Development of a demonstration device for vibrations Jonathan Gengenbach Bachelor’s thesis

Development of a demonstration device for vibrations Jonathan Gengenbach Bachelor’s thesis
Development of a demonstration device for
vibrations
Jonathan Gengenbach
Bachelor’s thesis
May 2015
Degree Programme in Mechanical and Production Engineering
1
Table of contents
1
Introduction and objectives..................................................... 6
2
Theoretical part ....................................................................... 7
2.1
Introduction and motivation.................................................................. 7
2.2
Laval Rotor ............................................................................................. 7
2.2.1
Modelling ......................................................................................... 7
2.2.2
Simulation ..................................................................................... 12
2.2.3
System behaviour .......................................................................... 13
2.2.4
Supporting ..................................................................................... 15
2.2.5
Weaknesses of the model .............................................................. 16
2.3
2.3.1
Introduction................................................................................... 18
2.3.2
Natural frequencies and natural modes of a continuous shaft ..... 18
2.3.3
Rotor with two attached disks ....................................................... 19
2.4
3
System with two and more degrees of freedom ................................... 18
Torsional vibrations ............................................................................ 24
Development of the demonstration device ..............................26
3.1
Introduction ........................................................................................ 26
3.2
Features of the device ........................................................................... 27
3.3
Concept ................................................................................................. 27
3.4
Drafting ............................................................................................... 28
3.4.1
Functional groups ......................................................................... 28
3.4.2
Dimensioning rotor ...................................................................... 29
3.4.3
Supporting .................................................................................... 32
3.4.4
Shaft hub joint ............................................................................... 37
3.4.5
Housing ......................................................................................... 40
3.4.6
Safety concept ............................................................................... 42
2
3.4.7
3.5
Safety bearing ................................................................................45
Elaboration .......................................................................................... 46
3.5.1
Introduction.................................................................................. 46
3.5.2
Bearing choosing ...........................................................................47
3.5.3
Supporting design.......................................................................... 51
3.5.4
Rotor design................................................................................... 55
3.5.5
Safety bearing design .....................................................................59
3.5.6
Shaft design .................................................................................. 60
3.5.7
Housing design .............................................................................. 67
3.5.8
Safety-cap design .......................................................................... 68
3.6
Summary and prospects ...................................................................... 69
4
Summary ............................................................................... 71
5
Discussion .............................................................................. 72
6
References ............................................................................. 73
7
Attachment ............................................................................ 75
3
Figures
Figure 1: Laval Rotor with blocked translatory motion ..................................... 8
Figure 2: Laval Rotor with equivalent Springs and damping ............................ 8
Figure 3: Laval Rotor with forces ....................................................................... 8
Figure 4: Bearing's influence of the equivalent stiffness .................................. 11
Figure 5: Unbalance ........................................................................................... 11
Figure 6: Simulink model equation (2.4) .......................................................... 12
Figure 7: Simulink model equation (2.2) .......................................................... 13
Figure 8: Simulink model .................................................................................. 13
Figure 9: Simulink model constant angular speed............................................ 13
Figure 10: Transfer function Laval Rotor .......................................................... 14
Figure 11: Orbits centre of shaft and mass ........................................................ 14
Figure 12: Laval Rotor with non-rigid bearings ................................................ 15
Figure 13: Comparison stiffness ratio of shaft to supporting............................ 16
Figure 14: Beam's three lowest mode shapes .................................................... 19
Figure 15: Rotor with two disks ........................................................................ 20
Figure 16: Bending line w(F,x) .......................................................................... 21
Figure 17: Force diagram at position I and II .................................................... 21
Figure 18: Rotatory motion of the disk............................................................. 23
Figure 19: Torsional system .............................................................................. 24
Figure 20: Diagram of moments ...................................................................... 24
Figure 21: Unbalanced caused force ..................................................................25
Figure 22: arrangement with a) one rotor b) two rotor c) cantilevered rotor . 28
Figure 23: Curves of constant masses and frequency ...................................... 30
Figure 24: max. rotor displacement according unbalanced mass and damping
ratio .................................................................................................................... 31
Figure 25: Bearing arrangement ...................................................................... 34
Figure 26: Design proposals bearing shifting ...................................................35
Figure 27: Design proposals shaft hub connection .......................................... 38
Figure 28: Ringspann Trontorque Mini (Ringspann GmbH, 2014) ................ 39
Figure 29: Design proposals housing ............................................................... 40
Figure 30: Safety bearing concept .................................................................... 46
4
Figure 31: non-rigid bearing axis .......................................................................47
Figure 32: self-aligning ball bearing ................................................................. 48
Figure 33: Force diagram disk .......................................................................... 48
Figure 34: Supporting ........................................................................................ 51
Figure 35: Bearing loading ................................................................................52
Figure 36: Fittings bearing ................................................................................52
Figure 37: Stress due to shrinkage fit ................................................................ 55
Figure 38: Stress due to dynamic loading ......................................................... 55
Figure 39: Inner rotor ........................................................................................56
Figure 40: Outer rotor .......................................................................................56
Figure 41: Safety bearing .................................................................................. 60
Figure 42: ROTEX coupling (KTR Kupplungstechnik GmbH , 2015) .............. 61
Figure 43: Bending line .................................................................................... 62
Figure 44: Shaft connection .............................................................................. 63
Figure 45: FE calculation strain caused by bending .........................................65
Figure 46: FE calculation combined stress .......................................................65
Figure 47: Stress cycle........................................................................................65
Figure 48: fatigue strength diagram ..................................................................67
Figure 49: Housing ........................................................................................... 68
Figure 50: Safety cap with several plates ......................................................... 69
Figure 51: Safety cap with bended plate ........................................................... 69
Figure 52: Complete device .............................................................................. 70
5
Tables
Table 1: Rotor dimensions ................................................................................. 31
Table 2: Decision matrix supporting .................................................................33
Table 3: Decision matrix bearing adjusting ..................................................... 36
Table 4: Decision matrix shaft hub joint .......................................................... 39
Table 5: Decision matrix housing ...................................................................... 41
Table 6: Harm evaluation categories (Mattilainen, 2015) ............................... 43
Table 7: Levels of risk (Mattilainen, 2015) ....................................................... 44
Table 8: Risks of the identified hazards ........................................................... 44
Table 9: Parameter bearing calculation............................................................ 49
Table 10: Bearing load ratings .......................................................................... 50
Table 11: Properties PA6 and POM-C ................................................................53
Table 12: Parameter for fit selection .................................................................. 57
Table 13: Main properties ROTEX coupling ..................................................... 61
Table 14: Properties Loctite 638 ....................................................................... 63
Table 15: Properties S355 ................................................................................. 66
6
1 Introduction and objectives
I got the possibility to finish my Automotive Engineering studies in the double
degree program which is provided in cooperation of my home institution, the
Hochschule Esslingen University of Applied Sciences and the JAMK University of Applied Sciences. Therefore I have spent the autumn semester 2014 and
the spring semester 2015 at the JAMK in Jyväskylä, Finland. Part of the double degree program is creating the bachelor’s thesis abroad. In the end of the
autumn semester 2014, Jorma Matilainen who is principal lecturer in the
School of Technology, proposed me to make the, in this document introduced,
development project within my thesis work. At this point I want to thank
Jorma for the opportunity to do the thesis work at the JAMK and for his supervision as his role as representative of the host organisation. Furthermore I
want to thank Matti Kurki who accompanies the thesis work as the supervisor.
The objective of this thesis work is to develop a laboratory device which
demonstrates imbalance excited bending vibrations of a shaft. The device’s
purpose is to illustrate the in the lecture taught theory about critical shaft revolution speeds and to sensitise the students for the thereby arising risks.
The thesis is separated into two main parts, the theoretical part (chapter 2)
and the actual development project (chapter 3). The theoretical part deals with
the modelling and mathematically description of the oscillating rotor and
serves as fundament for the designing of the device. Chapter 3 includes all
steps which are passed while developing the demonstration device.
7
2 Theoretical part
2.1 Introduction and motivation
Bending vibrations of a shaft which are caused by an unbalanced mass distribution have a big influence of the runnability and the stability of machines. On
bearings transmitted vibrations stress the bearings and reduce their life expectancy. The vibrations also can generate noises and rattling which are disturbing the machine users and the environment. The oscillating bending of the
shaft increase the strain in the shaft which can cause an earlier destruction.
High displacements of the shaft bring the risk of a collision between the shaft
and the housing. Because of the named hazards it is necessary to consider imbalances while designing and operating a machine with a turning shaft.
In the theoretical part the effect of this vibrations are illustrated. It basically
examines two different types of vibrations, the imbalanced excited bending vibrations (2.2, 2.3) and the imbalanced excited torsional vibrations which are
studied in brief in chapter 2.4. In chapter 2.2 the bending vibrations are deduced in the simplified one degree of freedom model. Based on the in chapter
2.2 gained knowledge the chapter 2.3 introduces shortly bending vibrations in
systems with two and more degrees of freedom.
2.2 Laval Rotor
2.2.1 Modelling
The simplest model to understand and calculate the effect of an imbalanced
rotor is the so called Laval Rotor which is in English-written literature often
introduced as Jeffcott-Rotor. The Laval Rotor was first described and patented
by the Swedish engineer Gustav de Laval (Gasch, Nordmann, & Pfützner,
2006, 15). The Laval Rotor consists of a bending flexible and massless shaft
with a disk attached on its half-length (Figure 1). The most elementary bearing
model are two bearings which are infinite stiff in vertical directions to the
shaft and torque-free for bending. In the technical mechanics this bearings are
called pinned-supporting.
8
Figure 1: Laval Rotor with blocked translatory motion
Because of the symmetric structure of the system the disk is only free to shift
vertically to the shaft (y- and z- direction) as well as to rotate around the shafts
axis (x-direction) and does not tilt around the y- and z- axis. Due the simplification of the shaft as massless and the fact that in this particular system the
disk is just moving in a plane the shaft is substitutable for a spring with an
equivalent stiffness k (Figure 2). Another simplification of the system is the
negligence of the gravity so that there is no static bending caused by the gravity force. In an imbalanced rotor the centre of the disk’s mass C is not the same
then its geometrical centre M. If the disk is not displaced its geometrical centre
is on the bearings centres’ connection line. To simplify the calculations it is
functional to place the origin of co-ordinates into the connection line of the
bearings.
Figure 2: Laval Rotor with equivalent Springs and damping
Figure 3: Laval Rotor with forces
9
Figure 3 shows the force-diagram of the in y- and z-direction displaced Laval
Rotor. zm and ym are the displacement of the rotor. zc and yc are the positions of
the disk’s centre of mass. e is the constant value of the disks eccentricity. The
determination of the mathematically determination of the eccentricity e is described in the equation (2.7). � is the current rotation angle of the disk. � is for
calculation reasons determined as counter clockwise (mathematical positive)
and begins on the z-axis. The active forces in this model of the Laval Rotor are
the inert forces
∙
∙
/
, the spring forces
which are caused by the damping
∙
∙
/
∙
/
∙
and the forces
. The damping in this model is
implemented as a viscose damping which is the mathematically simplest way
to describe damping. The accurateness and weakness of this damping model
will be discussed later. According to the Newton’s Law of Motion the rotor has
to be in balance of forces (2.1).
