a review - Acta Technica Corviniensis

The motion of the spring pendulum is described by
the following second order differential equation
system. They can be obtained by the aid of second
order Lagrangian-equation.
2
mr&
ϕ −s(
r −ro ) + mgcosϕ
& r =
,
m
M −mgrsinϕ
−2mrr
& & ϕ
&& ϕ =
2
mr
After two-time numerical integration of motion
equations the following kinematical functions can be
obtained.
50
40
30
20
10
-20
-30
-40
r"(t), m/s2
0
-10 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1
150
100
50
φ"(t), rad/s2
0,35
0,30
0,25
0,20
0,15
0,10
0,05
ACTA TECHNICA CORVINIENSIS – Bulletin of Engineering
r(t), m
0,00
0 0,2 0,4 0,6 0,8 1
1,5
1,0
0,5
-1,0
-1,5
-2,0
φ(t), rad
0,0
0
-0,5
0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1
Figure 2. The kinematical functions for Example 1:
Spring pendulum
EXAMPLE 2: LINK MECHANISM WITH
RECIPROCATING MOTION MASS (DOF: 2)
In Fig. 2 a sketch of a simple mechanical system (two
degrees of freedom) can be seen. There is a mass and
a link mechanism in between a spring.
0
0
-50
0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1
-100
-150
r'(t), m/s
2,0
1,5
1,0
0,5
0,0
-0,5 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1
-1,0
-1,5
-2,0
φ'(t), rad/s
15
10
5
0
0,0
-5
0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0
-10
-15
110
Figure 3. Sketch of link mechanism
Data: M = 2 Nm, J = 4 kgm 2 , r = 0.15 m, s = 1000 N/m,
m = 20 kg, x& o
= 0m/
s, x o
= 0m, ϕ& =3rad
s ,
o
/
ϕ
o
= 0rad, (time step: 0.005 s, time interval: 0 ≤ t ≤ 3
s). The motion of the link mechanism can be
described by the following second order differential
equation system.
srcosϕ
−kx&
−sx
&& r =
,
m
M −sr( x−rcosϕ
)
&
ϕ =
J
After two-time numerical integration of motion
equations the kinematical functions will be the
followings.
2013. Fascicule 2 [April–June]