Linköpings Universitet, Hållfasthetslära, IEI/IKP Tore Dahlberg ...

Tekniska Högskolan i Linköping, IEI/IKP
Tore Dahlberg
TENTAMEN i Mekaniska svängningar och utmattning, 071016 kl 14-18
EXAMINATION in Mechanical Vibrations and Fatigue
DEL 2 - (Problemdel med hjälpmedel)
5. En massa M hängs upp i tre fjädrar (styvhet
k
k 1 = k, k 2 =2k och k 3 =3k).
1 k 2 k1
k (a) Bestäm egenvinkelfrekvensen för systemet
M
3
k2
om massan och fjädrarna monteras enligt figur
k3
M
(a).
(a)
(b) (b) Vad blir egenvinkelfrekvensen om massan
och fjädrarna monteras enligt figur (b)?
English: 5. A mass M is mounted with three springs (stiffness k 1 = k, k 2 =2k,
and och k 3 =3k).
(a) Determine the (angular) eigenfrequency of the system if the mass and the
springs are mounted as shown in figure (a).
(b) What will the eigenfrequency be if mass and springs are mounted as in
figure (b).
Lösning/Solution:
(a) The equation of motion of the mass is Mẍ =−F 1
− F 2
+ F 3
(a)
where
This gives
which gives
(b) The equation of motion of the mass now becomes Mẍ =−∑F
For the two springs in series one obtains (same force F in the two springs)
Enter this into (b). It gives
and the eigenfrequency becomes
F 1
= k 1
x F 2
= k 2
x and F 3
=−k 3
x
Mẍ +(k 1
+ k 2
+ k 3
)x = 0
ω e
= √⎯⎯⎯⎯
(k 1 + k 2 + k 3 )
= √⎯6 √⎯ k M
M = 2.45 √⎯ k M
x = x 1
+ x 2
= F k 1
+ F k 2
giving k 1k 2
k 1 + k 2
x = F
Mẍ + ⎛ ⎜ ⎝
k 1 k 2
k 1 + k 2
+ k 3
⎞ ⎟⎠ x = 0
ω e
= √⎯⎯⎯⎯⎯
⎛ k 1 k 2
⎜
⎝ (k 1 + k 2 ) + k 3
Thus, the eigenfrequency in case (b) goes down, but not that much, because
the stiffest spring k 3 will dominate the behaviour of the mass in both cases.
⎞
⎟
⎠
(b)
1
M = √⎯ 11
3 √⎯ k M = 1.91 √⎯ k M
10