(28)-(29). These results are for Eq.(2) with parameter of = 0.05. Eq.(33) appears to be the front end of a geometric series. Assuming the series (33) actually is a reometric series, we can sum it:
Fig. 4 Numerical Hopf bifurcation curves (blue/solid) for Eq.(2) for € = 0.5, @ =0.05 and y= 1. Also show are the results of approach I, the analytical Hopf condition Eq.(22) (black/dashdot), and the results of approach IT, Eqs.(38),(39) (red/dashed)
Fig. 4 Numerical Hopf bifurcation curves (blue/solid) for Eq.(2) for € = 0.5, @ =0.05 and y= 1. Also show are the results of approach I, the analytical Hopf condition Eq.(22) (black/dashdot), and the results of approach IT, Eqs.(38),(39) (red/dashed)
Fig. 5 Numerical Hopf bifurcation curves for € = 0.1 for Eq.(40) obtained by using DDE-BIFTOOL .
Fig. 5 Numerical Hopf bifurcation curves for € = 0.1 for Eq.(40) obtained by using DDE-BIFTOOL .
Fig. 6 Numerical Hopf bifurcation curves (blue/solid) and approach I analytical Hopf condition Eq.(45) (black/dashdot) for ¢ = 0.1 for Eq.(40) .
Fig. 6 Numerical Hopf bifurcation curves (blue/solid) and approach I analytical Hopf condition Eq.(45) (black/dashdot) for ¢ = 0.1 for Eq.(40) .
Fig. 7 Comparison of numerical versus analytic results obtained by approach II for Eq.(40). Critical delay vs. the feedback magnitude k for € = 0.5 (left). Critical delay vs € for k = 2 (right). Red/solid curves: Eq.(58)-(59). Blue dots: numerical roots of Eqs.(51)-(52).
Fig. 7 Comparison of numerical versus analytic results obtained by approach II for Eq.(40). Critical delay vs. the feedback magnitude k for € = 0.5 (left). Critical delay vs € for k = 2 (right). Red/solid curves: Eq.(58)-(59). Blue dots: numerical roots of Eqs.(51)-(52).
Fig. 8 Numerical Hopf bifurcation curves (blue/solid) for Eq.(40), for e= 0.1 (left) and e= 0.5 (right). Also shown are the results of approach I, the analytical Hopf condition Eq.(45) (black/dashdot), and the results of approach IT, Eqs.(58)-(59) (red/dashed). Inserting Eqs. (53)-(54) in Eqs.(51)-(52), Taylor expanding the trig functions with respect to the small parameter € << 1, and equating terms of equal order of € we obtain:
Fig. 8 Numerical Hopf bifurcation curves (blue/solid) for Eq.(40), for e= 0.1 (left) and e= 0.5 (right). Also shown are the results of approach I, the analytical Hopf condition Eq.(45) (black/dashdot), and the results of approach IT, Eqs.(58)-(59) (red/dashed). Inserting Eqs. (53)-(54) in Eqs.(51)-(52), Taylor expanding the trig functions with respect to the small parameter € << 1, and equating terms of equal order of € we obtain:
Figure 10 shows a comparison between approach II, Eqs. (61),(62),(63),(64), and approach I, which again gives Eqs. (55),(56), and the numerical Hopf curves obtained by use of DDE-BIFTOOL. Fig.10 shows that approach II gives better results than approach I. However approach I still gives a good fit for the lower Hopf curve as in the case of Eq. (40).
Figure 10 shows a comparison between approach II, Eqs. (61),(62),(63),(64), and approach I, which again gives Eqs. (55),(56), and the numerical Hopf curves obtained by use of DDE-BIFTOOL. Fig.10 shows that approach II gives better results than approach I. However approach I still gives a good fit for the lower Hopf curve as in the case of Eq. (40).
Fig. 9 Numerical integration for x as a function of time ¢ in Eq.(40) for € = 0.5, k= 0.21 and delay T=0.4. Note that the motion grows large and there is no limit cycle. The origin has changed its nature. See Figure 8 and text.
Fig. 9 Numerical integration for x as a function of time ¢ in Eq.(40) for € = 0.5, k= 0.21 and delay T=0.4. Note that the motion grows large and there is no limit cycle. The origin has changed its nature. See Figure 8 and text.
Fig. 10 Numerical Hopf bifurcation curves (blue/solid) for Eq.(60) for e= 0.1 (left) and e= 0.5 (right). Also shown are the results of approach I, the analytical Hopf condition Eq.(45) (black/dashdot), and the results of approach IT, Eqs.(63)-(64) (red /dashed).
Fig. 10 Numerical Hopf bifurcation curves (blue/solid) for Eq.(60) for e= 0.1 (left) and e= 0.5 (right). Also shown are the results of approach I, the analytical Hopf condition Eq.(45) (black/dashdot), and the results of approach IT, Eqs.(63)-(64) (red /dashed).
From Figure 1, when T < 0.15 the, value for p lies between 0 < p< 1. So take T =0.1 and on substituting A and B in Grammaian operator, we get The nonlinear function satisfies the Lipschitz condition with the constant Nj; = 2. The numerical simulation for finding p by varying time T is given in Figure 1.
From Figure 1, when T < 0.15 the, value for p lies between 0 < p< 1. So take T =0.1 and on substituting A and B in Grammaian operator, we get The nonlinear function satisfies the Lipschitz condition with the constant Nj; = 2. The numerical simulation for finding p by varying time T is given in Figure 1.
From Figure 2, when T < 0.14, the value for g lies between 0 <q <1. So take T =0.1 and on substituting A and B in Grammaian operator, we get
From Figure 2, when T < 0.14, the value for g lies between 0 <q <1. So take T =0.1 and on substituting A and B in Grammaian operator, we get
Fig. 1 The steady-state PDF's of the van der Pol oscillator with the parameters B = 1 and D=0.25. (a) The PDF obtained with the method of stochastic averaging. (b) The PDF obtained with one application of the method of weighted residue using the weighting function in Mj; with n =6 and m= 100. (c) The PDF obtained with the Monte Carlo simulations. (d) The PDF in Equation (13), which is in excellent agreement with the one by Monte Carlo simulations in (c).
Fig. 1 The steady-state PDF's of the van der Pol oscillator with the parameters B = 1 and D=0.25. (a) The PDF obtained with the method of stochastic averaging. (b) The PDF obtained with one application of the method of weighted residue using the weighting function in Mj; with n =6 and m= 100. (c) The PDF obtained with the Monte Carlo simulations. (d) The PDF in Equation (13), which is in excellent agreement with the one by Monte Carlo simulations in (c).
Fig. 2 The steady-state PDFs of the van der Pol oscillator with B = 1.5 and D=0.25. (a) The closed-form PDF in Equation (14) with n = 6 and m= 100, and (b) The PDF obtained with the Monte Carlo simulations. The error Ap.. is 0.2076.
Fig. 2 The steady-state PDFs of the van der Pol oscillator with B = 1.5 and D=0.25. (a) The closed-form PDF in Equation (14) with n = 6 and m= 100, and (b) The PDF obtained with the Monte Carlo simulations. The error Ap.. is 0.2076.
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