Mechanical Vibration: Shaft Vibration Orbit Analysis

This research report investigates the effects of asymmetric bearing stiffness on shaft vibration orbits in rotating systems. The study examines various aspects, including damping, decay exponential signals, concentrated disc mass properties, and the gyroscopic effect, using literature reviews and simulations. It explores one-degree-of-freedom models, linearity assumptions, and steady-state sinusoidally forced systems, providing a comprehensive analysis of the vibration mechanisms. The report includes preliminary results on unbalanced systems, project management timelines, and a list of references. The methodology involves simulation calculations to depict the impacts of vibrations on the system, with graphs showing displacement, speed, amplitude, and frequencies against time. The findings highlight the significant effects of shaft impacts on both asymmetrical and symmetrical processes, contributing to a better understanding of designing systems with high workability and reduced efficiency impacts.

Mechanical Vibration Research Report 1
Mechanical Vibration Research Report on:
Effect of Asymmetric Bearing Stiffness on the Shaft Vibration Orbit
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Table of Contents
Abstract............................................................................................................................................4
Background......................................................................................................................................5
Research Objectives.........................................................................................................................6
Literature Reviews...........................................................................................................................6
Damping and Decay Exponential Signal.....................................................................................6
Concentrated Disc Mass Properties.............................................................................................8
Gyroscopic Effect Explanation....................................................................................................9
One-Degree-of-Freedom Model................................................................................................11
Linearity assumptions................................................................................................................12
Steady-State Sinusoidally Forced Systems................................................................................13
Self-Excited dynamic-Instability vibration................................................................................14
Simple linear Motion.................................................................................................................16
Jeffcott Rotor Model..................................................................................................................18
Research Methodology..................................................................................................................20
Preliminary Results and Discussions.............................................................................................23
Task 1: Weights of Unbalanced Symmetrical Systems.............................................................23
Station 1 Data.........................................................................................................................23
Task 2: Weight Asymmetrically Unbalanced............................................................................26

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Preliminary Conclusions................................................................................................................30
Project Management Timeline.......................................................................................................32
List of References..........................................................................................................................33
List of Figures
Figure illustrating the energy dissipated and damped decay per periodic motion cycle
Figures showing the trajectories for the gyroscopic effect
Figure Showing the Linearly Assumptions Models
Figure Showing the Unforced I-DOF Motion Type
Figure showing the growth exponential from makeable initial disturbance

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Abstract
This paper examines the effects of the asymmetrical and symmetrical effects of the bearing
shafts in line with the vibrations in the orbits. Various literature reviews on the past studies in the
same area mainly appraised and evaluated in-depth. Simulation calculations and the overall
program mainly used in depicting and as the paramount approach in establishing the impacts of
the vibrations impacts on the system. The results depicting, the displacements, speed, amplitude
as well as the frequencies against time in line with the system mechanisms mainly pose at the
end of the process. Various graphs in line with study were thereby generated and they showed
tremendous impacts in line with the effects of the shaft impacts on both the asymmetrical and
symmetrical processes.

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Effect of Asymmetric Bearing Stiffness on the Shaft Vibration Orbit
Background
Conversely, the concept of vibration damping has got a wide range of application in both the
engineering devices and in the parametric nature as a whole. The approach associated with the
vibration damping involves the implementation of the standard linear model. The model uses the
ratio of the drag force proportional to the makeable velocity magnitude. Notably, there are a
number of damping mechanisms which are often nonlinear. Some of the typical examples
include coulomb damping (Ebrahimi and Barati 2017 p.926). The coulomb damping mainly
refers to the systems which have hysteresis damping in the internal structure of the material.
However, the computation of the nonlinear damping involves mostly the utilization of the linear
models and thereby estimating for the values of the nonlinear using them. The approach utilizes
the concept of match the existing energy dissolute for every cycle (Kheibari and Beni 2017
p.576).
The method is efficient since the damping modest amount has little effect in line with the current
natural frequency. Therefore, the energy dissipated mainly illustrated as shown above. From the
analysis, it is evidential that the power mostly dissipated per cycle in line with the single-
frequency harmonic pedaling. The study of mechanical vibrations in rotating systems is not only
relevant and essential. In essence, the study helps in the designing system which has high
workability and reduced effects in line with the efficiency (Barati, Shahverdi and Zenkour 2017
p.988).
Mechanical designed and vibratory systems often classified as the under-damped group and this
is fundamentally important in ensuring that the modes associated with them primarily examined.
This examination provides that there is accuracy in handling the modal-coordinate space. The

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approach helps in the modern method of digitizing the data since the context ensures that the data
mainly process individually (Ebrahimi and Jafari 2018 p.213).
Research Objectives
The overall objective of this research project is to establish and determine the shaft vibration
mechanism in rotating systems. In essence, the research tends to establish the level of the
impacts and the overall impacts in both the asymmetrical and symmetrical systems.
Literature Reviews
Damping and Decay Exponential Signal
Preferably, the model facilitates the process of sorting the exponential decay signal from the
overall total transient. Then the individual values for the linear damping coefficient thereby
established using the log-detrimental method and this are essentially illustrated as shown below
(Benaroya, Nagurka and Han 2017).
Figure illustrating the energy dissipated and damped decay per periodic motion cycle

