^{*}

This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

Normally, the boundaries are assumed to allow small deflections and moments for MEMS beams with flexible supports. The non-ideal boundary conditions have a significant effect on the qualitative dynamical behavior. In this paper, by employing the principle of energy equivalence, rigorous theoretical solutions of the tangential and rotational equivalent stiffness are derived based on the Boussinesq's and Cerruti's displacement equations. The non-dimensional differential partial equation of the motion, as well as coupled boundary conditions, are solved analytically using the method of multiple time scales. The closed-form solution provides a direct insight into the relationship between the boundary conditions and vibration characteristics of the dynamic system, in which resonance frequencies increase with the nonlinear mechanical spring effect but decrease with the effect of flexible supports. The obtained results of frequencies and mode shapes are compared with the cases of ideal boundary conditions, and the differences between them are contrasted on frequency response curves. The influences of the support material property on the equivalent stiffness and resonance frequency shift are also discussed. It is demonstrated that the proposed model with the flexible supports boundary conditions has significant effect on the rigorous quantitative dynamical analysis of the MEMS beams. Moreover, the proposed analytical solutions are in good agreement with those obtained from finite element analyses.

Micro-beams [

Microfabrication methods and limitations can lead to boundary support conditions for suspended MEMS beams that are not rigidly clamped [

Hence, the boundary support conditions need to be theoretically quantified [

In accordance, the boundaries are assumed to allow small deflections and moments [

In this paper, a rigorous theoretical solution is presented for the case of flexible supports of microbeams. Equivalent deformation in the normal and tangential direction at the boundary of the microbeam were formulated by Boussinesq's and Cerruti's displacement equations [

The displacement of any point (_{s}_{s}

When the bending moment is acting on the beam, the non-uniform normal stress is correspondingly applied on the support, as shown in the rectangular region (

The displacement of any point (_{z}dmdn_{x}_{x}_{z}

The effect of non-ideal boundary conditions on the dynamics of the arch was investigated by Alkharabsheh _{t}_{R}

A similar experiment for the atomic force microscope (AFM) micro-cantilever probes was presented by Rinaldi _{t}

In this section, we formulate the problem for the forced vibration of a microbeam of nonideal supports. Rotational and transversal springs are added to the boundaries of the beam to model the compliant supports shown schematically in _{b}_{b}_{b}

For convenience, the following nondimensional variables are introduced:

The nondimensional parameters in

First, we study the effect of non-ideal boundary conditions on the resonance frequencies and mode shapes of the beam. These springs affect the stiffness of beam and, hence, its frequencies and mode shapes. The linearized undamped and unforced version of

The homogeneous solution of this fourth-order ordinary differential equation can be expressed as:
_{i}_{i}

The equivalent tangential stiffness and equivalent rotational stiffness of the flexible supports are listed in the second and third columns of

By solving the eigenvalue problem of

It is obvious that the qualitative and quantitative behaviors of the mode shape are different for different boundary conditions. The amplitude of the ideal supports is less than the flexible one as the position close the substrate. However, the situation is almost reversed as the position away from the substrate. Moreover, this trend will be more strengthen as the softer material (polysilicon) is used.

It is also easily observed that the actual modal is more close to the ideal modal when the supports' material performance approximates to the rigid body. Moreover, the softer the supports' material is, the greater the difference between the actual modals and ideal modals will be.

Next, we study the effect of the nonideal boundary conditions on the dynamics response of the beam. To solve the _{0}_{1}_{0}_{1}_{0}=∂/∂_{0}, D_{1}=∂/∂_{1}. By introducing the following variables _{1}=_{1}, _{T}_{T}_{R}_{R}^{0}^{1}

The general solution of first equation of _{1})=a(_{1})/2^{jβ}^{(}^{T}^{1)}, and cc denotes complex conjugate. Substituting

The solution is:
_{i}(x)

At order

The first part of the solution is the one corresponding to secular terms and the second is the one corresponding to non-secular terms. Substituting

Since the homogeneous problem has a non-trivial solution, the non-homogeneous problem

So the solvability condition of

Then by expanding the trigonometric functions, and separating real and imaginary parts, the secular terms yields two first order nonlinear ordinary-differential equations that describe the amplitude _{1}−_{1}a=D_{1}

Therefore, a function of the independent variables can be given by:

In ^{3}^{3}_{s}

From

In the published literatures [_{T}_{R}

However, with the equivalent stiffness we have derived, it is found from

The first four natural frequencies and mode shapes of the beam are obtained using the finite element software under rigid and flexible boundary conditions, respectively.

