rod24

SELF	24
S.B. Karavashkin and O.N. Karavashkina

4. The problem of a homogeneous elastic finite-cross-section rod
Basing on the results for a 1D semi-finite distributed elastic line studied above, we can model the processes occurring in a homogeneous elastic finite-cross-section rod. Note that the problem becomes 3D, because the cross-section is finite, so we have to transit correctly from 2D to 3D. Simplifying, consider a semi-finite homogeneous round-cross-section rod on whose free end there affects a strictly longitudinal harmonic force uniformly distributed over the rod butt. The same as in the previous problem, in the entire studied range of loads and frequencies we will suppose that the rod stiffness obeys the Hook law; so we will limit the investigation by the linear model. We will study this problem similarly to a problem of infinitely thin thread, finding the limit passings from a distributed line to a finite-cross-section elastic rod model. Note that it would be incorrect to use immediately the model constructed on a simple assemblage of an infinite set of 1D lines, since in a 1D approach we could not take into account the transversal deformation of elementary regions that is known to take place in real rods. The model of elastically connected thin disks having the diameter of a rod will be more correct in this case (see Fig. 2).
Fig.2. The model of elastic line by infinitely thin disks having finite diameter
As we consider a linear model of a rod and the load on its butt is strictly longitudinal and distributed uniformly over the butt, the introduced change does not effect on the solution (1), but at the limit passing we can now introduce a space density _v instead the linear density that we used before. Noting that the cross-section is round, conveniently do it so:
	(26)
where s is the rod cross-section at the studied point, and is the momentary radius of the rod. With it in the state of rest
	(27)
At the same time we have to transit from the elastic line stiffness T to the elasticity modulus (Young modulus) E [6],
	(28)
Given (27) and (28), (15) transforms so:
	(29)
We see from the obtained solution that the vibration amplitude in a finite-cross-section rod is reciprocal to its cross-section, density, Young modulus and force frequency. This is natural on one hand, however it is important that the vibrations lag the external force by / 2 ; just the exact analytical solutions show it.