∙
∙
+
+
∙
∙
+
∙
+
∙
=
(2.1)
=
Beside the forces the torques also have to be in balance. The moments which
act around C are the rotor’s moment of inertia � ∙ , the inertia of the rotor’s
mass multiplied by the effective eccentricity
the external driving torque .
∙ sin � ∙
∙
+
∙
∙ cos � ∙
− ∙ cos � ∙
+� ∙ −
=
/ ∙ cos � ∙
∙
+
∙
With help of the trigonometric functions and the two-times derivation
and
are expressible as a function of
/
and �.
and
(2.2)
10
=
+ ∙ sin �
=
− ∙ � ∙ sin � + ∙ � ∙ cos �
=
+ ∙ � ∙ cos �
=
+ ∙ cos �
=
− ∙ � ∙ cos � − ∙ � ∙ sin �
=
(2.3)
− ∙ � ∙ sin �
Equation (2.3) inserted in (2.1) makes a differential equations with the geo/
metrical centre (
(
/
,
terms with
/
) of the rotor and its first and second derivation
). For a better comprehension of the differential equation the
/
/
,
and
/
are brought on the left side of the equal-
ity sign and the equation is transposed so that the highest derivation (
is an own term.
+
+
∙
∙
+
+
∙
∙
= − ∙ � ∙ cos � + ∙ � ∙ sin �
=
/
(2.4)
∙ � ∙ sin � + ∙ � ∙ cos �
Equations (2.4) are inhomogeneous differential equations of second order.
The left side of the equality sign is the homogeneous part. The homogeneous
part is the differential equation of a free, damped oscillation. The undamped
natural frequency of those systems is � (2.5).
� =√
(2.5)
The differential equation’s right side is the perturbation function which describes the excitement of the system caused by the imbalance. To predict the
behaviour of a Laval Rotor the equivalent spring stiffness ( ) is needed. It depends on the geometry and material of the shaft and the kind of its bearing.
11
The equivalent spring stiffness is for small bending nearly constant and can be
approximated with equations (2.6). The influence of the kind of bearing ( ) is
listed in Figure 4.
describes the Young’s modulus, the second moment of
area and and are the radius and length of the shaft.
=
with:
=
∙
∙
∙
(2.6)
Pin-joints
=48
Fixed-joint
=192
Figure 4: Bearing's influence of the equivalent stiffness
The eccentricity ( ) can be calculated by the unbalanced mass (
position (
) and its
) according equation (2.7). Both parameters are known in the
case of the demonstration device, in which the imbalance is constructed.
Figure 5: Unbalance
12
=
∙
(2.7)
Due the appearance of � and its second derivation in non-linear terms like
sine, cosine and quadratic functions in the equation (2.4) and (2.2) the solutions of the differential equations are not trivial. Therefore the system is solved
in a numerical way, which is described in chapter 2.2.2.
2.2.2 Simulation
To solve the differential equations (2.2) and (2.4) the system is built in
Matlab/Simulink. Figure 6 shows the implementation of the equation (2.4). It
has the angle � and its derivations as input signals. The eccentricity ( ), the
damping coefficient ( ), the spring stiffness ( ) and the mass are constant values which are calculated outside the simulation. The model’s structure of the
part of equation (2.4) which contains
is similar to the described one.
Figure 6: Simulink model equation (2.4)
The torsional moment’s Simulink model is shown in Figure 7. There is the
torque ( ) the external control variable. The eccentricity ( ) and the inertia ( )
are constant values.
13
Figure 7: Simulink model equation (2.2)
Figure 8: Simulink model
Figure 8 contains the Simulink model which executes the correlations between
equations (2.4) and (2.2). To analyse the system’s behaviour by constant rotations speeds and to deduce the transfer function it is viable to replace the moment’s equation (2.2) with a constant angular speed � (Figure 9). The mo-
ment’s equation calculates out of the system’s state and the external torque the
momentary angle (�) respectively the angular speed (�) and its acceleration
(�).
Figure 9: Simulink model constant angular speed
2.2.3 System behaviour
The transfer function (Figure 10) illustrates the vectorial sum of the rotor’s
displacement (√
+
) plotted against the rotation speed by several
damping ratios. The revolution speed (�) is normalised to the system’s natural
frequency (� ) and the displacement to the eccentricity ( ). The transfer function shows that there is almost no displacement low angular speeds. Next to
the natural frequency the displacement ascends depending, on the damping
14
ratio to very high values. An undamped system reaches at its natural frequency
infinite deflexion. Because of the hazard of shaft fractures this angular speed is
called the critical speed. At higher angular speed the displacement stabilised to
the value of the eccentricity.
Figure 10: Transfer function Laval Rotor
The observation of the paths which the centre of the mass respectively the geometrical centre of the shaft describe are shown in Figure 11. This paths are
called also called orbits. At lower rotation speed the radius of the mass centre
orbit is bigger than the geometrical centre ones. By increasing the rotation
speed this ratio turns. At higher speeds, the centre of mass approaches to a
stable non-moving state and the centre of the shaft describes an orbit with the
radius of the eccentricity.
Figure 11: Orbits centre of shaft and mass
15
2.2.4 Supporting
The kind of the bearing has basically two factors of influence on the bending
vibration of the shaft. As opposed to the assumption made in the first model of
the Laval Rotor the bearings are not totally rigid in the radial direction and
also not totally non-rigid for bending. The stiffness of the supporting in the radial directions (y/z) can be seen as springs in a row to the equivalent springs of
the shaft (Figure 12).
Figure 12: Laval Rotor with non-rigid bearings
For an undamped system the result stiffness can be calculated with
equation (2.8). It is assumed that the supporting’s stiffness is different in the
y- and z- direction but symmetrically on both sides. This assumption reproduce very well the reality like it is in the demonstration device, where the same
bearing supporting is used on both sides.
,
,
=
=
ℎ
ℎ
ℎ
ℎ
∙
+
∙
+
,
,
,
(2.8)
,
Due the different stiffness in the z- and y- direction the orbits of the shaft’s geometrically centre and its centre of mass are an elliptical instead of circular.
Additionally to the position of the centre of mass and the geometrically centre
of the shaft it is interesting to examine the displacement of the bearing. According the Hooke’s law the deflexion is inversely proportional to the spring
16
stiffness. Referred to that the share of the displacement in the shaft and the
supporting depends on the ratio of the stiffness between those. Figure 13
shows exemplarily the bending of the shaft and the deflexion of its supporting.
In both cases are the totally stiffness the same so that the totally displacement
of the rotor is also the same. In the upper case the equivalent stiffness of the
shaft is much lower than the stiffness of the bearing. In the lower case both are
similarly stiff.
Figure 13: Comparison stiffness ratio of shaft to supporting
Since the total system stiffness is lower with a less stiff supporting, the natural
frequency decreases (2.5). The influence of the stiffness in bending directions
is more difficult to determine. It is contained in the constant number ( ) in
equation (2.6). The totally rigid bending support (Fixed-joints) leads to a fourtime higher equivalent spring than the totally non-rigid bending support
(Pinned-joints). Depending on the actual bearings the stiffness is somewhere
between. Self-aligning ball bearing for instance are nearly non-rigid in bending direction as against plain bearings are rather rigid.
2.2.5 Weaknesses of the model
The described model of the Laval-Rotor is just an approximation to an actual
system. Depending on the actual system characteristics, the model has different weaknesses with different impact to the system’s accurateness. In this
17
chapter are the negligence of the gravity, the influence of the damping, the
negligence of the shaft’s mass and the determination of the shaft’s stiffness
discussed.
The negligence of the gravity is just in case of a completely balanced shaft exact. In this case the bending vibrations are added to the static, of the gravitation force caused bending of the shaft. Imbalanced, horizontal mounted shafts
have according to Stodola (1918, 1-3) an additional critical speed. Is the rotor
central on the shaft this critical speed is the half of the shafts first critical
speed. The derivation of this new critical speed is with following concept possible (Stodola, 1918, 1-3). The view on the system should be from a with the
shaft speed (�) rotating co-ordinate system. Due the elastic vibration the
shafts centre follows a circle with the velocity of the critical speed (� ). The
motion seen by the rotating system has the relative speed � − �. The gravity
impacts the centre of mass with the rotating force
∙
∙ cos �
∙
∙ sin �
respectively
which generate vibrations with the angular frequency (�). The
gravity caused vibration and the natural vibration are in resonance if � − � =
�. In this case is � = � / and the amplitudes of both vibrations can be
added.
Another aspect which brings inexactness into the model is the assumption that
the damping is a totally viscose. This assumption simplifies the mathematical
model but reflects the reality not completely accurate. Furthermore even if the
system is precisely described with a viscose damping the determination of the
damping ratio is not easy. The most practical procedure to determinate the
damping ratio is to measure the decay time of the actual system which is obviously not feasible when the system is still in development.
The most-reaching simplification of the system is probably the consideration
of the shaft as massless. The accurateness of this simplification depends on the
ratio between the rotor mass and the shaft mass. A continuous shafts has its
own vibration behaviour. The determination of the influence of this vibrations
should not be done here because it requires a highly more complex model than
18
the model of the Laval-Rotor is. For more exactly outcomes with consideration
of the shafts mass and geometry it is recommendable to calculate the system
with the finite element method.
As earlier discussed the kind of the supporting has a big influence of the stiffness of the shaft. And even if the correct equation for the equivalent stiffness is
found it is valid for small bending magnitudes. For higher bending magnitudes
the shaft gets stiffer. Moreover the model assumes a linear-elastic material behaviour which is just valid in certain limits.
2.3 System with two and more degrees of freedom
2.3.1 Introduction
Each multi-degree of freedom system with n degree of freedoms has n natural
frequencies with the belonging n natural modes (Rao, 2010, 461). Due the continuous distribution of mass and rigidity on the shaft, the in the previous chapter introduced Laval-Rotor actually has an infinite number of natural frequencies. Basically this applies for all actual systems, but the influence of the higher
natural frequencies is different in different systems. In this chapter a system
similar to the Laval-Rotor but with two disks as well as a shaft without attached disks are discussed. Due the disks’ slim shape orthogonal to the shaftaxis, they have a high mass at a marginal length on the shaft. The shaft is contrary to the disks relative lightweight by its comparatively high length. Therefore the disk mass’ influence on the rotor’s vibration behaviour is much higher
than the shaft mass’. This is why systems with attached disks are more accurately describable with a lower number degrees of freedom then it is the case
of a rotor without attached disks.
2.3.2 Natural frequencies and natural modes of a continuous shaft
According to Rao (2010, 527) the shape of a continuous beam’s respectively a
shaft’s nth natural mode is described with the formula (2.9). is the length of
the beam and
the particular maximal deflexion. The determination of
which is depending on the beam’s material, dimensions and damping is not
19
described in this thesis. The natural frequencies of continuous beams which
are pinned-supported are computed in equation (2.10) (Rao, 2010, 526). Figure 14 shows the three lowest mode shapes according formula (2.9).
=
� =
∙ �
∙
∙
√
∙
�
(2.9)
(2.10)
Figure 14: Beam's three lowest mode shapes
2.3.3 Rotor with two attached disks
Basically the in the previous chapter 2.2 about the Laval- Rotor got findings
are also valid for the rotor with two masses (Figure 15). The mechanism of unbalanced gained vibrations are the same in the system with two rotors. So it is
also dangerous to run the rotor at rotation speeds which are nearby its natural
frequencies. Compared to the Laval- Rotor, a Rotor with two disks has two
natural frequencies which have to be considered. Both natural frequencies are
important for the dimensioning of the demonstration device.
