solution

```TMMI09 2013-08-23
TMMI09 2013-08-23.01
(Del I, teori; 1 p.)
1. En konsolbalk med en punktmassa
p&aring;verkas av en st&ouml;rkraft
.
Den resulterande station&auml;rsv&auml;ngningens
amplitud kan skrivas
, d&auml;r
&auml;r balkens utb&ouml;jning vid statisk
belastning med massan . Rita i figuren till
h&ouml;ger in principutseendet av
.
1. A cantilever beam carrying a discrete mass
is loaded by an external force
. The amplitude of the resulting
stationary vibration can be expressed as
, where
is the
the mass . Sketch
in the figure
template to the right.
------------
SOLUTION
------------------------------
Se figuren!
TMMI09 2013-08-23.02
(Del I, teori; 1 p.)
2. Figuren visar en massl&ouml;s balk med tv&aring; punktmassor. Balken har i fri sv&auml;ngning tv&aring;
sv&auml;ngningsmoder, svarande mot grundton och en &ouml;verton. Rita dessa b&aring;da sv&auml;ngningsmoder. Ange
vilken av de uppritade moderna som svarar mot grund- resp. &ouml;verton.
2. The figure shows a massless beam carrying two discrete masses. In free vibration, the beam has
two eigenmodes, corresponding to principal and overtone.. Draw these two eigenmodes. Mark which
of them is principal and overtone, respectively.
TMMI09 2013-08-23
------------
SOLUTION
-----------------------------Principal
Overtone
TMMI09 2013-08-23.03
(Del I, teori; 1 p.)
3. Detaljer som ska dimensioneras mot utmattning inneh&aring;ller ofta lokala sp&auml;nningskoncentrationer,
till exempel p.g.a. h&aring;lk&auml;lar, area&ouml;verg&aring;ngar och liknande. Den maximala sp&auml;nningen i sj&auml;lva k&auml;len &auml;r
, d&auml;r i extrema fall kan vara
. I utmattningsdimensionering beh&ouml;ver man emellertid
s&auml;llan r&auml;kna med hela utan kan anv&auml;nda ett reducerat v&auml;rde
F&ouml;rklara varf&ouml;r!
3. Components to be designed against fatigue often contains local stress concentrations, for instance
caused by notches, fillets, ... The maximal stress in the notch is then
, where in extreme
cases can be &gt; . In fatigue design, however, one seldom needs to take this whole
into account but
can use a reduced value
Explain why!
------------
SOLUTION
------------------------------
- The fatigue failure will be the result of a crack starting at a defect in the material, and
- the larger the highly stressed volume is, the more likely is it that there exists an extremely bad
defect.
The fatigue data are always measured on a test specimen which is designed to give the same high
stress over a rather large volume of material. The fatigue risk in an actual component with the same
maximimm stress but over only the small volume surrounding the notch is therefore less, and the use
of a reduced value
TMMI09 2013-08-23
TMMI09 2013-08-23.04
(Del I, teori; 1 p.)
4. En komponent i en maskin uts&auml;tts f&ouml;r en upprepad typisk lastsekvens enligt fig. 4a. F&ouml;r materialet
g&auml;ller W&ouml;hlerdiagrammet i fig. 4b. Anv&auml;nd Palmgen-Miners linj&auml;ra delskadeteori f&ouml;r att ber&auml;kna hur
m&aring;nga s&aring;dana lastsekvenser komponenten kan f&ouml;rv&auml;ntas t&aring;la.
4. A component of a machine is repeatedly loaded by the typical load sequence of Fig. 4a. The
W&ouml;hler diagram of the material is shown in Fig. 4b. Use the linear Palmgren-Miner theory to compute
how many such load sequences the component can be expected to stand.
Fig. 4a
Fig. 4b
------------
SOLUTION
1st cycle (
------------------------------
corresponds to
The 100 cycles following (
The load sequence therefore gives a damage
therefore stand
correspond to
, and the component can
TMMI09 2013-08-23
TMHL09 2013-08-23.05
(Del II, problem; 3 p.)
5. En massl&ouml;s, fritt upplagd balk b&auml;r p&aring;
mitten en punktformig massa . P&aring; masssan
angriper en st&ouml;rkraft
, och
dessutom ansluter d&auml;r en linj&auml;r d&auml;mpare med
d&auml;mpkonstant . Data:
, och
, d&auml;r
&auml;r egenvinkelfrekvensen
. Best&auml;m sv&auml;ngningsamplituden i station&auml;r sv&auml;ngning.
5. A massless beam has moment-free supports in both ends and carries a discrete mass at its midpoint.
The mass is acted on by a periodic force
, and in addition there is also a linear
dashpot damper (damping constant ). Data:
, and
, where
is the
eigenfrequency
. Compute the amplitude of the vibration of the mass in the stationary
state.
------------
SOLUTION
-------------- ---------------I Equation of motion
See free-body diagram to the left!
II Beam stiffness equation
Formula table cases directly give
III Vibration equation
IV Solution
We seek the stationary solution and make the following ansatz
Insertion into Eq. (4):
TMMI09 2013-08-23
Eq. (6) gives separate equations for the sine and the cosine terms:
We can gain compactness by inserting the known expressions
gives
och
. This
with the solution
What remains now is to compute the amplitude. Since we have a
bo in nd co in ,
’ o l’ amplitude becomes
TMHL09 2013-08-23.06
ansatz [Eq. (5)] which contains
(Del II, problem; 3 p.)
