Download - Example Lagrange
8/3/2019 Example Lagrange
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Q1 Figure Q1 shows a line of delivery container system which is connected each other with the
spring arranged in a serial orientation. By considering that the system has the mass of m1 =
m2 = m, m3 = 2m and the spring stiffness of k 1 = k 2 = k, k 3 = 2k.
(a) Calculate kinetic energy of the system shown
(b) Calculate potential energy of the system shown
(c) Derive the equation of motion of the system using Lagrange’s Equation by treating the
linear displacement of the components as generalized coordinates.
(d) Express the equation of motion obtained in matrix form.
m1 m2 m3
x3 x2 x1
k1 k2 k3
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a. Calculate kinetic energy of the system shown
b. Calculate potential energy of the system shown
c. Derive the equation of motion of the system using Lagrange’s Equation by
treating the linear displacement of the components as generalized coordinates
i. State the Lagrange’s equations
where
ii. External forces
2
33
2
22
2
11
2
1
2
1
2
1 xm xm xmT
2
3
2
2
2
12
1
2
1 xm xm xmT
2
34
2
233
2
122
2
112
1
2
1
2
1
2
1 xk x xk x xk xk V
223
2
12
2
1
2
1
2
1 x xk x xk kxV
n jQ
q
V
q
T
q
T
dt
d n
j
j j j
...,,2,1 ,
k j
k
zk
j
k
yk
j
k
xk
n
jq
zF
q
yF
q
xF Q
3
3
3
3
3
2
2
3
1
13
2
2
3
3
2
2
2
2
1
12
1
1
3
3
1
2
2
1
1
11
F x
xF
x
xF
x
xF Q
F x
xF
x
xF
x
xF Q
F x
xF
x
xF
x
xF Q
n
n
n
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iii. Differentiating Kinetic Energy, T with respect to velocity and then with respect to time
1111
1
xm xmdt
d
x
T
dt
d
2222
2
xm xmdt
d
x
T
dt
d
3333
3
xm xmdt
d
x
T
dt
d
iv. Differentiating Kinetic Energy with respect to displacement
0321
x
T
x
T
x
T
v. Differentiating Potential Energy, V with respect to displacement
2212112211
1
xk xk k x xk xk x
V
(use chain rule of calculus)
3323212233122
2
xk xk k xk x xk x xk x
V
3323233
3
xk xk x xk x
V
2
33
2
22
2
112
1
2
1
2
1 xm xm xmT
2233
2
122
2
11
2
1
2
1
2
1 x xk x xk xk V
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vi. Substituting all the derivatives into Lagrange’s equations to obtain
12212111
F xk xk k xm
2332321222
F xk xk k xk xm
3332333
F xk xk xm
d. Express the equation of motion obtained in matrix form
3
2
1
3
2
1
33
3322
221
3
2
1
3
2
1
0
0
00
00
00
F
F
F
x
x
x
k k
k k k k
k k k
x
x
x
m
m
m
Since mass of m1 = m2 = m, & m3 = 2m and the spring stiffness of k 1 = k 2 = k, & k 3 = 2k
therefore ;
3
2
1
3
2
1
3
2
1
220
23
02
200
00
00
F
F
F
x
x
x
k k
k k k
k k
x
x
x
m
m
m
