Asymptotic analysis of a noncontact AFM microcantilever sensor with external feedback control

Abstract

An external feedback control is inserted in a nonlinear continuum formulation of a noncontact AFM model. The aim of the feedback is to keep the system response to an operationally suitable one, thus allowing reliable measurement of the sample surface by avoiding possible unstable microcantilever sensor motions. The study of the weakly nonlinear system dynamics about the desired fixed point close to primary resonance is carried out via multiple-scale asymptotics, whose outcomes are validated via numerical simulations of the original system equations of motion. The latter include controllable periodic dynamics and additional periodic and distinct quasiperiodic solutions that appear beyond the asymptotic stability thresholds. The results highlight the effectiveness of the applied feedback control technique and also enable the derivation of a comprehensive system bifurcation structure highlighting the stability thresholds for robust controllable AFM dynamics.

Appendices

Appendix 1

The integral expressions in (15) are

$$\begin{aligned}&I_1=\left( I_{11}-\mu I_{12}\right) , \quad I_4=\left( I_{41}-\mu I_{42}\right) , \nonumber \\&I_{11} = \int _0^1\varPhi _1^2\,\mathrm {d}s=1,\nonumber \\&I_{41} = \int _0^1\varPhi _1\left( {\varPhi _1}_s\int _1^s\int _0^s{\varPhi _1}_s^2\,\mathrm {d}s\,\mathrm {d}s\right) _s\,\mathrm {d}s,\nonumber \\&\quad \int _0^1\varPhi _1 {\varPhi _1}_{ssss}\,\mathrm {d}s=\omega _1^2 \int _0^1\varPhi _1^2\,\mathrm {d}s=\omega _1^2 I_{11},\nonumber \\&I_{42} = \int _0^1\varPhi _1\left( {\varPhi _1}_s^3\right) _s\,\mathrm {d}s,\nonumber \\&I_{12} = \int _0^1\varPhi _1 {\varPhi _1}_{ss}\,\mathrm {d}s,\nonumber \\&I_5 = \int _0^1\varPhi _1\left( {\varPhi _1}_s\left( s-1\right) \right) _s\,\mathrm {d}s=\int _0^1{\varPhi _1}_s^2\left( 1-s\right) \,\mathrm {d}s,\nonumber \\&I_2 = \int _0^1\varPhi _1\,\mathrm {d}s,\nonumber \\&I_6 = \int _0^1\varPhi _1\left( {\varPhi _1}_s^3\left( s-1\right) \right) _s \,\mathrm {d}s=\int _0^1{\varPhi _1}_s^4\left( 1-s\right) \,\mathrm {d}s,\nonumber \\&I_3 = \int _0^1\varPhi _1\left( {\varPhi _1}_s\left( {\varPhi _1}_s{\varPhi _1}_{ss}\right) _s\right) _s\,\mathrm {d}s\nonumber \\&\quad \,\, =2\int _0^1\left( {\varPhi _1}_s{\varPhi _1}_{ss}\right) ^2\,\mathrm {d}s,\nonumber \\&I_7 = \int _0^1\varPhi _1\left( {\varPhi _1}_s\int _1^s\left( \int _0^s{\varPhi _1}_s^2\,\mathrm {d}s\right) _{ss}\,\mathrm {d}s\right) _s\,\mathrm {d}s. \end{aligned}$$

(45)

Appendix 2

1.1 MSM: second-order solution

Substitution of $p_1$ (36) in the second equation of (33), and elimination of secular terms yield

$$\begin{aligned} D_1B=0 \end{aligned}$$

(46)

so that

$$\begin{aligned} B=B(T_2,T_3),\quad p_2=0 \end{aligned}$$

(47)

Using (36) and (47), and remembering that $\tilde{x}_{ \textit{ref1} }= A_{un}(T_1,T_2,T_3)e^{i\omega _1T_0}+c.c.$ is the solution of amplitude $A_{un}(T_1,T_2,T_3)$ of the first-order uncontrolled system, the first equation of (33), without $\tilde{x}_{ \textit{ref2} }$ and $N_{22}(\tilde{x}_{ \textit{ref1} })$ terms, becomes

$$\begin{aligned}&D_0^2y_2+\omega _1^2y_2\nonumber \\&\quad =-2C_{214}\left( A\bar{A}+A\bar{A}_{un}\right) -C_{212}B^2\nonumber \\&\qquad {}-e^{i\omega _1T_0}\left( C_{213}\left( A+A_{un}\right) B+2i\omega _1D_1A\right) \nonumber \\&\qquad {}-C_{211} e^{2i\omega _1T_0}\left( A^2+2AA_{un}\right) +c.c. \end{aligned}$$

