Cantilever-beam formulas

The force and moment ia a constrained system can be estimated by the cantilever formula. Leg MB is a cantilever subject to a displacement of and leg CB subject to a displacement Av. Taking leg CB, for example, the task has become the problem of a cantilever beam with length E and displacement of Av. This problem caimot be readily solved, because the end condition at is an unknown quantity. However, it can be conservatively solved by assuming there is no rotation at poiat B. This is equivalent to putting a guide at poiat B, and results ia higher estimate ia force, moment, and stress. The approach is called guided-cantilever method. 

Thus, if 5 = 1mm, / = 15.8 Hz. This very simple result is quite useful for approximately evaluating the gravity driven deflections of a stmeture given its natural frequency, or visa versa. Of course this was derived for a specific and very simple system, so it does not perfectly apply to more complex systems. Still it is a very useful rule of thumb. For a mass on the end of a cantilever beam, the above formula is correct. The lowest natural frequency of a massive cantilever beam is about 1.2 x the prediction of the above formula. 

Microcantilever deflection changes as a function of adsorbate coverage when adsorption is confined to a single side of a cantilever (or when there is differential adsorption on opposite sides of the cantilever). Since we do not know the absolute value of the initial surface stress, we can only measure its variation. A relation can be derived between cantilever bending and changes in surface stress from Stoney s formula and equations that describe cantilever bending [15]. Specifically, a relation can be derived between the radius of curvature of the cantilever beam and the differential surface stress ... 

The maximum deflection Y occurs at x=L, i.e., Y =FVI2> El nd this equation could be used to determine E from the beam deflection. If one inspects Fig. 4.5, the three-point geometry can be viewed as two attached cantilever beams. Replacing Fhy FI2 and L by Ul in the cantilever beam deflection formula, the maximum deflection in three-point bending can be determined, i.e., Tp=FZ,V48 /. The resistance of a beam to bending depends on El (Eq. (4.9)), which is termed the flexural rigidity. 

Fig. 8. Supeiposition formulae for deflections, v(,v), of representative cantilever beams (subject to assumptions listed in text).

When looked at as a ordinary coil spring, the a-helix is one of the smallest and it will be interesting to see how the well known formula to estimate the spring constant (stiffness) of a coil spring can be applied. To start with, we realize that the spring constants (stiffness) of cantilevers, beam springs and coil springs are not scale free. 

Beams are often subject to different boundary conditions and forces in MEMS design. A comprehensive reference for the bending of beams in different situations is Roark s formulas for stress and strain [4]. We considered a fixed-free cantilever beam that is subjected to a point load at its free end in the spring analysis above. The fixed-free boundary condition means that one end is fixed, so that both its displacement and slope do not change under the applied force. The other end of the beam is free to both move and change its slope in response to a point load F, as shown in Figure 2.7. In that case the deflection of the beam y(j ) as a function of position x is given by... 

Regarding the measurement of the adhesive fracture energy, Gc, under impact (or, in general, dynamic) conditions, the double cantilever beam test and the tapered double cantilever beam test (O Fig. 21.15) must be adapted to account for the effect induced by the loading rate. Indeed, in statics the base for the measurement is the formula (Williams 1984)... 

Fig. 31 Mechanical actuation of a gold-coated microcantilever by molecular muscles [227]. (a) Structural formula of a palindromic, bistable [3]rotaxane with gold-binding dithiolane groups attached to the cyclophanes. (b) Reversible bending up and down of a cantilever by actuation of a monolayer ( 8 billion molecules) of the rotaxanes on its surface. The gold surface bends when the rotaxanes contract under the influence of an electrochemical oxidation that causes the cyclophanes to shuttle inward from the periphery of the molecule, (c) Electrochemical cell (Ag/AgCl, Pt, and the cantilever are the reference, counter, and working electrodes, respectively) and combined AFM device used to measure the bending by detecting a laser beam reflected off of the cantilever s surface...

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