20
Figure 15: Rotor with two disks
To identify the natural frequencies the equation of motion has to be determined (2.11). The introduced system is undamped. Equation (2.11) is a differential equation written in matrix form with
as mass matrix and
as stiff-
ness matrix. Solving equation (2.12) gives the natural frequencies.
̅∙
+ ̅∙
(2.11)
=
det[ ̅ − � ̅ ] =
With:
̅=[
], ̅ = [
(2.12)
]
−
are so called factors of influence and are measuring units of the system’s
stiffness relating to the disk’s positions. Technically they are the shaft’s deflexion on position w caused by a force (F =
N) applied on position v. To deter-
mine the factors of influence, the bending line
(Figure 16) caused by the
force F is calculated. The correlation between the bending line and
=
and
rotor, the equations
=
=
=
and
is
. Since the symmetrical structure of the
=
are valid.
=
21
Figure 16: Bending line w(F,x)
The approach to calculate the bending is the Bernoulli beam theory (2.13).
∙ ∙
To determine
=−
(2.13)
the beam is cut twice (Figure 17). At their cross section the
equations of the moments (2.14) is formulated.
Figure 17: Force diagram at position I and II
�
��
� =
=
=
∙
(2.14)
−
∙
22
The moment’s equations (2.14) inserted into the Bernoulli beam theory (2.13)
and two times integrated gives the equation of the bending line (2.15) with the
integration constants
,
,
and
=−
∙ ∙
∙
=− ∙
∙ ∙
=− ∙
∙ ∙
∙ ∙
=
∙ ∙
=
∙ ∙
.
=
∙
∙
∙
∙
−
∙
−
∙
+
∙
∙ ∙
− ∙ ∙
+
∙
∙
+
∙
+
+
(2.15)
∙
+
There are four boundary conditions needed to determine the four integrations
constants. Boundary conditions
and
result from the supporting at both
ends of the shaft which do not allow displacements. Boundary conditions
and
� are formed out of the fact that
= /
and
=
are the same
point and therefore their displacement and their sloping are the same.
�
= /
= /
= /
=
=
=
=
(2.16)
=
=
The four boundary conditions are an equations system with four unknowns.
Its solution is not shown here. The determined integrations constants inserted
in equation (2.15) gives the equation of the bending line (2.17).
23
=− ∙
∙ ∙
=
∙ ∙
∙
∙
−
+
∙
∙
∙
+
∙
(2.17)
∙
+
Out of the bending line’s equation (2.17) the factors of influence
and
,
,
are determined with the above described conditions.
=
=
=
=
(
(
=
=
,
,
=
=
)=
)=
∙
∙
∙
∙
∙
(2.18)
∙
The in chapter 2.2.5 described weaknesses of the Laval-Rotor which are the
negligence of the gravity, the inadequacy of the damping model, the negligence
of the shafts’ mass and the inexactness in the determination of the system’s
stiffness are also valid for the rotor with two attached disks. Beside those there
is a further influence factor. Due the bending of the shaft the rotors do not
only move translational and rotatory around the x-axis but also in small angles
rotatory around the z-axis (Figure 18).
Figure 18: Rotatory motion of the disk
According the law of conservation of angular momentum or colloquially called
the gyro effect the rotating disk pursues because of its inertia to stay in its actual state. For the described system, that means that depending on the inertia
around the shaft axis and the rotating speed there arises a moment which is
opposite directed to the slope of the shaft (�). This moment acts against the
24
shafts deflexion and stabilize the system against the vibration. A numeral determination of this moment is not part of this thesis but should be considered
by fast running systems with heavy hubs.
2.4 Torsional vibrations
Beside the in the previous chapters described motions and vibrations in translational directions a shaft also has a rotatory behaviour which should not be
neglected. This chapter deals with the determination of the torsional vibrations of a shaft with two rotors.
Figure 19: Torsional system
Figure 20: Diagram of moments
The system shown in Figure 19 is a shaft with two attached masses and a neglected inertia. Since both disks have generally different rotation angles, they
are cut free (Figure 20). On both disks act their respective moment of inertia
as well as the moment emerging by their torsion against each other. The oscillating differential equations (2.19) are derived out of the principle of the moment equilibrium.
� +
� +
∗
∗
� −�
� −�
=
=
(2.19)
The oscillating differential equations (2.19) written in matrix-form makes the
equation (2.20). The natural frequencies are determined by solving the equation (2.21).
25
With:
�
̅ ∙ [� ] + ̅ ∙ [ ] = [ ]
�
�
̅=[
]; ̅ = [
−
∗
∗
−
∗
∗
(2.20)
]
det[ ̅ − � ̅ ] =
(2.21)
The torsional rotating system responses analogue to the in chapter 2.2.3 described behaviour of the Laval-Rotor on external oscillating excitations. If the
excitation frequency is nearby a natural frequency of the rotating system, its
torsional vibrations gets high amplitudes. A kind of excitement are like in the
case of the Laval-Rotor, imbalanced caused forces.
Figure 21: Unbalanced caused force
Figure 21 shows the, by the off-centred mass caused force ( ∙
). This force
causes with the on the rotors angular position (�) depending lever arm a mo-
ment ( ∙
∙
∙ cos �). Since generally both rotors are imbalanced, the oscil-
lating differential equations (2.19) changes to the non-linear differential equation (2.22).
With:
�
̅ ∙ [� ] + ̅ ∙ [ ] = [
�
�
̅ =[
]; ̅ = [
−
cos �
cos � ]
∗
∗
−
∗
∗
]
(2.22)
26
3 Development of the demonstration device
3.1 Introduction
Based on the in the previous chapters generated knowledge about bending vibrations a demonstration device is developed. The demonstration device
should be used in the context of lectures and laboratory-works to visualize the
imbalanced-caused bending vibrations and the possibilities to influence them
by changing the system properties. This work contains all steps of the product
development from the request to the finished designed and documented product. The process is split in four main stages: the planning-, the concept-, the
drafting- and the elaboration-stage. The planning-stage contains mainly the
elaboration of the list of requirements in agreement with the customer. The
list of requirements consists all features which are required of the finished
product. The scheduling is also part of the planning. It determines periods of
time for the different development stages and also consists milestones to set
specific project-events. Based on the list of requirement the concept is described. During the concept-phase an abstract working principle is developed.
It describes the particular functions of the system without specifying its technical realization. The technical realization is developed in the drafting-stage.
The overall design is executed and the system is divided in functional groups.
Additionally main attributes like functional-dominating dimensions or the
speed-range are defined during the drafting-stage. The elaboration-stage is
mainly the construction respectively designing of the particular parts and the
producing of the technical drawings for manufacturing. A complete product
documentation which contains an operational manual as well as an assembly
guidance is done in the developing process. The manufacturing of the parts
will be done in the JAMK’s mechanical workshop and is not part of this thesis
work.
27
3.2 Features of the device
The list of requirements is elaborated together with Jorma Mattilainen and
Matti Kurki. The requirements are derived from the demand of the device as a
lecture and laboratory-work supporting apparatus. It should be able to show
unbalanced excited bending vibrations with amplitudes which are high enough
to see with the naked eye. The systems arrangement should be changeable in
the position and amount of the masses. It also should be able to influence the
bending stiffness of the shaft. Since of the purpose of the device, it should be
easily moveable and the operating should be fast, safety and easy.
3.3 Concept
The working concepts for the different rotor arrangements are directly derived
out of the list of requirements. There are basically three different kinds of arrangement which are shown in Figure 22. All of those modes are controlled by
a by the user predetermined revolution speed. The shaft with the rotor respectively rotors should be as good as possible uncoupled from the environment so
that the bending vibrations are as free and unaffected as possible. Figure 22 a)
shows the working mode with one imbalanced rotor between the two bearings.
The magnitude of the imbalance, the position of the rotor as well as the effective length of the shaft are adjustable. The second working mode (Figure 22
b)) has a second rotor with adjustable position and imbalance. The last mode
as shown in Figure 22 c) has a cantilevered adjustable rotor.
a)
b)
28
c)
Figure 22: arrangement with a) one rotor b) two rotor c) cantilevered rotor
3.4 Drafting
The objective of the drafting is to gather, on the basic of the overall concept,
the functionalities of the different parts and their main attributes. Therefore
the concept is separated into different functional groups (chapter 3.4.1). By
analysing the functional groups, the overall design of the different components
is made. The first step is the specification of the rotor’s dimensions (chapter
3.4.2) because they determine the dimensions of every other component. An
important part of the drafting is the development of the safety concept which
is made in chapter 3.4.6.
3.4.1 Functional groups
The device is divided into different functional groups which are under certain
limitations independently designable. The first functional group is the oscillating system. It contains all parts which are directly involved in the bending vibration. The second functional group, the supporting includes all parts that
are necessary to support the shaft and adjust the shaft’s effective length. In the
functional group driveline are all parts integrated which are necessary to provide the torque for operating the system by a certain speed. The housing contains all the equipment needed to keep the systems parts in position, hold up
the forces and torques and also the mechanism to avoid accidents by touching
rotating parts.
29
Functional group: Oscillating system




Shaft
Rotor
Unbalanced mass
Mechanism for adjusting rotor’s position
Functional group: Supporting



Bearings
Bearing supporting
Mechanism for adjusting bearing’s position
Functional group: Driveline



Motor
Controlling
coupling – motor to shaft
Functional group: Housing



Carrier, ground plate
Vibration absorber system
Safety equipment
3.4.2 Dimensioning rotor
Important attributes to get the desired natural frequencies of the rotor, are beside the shaft’s length, diameter and its modulus of elasticity the disk’s mass
and the bearing type. The calculations are based on the in the theoretical part
gathered formulas. The dimensioning process is described below.
1. Determine maximal rotating speed depending on the maximal motor
speed and safety issues.
�
=
/min
30
2. Determine second natural frequency lower then maximal rotating
speed.
� ≪�
→� =
/min
3. Determine material (young’s modulus) and bearing type.
Young’s modulus: 200 GPa (Steel)
Bearing types: self-aligning ball bearings (non-rigid bending support)
4. Choose shaft length, diameter and disk mass. Therefore curves with a
constant second natural frequency and modulus of elasticity are plotted
(Figure 23).
Figure 23: Curves of constant masses and frequency
In Table 1 the in agreement with the thesis supervisor chosen attributes and
the expected natural frequencies are listed.
31
Mass
1 kg
Length
0,6 m
Diameter
9,982 mm
-> 10 mm
f1
1030 1/min
f2
2007 1/min
fLaval
1410 1/min
Table 1: Rotor dimensions
Determination unbalanced mass
The unbalanced mass and its lever arm causes the excitement of the vibrations. For the above given shaft dimensions, the maximum displacement at its
first natural frequency of a Laval-Rotor is shown in Figure 24. The displacement is plotted against the unbalanced mass by a constant lever arm of 0,1 m.
The displacement depends also on the damping ratio. Since the damping ratio
is not accurately definable, it is practicable to choose the unbalanced mass experimentally on the finished demonstration device.
