2.1.3 Deflections under the longitudinal force
In this section we determine the magnitude and direction of the deformation produced by the axial force
. Solution to this problem will give the middle column (2) of tensor
(see (2) in chapter 2.1.1).
![](/data/media/images/spm_basics/scanning_force_microscopy_sfm/cantilever/deflections_under_longitudinal_force/img03.gif)
(1)
![](/data/media/images/spm_basics/scanning_force_microscopy_sfm/cantilever/deflections_under_longitudinal_force/img04.gif)
(2)
![](/data/media/images/spm_basics/scanning_force_microscopy_sfm/cantilever/deflections_under_longitudinal_force/img05.gif)
(3)
The force
acting in the cantilever axis direction produces moment
that results in deformation called here the vertical bending of y-type (Fig. 1).
Fig. 1. Vertical deflection of the y-type.
In spite of the formal resemblance to vertical bending of z-type (see chapter 2.1.2), the deformation profile in this case is quite different. The equation describing the y-type bending reads (compare with (7) in chapter 2.1.2):
![](/data/media/images/spm_basics/scanning_force_microscopy_sfm/cantilever/deflections_under_longitudinal_force/img08.gif)
(4)
Boundary conditions remain the same:
and
. For the solution we find:
![](/data/media/images/spm_basics/scanning_force_microscopy_sfm/cantilever/deflections_under_longitudinal_force/img11.gif)
(5)
Thus, the tip vertical deflection due to this type of deformation is as follows:
![](/data/media/images/spm_basics/scanning_force_microscopy_sfm/cantilever/deflections_under_longitudinal_force/img12.gif)
(6)
Comparing (6) and (3) and taking into account the expression for the common multiplier
(see (12) in chapter 2.1.2), we get:
![](/data/media/images/spm_basics/scanning_force_microscopy_sfm/cantilever/deflections_under_longitudinal_force/img14.gif)
(7)
The angle of the beam end deflection
is given by the following formula:
![](/data/media/images/spm_basics/scanning_force_microscopy_sfm/cantilever/deflections_under_longitudinal_force/img16.gif)
(8)
From formula (8) and diagram for the beam vertical bending of y-type (Fig. 1) it is easy to derive the tip deflection
induced by the force
application:
![](/data/media/images/spm_basics/scanning_force_microscopy_sfm/cantilever/deflections_under_longitudinal_force/img18.gif)
(9)
From (2), (7) and (9) it is easy to obtain:
![](/data/media/images/spm_basics/scanning_force_microscopy_sfm/cantilever/deflections_under_longitudinal_force/img19.gif)
(10)
Taking into account that
, we get:
![](/data/media/images/spm_basics/scanning_force_microscopy_sfm/cantilever/deflections_under_longitudinal_force/img21.gif)
(11)
Finally, we calculate the components of the matrix (3) from chapter 2.1.1 second column. From expressions (6–8) it follows that
![](/data/media/images/spm_basics/scanning_force_microscopy_sfm/cantilever/deflections_under_longitudinal_force/img22.gif)
(12)
Because under the influence of the force
the top cantilever surface does not bend in the
direction, then
![](/data/media/images/spm_basics/scanning_force_microscopy_sfm/cantilever/deflections_under_longitudinal_force/img24.gif)
(13)
Summary.
- The y-type deflection is a result of the axial bending force action.
- To find the components of the inverse stiffness tensor corresponding to the y-type deflection, one should solve the problem of the beam static deflection which is reduced to the ordinary differential equation of the second order.
- The axial force results in the tip deflection not only in the longitudinal but also in vertical direction
and in the deflection angle
appearance.