Started on Tuesday, 9 June 2020, 2:00 PM
State Finished
Completed on Tuesday, 9 June 2020, 3:30 PM
Time taken 1 hour 29 mins
Grade 48 out of 60 (80%)
Question 1
Complete
Not graded
Please copy the following statement onto the rst page of your written material that will be uploaded:
"I do not take any unauthorised help during the exam."
I
agree
with the statement.

(Select the "agree" option, otherwise your exam will not be evaluated.)
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Question 2
Correct
Mark 1 out of 1
What is the result of the Lagrangian equation of the second kind?
Select one:
a. Second order partial di erential equation.
b. Second order ordinary di erential equation.

c. First order ordinary di erential equation.
d. First order partial di erential equation.
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Question 3
Correct
Mark 1 out of 1
At the time instant of throwing the javelin, the following can be stated about the dynamic loads of the javelin:
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Select one:
a. the normal stress has its maximum at point P of the javelin.
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b. the dynamic load causes bending moment only.
c. the normal force along the length of the javelin is constant.
d. the dynamic load causes shear force and bending moment.
Question 4
Correct
Mark 1 out of 1
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What quantities appear on the axes of the Campbell-diagram in the gure?
What quantity appears on the horizontal axis?
The angular velocity or the RPM of the rotor.
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What quantity appears on the vertical axis?
The natural angular frequencies of the rotor.
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Question 5
Correct
Mark 1 out of 1
What is Rayleigh's principle?
Select one:
a. During the vibrations of a conservative system, the kinetic energy is equal to the potential energy.
b. During the vibrations of a conservative system, the kinetic energy and the potential energy are maximal at the same time instant.
c. During the vibrations of a conservative system, the sum of the kinetic and the potential energy is constant in time.
d. During the vibrations of a conservative system, the maximum value (in time) of the kinetic energy is equal to the maximum value (in
time) of the potential energy.
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Question 6
Correct
Mark 1 out of 1
If the longitudinally vibrating beam is subjected to a linear impulse excitation, then the wave front travels
Select one:
a. with the speed of the impact exerted on the beam.
b. always slower than the speed of sound in the beam.
c. with the speed of sound in the beam.
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d. with the half of the speed of sound in the beam.
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Question 7
Correct
Mark 1 out of 1
In the Newtonian rigid body impact model, we assume that
Select one or more:

nite. 
a. the impact force during the collision is in nitely large.
b. the linear impulse generated by the collision is
c. the linear impulse generated by the collision is in nite.
d. the collision time is in nitely short.

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Question 8
Correct
Mark 1 out of 1
The motion of an old-type tramcar is investigated. The tramcar has rigid wheel-sets. The impact model shown in the gure is used to analyse
the change of the state of the motion due to the sudden variation of the rail’s curvature. The e ect of the adjoined tramcars and the e ect
of the contact forces on the wheels are not considered. Since the rail is xed rigidly, its mass can be assumed to be in nite in the model.
Where is the centre of impact of the tramcar? (Drag the marker below the gure to the appropriate place!)
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Talppont
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Question 9
Incorrect
Mark 0 out of 3
The balls with masses m1 and m2 su er an impact as shown in the gure. The coe cient of percussion is e
of the ball with mass m1 after the impact!
Data: m1
= 0.7
. Determine the velocity
= 3.4 kg, m 2 = 6.2 kg, c S1n = 6 m/s, c S2n = −0.2 m/s

v S1n =? m/s
Answer: 
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Question 10
Correct
Mark 4 out of 4
The dynamic load of the depicted airplane wing is investigated. In the time instant, the acceleration a of the centre of gravity of the airplane
point toward the longitudinal direction of the airplane. The angular velocity and the angular acceleration of the airplane are zeros. The
speci c mass of the wing varies by the function ρA(x) = c − dx2 .

Give the function of the acceleration a(x)
= ax (x)i + ay (x)j + az (x)k
along the length of the wing with respect to the coordinate x!
ax (x) =

0
2
−a
a
xω
−a
a
xω
2
−xω
xε
−xε
xε
−xε
xε
−xε
ay (x) =

0
2
2
−xω
az (x) =
0
−a
Give the distributed force p(x)

a
2
xω
= px (x)i + py (x)j + pz (x)k
2
−xω
corresponding to the dynamic load, which acts on the wing!
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px (x) =

0
2
−a(c − dx )
2
a(c − dx )
2
2
xω (c − dx )
2
2
−xω (c − dx )
2
xε(c − dx )
2
−xε(c − dx )
py (x) =