Figure 24: max. rotor displacement according unbalanced mass and damping ratio
32
3.4.3 Supporting
Derived from the concept, the supporting has to execute different tasks respectively it has to meet different properties. First it has to support the bearings
forces properly. Part of this is to isolate the shaft’s vibrations from the device’s
housing. This isolation is on the one hand necessary to avoid an oscillating of
the device’s housing and on the other hand to keep the shaft’s vibration as free
from external influences as possible. The supporting also has to provide the
possibility to change the effective shaft length by varying the bearing distance.
Summarized, the supporting has three different main tasks: supporting the
bearing, isolating the vibrations and adjusting the length. Since this three
tasks demand quite different properties of the supporting, they are designed
separately.
Bearings supporting
The supporting has to hold the static load of the rotor which is caused by the
gravitation and the forces which are caused by the oscillating of the rotor. It
also has to determine the shaft’s axial position. Basically there are two different main solutions to support the bearing. Either to use standardized bearing
housings or to design and manufacture custom housings. This decision is
made with the following decision matrix (Table 2) and in dialog with the projects supervisors. The criteria which are used for the decision are:
-
Overall dimensions:
Since the standardized housings for bearings are designed for about the
maximum bearing load, they have presumably way bigger overall dimensions than special for the demonstration devices designed ones.
-
Weight:
Because of the same reason as for the overall dimensions, the weight of
a custom housing will be presumably way lower than a standardized
housing.
33
-
Costs:
Compared to the purchase price of standardized housings, the manufacturing which includes material costs, machining costs and working
times is most probably more expensive.
Factor
Standardized housing
Custom housing
rating 1)
weighted
rating
weighted
Dimensions
5
2
10
5
25
Weight
2
2
4
5
10
Costs
4
5
20
1
4
Sum
34
39
1)
1 - very poor; 5 –very good
Table 2: Decision matrix supporting
According the decision matrix a custom manufactured housing is the more
matching solution.
Adjusting of the effective length
To adjust the shaft’s effective length at least one of both bearings has to be
moveable. Since the shaft’s axial positions has also to be fixed by the supporting and it is difficult feasible to have an adjustable bearing which also should
fix the shaft’s axial position, it is practicable to separate those two tasks onto
both bearings. This is realized by setting the motor-sided bearing as static and
the other one as shiftable. The static, motor-sided bearing has to fix the shaft’s
axial position. This layout of the supporting is a so called locating and non-locating bearing arrangement (Figure 25).
34
Figure 25: Bearing arrangement
There are several different possibilities to realize the shifting and fixing of the
adjustable bearing. There are three basically different solutions gathered. Figure 26 a) shows the first solution. It is based on the mechanism like it is used
to move the slide of a milling-machine or a bench vice. It has a turnable-supported threaded bar which is as long as the adjusting range of the bearing
should be. The threaded bar is on one side axially fixed and has on the other
side a handle. The bearing supporting is located on a slide which is routed on
two guide rails. The slide-supporting unit has an inside thread in which is the
threaded bar stuck. Thereby the slide moves by turning the threaded bar. On
Figure 26 b) and c) are the second solution for shifting and fixing the bearing
shown. They consist out of four different main units. A slide-supporting unit,
guide rails, a clamping plate and clamping screws. The slide is axially free supported on the guide rails and clamped with the clamping plate and screws
against them. The slide-supporting unit is manually shiftable into the desired
position after loosening the clamping screws. The third solution (Figure 26 d))
acts similarly to the previous one. The clamping is performed by screws which
push against the upper face of the rail. The slide and the rail are shaped in a
way that they are clamped to each other by tightening the screws.
35
a)
b)
c)
d)
Figure 26: Design proposals bearing shifting
The selecting of the adjusting and fixing principle is executed by using a decision matrix. Therefore the solutions are evaluated by the following criteria:
-
Usability
Since the solution a) just requires a turning of the threaded bar to adjust the bearing’s positions, it is quite easy to use. Especially the finetuning is comfortable but a larger changing takes a quite long time. The
usage of the solution b)/c) and d) are similar. They require a loosening
and tightening of the screws for the positioning of the bearing. A larger
changing of the positioning is compared to solution a) much faster.
-
Costs
The manufacturing of solution a) is pretty extensive. Especially the
threaded bar and its supporting is expensive. Since the rails of solution
d) have a complex shape, they are not realizable with standard profiles
but have to be manufactured out of solid profiles which leads to higher
36
costs. For the rails of solution b)/c) can be standardized profiles used.
The solution b)/c) has, compared to solution d) an additional part.
-
Dimensions and weight
The solutions b)/c) and d) have comparable dimensions and weights.
Since of the threaded bar and its supporting, the solution a) is heavier
and needs additional space.
-
Clamping capability
Since of the threaded bar’s self-locking, it has a good clamping capability. The solution d) has a between the screws and the rails just a small
contact surface a therefore a lower axial fixation. The solution’s a) and
d) fixation in the plane vertically to the shaft is because of the rail’s
shape very strong. The clamping of solution b)/c) has to be supplied by
its screws initial tension and has therefor a lower clamping capability.
Solution
Factor
a)
b)/c)
d)
r1)
w2)
r
w
r
w
Usability
4
3
12
4
16
4
16
Costs
5
2
10
5
25
4
20
3
2
6
5
15
5
15
2
5
10
3
6
4
8
Dimensions
and weight
Clamping capability
Sum
38
1)
62
59
rating: 1 - very poor; 5 –very good; 2) weighted rating
Table 3: Decision matrix bearing adjusting
37
According the decision matrix the solutions b) and c) are the concepts with the
highest potential. Since the solution b) has because of its triangular rail a selfaligning while clamping, it is used as concept for the further elaboration.
Vibration isolation
Since the bearing housing has to be stiff enough to support the bearings outer
ring, it has nearly no damping capability. To provide the necessary damping
capability the supporting’s clamping unit should be quite solid and out of a
material with a high internal damping.
3.4.4 Shaft hub joint
According the concept the disks have to be shift- and removable attached onto
the shaft. Since there appears between the disk and the shaft only the torque
which results of the disk’s inertness, the holding force does not affect the
choosing of the shaft hub joint much. Beside the fulfilment of its main tasks,
the shaft hub joint has to satisfy different demands. It should be fast to strain
and loose, around the shaft axis balanced and as low priced as possible. Since
the connection has to be re-adjustable, it should to be friction-locked. There
are several different principles for the shaft hub connection. Proposal a) (Figure 27 a)) is to separate the disk into two part which are clamped with stretching screws and an interference fit onto the shaft. The manufacturing of this solution is complicated because the accurateness of the fit is difficult to achieve
on the open hole. Since the point symmetrical shape, the solution a) is balanced. Its loosing and straining is quite fast. Figure 27 b) shows the second
proposal for the shaft hub connection. It is a screw-strained cone clamping
connection. It has on one or both sides of the hub a loose element with a conical plane which clamps due tightening against a further conical plane on the
inner-side of the hub. This kind of shaft hub joint is as standardized part available and has therefore low manufacturing costs. Since this shaft hub joint has
several screws to strain, it is more time-intensive to adjust the position of the
disks than by systems with fewer screws. The last (Figure 27 b)) solution which
is taken into account is also a cone clamping connection and has basically the
38
same working principle like the previous one. Instead of multiple screws which
strains the conical faces against each other it has just a single nut. This brings
the advantage of a fast adjusting of the disk by just loosing and straining one
nut. This kind of element is also available as standard purchasing part and also
needs no additional machining.
a)
b)
c)
(Ringspann GmbH, 2014)
Figure 27: Design proposals shaft hub connection
The selection of the used system is made with a decision matrix. Evaluated are
the properties:
-
Costs
Considered are the purchasing costs as well as the manufacturing costs.
39
-
Ease of use
The ease of use is mostly depending on the time which is needed for adjusting the time. Solution a) has compared to the other solutions the advantage that the rotor is completely attach- and removable at any position of the shaft. The other solutions has to be putted on the end of the
shaft.
Solution
Factor
a)
b)
d)
r1)
w2)
r
w
r
w
Costs
1
2
2
4
4
4
4
Ease of use
1
3
3
2
2
4
4
Sum
5
1)
6
4
rating: 1 - very poor; 5 –very good; 2) weighted rating
Table 4: Decision matrix shaft hub joint
According the decision matrix the cone clamping element which is strained
with a single nut is the best solution. The in the device used clamping element
is the Ringpspann Trantorque Mini. It is available for shaft dimension between 3 mm and 16 mm and supports a maximum transmissible torque of 68
Nm which is way more than needed. Its outer diameter of the clamping element is 23 mm and the seat length for the hub is 13 mm. This dimensions has
to be considered by designing the disk.
Figure 28: Ringspann Trontorque Mini (Ringspann GmbH, 2014)
40
3.4.5 Housing
The housing’s main function is to keep all the different parts like the motor,
the bearing supporting and the safety-cap in position and provide them a sufficient supporting as well as the needed space to execute their functions. The
housing has to be stiff enough to keep in shape while carrying but also resistant against stimulation of the shaft’s vibration. It should be easy to
transport and therefore lightweight. Since an usage of standard parts is mostly
more cost-effective than the manufacturing of custom-parts, they are preferred. The ease of assembling should also be considered. The first developed
concept (Figure 29 a)) for the housing consists of a base frame assembled out
of standard aluminium profiles and a plastic cover plate where the different
units are mounted on. At Matti Kurkis suggestion a concept (Figure 29 b))
with a massive base plate out of plastic is considered, too. The different parts
are mounted directly onto the base plate.
a)
b)
Figure 29: Design proposals housing
The choosing of the housing concept is executed by using a decision matrix.
Therefore the solutions are evaluated by the following criteria:
41
-
Costs
The costs of proposal a) are mostly determined by the material costs of
the standard profiles and their connecting elements. A roughly estimated price by using parts of the company Item is about 270 € (item
Industrietechnik GmbH, 2015). A massive 50 mm PA6 plate is for comparable dimensions about 320 €. Additionally to the higher material
costs the machining of proposal b) is more extensive.
-
Weight
The estimated weight of the aluminium frame and the cover plate (proposal a)) is about 14 kg. The weight of a comparable PA6 plate is about
22 kg (B & T Metall- und Kunststoffhandel GmbH, 2015).
-
Stiffness and damping capability
The damping capability of proposal a) is probably for the same stiffness,
compared to all in proposal b) applicable plastics much lower.
Factor
Solution
a) Al-Profiles
b) Solid plate
r1)
w2)
r
w
Costs
4
4
16
3
12
Weight
3
4
12
3
9
Stiffness and damping
2
3
6
5
10
Sum
34
1)
31
rating: 1 - very poor; 5 –very good; 2) weighted rating
Table 5: Decision matrix housing
According the decision matrix the aluminium-frame solution is the better concept for the requirements and serves as draft for the further development.
42
3.4.6 Safety concept
The elaborated safety concept is based on the approach of risk assessment like
it is described in ISO- Norm 12100 (Mattilainen, 2015). Therefore, the possible
hazards are identified. For this purpose, all tasks which are proceed by using
the device have to be considered. After the identification, the risks of the hazards are evaluated. Depending on the risks their countermeasures are elaborated.
Identified hazards
-
Turning parts
The fast turning shaft during the operation involves the risk of injuries
if a part of the body comes into contact with it. Also clothes or other
fabrics can be dangerous if they winding up on the turning shaft.