(48)

and the solvability condition implies that

$$\begin{aligned} D_1A=\frac{iC_{213}}{2\omega _1}\left( A+A_{un}\right) B \end{aligned}$$

(49)

For the uncontrolled system, it is $D_1A_{un}=0$ and thus $A_{un}=A_{un}(T_2,T_3)$. The particular solution at this order is

$$\begin{aligned} y_2&= \frac{C_{211}}{3\omega _1^2}\left( A^2+2AA_{un}\right) e^{2i\omega _1T_0}\nonumber \\&\quad -\frac{2C_{214}}{\omega _1^2}\left( A\bar{A}+A\bar{A}_{un}\right) -\frac{C_{212}}{\omega _1^2}B^2+c.c. \end{aligned}$$

(50)

while the solution of the uncontrolled system is $\tilde{x}_{ \textit{ref2} }=\frac{C_{211}}{3\omega _1^2}A_{un}^2 e^{2i\omega _1T_0}-\frac{2C_{214}}{\omega _1^2} A_{un}\bar{A}_{un}+c.c.$ For the expression of the $C_{ijk}$ coefficients, see “Appendix 3” of [37].

1.2 MSM: third-order solution

At the third order, by means of the obtained results, the second of (34) becomes

$$\begin{aligned} D_0p_3=-D_2B+\hat{k}_gC_{11}B/\omega _1^2-\hat{k}_gAe^{i\omega _1T_0}+c.c. \end{aligned}$$

(51)

and the secular terms elimination, providing

$$\begin{aligned} D_2B=\hat{k}_gC_{11}B/\omega _1^2, \end{aligned}$$

(52)

permits to obtain

$$\begin{aligned} p_3=\frac{i \hat{k}_gAe^{i\omega _1T_0}}{\omega _1}+c.c. \end{aligned}$$

(53)

Here, $\tilde{x}_{ \textit{ref3} }$ and $N_{32}(\tilde{x}_{ \textit{ref1} },\tilde{x}_{ \textit{ref2} })$ terms, together with terms related to the horizontal and vertical excitations, are present also in the uncontrolled system, so that they can be neglected; using (36), (46), (47), (49), (50), (52), (53), and being $D_1A_{un} = 0$ (“Appendix 2” of [37]), the first equation of (34) hence becomes

$$\begin{aligned}&D_0^2y_3+\omega _1^2y_3\nonumber \\&\quad =\gamma _{31} e^{i\omega _1T_0}+\gamma _{32} e^{2i\omega _1T_0}+\gamma _{33} e^{3i\omega _1T_0}+\gamma _{35} B^3\nonumber \\&\qquad {}+\gamma _{36}\left( A\bar{A}+\bar{A}A_{un}+A\bar{A}_{un}+A_{un}\bar{A}_{un}\right) B\nonumber \\&\qquad {}+\gamma _{37} \left( \bar{A}+\bar{A}_{un}\right) D_1A+c.c. \end{aligned}$$

(54)

with $\gamma _{ij}$ defined in “Appendix 3” of [37]. Note that $\gamma _{31}$ depends on $D_2A$, so that the solvability condition of (54) provides

$$\begin{aligned}&D_2A=-\frac{C_{35} \omega _1^2+C_{11} \hat{k}_g}{2 \omega _1^2}A\nonumber \\&{\quad }{}+\,i\frac{C_{301}\left( A^2\bar{A}+2A\bar{A}A_{un} +\bar{A}A_{un}^2+A^2\bar{A}_{un} +2AA_{un}\bar{A}_{un}\right) }{2\omega _1}\nonumber \\&{\quad }{}+\,i\frac{C_{302}\left( AB^2+A_{un}B^2\right) }{2\omega _1} \end{aligned}$$

(55)

