Machine Dynamics Problems

70 B. Dyniewicz, Cl. Bajer
If we neglect next terms of the series, we assume higher powers of au(x,t)/fJx to be
nearly equal to zero. The equation (15) can be applied to the problem of small
displacements of the string only. Finally the potential energy of the system, i.e. the
string and the moving constant force P gains a form
E p =~NI[
au~:,t)
rdx - PU(UI,t) (16)
The examined string has a finite length. It is convenient to use standing waves for
description of its displacements. We assume the solution in the following form:
co
utx, I) = LUi(x)~i(t)
i=1
(17)
~i (I) are the generalized coordinate functions. In order to compute both the kinetic
and the potential energy and to determine its derivatives required, we express them
by generalized coordinates. We derive first the equation (17) with respect to 1
(18)
and with respect to spatial variable x
The displacement
by the equation.
au(x, t)
ax
co
LU:(x)~i(t)
i=1
(19)
of the string in the contact point with a travelling mass is given
co
U(UI,/) = LUi
i=1
(UI)~i (I)
The transverse velocity of the moving mass is expressed by a composite derivative.
It expresses the load travelling along the string
(20)
Ou\~:,t)= UtVli(X)~i(t)\x=Lt +tv
i(X)~i(t)\x=1A (2\)
A.ccording to the above equation th.e ve\ocit'y ou(vt,t)/ot is expressed as a function
of both generalized coordinates and the derivative of generalized coordinates with
respect to time