The results were list in

It can be easily found from

The maximum amplitude of oscillation is reached when the magnitude under the square root in _{max}=

In the above relation, the second term is due to the non-ideal boundary conditions and the third term is due to the nonlinearity mechanical spring. It can be obviously seen that the flexible supports conditions would decrease the resonance frequency while the nonlinearity mechanical spring increase the frequencies. However, it is also apparent of

The resonance frequencies increase linearly with the beam thickness. Moreover, the resonance frequencies of the flexible resonators are smaller than the ones of the ideal support resonators no matter what value the beam thickness is.

In fact, it is revealed that the material performances and the geometric sizes of the supports conditions not only influence the system stiffness and the resonant frequency shift, but also affect the system vibration amplitude. That was presented in the paper [

In this paper, we have quantitatively studied the effect of the flexible supports boundary conditions on the dynamic characteristics of MEMS beams. Utilizing the tangential and rotational equivalent stiffness formulations derived by employing Cerruti's and Boussinesq's displacement equations and the principle of energy equivalence, rigorous theoretical dynamic analytical models are presented.

It is of great significance to investigate the rigorous variation of the resonant frequency and dynamic response due to the equivalent stiffness of the flexible supports, where the nonlinearity mechanical spring increases the frequencies while the flexible supports conditions decrease the resonance frequency. It is also demonstrated that the support material property has an important influence on the equivalent stiffness, dynamic response and the resonant frequency shift. The advantage of the proposed solution is that no approximated displacement and force fields are introduced during the derivation of the equivalent stiffness. Moreover, the proposed analytical solutions are in good agreement with the results obtained from finite element analyses. Based on the proposed solutions, it is convenient to quantitatively and accurately analyze the dynamics problem of the MEMS beams with the flexible support boundary conditions.

This work was supported by the National Natural Science Foundation of China (No. 11322215 and 11342001) and the 973 Program (National Basic Research Program of China) (No. 2011CB706502), and sponsored by Shanghai Rising-Star Program under Grant No.11QA1403400.

The authors declare no conflict of interest.

(

Schematic representation of an electrically actuated beam with compliant supports.

First four mode shapes corresponding to the natural frequencies of

First four resonance response curves. (Black) Ideal boundary conditions. (blue), (red) non-ideal boundary conditions with the supports material of silicon carbide and polysilicon, respectively. (

Modal vectors of the first and fourth mode shapes solved by FEM. (

(

The resonant frequency shift to the beam thickness with respect to the different flexible supports materials.

The comparison of the equivalent stiffness between the proposed results and the reported experimental data [

_{R}^{−8} | |
---|---|

Experiment data 1 of Alkharabsheh |
9.0396 |

Proposed results | 7.4270 |

Experiment data 2 of Alkharabsheh |
10 |

Proposed results | 8.3001 |

The comparison of the equivalent stiffness between the proposed results and the reported experimental data [

_{t} |
_{R}^{−8} | |
---|---|---|

Experiment data 1 of Rinaldi |
9.8018 × 10^{7} |
13.284 |

Proposed results | 1.9743 × 10^{7} |
16.567 |

Experiment data 2 of Rinaldi |
1.6620 × 10^{8} |
13.521 |

Proposed results | 1.9801 × 10^{7} |
17.120 |

The equivalent tangential stiffness and equivalent rotational stiffness of the flexible supports and the first four natural frequencies of the beam. The second and third rows relate to the supports material [^{3}, v = 0.192) and polysilicon (E = 150 GPa, density = 2,300 Kg/m^{3}, v = 0.226), respectively.

_{T}^{6}N/m) |
_{R}^{−8}N.m/rad) |
_{1}(10^{6}Hz) |
_{2}(10^{6}Hz) |
_{3}(10^{6}Hz) |
_{4}(10^{6}Hz) | |
---|---|---|---|---|---|---|

Ideal boundary conditions | - | - | 1.1591 | 3.1950 | 6.2635 | 10.354 |

Non-Ideal boundary conditions (SiC) | 5.4996 | 17.895 | 0.95669 | 2.7104 | 5.4261 | 9.1206 |

Non-Ideal boundary conditions (PolySi) | 1.9929 | 6.5648 | 0.80512 | 2.4305 | 5.0381 | 8.6479 |

The first four natural frequencies of the beam solved by FEM.

_{1}(10^{6}Hz) |
_{2}(10^{6}Hz) |
_{3}(10^{6}Hz) |
_{4}(10^{6}Hz) | |
---|---|---|---|---|

Rigid | 1.0885 | 3.0365 | 6.0639 | 10.277 |

Flexible | 0.9461 | 2.8851 | 5.8395 | 9.9402 |

The boundary displacements of first four modal vectors.

^{−6}) |
^{−6}) |
^{−6}) |
^{−6}) | |
---|---|---|---|---|

x = 0 | 6.9028 | 35.689 | 103.53 | 227.74 |

x = L | 6.9028 | −35.689 | 103.53 | −227.74 |