The particular solution at the third order for the controlled system results

$$\begin{aligned} y_3&= C_{306}B^3+C_{303}\left( A^3+3A^2A_{un} +3AA_{un}^2\right) e^{3i \omega _1 T_0}\nonumber \\&\quad {}+C_{304}\left( A^2B+2AA_{un}B+A_{un}^2 B\right) e^{2i \omega _1 T_0}\nonumber \\&\quad {}+C_{305}\left( A\bar{A}B+A_{un}\bar{A}B +AB\bar{A}_{un}+A_{un}B\bar{A}_{un}\right) \nonumber \\&\quad {}+c.c. \end{aligned}$$

(56)

For what concerns the uncontrolled system, the solvability condition at the third order yields

$$\begin{aligned} D_2A_{un}&= -\frac{C_{35}}{2}A_{un}+i\frac{C_{301}}{2\omega _1}\left( A_{un}^2\bar{A}_{un}\right) \\&\quad {}+\frac{C_{sv}}{4\omega _1}e^{i \sigma _v T_2}+\frac{C_{su}}{4\omega _1}e^{i\left( \sigma _u T_2+\phi _u\right) }\nonumber \\&\quad {}+i\frac{C_{cu}}{4\omega _1}e^{i\left( \sigma _u T_2+\phi _u\right) } \end{aligned}$$

whose expression is later on needed, and the particular solution results $\tilde{x}_{ \textit{ref3} }=C_{303} A_{un}^3 e^{3i \omega _1 T_0}+c.c.$

1.3 MSM: fourth-order solution

By means of the previous results, the second equation of (35) at the fourth order becomes

$$\begin{aligned}&D_0 p_4\nonumber \\&\quad =-D_3 B-i \frac{\hat{k}_g}{\omega _1} D_1A e^{i \omega _1 T_0}\nonumber \\&\qquad {}+\frac{2C_{214}\hat{k}_g}{\omega _1^2}\left( A\bar{A}+A_{un}\bar{A}+A\bar{A}_{un}\right) +\frac{C_{212}\hat{k}_g}{\omega _1^2}B^2\nonumber \\&\qquad {}-\frac{C_{211}\hat{k}_g}{2\omega _1^2}\left( A^2+2AA_{un}\right) e^{2i \omega _1 T_0}+c.c. \end{aligned}$$

(57)

with the vanishing of secular term providing

$$\begin{aligned} D_3 B&= +\frac{2C_{214}\hat{k}_g}{\omega _1^2}\left( A\bar{A} +A_{un}\bar{A}+A\bar{A}_{un}\right) \nonumber \\&\quad {}+\frac{C_{212}\hat{k}_g}{\omega _1^2}B^2 \end{aligned}$$

(58)

and the particular solution resulting

$$\begin{aligned} p_4&= -\frac{iC_{213}\hat{k}_g}{2\omega _1^3}\left( AB+A_{un}B\right) e^{i \omega _1 T_0}\nonumber \\&\quad {}+\frac{iC_{211}\hat{k}_g}{3\omega _1^3}\left( A^3+2AA_{un}\right) e^{2i \omega _1 T_0}+c.c. \end{aligned}$$

(59)

The first equation of (35) can thus be rewritten as

$$\begin{aligned}&D_0^2y_4+\omega _1^2y_4\nonumber \\&\quad =-A^2 \left( \gamma _{47}\bar{A}^2 +2\gamma _{47}\bar{A} \bar{A}_{un} +\gamma _{47}\bar{A}_{un}^2 \right) \nonumber \\&\qquad {} -4\gamma _{47}A A_{un} \bar{A} \bar{A}_{un} -A\left( B^2 \left( \gamma _{45}\bar{A} +\gamma _{45}\bar{A}_{un} \right) \right. \nonumber \\&\qquad {} \left. +\,\gamma _{49}\bar{A}_{un}+\gamma _{54} e^{-i \sigma _v T_2}+\gamma _{55} e^{-i \sigma _u T_2-i \phi _u}\right) \nonumber \\&\qquad {} - 2\gamma _{47}A_{un}^2 \bar{A} \bar{A}_{un}-\gamma _{45}A_{un} B^2\bar{A}_{un} \nonumber \\&\qquad {} -B \left( \gamma _{48}\bar{A} D_1A+ \gamma _{48}\bar{A}_{un} D_1A \right) \nonumber \\&\qquad {} -\gamma _{57}\bar{A} D_1^2A -\gamma _{56}\bar{A} D_2A \nonumber \\&\qquad {} -\gamma _{56}\bar{A} D_2A_{un}-\gamma _{57}\bar{A}_{un} D_1^2A\nonumber \\&\qquad {} -\gamma _{56}\bar{A}_{un} D_2A -\gamma _{50}D_1A D_1\bar{A} \nonumber \\&\qquad {} -\gamma _{46}B^4 -\gamma _{41} e^{i \omega _1 T_0}-\gamma _{42} e^{2 i \omega _1T_0}\nonumber \\&\qquad {} -\gamma _{43} e^{3 i \omega _1 T_0}-\gamma _{44} e^{4 i \omega _1 T_0}+c.c. \end{aligned}$$