0
2
−a(c − dx )
2
a(c − dx )
2
2
xω (c − dx )
2
2
−xω (c − dx )
2
xε(c − dx )
2
−xε(c − dx )
pz (x) =
0
2
−a(c − dx )

2
a(c − dx )
2
2
xω (c − dx )

2
2
−xω (c − dx )
2
xε(c − dx )
2
−xε(c − dx )

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Question 11
Correct
Mark 9 out of 9
The cantilever beam shown in the gure is subjected to a dynamic load, which gives the following distributed loads:
px (x) = ρAa ,
py (x) = −ρAεx .
The reaction forces acting on the beam from the wall (at x
O x = −ρAal ,
Oy =
1
2
2
ρAl ε ,
MOz =
1
3
= 0
) are:
3
ρAl ε .
Determine the stress resultant functions for the beam and choose the appropriate shape of the stress resultant functions from the gures
below! Mh refers to the bending moment function. (Drag the gure and drop it to the appropiate location! Each gure can be used multiple
times!)
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Question 12
Partially correct
Mark 4 out of 8
The beam shown in the gure performs bending vibrations. By neglecting the mass of the beam, the rst natural angular frequency has to
be estimated.
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Data:
2
2
θ1 = 4 kgm , θ2 = 2 kgm , K = [
2
−1
−1
2
] × 10
4
Nm
Tasks:
Choose the rst estimate mode shape by which a better approximation can be given for the exact rst mode shape: A 1(0) = [1 0,5]
T

. Justify your answer by a drawing!
Determine the mass matrix of the system and give it numerically:
M = [


] kgm
2
Perform a one-step Stodola-iteration by using the estimate mode shape and give the result:
A
T
1(1)
= [1 ,
]
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Question 13
Incorrect
Mark 0 out of 4
The mass and compliance matrix of a system are the following:
2
0
0
5
M = [
] kg
,
375
125
125
375
W = [
] × 10
−6
m/N
.

Give a lower approximation of the rst natural angular frequency!
ωn1
> ? rad/s
Answer: 
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Question 14
Correct
Mark 7 out of 7
The prismatic beam shown in the gure performs bending vibrations. We would like to estimate the rst natural frequency. Let's investigate
the appromation of the rst mode shape function!

Choose the boundary conditions corresponding to the gure!
y(0, t) = 0
′
y (0, t) = 0
y
y
′′


(0, t) = 0
′′′
(0, t) = 0
y(l, t) = 0

′
y (l, t) = 0
y
y
′′
(l, t) = 0
′′′


(l, t) = 0
We would like to use the following function to approximate the rst mode shape:
5
Y (x) = 2x
4
− 5lx
3
2
+ 10l x
5
− 3l
.
Mark the boundary conditions that are satis ed by this function!
y(0, t) = 0
′
y (0, t) = 0
y
y
′′

(0, t) = 0
′′′
(0, t) = 0

y(l, t) = 0

′
y (l, t) = 0
y
y
′′
(l, t) = 0
′′′


(l, t) = 0
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Question 15
Correct
Mark 8 out of 8
A beam of length l = 1 m and bending sti ness 2000 Nm2 performs bending vibrations. The speci c mass along the length of the beam is
4
3
4 kg/m . The rst mode shape function of the beam is approximated by Y (x) = x + 7x . Give the upper estimate of the rst angular
natural frequency of the beam!
ωn1
< ? rad/s
Answer:

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Question 16
Partially correct
Mark 9 out of 10
The lateral vibration of a crane’s cable is investigated. The cable is loaded by the tension force F , which is smaller than the payload weight.
The crane trolley can freely move along the crane rail, and its mass is neglected. The bending sti ness of the cable is negligible.

Choose the boundary conditions from the list below!
y(0, t) = 0
′
y (0, t) = 0
y
′′

(0, t) = 0
y(l, t) = 0

′
y (l, t) = 0
y
′′
(l, t) = 0

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Determine the frequency equation analytically and choose the correct one from the below ones!
cos(ωn l/c) = 0

sin(ωn l/c) = 0
sin(μl) = 0
cos(μl) = 0
sinh(μl) = 0
tanh(μl) = − tan(μl)
mcωn /(AE) = cot(ωn l/c)
mcωn /(AE) = tan(ωn l/c)
Determine the formula of the jth natural angular frequency if j
c((j − 1)π + π/2)/l
→
∞
:


cjπ/l
c(j − 1)π/l
cjπ/2/l
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