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However, the coefficients used in the equation mainly computed as shown in the equations
below
The log-decrement method has been an approach which has been used over the decades in the
computation of the damping. The approach utilizes the transient decay and initial motion
displacement. However, in the plan, the system must be in an unforced manner. On the other
hand, the method of half power and bandwidth test applies the steady-state response in
computing the dissipated energy (Avallone and Baumeister 2017).
Moreover, this approach incorporates the harmonic excitation forces in the mechanism.
Therefore, if the steady-state linear and single frequency ratio are used; then the equation below
can be used to compute for the overall outcome. Thus, the single DOF model is mainly given by
the equation below (Naudascher 2017).
Using the above equations, the two graphs below can be obtained, and these diagrams essentially
show the trends in the system (Ebrahimi and Barati 2017 p.433).

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Figure 1.5a mainly indicates the trends for the underdamped plots in which the frequencies are
given as
In the computation, it is important to note that the frequencies utilized mainly taken in line with
the horizontal lines intersects (Inman 2017). This is illustrated as
Conversely, the particular amplitude against the frequency mainly plotted where the overall
amplitude vibrations peak established as
Moreover, the term Q mainly derived from quality and therefore, defined as the measure of
quality in line with the electrical resonance circuit. Conversely, the high Q is termed associated
with the systematic low damping concept.
Concentrated Disc Mass Properties
The element is also another vital phenomenon which is incorporated in this shaft design. The
axially overall symmetry in line with mass specified mainly given by the below equation

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The effect of the energy recorded per unit cycle mainly examined using the harmonic motion.
The norm has to be imparted in the overall rotor as a skew-symmetric and in line with the mass
matrix. This result in the equation formulated as indicated below;
The sigma factor essentially obtained from the computation of the two, x and y sinusoidal
motions value. These results in the equation demarcated as shown in the figure below
(Benaroya, Nagurka and Han 2017).
Gyroscopic Effect Explanation
The understanding of the gyroscopic effect involves the overall evaluation and appraisal of the
pivot force. This is important since the pivot force assists in transverse precession and
contemporary spin angular velocities studies. In essence, the illustrations obtained from the
process tends to be clearer if the norm is adapted well. Notably, pivot force used in the process
has to have masses for the two points. These are recorded as m1 and m2. The mass continuum
carried in the disc is the same as the varying degree (Jing et al. 2018 p.11). In essence, the
moment coupling yield required mainly expressed in line with Newton’s law as follows

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In the equation, the mass points mainly viewed as trajectories. The illustrations for the
gyroscopic effects primarily summarized as indicated in the figures below
Figures showing the trajectories for the gyroscopic effect (Sonoda et al. 2018)
Lateral shaft rotor vibration mainly results from the unbalanced, instability as well as the action
of other related forces acting in the system. They often stimulate and hasten the rotational system
rate (Piersol and Harris 2017). Therefore, it is essential to calculate the overall critical speeds,
estimate vibrations amplitudes and frequencies as well as establish recommendations for the
global mechanism applied in reducing the vibration risks. The diagram below represents the
actions for the lateral shaft rotor vibration (Benaroya, Nagurka and Han 2017).
The feature analysis for the above system mainly discussed as indicated in the following sections

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One-Degree-of-Freedom Model
Preferably, the mass-spring-damper model plays an essential role in the overall mechanical
vibration system. In essence, thorough and decisive understanding of the vibration characteristics
is not only crucial but also necessary when it comes to the appraisal of the rotating vibrations and
the associated vibration fields (Bies, Hansen and Howard 2017). Notably, Newton's second law
of moments is the backbone and fundamental player in the vibration field. The law states that
the same of the sum of the overall forces acting in the systems equals the product of acceleration
and the mass of the system. The equation derived from the process is a vector since both the
effect as well as the accelerations yield from the law is vector parameters. The equation for
Newton's second law mainly expressed as given below
F= Ma
From the comparison, F refers to the overall sum of the parametric forces where A and M refer
to the acceleration and mass respectively. However, for the system, the yield in line with the
motion mainly expressed as
This equation motion defined as yield motion differential equation. Conversely, there are a
number of forces which acts upon the designated mass. Some of the forces include applied
external forces which are time-dependent, forces resulting from the action of the damper and
spring motion-dependent connections. Notably, the weights, as well as the forces acting in the
spring, tend to cancel each other. That is static deflection forces and the parametric gravity of the
spring. Furthermore, the equations associated with the actions of the springs mainly expressed

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and written regarding the elementary static equilibrium and often do not include the weights of
the makeable springs as well as the deflection forces.
Linearity assumptions
This is another important and essential element often considered in both the analysis and the
designing of the system with the mechanical vibration. The pictorial diagram used in the
evaluation and the appraisal of the linear assumption mainly expressed as indicated in the
description below
Figure Showing the Linearly Assumptions Models
Preferably, linearly assumption mainly grounded on the assumption since the consideration is
taken that the systems tend to vibrate linearly although the actualization of the same on the actual
world is not possible.