6. En flygplansvinge b&auml;r upp 2 likadana motorer, som (f&ouml;r en &ouml;verslagsanalys) kan anses
vara punktformiga massor . Vingens styvhet
varierar utefter l&auml;ngden, men vi kan (f&ouml;r &ouml;verslagsanalysen) anta att den &auml;r styckevis konstant enligt figuren, d&auml;r &auml;ven l&auml;ngdm&aring;tten &auml;r
angivna.
Ber&auml;kna vingens egenvinkelfrekvenser.
6. The wing of an aicraft carries 2 equal engines, which (for a first analysis) can be considered to be
discrete masses . The stiffness of the wimg varies along its length, but we can (again, for this first
analysis) assume that it is piecewisely constant with the measures given in the figure.
Compute the eigenfrequencies of the wing.
TMMI09 2013-08-23
------------
SOLUTION
------------------------------
I. Free-body diagram
L
(Note that this solution stems from an examination paper in a course, where the mass displacements
were generally called instead of and the internal forces instead of )
II. Equations of motion
III. Stiffness equations (relations
Study the beam solutions in the formula table:

)
TMMI09 2013-08-23
and
IV. Vibration equation
V. Eigenfrequencies
TMMI09 2013-08-23
TMHL09 2013-08-23.07
(Del II, problem; 3 p.)
7. En cirkul&auml;r st&aring;ng belastas med en tidvarierande axiallast
. M&aring;tt och &ouml;vriga data
framg&aring;r av fig. 7 samt tabell. G&ouml;r en utmattningsanalys och ange om komponenten &auml;r O.K. ur
utmattningssynpunkt.
. Data are given in Fig. 7 and
the table. Perform a fatigue analysis and determine whether or not the component is O.K. from a
fatigue point of view.
= 50103 N
Material
= &plusmn; 250 MPa
D = 40 mm
200 &plusmn; 200 MPa
d = 20 mm
= 700 MPa
= 2 mm
= 300 MPa
------------
SOLUTION
------------------------------
Haigh diagram reductions
1st comment: Since
, w know
w will k p ric ly on
‘y xi ’ of
and we can do all necessary computations without formally drawing it.
Polished surface, no volume influence (other than that intrinsic in
Stress analysis
r
conc n r ion f c or , circ l r b r wi
fill
below)
H ig di gr m,
no reductions needed.
TMMI09 2013-08-23
We therefore have
O.K:?
(already without taking any safety factor into account)
TMHL09 2013-08-23.08
not O.K.
(Del II, problem; 3 p.)
8. En konstruktionsdetalj, som kan f&ouml;renklas till en plattstav med rektangul&auml;rt tv&auml;rsnitt, har en
area&ouml;verg&aring;ng med h&aring;lk&auml;lsgeometri enligt fig. 1. Belastningen &auml;r en tidvarierande j&auml;mnf&ouml;rdelad
sp&auml;nning
, som &auml;r s&aring; h&ouml;g att det finns risk f&ouml;r LCF. Komponenten ska dimensioneras f&ouml;r en LCF-livsl&auml;ngd om 3000 cykler. Best&auml;m den sp&auml;nningsamplitud som d&aring; kan till&aring;tas!
G&ouml;r p&aring; f&ouml;ljande s&auml;tt:
1. An &auml;nd Morrow’ ekvation f&ouml;r att ur kr&auml;vd LCF-livsl&auml;ngd best&auml;mma till&aring;ten lokal
t&ouml;jningsamplitud .
2. Anv&auml;nd materialets Ramberg-Osgood-kurva (fig. 2) f&ouml;r att best&auml;mma motsvarande lokala
sp&auml;nningsamplitud .
3. Anv&auml;nd Neubers ekvation f&ouml;r att med k&auml;nnedom om
hos den utifr&aring;n p&aring;lagda sp&auml;nningen.
och
ber&auml;kna till&aring;ten amplitud
8. A component, which for simplicity can be treated as a flat specimen with a rectangular cross
section, has an area reduction with notch geometry; see Fig. 1. The load is a time-varying, evenly
distributed stress
, which is high enough that there is a risk of LCF. The component
is to be designed for an LCF life of 3000 cycles. Compute the stress amplitude that can be allowed!
Instructions:
1. Knowing the required LCF life, use the Morrow equation to find the allowable local strain
amplitude .
2. Use the Ramberg-Osgood curve diagram (Fig. 2) to find the corresponding local stress amplitude .
3. Use the Neuber equation with the and
tude of the externally applied stress.
known from above to compute the allowed ampli-
TMMI09 2013-08-23
Fig. 1
Morrow data:
Fig. 2
-----------Morrow’s equation
SOLUTION
------------------------------
Ramberg-Osgood
Reading in the diagram of Fig. 2 gives
(Accurate numerical solution would have given
MPa)
Neuber
We need
Accurate reading in the table on top left of p. 64 gives
Since we have no
information on the tensile strength of the material, we cannot compute any notch sensitivity factor