Idea about citation of Referee 3’s papers

Some of these papers (1, 3, 10?, 14?, 17, 19, 20, 22) can be cited in section 1(c) as one example of the previous research about the vibration of microbeams.

whereas in the modified theory used by Tilmans et al., the structure additionally transmits a constant tensile force.

Ghayesha et al. [1, 3, 17, 19, 20, 22] and Farokhi et al. [10, 14] studied nonlinear dynamics of microbeams by considering the size effect. They obtained size-dependent frequency–response curves of both Euler-Bernoulli beams and Timoshenko beams through Galerkin and pseudo-arclength continuation techniques. The geometric nonlinearity in the beam model comes from the mid-plane stretching effect [1, 3, 10, 19, 22].

Zhang et al. [45] studied the forced vibration of an adhered MEMS microbeam.

Referee 3’s list of literatures:

1. Nonlinear behaviour of electrically actuated MEMS resonators
2. Nonlinear dynamics of a geometrically imperfect microbeam based on the modified couple stress theory
3. Nonlinear forced vibrations of a microbeam based on the strain gradient elasticity theory
4. Nonlinear dynamics of microplates
5. Thermo-mechanical dynamics of perfect and imperfect Timoshenko microbeams
6. Chaotic motion of a parametrically excited microbeam
7. Size-dependent parametric dynamics of imperfect microbeams
8. Nonlinear mechanics of electrically actuated microplates
9. Motion characteristics of bilayered extensible Timoshenko microbeams
10. Nonlinear resonant response of imperfect extensible Timoshenko microbeams
11. Oscillations of functionally graded microbeams
12. Nonlinear oscillations of functionally graded microplates
13. Viscoelastically coupled size-dependent dynamics of microbeams
14. Large-amplitude dynamical behaviour of microcantilevers
15. Nonlinear dynamical behaviour of geometrically imperfect microplates based on modified couple stress theory
16. Vibration analysis of geometrically imperfect three-layered shear-deformable microbeams
17. In-plane and out-of-plane motion characteristics of microbeams with modal interactions
18. In-plne nd out-of-plne nonliner size-dependent dynmics of micropltes
19. Nonlinear dynamics of a microscale beam based on the modified couple stress theory
20. Size-dependent performance of microgyroscopes
21. Supercritical nonlinear parametric dynamics of Timoshenko microbeams
22. Three-dimensional nonlinear size-dependent behaviour of Timoshenko microbeams

Comments

All of the above papers are actually very similar to each other. The main idea is nonlinear dynamic analysis of a micro-structure (either beam or plate). The nonlinearity comes from the size effect of the micro-scale structure. They use Hamilton’s principle to derive the nonlinear PDEs. Then they discretize the PDES into ODEs by Galerkin method and solve them numerically employing the pseudo-arclength continuation technique. The difference between these papers are:

• The size-dependent effect are considered through either strain gradient theory (3) or modified couple stress theory (all except 3).
• The structure can be modeled by Euler-Bernoulli beam or Timoshenko beam (5, 9, 10, 21, 22) or von Kármán plate (3, 12, 15) or Kirchhoff’s plate (8).
• The material of the structure can be isotropic elastic or functionally graded (11, 12) or two-layer (9) or three-layer (16 )or viscoelastic (13) or temperature-dependent (5).
• The geometry of the structure can be imperfect (2, 5, 7, 10, 15, 16).
• The boundary condition of the beam can be hinged-hinged or clamped-clamped (1, 13) or cantilever (14, 20).
• The loading condition can be free vibration or forced-vibration (3) or electrically actuated (1, 8). The loading direction can be transverse or/and axial (6, 7, 9, 11, 21) for beam or in-plane or/and out-of-plane for plate.
• The problem to be investigated can be frequency analysis or buckling bifurcation analysis (5, 21).
• The deformation can be small or large (14). The motion to be considered can be transverse and/or longitudinal and/or rotational (9, 22).