Mechanical structures have been widely studied from both static and dynamic points of view. As it was mentioned in chapter 1, MEMS and NEMS are widely used for applications in RF-communications, sensors and actuators. which demands a high control of the mechanical characteristics of the mobile part in terms of geometric parameters and material specifications. A first overview of the spring-mass model will be held in this section and will be taken up again in the first lines of the third section of this chapter.
Considering the bar depicted in Fig. 2.1, the arc defined by the angular differential, d, can be expressed as a function of the coordinate z accomplishing that its length does not varies when the bar is bended along the dashed white line (neutral plane):
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(2.1b)
Where dl stands for the length variation of the bar, r the radius of the bending, z is the position where the change in length is measured and dx is the projection of dl in the x- axis. Therefore, for z>0 the bar is stretched while for z<0 a compressive strain is produced.
Fig. 2. 1. Segment of a beam under deformation caused by a force, F, provoking a bending moment, M. The neutral plane represented by a dashed white line is under no stress.
The slope of the beam at any point for a certain deformation defining an angle, , is
given by:
(2.2)
And the relation between the bending radius and the bending moment can be expressed as:
(2.3)
Where l, w and t are defined in Fig. 2.2 and Y is de Young’s modulus. The second moment of area is defined as I=t3w/12. Both, I and Y, are normally assumed to be
constant. Combining equations 2.1, 2.2 and 2.3 one can obtain the differential equation for a beam under deformation:
23 Once M is determined the deflection at any point is known integrating twice equation 2.4, taking into account particular boundary conditions.
The power of this theory is its ability of describing correctly the majority of the features of beams, regardless of the particular conditions, including composite structures and rare geometries. However, the interest of this work is focused on what is known as “clamped-free beams”, or “cantilevers”, and “clamped-clamped beams”, or its abbreviation “cc-beam”, with constant rectangular cross-section.
Therefore, if a cantilever of length, l, width, w, thick, t, with u(0)=0 and as boundary conditions is considered (Fig. 2.2), equation 2.4 can be solved directly to obtain:
(2.5)
where u(x) represents the deflection of the beam at some point x. Here the force, F, is assumed to be a point load acting at the free end of the cantilever and, therefore, the deflection of the free end can be expressed in a very compact manner as:
(2.6)
Fig. 2. 2. Representation of a point load, F, acting at the free end of a cantilever provoking a deformation, u.
The whole structure is treated as a system containing two elements: a spring governed by the Hook’s law and a mass. If equation 2.6 is compared with the Hook’s law, F=kx, it seems reasonable to state that the mechanical system behaves as a spring with elastic constant:
24 The same development could be done in order to describe the deflection of a cc-beam (Fig. 2.3), which is the other structure concerning the present dissertation. In this case a point load acting on the center of the beam is considered, and boundary conditions described by u(0)=u(l)=0 and
. This time the displacement of the
center of the beam is expressed as:
(2.8)
Fig. 2. 3. Representation of a point load, F, acting at the center of the structure, x=l/2, of a cc-beam provoking a deformation, u.
And the expression for its elastic constant may be written as:
(2.9)
Once the statics of these two cases have been determined in terms of material and geometric parameters, the next step is to determine the dynamics, that is, determining the natural frequency of the mode n, n=2fn. Considering the principle of minimum
action, derived from the Hamilton principle [3], the governing equation is obtained for a dynamic deflection:
(2.10)
Where stands for the longitudinal mass density, ·w·t. Equation 2.10 can be solved
assuming variable separation, , with and considering free vibrations, thus, no external force is applied, F=0. Therefore:
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(2.11)
This equation has four solutions which can be written as: (2.12a) (2.12b) (2.12c) (2.12d)
These expressions determine the dependence of the resonant frequency for each mode of vibration in terms of a parameter, kn:
(2.13)
In other words, the solution of equation 2.11 can be expressed as a sum of trigonometric functions as follows:
(2.14) Once again, the boundary conditions must be here applied in order to obtain the different coefficients showed on equation 2.14. For a cantilever beam, the Di
coefficients can be expressed as:
(2.15a)
(2.15b)
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(2.15d)
These equations lead to the transcendental equation for the cantilever beam which is:
(2.16)
For a cc-beam, as the boundary conditions are not the same than those for a cantilever beam, the equations for the coefficients are:
(2.17a) (2.17b) (2.17c) (2.17d)
For these kinds of structures, then, the transcendental equation becomes:
(2.18)
The valid solutions for kn are listed in Table 2.1 for both kind of beams.
Cantilever beam Clamped-clamped beam
First mode 1.87 4.73
Second mode 4.69 7.85
Third mode 7.85 10.99
Table 2. 1. Solution parameters to the transcendental equations 2.13 and 2.15 for a rectangular cross-section cantilever and a cc-beam for the three first vibration modes.
Fig.2.4 plots the three first modes for both cantilever (a) and cc-beam (b). For the sake of simplicity the first mode is assumed to be the only relevant [4].
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Fig. 2. 4. Shape for the three first modes, , for a (a) cantilever and (b) cc-beam. For a cantilever it presents 1, 2 and 3 nodes while for a cc-beam it has 2,3 and 4 nodes.