-
Unintentional restart
An unintentional start of the device during the adjusting, cleaning and
maintaining can lead to a physically contact with fast turning parts.
-
High shaft deflexions
If the device is operated next to the shaft’s critical speed, the deflexion
of it can be improper high. This can cause a fracture of the shaft or
other components.
-
Loosening of imbalance-screw
If a screw which are attached onto the rotating disk for apply an imbalance get loose while operating, it can become a dangerously projectile.
43
Risk evaluation
The evaluation of the risk is done by rating the severity of the harm and the
probability of its occurrence. The severity of the harm is grouped into the categories slight (normal reversible) injury or health hazards, serious (normal irreversible) injury or health hazards and death. The probability of the harm’s occurrence is evaluated by the user’s expose to the hazard, the probability of occurrence of a hazardous event and the possibility to avoid or limit the harm
(Mattilainen, 2015).
1) slight injury or health hazards
Severity of
harm
A
2) serious injury or health hazards
3) death
B
PROBABILITY OF OCCURRENCE
OF HARM
C
D
Person’s exposure to hazard
1) seldom … quite often
2) regularly … constantly
Probability of oc- 1) low
currence of a haz- 2) average
ardous event
3) high
Possibility to
limit the harm
1) possible under certain conditions
2) seldom possible
Table 6: Harm evaluation categories (Mattilainen, 2015)
After evaluating each hazard according Table 6 the Level of risk can be read
out of Table 7. The higher the number the more dangerous is the rated hazard.
The result of the risk level affects the priority of the countermeasure.
44
B
C
1)
1)
2)
3)
2)
3)
1)
1*
2
3
4
5
6
1)
3
4
5
6
7
8
2)
5
6
7
8
9
10
1)
7
8
9
10
11
12
2)
9
10
11
12
13
14
1)
2)
1)
2)
1)
2)
D
D
Level of risk
A
D
*1… very safe – 14… highly unsafe
Table 7: Levels of risk (Mattilainen, 2015)
Table 8 shows the evaluated criteria and the out of Table 7 derived risk level of
the identified hazards. According this the hazard of turning parts and the loosening of an imbalance-screw involve the highest danger. But every other single
hazard is also not to neglect and it also has to be a countermeasure developed
against them.
Hazards
Turning
parts
Unintentional restart
High shaft
deflexions
Loosening
of screw
A) Severity of
harm
B) Person’s
exposure to
hazard
C) Probability of occurrence
D) Possibility
to limit the
harm
Level
of risk
2
2
3
1
9
2
1
2
1
5
2
1
2
1
5
3
2
1
1
9
Table 8: Risks of the identified hazards
45
Countermeasures
To avoid the danger of an accidently grabbing into turning parts or other contact with it, a safety-cap is practicable. The safety-cap has to be transparent to
not restrict the purpose of the device. It also has to be removable or has to be
able to open to adjust the arrangement of the system. This safety-cap also
serves as a barrier for loosening part of the rotor.
To prevent an unintentional restart while working on the device a safety circuit, which cuts of the current if the safety-cap is open, is needed. Therefore a
switch is required which cannot be accidently pressed. Since the electrical
work is not part of this thesis, this countermeasure has to be considered by the
person which does the wiring of the motor.
To ensure that there are no improper high shaft deflexions the maximum deflexion of the shaft has to be limited. Therefore a safety bearing is used. It restricts the shaft deflexion as a mechanical barrier.
3.4.7 Safety bearing
In chapter 3.4.6, the safety bearing as a countermeasure against too high shaft
deflexions is introduced. It restricts the deflexion to a maximal amplitude of 5
mm (Figure 30 a)). If the shaft deflexion reaches 5 mm the rotor collides with
the safety bearing and stops the swinging up. The safety bearing has to be out
of a plastic material which does not damage the rotor while touching. The
safety bearing has beside the restriction of the shaft deflexion a second purpose. It serves as an additional supporting if the shiftable bearing is removed.
Therefore the rotor has an additional shoulder with the same diameter as the
inner diameter of the safety bearing. The shaft can be supported due sliding
the safety bearing onto the rotors shoulder (Figure 30 b)). The adjustment of
the safety bearing position is similar to the of the supporting.
46
a)
b)
Figure 30: Safety bearing concept
3.5 Elaboration
3.5.1 Introduction
Chapter 3.5 deals with the elaboration of the in the drafting specified concepts.
In the elaboration are the custom components designed and standard parts selected. Part of the elaboration are also the material selection of the custom
parts and, if required, their strength calculation. The order of the subchapters
are in accord with the order the elaboration process is made. Basically the
elaboration is made “from inside to outside”. This means that firstly the bearing is selected. The dimensions of the bearing sets the dimension of the support which affects the housing and the safety-cap. Obviously, this process cannot be made completely sequential since sometimes loops are necessary. Part
of the results of the elaboration process are the manufacturing and assembly
drawings which can be found in the attachment.
47
3.5.2 Bearing choosing
Since the dimension of the bearings determines the design of the complete
supporting, it is chosen at the beginning of the elaboration process. The bearings has to fulfil different requirements in its application at the demonstration
device. The most apparently requirements are to hold the forces which are
transferred from the shaft and enable the shafts almost torque free rotation.
Beside these, the bearing should be as non-rigid as possible against angulardisplacements in the shafts direction (Figure 31). This allows the shaft a nearly
free bending without high restraints caused by the supporting. The bearings
should also be almost maintenance-free and should have a life-cycle which is
at least as long as the demonstration device is in business.
Figure 31: non-rigid bearing axis
Because of the requirement of the bearing as mostly non-rigid against turning
in the shaft direction, the selection of possible bearing-types is limited. Every
type of plain bearing restricts this angular movement too much. The for this
angular displacement most non-rigid rolling-element bearings are the so
called self-aligning ball bearings (Figure 32). Self-aligning ball bearings are
bearings with two ball rows, which are running in two grooves on the inner
ring. The contact surface on the outer ring is spherically shaped and allows the
balls to move some degrees around the bearing centre (Figure 32b)). Typically,
self-aligning ball bearings allow an angular displacement of about 4 to 7 degrees (NSK Ltd., 2015).
48
Figure 32: self-aligning ball bearing
The maximal load is the decisive parameter for choosing a suitable bearing.
The radial load is a combination of the static gravity force and the rotary force.
During a proper operating of the demonstration device there does not occur
noteworthy axial forces. The rotary forces are centrifugal forces which results
from the rotation of the disks with a particular displacement at a particular angular speed. Figure 33 shows the force diagram with the gravity force and the
centrifugal force marked.
Figure 33: Force diagram disk
The maximal centrifugal forces occur at the second natural frequency of the
system, with both disks attached and their maximal allowable displacement of
5 mm (3.1).
49
∙
,
∙
with:
,
�
=~ ∙
�=
=
∙
,
�
,
�
∙
,
,
�
�
≈
=
=
∙
�
≈
�
�
≈
∙
≈
∙ ,
≈
≈
≈
=
=
�
−
−
≈
∙
≈
�
(3.1)
∙
−
≈
N
The determination of the required dynamic ( ) and static (
) basic load rat-
ings are done according the Roloff/Matek guideline (Wittel, Muhs, Jannasch,
& Voßiek, 2013, 506) in (3.2). The used parameters are listed in Table 9. All
parameters are selected either out of Roloff/Matek or, if not named, calculated
in this thesis.
Symbol
P
p
n
L1oh
P0
S0
Parameter
Dynamic load
Life-time exponent
Rotating speed
Desired life-time
Static load
Static safety factor
Value
0,22 kN
3
2000 min-1
2ooo h
0,01 kN
1,5
Table 9: Parameter bearing calculation
Source
Roloff/Matek 14.2.6
Roloff/Matek TB 14-7
Roloff/Matek 14.2.6
50
�
√
≥
≥ ,
,
,
6
ℎ
6
√
≥P ∙ S
≥ ,
(3.2)
/
i
6
ℎ
= ,
∙ , = ,
kN
kN
The, by the leading ball bearing manufacturers SKF and NSK offered selfaligning ball bearings with the lowest load ratings and an inner diameter of 10
mm are listed in Table 10.
Manufacturer
SKF
NSK
Type
1200 ETN9
1200
C
5,53 kN
5,55 kN
C0
1,18 kN
1,19 kN
Table 10: Bearing load ratings
The actual dynamic ( ) and static ( ) basic load ratings of the bearings are
way higher than the required basic load rating (
,
,
) and are therefore
not in danger to be destroyed by the load. The bearing life-time and the minimum load of the bearing is also calculated with the SKF Bearing Calculator
which is available on www.skf.com. The SKF Bearing Calculator attests a lifetime (L1oh) of 132400 hours which exceeds the desired life-time multiple. To
ensure an operation without slippage a minimum radial load is needed. Especially at high angular accelerations and at high revolution speeds ball bearings
tent to slip what could cause damages of the bearings. The risk of slipping increases with the viscosity of the lubricant. To keep the risk of slipping down
the demonstration device should not be operated at high revolution speeds if
there is no disk attached. A fast raise of the motor should also be avoid.
51
3.5.3 Supporting design
The supporting is separated into two main units: the bearing housing and the
clamping unit. The bearing housing is a milled aluminium part. It has to support the bearing with an appropriate fit. The clamping unit consists of two
plastic plates which are strained by four screws and nuts on a guidance. The
upper plate has a milled pocket where the housing is bolted into. The guidance
are two triangular Item aluminium profiles and two aluminium L-profiles
which are mounted onto the baseplate.
Figure 34: Supporting
Selection of fit
Since of the imbalanced excited force, the load on the bearing rotating inner
ring is a point loading (Figure 35). The stationary outer ring has a circumferential load. The bearing manufacturer recommend for this type of load a transition fit at the outer ring and clearance fit at the inner ring (Haberhauer & Bodenstein, 2014, 134). The specific load is low (P / C = 0,22 kN / 5,53 kN=
0,04) and therefore the more loose recommended fits allowed. The used bore
tolerance is K7 and the shaft tolerance is g6 (Figure 36).
52
Figure 35: Bearing loading
Figure 36: Fittings bearing
For a forceless joining of the bearing housing and the bearing, the housing has
to be heated. Since of the thermal expansion the inner diameter of the housing
increases depending on its temperature. The required joining temperature
(
ℎ
�
) of the housing is calculated in formula (3.3) (Haberhauer & Boden-
stein 2014, 134).
with:
ℎ
�
ℎ
�
ℎ
�
=
=
=
+Δ
�
,
° +
,
+
, °
DΔ = ,
=
= ,
�=
=
−6
+ ,
− ·
(3.3)
(join clearance)
°
,
(ambient temperature)
mm
−6
(maximum clearance)
−
(thermal expansion)
(nominal Diameter)
Material selection of the plastic parts
The material for the clamping unit has to fulfil different requirements. Firstly,
it has to be good to machine. It also has to have a sufficient mechanical
53
strength and good damping properties. In the group of the established structure plastics Polyamide 6 (PA6) and Polyacetal (POM) meets this properties
best. PA6 has a slightly higher mechanical strength and also a higher ductility
as well as better damping characteristics and suits therefore better for the application.
