Machine Dynamics Problems

Discontinuous Trajectory of the Mass Particle Moving on a String or a Beam 75
We can observe the same properties of the solution in the case of inertial string.
Comparative plot is presented in Fig. 2. We can emphasise that in the case of lower
m/pAI ratio the coincidence of each pair of curves is higher. However, analytical
proof of discontinuity in the case of inertial string is impossible to be obtained,
because of the numerical integration stage.
2.4. The beam under the moving mass
The motion of the Bernoulli-Euler
equation
beam under the moving mass is described by the
El a4u(x,t) .,1 a 2 u(x,t) _ 5:( )P 5:( ) a 2 u(vt,t)
-~.:........:... + p/1. - U X - vt - u X - vt m -~__:__;_
&4 &2 &2
(45)
with boundary conditions
u(O,t) =0, u(/,t) = 0, =0, =0 (46)
and initial conditions
x=o x=1
u(x,O) = 0, au(x,t)1 =0
(47)
at /=0
The Fourier transform method carried on in a way as in the case of the string
results in he following equation
where
00 00
V(j, t) + a "LV(k, t) sin {i)kt sin (i)/ + 2a"L (i)kV(k, t) cos (i)kt sin (i)/ +
k=1 k=1
+ Q 2 V(j,t) - f.{i)iV(k, t)sin {i)kt sin (i)/ = ~sin (i)/ (48)
k=1 pA
k,W
ca, =-1-'
j1W
(i). =--
} I'
2m
a=--
pAl
(49)
The equation (48) can not be easily solved and we must integrate it in a numerical
way. We use the matrix notation here
V(l,t) Vel, t)
[ V(l,t)
M V(2,t) +C V(2,t) +K V(2,t) . =p (50)
V(n,/) V(n, I) V(n, t)