(60)

The secular terms elimination requires that $\gamma _{41}$, which depends on $D_3A$, identically vanishes. Using Eqs. (46),(49),(52),(55), and assuming that $2 D_1D_2A=\frac{ d D_1A}{ d T_2}+\frac{ d D_2A}{ d T_1}$ [39], it results

$$\begin{aligned} D_3A&= B^3 \left( \gamma _{402}A +\gamma _{402}A_{un} \right) \nonumber \\&\quad {} +\,B \left( \gamma _{401}A^2 \bar{A} + \gamma _{401}A^2 \bar{A}_{un}\right. \nonumber \\&\quad {} \left. +\,2\gamma _{401}A \bar{A} A_{un}+2 \gamma _{401}A A_{un} \bar{A}_{un}\right. \nonumber \\&\quad {} \left. +\,\gamma _{404}A +\gamma _{403}A_{un} \right. \nonumber \\&\quad {} \left. +\,\gamma _{401}\bar{A} A_{un}^2 +\gamma _{401}A_{un}^2 \bar{A}_{un}\right. \nonumber \\&\quad {} \left. +\,\gamma _{405} e^{i \sigma _u T_2+i \phi _u}+\gamma _{406} e^{i\sigma _v T_2}\right) \end{aligned}$$

(61)

Appendix 3

Terms $c_{hk}$, $h,k=1,...,5$ of the Jacobian matrix (42) are

$$\begin{aligned} c_{11}&= \frac{1}{2} (b (2 \beta _5-\beta _1 (j+j_{un}) (n+n_{un}))\nonumber \\&\quad {}-\beta _4 (j (n+n_{un})+j_{un} n))\nonumber \\ c_{12}&= \frac{1}{4} (-4 \beta _2b^3 -4 \beta _3b^2 -b (\beta _1 ((j+j_{un})^2\nonumber \\&\quad {}+3 (n+n_{un})^2)+4 \beta _7)\nonumber \\&\quad {}-\beta _4 (j^2+2 j j_{un}+3 n (n+2 n_{un})))\nonumber \\ c_{13}&= \frac{1}{2} (b (2 \beta _6-\beta _1 (j+j_{un}) (n+n_{un}))\nonumber \\&\quad {}-\beta _4 (j+j_{un}) (n+n_{un})+2 \beta _9)\nonumber \\ c_{14}&= \frac{1}{4} (-4 \beta _2b^3 -4 \beta _3b^2\nonumber \\&\quad {}-b (\beta _1 ((j+j_{un})^2+3 (n+n_{un})^2)+4 \beta _7)\nonumber \\&\quad {}-\beta _4 ( (j+j_{un})^2+3 (n+n_{un})^2) +4 \omega _1 (\varOmega _u-1))\nonumber \\ c_{15}&= \frac{1}{4} (-(n+n_{un}) ( 12 \beta _2b^2 \nonumber \\&\quad {}+8 \beta _3b +\beta _1 ((j+j_{un})^2+(n+n_{un})^2)\nonumber \\&\quad {}+4 \beta _7)+4 \beta _5 j_{un}+4 \beta _6 j+8 \beta _8) \end{aligned}$$