Properties
PA6
POM-C
Density
1.14 g/cm3
1,41 g/cm3
Modulus of elasticity
3300 MPa
2800 MPa
Tensile strength
79 MPa
67 MPa
Elongation at break
130 %
32 %
Damping characteristics
Very good
good
(Ensinger GmbH, 2015)
Table 11: Properties PA6 and POM-C
Stress analyses bearing housing
The stress which acts in the bearing housing is a combination of the static
stress resulting of the shrink fit between the bearing and the housing and the
dynamic, rotating load. The stress analyses is made with the finite element
tool of CATIA V5. The following calculations have the purpose to get a rough
scale of the stress in the housing. If the stress is in a critical range the analyses
has to be done more precisely. To calculate the stress, the load of the fit and
the dynamic load are needed. As dynamic load the number which is calculated
for the bearing dimensioning is used (3.1). For a rough calculation of the pressure in the shrink fit, the bearing is assumed as a solid part. This assumption
reduces the shrinkage of the bearing and increases the result of the calculation
and is therefore uncritical. The bearing housing is assumed as a hub with an
outer diameter appropriate the upper part of the housing (
). The maximal
possible pressure results of the highest interference ( ℎ ), the bearing’s and
housing’s elasticity ( � ;
), Poisson’s ratio (�� ; � ) and the housing’s geometry.
54
�=
�=
�=
�=
ℎ
,
,
∙
∙
�
∙
,
/
∙
∙ ,
∙ ,
/
/
(3.4)
with:
=
=
�
=
− �� +
�
=
/
/
+
−
�
�
− ,
= ,
+�
+
+ ,
− ,
+ ,
= ,
The fixation which are simulated in the finite element analyses are not like in
the actual case made with bolts and surface contacts. The bolts are neglected
and the fixation is realised with three so called surface slider on the bearing
housings base (Figure 38). Surface slider lock the displacement vertically to
the restrict surface. This simplification does affect the result of the finite element result just insignificantly, as the critical locations are in the upper part of
the housing. The load of the shrinkage fit is simulated as a surface pressure on
the contact face to the bearing. This loading type conforms to the actual loading precisely. The equivalent stress (Von Mises stress) resulting from the
shrinkage fit has a local maximum of about 50 MPa at the flange of the bearing
contact surface. This enhancement of the stress evolves from the notching effect on this position. This maximum stress is uncritical because it is far lower
than the tensile strength of every aluminium alloy. If commercially available
pure aluminium (tensile strength: ~90 MPa, yield strength: ~35 MPa) is used,
local non-critically plasticity can occur. The stress caused by the dynamic load
is computed statically in different directions. In that way the maximal stress of
the rotating load can be found. The maximum Von Mises stress is about 4 MPa
55
and therefore negligible. Since of the low dynamic stress, a fatigue strength
calculation is not made.
Figure 37: Stress due to shrinkage fit
Figure 38: Stress due to dynamic loading
3.5.4 Rotor design
The rotor’s purpose is to be the “mass” in the mass-spring system. Beside this
it has to provide the possibility to attach the imbalance. The major property of
the rotor is its weight which is determined in chapter 3.4.2. It is clamped with
the in chapter 3.4.4 selected shaft hub joint on the shaft. Therefore it needs a
suitable fit. Additionally the rotor has a shoulder that limits together with the
safety bearing the maximal shaft deflexion. This function is described in chapter 3.5.5. The rotor is divided into two parts. The outer rotor (Figure 39) which
is a disk with a constant thickness and the inner rotor (Figure 40) that has the
shoulder for the safety bearing, the fit for the shaft hub joint and a fit for the
outer rotor. The separation into two parts saves a lot of material and machining cost as the outer part can be made out of a sheet with a way lower thickness than the inner part needs. The outer rotor has threaded holes where bolts
can be mounted, to apply the desired imbalance. The holes are on four different diameter, so that is it possible to adjust four different magnitudes of imbalance with one bolt. A combination of several bolts increases the possible
56
steps of the imbalance’s magnitude. Since the rotors are in touch with skin
while adjusting their position, they are exposed to sweat. To avoid rusting, the
rotors material is stainless steel. Since there are no highly requirements on the
material strength, the used stainless steel can be chosen by its cost and availability. Because of its cheap price, the well-established stainless steel V2A steel
(x5CrNi18-10) is selected.
Figure 40: Outer rotor
Figure 39: Inner rotor
Fit calculation
The connection between the outer and the inner rotor is made with an interference fit. The calculation are made according Roloff / Matek (Wittel, Muhs,
Jannasch, & Voßiek, 2013, 389 – 395). The maximum requried transferable
torque is the torque which is needed to tight the shaft hub joint ( =
Symbol
�
�
�
Parameter
coefficient of adhesion
Adhesion safety factor
Yielding safety factor
Value
0,19
Roughness inner rotor
25 �
1,5
1,2
).
Source
Roloff/Matek TB 12-6 (steel –
steel)
Roloff/Matek 12.3.1 (static
loading)
Roloff/Matek 12.3.1 (ductile
material)
Roloff/Matek TB 2-12 (milled
surface)
57
,
�
�
��
�
Roughness outer rotor
Yield strength inner
rotor
Modulus of elasticity
Diameter of fit
Outer diameter outer
rotor
Inner diameter inner
rotor
Length of fit
25 �
190 MPa
Roloff/Matek TB 2-12 (milled
surface)
V2A (x5CrNi18-10)
200 GPa
35 mm
114 mm
V2A (x5CrNi18-10)
Drawing 2
Drawing 2
23 mm
Drawing 3
11 mm
Drawing 2
Table 12: Parameter for fit selection
First the lowest required joint compression is calculated (� ) (3.5). It results of
the fit’s surface (�� ) and diameter (
required transferable torque ( ).
With:
� =
=
�� =
∙
�
�� ∙ �
�/
∙
�
�
the coefficient of adhesion (�) and the
∙ ,
∙ ,
=
=
∙
� ),
, ∙
=
∙
−
=
∙
=
,
/
(3.5)
=
With the lowest required joint compression, the lowest required allowance of
interference can ( ) be calculated (3.6). The grading ( ) of the fit surface
caused by the compression is also considered.
58
=
� ∙
With:
=
�
=
=
�
∙
+
+
−
��
�
�
�
=
�
= , ∙
�
=
,
=
+
−
+
�
∙
+ ,
− ,
=
= ,
= ,
= , ∙
+
�
∙ ,
+
+
+ ,
− ,
+
�
�
=
�
= ,
=
(3.6)
�
The highest allowed joint compression (�ℎ ) is determined by the strength of
the used material of the inner and outer rotor and is calculated in (3.7). The
highest allowed joint compression sets the highest allowed interference ( ℎ )
(3.8).
,
�ℎ =
,
�ℎ� =
ℎ
=
�ℎ ∙
∙
∙
−
√
−
√
�
=
=
∙
,
∙
,
�ℎ� < �ℎ ⇒ �ℎ = �ℎ� =
�
∙
+
=
∙
− ,
=
√
− ,
√
∙ ,
/
/
=
+
(3.7)
/
�
=
�
(3.8)
The difference of the highest and lowest allowed allowance of interference determines the possible dimension variation ( ) of the fit. The dimension variation is separated into the tolerance of the inner and outer diameter ( � ,
). The
tolerances are chosen according the ISO- tolerance table (ISO 286). Because of
machining reasons a hole basic fit is chosen.
59
=
ℎ
⇒inner dimension:
−
=
�
−
�
=
�
H6 (EI=0 µm; ES=16 µm)
⇒outer dimension:
�=
=
(3.9)
+
+
ℎ
=
µm +
=
+
u6 (ei=60 µm; eo=76 µm)
µm =
µm =
µm
µm
To join the parts forceless, the inner rotor has to be cooled down or the outer
rotor heated up. The calculation of the required temperature difference is mad
in (3.10) (Haberhauer & Bodenstein, 2014).
Δ
Δ
=
=
+Δ
�
,
, ∙
−6
+ ,
− ·
=
°
(3.10)
with:
DΔ = ,
= ,
�=
=
, ∙
mm
(join clearance)
mm
(maximum clearance)
−6
−
thermal expansion)
(nominal Diameter)
3.5.5 Safety bearing design
The safety bearing consists out of the bearing itself, a strain plate, two metal
plates and a countersunk bolt. Since the safety bearing does not have to support high loads it is strained with just one bolt. To hold the strain plate while
straining the bolt, it has two shoulders which are in contact to the ground
plate. The metal part which is inserted into the strain plate is threaded and
transmits the bolts stress into the strain plate. The material requirements for
the bearing and the strain plate are similar to them for the supporting’s plastic
60
parts. Therefore the in chapter 3.5.3 selected material is also used for the
safety bearing.
Figure 41: Safety bearing
3.5.6 Shaft design
The main properties of the shaft are determined in chapter 3.4.2. Its diameter
is 10 mm and the maximal effective length which means the distance between
the bearings is about 600 mm. The shaft’s material is in drafting defined as a
steel material. To adjust or remove the supporting, safety bearing and the rotor, the shaft has to have a constant diameter on his complete length. This is
also necessary to meet the theoretical assumptions which are made for transforming the shaft into an equivalent spring (chapter 2.2.1). The axial fixation
of the shaft is done in the motor-sided supporting. Therefore the bearing is
fixed with a retaining ring on the shaft.
Coupling
The shaft is driven by the electric motor. The electric motor is already at hand
and therefore the shaft has to be adapted to the given dimensions. The connection between the shaft and the motor should be as non-rigid against bending
61
as possible to get a free bending of the shaft. Beside this, the connection
should isolate vibrations and rotation oscillating. A shaft coupling with a rubber element is practicable and common to satisfy this requirements. Since the
required transferable torque is quite low, it plays just a minor role for the selecting of the coupling. There are lot of different supplier which offer very similar solutions. The chosen coupling are the ROTEX torsionally flexible coupling of the company KTR (Figure 42, Table 13).
Figure 42: ROTEX coupling (KTR Kupplungstechnik GmbH , 2015)
Property
Value
Maximal angular displacement
0,9°
Maximal radial displacement
0,85 mm
Nominal torque
10 Nm
Shaft hub joint
Key connection
(KTR Kupplungstechnik GmbH , 2015)
Table 13: Main properties ROTEX coupling
The coupling is non-rigid against bending until an angle of 0,9 °. To check if
the coupling is non-rigid for the complete displacement of the rotor the bending line of the shaft is as parabolic approximated. For an exemplary calculation (3.11), the distance between the bearings is 600 mm and the maximal allowed rotor displacement is centric of the shaft (Figure 43).
62
Figure 43: Bending line
The shaft bending angles which are at the bearing’s position are about the
same like at the coupling. For a free bending the bending angle at the bearing
is 1,9 °. According this the stiffness of the system will increase for higher shaft
deflexion than 0,9 °.
=
=
=
With:
α
=
∙
+
∙
∙
∙
= arctan(
−
= /
) = arctan (
=−
⇒
⇒
=−
=
= /
(3.11)
)= , °
63
Figure 44: Shaft connection
Since the diameters of the shaft and the motor’s output shaft are different, an
adapter needed (Figure 44). This adapter is connected with a key to the ROTEX coupling. The connection between the adapter and the shaft is made with
a glued joint. A glue joint is the favourable connection because it does not require an additional machining of the shaft. An interference fit is not used because it would either require a second diameter of the shaft or the hole of the
adapter cannot made according the hole basic system (H). Usual industrial
metal glues can hold high shear forces up to 25 N/mm². A suitable glue is the
Loctite 638. For this glue connection the maximal transferable torque is about
100 Nm. The used fit between the adapter and the shaft is a H7/g6. This fit is a
clearance fit with a clearance of 5 to 29 µm which features an optimal bonded
joint and a force-less mounting.
