Machine Dynamics Problems

Discontinuous Trajectory of the Mass Particle Moving on a String or a Beam 69
OJk =-,-,
klru
j7ru
OJ· =--
2.2. The Lagrange equation
) "
2m
a=-,
(8)
Let us consider a string of the length I, cross-sectional area A, mass density p,
tensile force N, subjected to a mass m accompanied by a force P (Fig. 1), moving
with a constant speed v. The motion equation of the string under moving inertial
load with a constant speed v has a form
N a2u(x,t) .A a2 u(x,t) S:( )P S:( ) a 2 u(vt,t)
- + P£1 =u x-vt -u x-vt m---'-___:'_":'"
ax 2 at 2 at 2 (9)
We impose boundary conditions
u(O,t)=O, u(l,t)=0 (10)
and initial conditions
u(x,O) = 0', 8u(X,t)1 = °
(11)
at (;0
The kinetic energy of a string and a travelling mass is described by the equation
(12)
where
E =_!_m[8u(vt,t)]2
km 2 at
(13)
contributes the kinetic energy of the moving mass. The potential energy of the
string can be determined by computing of the ox to os change of its infinitesimal
segment. The work N(& - &) integrated III space allow us to compute the
potential energy of the string
'{
Ep = , IN(&-&)=NJ
o 0
We apply the expansion of (14) into the Maclaurin series and we consider only the
first term of it
(14)