Mechanics of Microelectromechanical Systems n lobontiu and e garcia docx

[...]... linear member by calculating the sum of all relevant components that are applied between one end point of the member (the free end of Fig 1.12 is a convenient choice because it does not introduce any reactions, which are usually unknown amounts) and the specific point, as given in the equations: Example 1.4 Determine the axial, shearing and bending moment equations for the fixed-free microcantilever, which... loading In the general case where the material properties are non-linear (as indicated by the force-displacement curve of Fig 1.15), two energy types can be defined, namely: the regular strain energy, which is: 22 Chapter 1 and the complementary energy: If the strain energy can be expressed as a function of solely the displacement function, as: then the variation in the strain energy can be written as:... and moment-produced bending slopes (rotations) of beams Both situations presented here, the linear spring under axial load and the rotary spring under a torque, define the stiffness as being the inverse to the corresponding compliance There is however the case of a beam in bending where a force that is applied at the free end of a fixed-free beam for instance produces both a linear deformation (the... structure in a clockwise direction, whereas in bending, a component of the bending moment (either force or moment) produces a positive bending moment if the analyzed structural segment deforms in a sagging manner (by compressing the upper fiber) All these situations are sketched in Fig 1.12 The normal force N, shearing force S, torsion moment and bending moment are defined at a specific point on the linear... membranes or plates) and threedimensional (such as blocks) For each of them, specific equations that describe the state of deformation or stress apply There are four different types of loading/deformations, namely: normal, torsion, shearing and bending They are briefly characterized here in terms of stresses, deformations and strain energy for one-dimensional members 4.1 Normal Loading In the case of. .. Determinate/Indeterminate Systems Figure 1.14 shows the most frequently encountered boundary conditions in one-dimensional members In a given member under load, each boundary condition introduces a number of reactions, which are unknown initially A pinned end, such as the one shown in Fig 1.14 (a) introduces one reaction force, which is normal to the support direction, a guided end – pictured in Fig... folded beams, and spiral springs (with either small or large number of turns) All these flexible components are treated in a systematic manner by offering equations for both the main (active) stiffnesses and the secondary (parasitic) ones Chapter 4 analyzes the micro actuation and sensing techniques (collectively known as transduction methods) that are currently implemented in MEMS Details are presented... considered that the member’s cross-section has two principal directions (it possesses two symmetry axes, and therefore a symmetry center) and that bending moments and shearing forces act about these axes Similarly, the complementary energy can be expressed in terms of loading, and in the case of a linear material this energy is: which has been obtained by collecting individual strain energy terms from... its length is: 18 Chapter 1 For relatively short beams, as already mentioned, the shearing effects are important, and the shearing stresses are given by the equation: where S is the shear force and the integral (the statical moment of area) is taken for the area enclosed by an arbitrary line, parallel to the y-axis, situated at a distance z measured from the cross-section center and one of the external... forces of the previous case It should be noted that for a line member, three equilibrium equations can be written, and therefore the boundary conditions should introduce three unknown reactions only, in order for the system to be statically determinate When less than three reactions are present, the respective system is statically unstable (it is actually a mechanism) For more than three unknown reactions, . the efficiency of modeling and designing reliable and desirably-optimized microsystems. The work represents an attempt at both extending and deepening the mechanical-based approach to MEMS in. situations presented here, the linear spring under axial load and the rotary spring under a torque, define the stiffness as being the inverse to the corresponding compliance. There is however the case. rotary) and one crossed, define the elastic response at the free end of a cantilever. More details on the spring characterization of fixed-free microcantilevers that are subject to forces and moments