Property
Value
Maximal shear forces
> 25 N/mm²
Maximal gap
250 µm
Curing time (hand-tight)
4 min
(Henkel AG, 2015)
Table 14: Properties Loctite 638
64
Strength calculation
The strain in the shaft is caused by two different effects. The bending of the
shaft and the contact pressure of the Ringspann shaft hub joint. The tension
caused by the bending is oscillating and the contact pressure static. Since there
is an oscillating load, the shaft is verified on fatigue strength. The calculations
are made by hand with supporting by finite elements methods. For the calculation the rotor is centric assumed. In this case the highest shaft deflexion is on
the same length as the contact pressure of the shaft hub joint. The strain
caused by the bending (�
With:
=
=
∙
� =
∙
∙
∙
∙
∙
=
is calculated in (3.12).
∙
=
=
∙
∙
∙
∙
=
=
∙
(3.12)
=
Its solution is evaluated with a finite elements calculation (Figure 45). The restraints in the finite elements calculation are blocked translator deflexions of
the shaft ends. The load is a forced deflexion at the centre of the shaft. This
constraints causes a parabolic bending of the shaft. The solution of the maximal equivalent stress is 169 MPa and therefor about the same than the maximal stress calculated by hand (176 MPa).
65
Figure 45: FE calculation strain caused by bending
The static stress caused by the clamping force of the hub joint connection is for
the lowest recommended tightening torque of the damping nut 197 MPa
(Ringspann GmbH, 2014). According the finite element solution, the combined stress is on the stretched side of the shaft (�ℎ ) 358 MPa and on the compressed side (� ) 22 MPa (Figure 46).
Figure 46: FE calculation combined stress
Figure 47: Stress cycle
The fatigue strength calculation is made according the strength of materials 2
script (Apel, 2012) which is based on the FKM- guidelines (research curatorship mechanical engineering). The fatigue strength of the material depends on
66
the nature of the load. A mean stress (� ) reduces the fatigue strength (� ).
The fatigue strength for a pure alternating load (R=-1) can be estimated with
its tensile strength (
) and its ductility. The FKM-guidelines gives experi-
mental determined ratios between the alternating fatigue strength and the tensile strength of certain material sorts ( ). The influence of the stress ratio (R)
can be read in the fatigue strength diagram (Figure 48). The rate of the stress
ratio’s influence depends on the materials tensile strength and is specified in
mean stress sensitivity (
less steel S355 (Table 15).
Tensile strength
� ).
The calculation is made for the proposed stain-
Mean stress sensitivity
−
, ∙
490..630 MPa
∙
− ,
Alternating strength factor
�
0,45
(Apel, 2012)
Table 15: Properties S355
=
� =
� =
�
�
=
�ℎ
�ℎ − �
∙
= , ∙
= ,
=
= ,
−
[
∙
−
≈
=
]− , = , ∙
The fatigue strength for the given stress ratio �
tigue strength diagram (Figure 48). It is �
for a fatigue fracture is
�=
(3.13)
−
�=
=
∙
− , = ,
can be read out the fa. The safety factor
= , (3.14). This result is just a roughly expectation
and is shows just a tendency of the actual fatigue strength. Since the calculations are made with the lowest specified tensile strength the actual fatigue
strength is probably higher. The fatigue strength is defined as the stress which
destroys the material after 106 stress cycles. 106 stress cycles are at a mean revolution speed of 1500 1/min a run-time of about 700 minutes. The run-time by
maximum shaft deflexion will be just several seconds or at least few minutes.
Therefore the probability of a collapse is even less.
67
Figure 48: fatigue strength diagram
=
�
�=
�
=
= ,
(3.14)
3.5.7 Housing design
The housing (Figure 49) consist mainly out of an aluminium profile frame on
which are two plastic plates mounted. The plates serve as basis for the motor
and the supporting rails. The aluminium frame has in combination with supporting rails the needed stiffness for carrying the device and for supporting the
motor and the rotor. The plates provide an additional damping between the
rotor supporting and the frame as well as between the motor and the frame.
This reduces a vibrating of the housing. A low vibration of the complete device
is necessary to ensure an undisturbed oscillating of the rotor. The requirements of the plate material is alike those of the supporting’s plastic parts.
Therefore the in chapter 3.5.3 selected material can also be used for the plates.
68
Figure 49: Housing
3.5.8 Safety-cap design
One of the in chapter 3.4.6 defined safety measures is the safety-cap. It has to
avoid a harmful contact between the rotating parts of the device and the user.
Additionally, it provides a barrier if rotating parts get loose or the shaft collapses. To not limit the view onto the rotor, the safety-cap is made out of Plexiglas (acrylic glass). An established glass thickness for applications like this is 8
mm. This thickness is adopted by commercial safety-caps for machine tools.
For a good torsion stiffness and to close the cap on both ends, there are endcaps out of plastics (PA6 or POM). The cap is fixed with two quick clamps on
each end to fastening plates. The Plexiglas cover can be either bended to shape
or pieced out of several plates. The bended cover (Figure 51) needs threaded
holes on its faces to join it to the cap ends. This threaded holes are critical to
produce because of the occurring notch effect. There is a high risk that the
Plexiglas cover cracks while cutting the tread or straining the bolts. The bending of Plexiglas is with a sufficient heating possible. If the cap will be made
with a bended Plexiglas cover the processing instructions of the material has
be followed. Since of the manufacturing difficulties, a cap is elaborated which
does not requires threads in the Plexiglas (Figure 50). It has instead of one
bended plate five several plates with milled chamfers on their long sides. The
cap ends have milled slots in which the Plexiglas plates are bonded. To achieve
an additionally stiffness, the cap ends are connected with two bars.
69
Figure 50: Safety cap with several plates
Figure 51: Safety cap with bended plate
3.6 Summary and prospects
The complete demonstration device is shown in Figure 52. The natural frequencies of the rotor can be varied by adjusting the free supporting’s position
as well as by adjusting the position of the disks. To remove or attach a disk the
free supporting has to be slide of the shaft so that the disk and its safety bearing can be pulled out the shaft. To keep the shaft horizontally while the free
bearing is taken-off, the second safety bearing is supposed to be put onto its
corresponding disk. The disks can also be mounted on the cantilevered side of
the shaft if desired. The combination of the thin shaft, the comparatively heavy
disks and the, to a certain extend non-rigid supporting should demonstrate the
critical speeds and their characteristic oscillating modes well. A sufficient operational safety is given by the safety-cap and the safety bearings. A detailed
70
operation manual and the assembly instructions for the device can be found in
the attachment.
Figure 52: Complete device
Since in this work is just the mechanical design of the device made, the controlling of the motor has to be elaborated in a further work. Unfortunately the
manufacturing of the custom parts is not done during my stay in Jyväskylä, so
the assembling and putting into service has to be made later.
71
4 Summary
The introduced Laval Rotor is a simple model of a shaft-disk combination. The
negligence of the shaft’s mass and the assumption of a symmetrically system
permits to transfer the model into a plane mass-spring system which can be
simply descript with a usual linear oscillating differential equation. An imbalance combined with a turning of the rotor excites the oscillating. This excitement makes the differential equation inhomogeneous and is described in the
perturbation function. Since of the imbalance the differential equation got
non-linear. The solving of the model is made numerical in the software
Matlab. Analysing the transfer function shows that the system’s maximum deflexion is on its natural frequency. This deflexion depends on the damping and
reaches in an undamped system infinite. Because of the risk of destruction, the
natural frequency of a shaft is called critical speed. The Laval Rotor has because of its simplification as a one-mass system just a single degree of freedom
and so just a single natural frequency. A distributed of the mass raises the
number of degrees of freedom and also the number of natural frequencies.
Therefore a shaft with several disks has several critical speeds. A rotor oscillates beside translational also rotational which also should be taken in account
while constructing a shaft.
The development project is separated in the planning phase, the conception
phase, the drafting and the elaboration. In the conception are the device’s
rough functions gathered. The concept describes that shaft length as well as
the disks amount and positions should be adjustable. The drafting improves
this concepts and specifies sub-functions and their operation methods. In the
elaboration the designing process is completed and provides every documents
which are necessary to manufacture and operate the device.
72
5 Discussion
A main objective of this thesis-work was on the one hand the gathering of the
theoretical basis which is necessary to understand the effect of imbalanced excited bending vibrations of rotors and to determine the concept of the demonstration device. Furthermore, the development of an executable demonstration device was the actual objective of this work.
The theoretical basis is compiled in the theoretical part of this thesis. It provides an extensive overview and a comprehensively mathematical description
of the phenomenon of imbalanced excited vibrations. The theoretical basis exceeds the, for the development demanded knowledge and deepened the author’s skills in the theory of oscillations and simulation.
The development of the demonstration device for imbalanced excited bending
vibrations has be done during the thesis-work. The demands on the device are
mainly fulfilled. Since the designing of the motor’s control circuit is not made
in this thesis-work, the demonstration device is not ready for use yet. Unfortunately, the manufacturing of the custom designed parts is not made during the
author’s stay in Jyväskylä, so the assembling as well as the bringing into service cannot be done by the author. The development process demands capabilities in different subjects but primarily in the field of mechanical engineering
design. The major challenge of the project was to meet the timetable.
The creating of the thesis was interesting, challenging and not least, because of
the results satisfactory. Once more, I want to thank Matti and Jorma for supervising the thesis and the frequent, fruitful meetings.
73
6 References
Gasch, R., Nordmann, R., & Pfützner, H. 2006. Rotordynamik [Rotor
dynamic]. Berlin: Springer-Verlag.
Rao, S. 2010. Mechanical Vibrations. New Jersey: Prentice Hall.
Wittel, H., Muhs, D., Jannasch, D., & Voßiek, J. 2013. Roloff/Matek
Maschinenelemente [Roloff/Matek machine elements]. Wiesbaden: Springer
Vieweg.
Haberhauer, H., & Bodenstein, F. 2014. Maschinenelemente [Machine
elements]. Berlin: Springer Verlag.
Stodola, A. 1918. Eine neue kritische Wellengeschwindigkeit [A new critical
shaft speed]. Dinglers Polytechnisches Journal, 12. January 1918.
Apel, N. 2012. Festigkeitslehre 2 [Strength of materials 2]. Hochschule Esslingen.
Mattilainen, J. 2015. Machine Safety & Ergonomics. JAMK University of
Applied Sciences.
B & T Metall- und Kunststoffhandel GmbH. 2o April 2015. Prices of PA6 and
POM-C. Retrieved from http://www.metallkunststoffhandel.de/shop/
Henkel AG. 12 May 2015. Loctite 638 Data sheet. Retrieved from
http://tds.henkel.com/tds5/docs/638-EN.PDF
item Industrietechnik GmbH. 20 April 2015. Prices of item products.
Retrieved from http://www.item24.com/
KTR Kupplungstechnik GmbH. 10 April 2015. ROTEX catalogue. Retrieved
from http://ktrinternational.com/root/img/pool/pdf/produktkataloge/en/
en_gesamt/001_rotex_en.pdf
74
NSK Ltd. 15 April 2015. Information about self-aligning ball bearings.