(62)

$$\begin{aligned} c_{21}&= \frac{1}{4} (4 \beta _2 b^3+4\beta _3 b^2 \\&\quad {}+b (\beta _1 (3 (j+j_{un})^2+(n+n_{un})^2)+4 \beta _7)\\&\quad {}+\beta _4 (3 j (j+2 j_{un})+n (n+2 n_{un})))\\ c_{22}&= \frac{1}{2} (b (\beta _1 (j+j_{un}) (n+n_{un})+2 \beta _5)\\&\quad {}+\beta _4 n (j+j_{un})+\beta _4 j n_{un})\\ c_{23}&= \frac{1}{4} ( 4 \beta _2b^3 +4 \beta _3b^2 \\&\quad {}+b ( \beta _1 ( 3 (j+j_{un})^2+(n+n_{un})^2)+4 \beta _7)\\&\quad {}+\beta _4 ( 3 (j+j_{un})^2+(n+n_{un})^2) -4 \omega _1 (\varOmega _u-1))\\ c_{24}&= \frac{1}{2} (b (\beta _1 (j+j_{un}) (n+n_{un})+2 \beta _6)\\&\quad {}+\beta _4 (j+j_{un}) (n+n_{un})+2 \beta _9)\\ c_{25}&= \frac{1}{4} ((j+j_{un}) ( 12\beta _2 b^2 +8 \beta _3b \\&\quad {}+\beta _1 ((j+j_{un})^2+(n+n_{un})^2)\\&\quad {}+4 \beta _7)+8 \beta _{10}+4 (\beta _5 n_{un}+\beta _6 n))\\ \end{aligned}$$

$$\begin{aligned} c_{31}&= \frac{C_{214} k_g j}{\omega _1^2}\\ c_{32}&= \frac{C_{214} k_g n}{\omega _1^2}\\ c_{33}&= \frac{1}{\omega _1^2}C_{214} k_g (j+j_{un})\\ c_{34}&= \frac{1}{\omega _1^2}C_{214} k_g (n+n_{un})\\ c_{35}&= \frac{1}{\omega _1^2}k_g (2 C_{212} b +C_{11})\\ c_{41}&= -\frac{1}{4}\left( \frac{C_{301} j_{un} n_{un}}{\omega _1}-2 C_{35}\right) \\ c_{42}&= \omega _1 (\varOmega _u-1)-\frac{C_{301} (j_{un}^2+3 n_{un}^2)}{8 \omega _1}\\ c_{51}&= \frac{C_{301} (3 j_{un}^2+n_{un}^2)}{8 \omega _1}-\omega _1 (\varOmega _u-1)\\ c_{52}&= \frac{1}{4} \left( \frac{C_{301} j_{un} n_{un}}{\omega _1}-2 C_{35}\right) \end{aligned}$$

Appendix 4

Coefficients $f_{hkl}$ of the steady-state responses (41) are

$$\begin{aligned}&f_{y0}=-\frac{C_{214} (j^2+2 j j_{un}+n^2+2 n n_{un})}{2 \omega _1^2}\\&{\quad }{}+C_{306}b^3 -\frac{C_{212}}{\omega _1^2}b^2\\&\quad +\Bigl ( \frac{1}{4} C_{305} ((j+j_{un})^2 +(n+n_{un})^2)-\frac{C_{11}}{\omega _1^2}\Bigr ) b \\&f_{yc1} = j\\&f_{ys1} = -n\\&f_{yc2} = +\,\Bigl (\frac{C_{211} (j^2+2 j j_{un}-n (n+2 n_{un}))}{6 \omega _1^2}\\&\qquad \qquad {} +\frac{1}{2} C_{304} (j+j_{un}-n-n_{un}) (j+j_{un}+n+n_{un})b \Bigr )\\&f_{ys2} = -\Bigl (C_{304} (j+j_{un}) (n+n_{un})b\\&\qquad \qquad {} +\frac{C_{211} (j (n+n_{un})+j_{un} n)}{3 \omega _1^2}\Bigr )\\&f_{yc3} = +\frac{1}{4} C_{303} \Bigl (j^3+3 j^2 j_{un}+3 j (j_{un}^2-(n+n_{un})^2)\\&\qquad \qquad {} -3 j_{un} n (n+2 n_{un})\Bigr )\\&f_{ys3} = +\frac{1}{4} C_{303} \Bigl (-3 j^2 (n+n_{un})-6 j j_{un} (n+n_{un})\\&\qquad \qquad {} +n (-3 j_{un}^2+n^2+3 n n_{un}+3 n_{un}^2)\Bigr )\\&f_{z0} = b\\&f_{zc1} = -\frac{k_g n}{\omega _1}\\&f_{zs1}=-\frac{k_g j }{\omega _1} \end{aligned}$$