Retrieved from http://www.nsk.com/products/ballbearing/selfaligning/
Ringspann GmbH. 15 December 2015. Installation and Operating
Instructions for Cone Clamping Elements. Retrieved from http://www.ringspann.com/files_db/1417603013_46__14.pdf
75
7 Attachment
1. Part list
Manufacturing drawings:
2. Supporting (DWR001)
3. Outer rotor (DWR002)
4. Inner rotor (DWR003)
5. Shaft (DWR004)
6. Hull (DWR005)
7. Supporting plate (DWR006)
8. Support strainplate (DWR007)
9. Safety-bearing (DWR008)
10. Safety-bearing strainplate (DWR009)
11. Safety-bearing mountingplate (DWR010)
12. Mountingplate strainplate (DWR011)
13. Groundplate (DWR012)
14. Corner 1 (DWR013)
15. Corner 2 (DWR014)
16. Cap plexiglass corner (DWR015)
17. Cap plexiglass side (DWR016)
18. Cap face (DWR017)
19. Cap face motor side (DWR018)
20. Cap rail (DWR019)
21. Cap shoulder (DWR020)
22. L-profile (DWR029)
Assembly drawings:
23. Safety-cap (DWR021)
24. Rotor (DWR022)
25. Supporting (DWR023)
26. Frame (DWR024)
27. Housing (DWR025)
28. Safety-bearing (DWR026)
29. Complete device top-view (DWR027)
30. Complete device side-view (DWR028)
Manuals:
31. Assembly instructions
32. Operation manual
PART LIST
Custom parts
Part Nr.
PART001
PART002
PART003
PART004
PART005
PART006
PART007
PART008
PART009
PART010
PART011
PART012
PART013
PART014
PART015
PART016
PART017
PART018
PART019
PART020
Drawing Nr.
DWR001
DWR002
DWR003
DWR004
DWR005
DWR006
DWR007
DWR008
DWR009
DWR010
DWR011
DWR012
DWR013
DWR014
DWR015
DWR016
DWR017
DWR018
DWR019
DWR020
Name
Bearing housing
Rotor outer
Rotor inner
Shaft
Hull
Supporting plate
Supporting strainplate
Safetybearing
Safetybearing strainplate
Mountingplate Safetybearing
Mountingplate Strainplate
Groundplate
Corner 1
Corner 2
Cap plexiglass corner
Cap plexiglass side
Cap face
Cap face motorside
Cap rail
CAP shoulder
Purchasing parts
PART101
PART102
PART103
PART104
DWR029
PART105
PART106
PART107
PART108
PART109
PART110
PART111
PART112
Item profile 40x40 45° x820mm
Item profile 40x40 x180mm
Item profile 40x40 x1400mm
Item L-profile 40x40x4 x820mm
Ringspann shaft-hub-coupling
Rotex coupling
Self-alligning ball bearing
Item T-Slot Nut 8 St M4
Item T-Slot Nut 8 St M6
ItemAutomatic-Fastening Set 8
quick fastener
Motor
Standard pts.
PART201
PART202
PART203
PART204
PART205
PART206
PART207
PART208
PART209
PART210
Name
Countersunk Screw M4x22
Countersunk Screw M6x25
Countersunk Screw M8x80
Cylinder Screw M8x25
Hex nut M8
Hex Screw M10x40
Hex nut M10
Cylinder Screw M6x14
Hex Nut M5
Countersunk Screw M5x25
Norm
ISO 10642
ISO 10642
ISO 10642
ISO 4762
ISO 4032
ISO 4017
ISO 4032
ISO 4762
ISO 4032
ISO 10642
Amount
2
2
2
1
1
2
2
2
2
2
2
2
2
2
3
2
1
1
2
2
2
3
2
2
2
1
2
44
12
12
4
1
40
12
10
8
16
4
4
4
4
4
Material
Aluminium
Stainless Steel (V2A)
Stainless Steel (V2A)
Structural Steel (S355)
Structural Steel (S355)
Plastics (POM-C/PA6)
Plastics (POM-C/PA6)
Plastics (POM-C/PA6)
Plastics (POM-C/PA6)
Structural Steel (S235)
Structural Steel (S235)
Plastics (POM-C/PA6)
Plastics (POM-C/PA6)
Plastics (POM-C/PA6)
Acryl glass (Plexiglass)
Acryl glass (Plexiglass)
Plastics (POM-C/PA6)
Plastics (POM-C/PA6)
Plastics (POM-C/PA6)
Plastics (POM-C/PA6)
Order Number
See Item offer
See Item offer
See Item offer
See Item offer
4202-010100-000000
see Rotex drawing
SKF 1200 ETN9
See Item offer
See Item offer
See Item offer
Weight [kg]
0,16
0,93
0,12
0,42
0,08
0,27
0,17
0,16
0,05
0,1
0,11
2,26
0,08
0,08
0,81
0,85
0,49
0,44
0,5
0,16
1,62
0,44
3,46
0,66
0,05
0,35
0,034
0,011
0,01
0,035
13,2
Position
Groundplate, Corners
L-Profile/45°-Profile
Supporting/Safetybearing
Supporting
Supporting
Motor
Motor
Rotor
Cap
Cap
0,002
0,005
0,034
0,014
0,004
0,032
0,01
0,005
0,003
0,002
PART213
PART214
PART211
PART212
ISO 4762
DIN 125A
DIN 6885
DIN 471
Cylinder Screw M4x40
Washer M10
Feather key - Form A 8x7x18
Retaining ring - A 10x1
4
8
2
1
Cap shoulder
Motor
Device weight:
GROUP LIST
GROUP Nr.
GROUP001
GROUP002
GROUP003
GROUP004
GROUP005
GROUP006
GROUP007
Drawing Nr.
DWR021
DWR022
DWR023
DWR024
DWR025
DWR026
DWR027
DWR028
Name
Safety cap
Rotor
Supporting
Frame
Housing
Safetybearing unit
Complete device
0,003
44 kg
The asse l of the shat has to e do e i a ertai
order. Firstl , the feather ke  has to e i serted i to 
the adapter  a d the retai i g ke  ´put o to the
shat  . Ne t, the eari g  a e put o to the

shat ith o ta t to the retai i g ri g. Ater ards, the
adapter has to e glued o to the shat. Therefor Lo ite
8 or a si ilar glue has to e used. The adapter should
e put ith o ta t to the eari g so that the eari g
does ot ha e a a ial pla . The glue ust e used a ordi g the i stru io s.
The ITEM proiles , are o e ted o ea h li k ith t o
ITEM faste i g sets  a ordi g the ITEM i stru io s.
The grou dplates  are ou ted o the
fra e  ith M
 ou tersu k
s re s a d ITEM M T-Slot uts  . Therefor the ITEM uts ha e to e posiio ed i to
fra e eforeha d.
The L-proiles  ha e to e alig ed ith
the la k of the grou dplates a d pla ed
up to o ta t ith the ross proile.
Ne t, the posiio s for the oreholes i
the grou dplates should e arked a d
drilled ith ,
dia eter.
The eari g housi gs
 should o e
ou ted o the
ou i g plates  .
Therefor M8
li der s re s a d M8
uts are used.
No , the eari gs a d housi gs a e joi ed. To joi the
eari gs  a d their housi gs  for eless the housi gs ha e to e heated o
a out 8 °C. The heai g a
e either do e ith a hot air
gu or ith a o e .


 
8

No , the tria gular ITEM proiles  a e ou ted ith
M
ou tersu k s re s a d ITEM M T-Slot uts as ell as
the or er parts  ith M
ou tersu k s re s IM slot
uts. The safet - ap shoulder  ha e to e i ed ith M
li der s re s a d M slot uts.
The se o d eari g a o also e put o to the
shat. Ne t, the shat-suppori g u it has to e pla ed
o the tria gular proiles.
The suppori g plates are
i ed together ith their
strai plates  . Therefor
M8 8 ou tersu k s re
a d M8 uts are used.

To joi the i er rotor  a d the
outer rotor  for eless, the
eed a te perature difere e of 
a out
°C. The shoulder’s fa e
of the i er rotor a d the outer
rotor should e i o ta t ater
joi i g.
Asse

li g i stru io
/
No , the rotors  a d safet - eari gs 
a e ou ted. Therefor the free suppori g
has to e slide of the shat. The shat ust e
keep horizo tall . The rotors are la ped
ith the Ri gspa
oupli g o to the shat.
The re uired tor ue is N a d should ot
e e eed. The safet - eari gs are ou ted
ith their etal plates a d a M8 8 ou tersu k s re ea h.
The se o d feather ke has to e put o to the
otor’s shat. Ne t, the Rote oupli g  a
e put o to the rotor’s shat. The otor ith
the Rote oupli g ust e put ge tl o the
shat. The otor’s eri all posiio should e
set ith thi spa er dis s. If the otor’s i al
posiio is fou d, the holes for the la pi g
olts a e arked a d
holes drilled.
The otor is i ed ith four M
olts a d
uts.



No , the ap-rails  should
e ou ted o the ap-e d
 ith a M
ou tersu k s re s a d M ut ea h.
Ater ards, the se o d ape d a e glued o Ple iglas
plates a d strai ed ith the
ap-rails.
The Ple iglas plates of the
ap ha e to e glued i to
o e of the ap-e ds. If the
plates do ot it i to the slot,
the slot should e ide
ith sa dpaper. The applied
glue should e a suita le
glue for plasi aterials.
The ui k faste ers 
should e ou ted i that
a that there is a slight
te sio et ee the ap
a d the ap shoulders.




Asse
li g i stru io
/
Operation manual:
Read operation manual before using the device!
While shaft rotating, the safety-cap must not be removed!
While shaft rotating, the safety-bearings must always be at the disks!
To change the rotor’s layout, open the quick fasteners and remove the safety-cap.
The oscillating system consists of the shaft (1), the free supporting (2), the fixed supporting (3), two
disks (4), two safety-bearings (5) and imbalance-screws (6). The free support, the disks with their
safety-bearings and the imbalance-screws can be adjusted.








To adjust the position of the free supporting, loose the four straining bolts
(internal hexagon 5 mm). The fixed supporting must not be shifted.
To shift the disk, the nut of the shaft hub connection has to be loosen.
Therefore, one person holds the disk and another person carefully loose the nut
with an open-ended wrench (19 mm). The tightening torque must not exceed
31 Nm.
Operation manual 1/2
Before starting the device, the safety-bearing always have to be shifted on the disk’s
rim (1). The safety-bearing restricts the maximal deflexion of the shaft. The safetybearing should be in position, that it barely not clamps the disk’s shoulder (2). The
safety-bearing is strained with 5 mm internal hexagon 5 wrench.



Before removing the free supporting from the shaft, the safety-bearing must be put
onto the disk’s shoulder. Now, the safety-bearing serves as an additional supporting
of the shaft and keeps the shaft horizontally if the free supporting is removed.
To remove or attach a disk and safety-bearing, the free supporting has to
be shifted to the end of the rails.
To remove the safety-bearing its strainplate has to
be 90° rotated. Therefor its bolt has to be loosen as
far as required to turn the strainplate. Now, the
safety-bearing can be taken.
The layout can be put in every desired way.
Before starting the device make sure that
each screw and the disks are nuts are
tightened. Check if the safety-bearings are
properly positioned to the disks.
Operation manual 2/2
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