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 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).

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.

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

Research Objectives

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).

Figure: a mainly indicates the trends for the underdamped plots in which the frequencies are given 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.

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

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).

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

Literature Reviews

In the equation, the mass points mainly viewed as trajectories.

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

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.

From the comparison, F refers to the overall sum of the parametric forces where A and M refer to the acceleration and mass respectively.

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 and written regarding the elementary static equilibrium and often do not include the weights of the makeable springs as well as the deflection forces.

This is another important and essential element often considered in both the analysis and the designing of the system with the mechanical vibration.

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.

One-Degree-of-Freedom Model

Dynamically stable systems tend to tolerate positive damped vibrations when they are subjected to the parametric long-term forcing and mechanism. Preferably, there is the presence of the mass unbalanced makeable distribution, and this cannot be adequately be reduced and eliminated in the rotating machinery completely.  Moreover, the unbalance rotor mass mainly rectified via the application of the equivalent forces often fixed in the parametric rotor. The troubleshooting in the rotor vibrations system is also fundamental and equally important at all time. The designers take into account the concerns and any elementary parameters which have the likelihood of affecting the rotating mechanisms and the concept in the rotor machinery. The troubleshooting mainly appraised in line with the long periods and duration of the vibration exposures as well as the related levels (Newland 2013). 

The failures appraised mainly results from the physical fatigues as well as the related considerations which have the likelihood of affecting the operations of the system as a whole. The analysis incorporates the evaluation and examination of the maximum peak and the amplitude vibration. The amplitude tends to pas via the forced resonances and must operate within the given certain range of the system. In this process, the steady-state solution must be included and analyzed well in the process, and this analytical evaluation consists of the commonly extracted digits. Conversely, the system tends to have the linear motion in both cases, and therefore, the only parameter often considered in the process is the frequency. The analysis of the spectrum includes the steady-state and particular solution appraisal.

From the equation, both the steady-state and the peak vibration for the amplitude as well as the phase angle mainly computed as a result of the sinusoidal variation in line with the force magnitude in the process (Scarpa et al. 2016). Moreover, the frequency, spring stiffness, mass as well as damper coefficient mainly included in the appraisal as the functions of sinusoidal elements, in solving the equation. Notably, the design engineer has the critical role of ensuring that the process is appraised and evaluated decisively at all times and in a parametric manner without underestimated any of the outlined elements. This is important as the approach helps in increasing both the workability and the frequency of the operations in the process (Wang and Wang 2016). 

This is both an essential and fundamental concept mainly used in the provision of the mechanisms of vibration analysis for the dynamic stability. In the practical aspect in the element is applied, the assumption mostly is taken that the c ≥0. Conversely, the utilization of the overall negative damping is essential in handling the dynamic model interactions in which there is an energy source. This analysis mainly termed as the self-excited concept.  Moreover, there is the application of the multi-DOF models in the computation and the evaluation of the rotor – dynamic systems (Ansari, Gholami and Norouzzadeh 2016 p.177).

Concentrated Disc Mass Properties

According to Lallart et al. (2016) Self-Excited dynamic-Instability vibration design and the approach mainly applied in handling the computations regarding the operation conditions of the rotor as well as in exciting the vibrations associated with them. This is not only important but also essential in the designing and the daily operations of the magnetic rotor in the long run.  In the examining of the parts, the real values and the parts mainly result in the eigenvalue, and thus, the elements often effect into the positive, genuine pieces. The established equation is known to have the same values as those of the 1-DOF systems. The range for the response in line with this equation mainly indicated as shown below (Ashworth et al. 2016 p.63).

C<0and c²<4km

The equation above mainly termed as classical and it indicates the self-excited vibration case. However, the vibratory motion primarily depicted to contain exponential growth, and this is believed to have evolved contrary to the exponential decay which tends to change as indicated in the diagram below (Le Guen et al. 2016 p.28).

Notably, the keen note has to be taken in line with a safety factor for the rotor vibration and mechanism. Thus, the preclusion of the arrangement is essential and this mainly appreciated in the figure above. This factor is not only crucial in the rotor machinery analysis but also indispensable in the daily operations, and therefore the effects should not be underestimated at all cost.  Mostly, the norms have got a broader application in the dynamic instabilities systems of the rotor mechanisms.

This model gives the representation of the makeable radial –plane orbital which is often encompassed in the rotor motion.

From the figure, it is evidential that the radial plane x-y allows for the translation of the mass.  This system mainly designed in such a manner that there is a connection between the spring as well as the dampers. This connection primarily carried via the application of the radial forces and the related components (Gatti 2014). Moreover, the commonly used ingredients in the process include the unbalance forces and this mainly indicated in the figure above.

The critical role of the dampers and the springs mainly include their usability in the radially isotopes as well as in designing the shaft flexibility. This designing primarily worked in the parametric series and line with the pertinent damping and has to be in parallel flexibility. Notably, two motions equation tend to be decoupled with each other. Conversely, if there is no decoupling of y-motion, then the overall y-x orientations tend to have little alternation and disturbance. This results in physically unchanged models and often known as the modal coordinates. Naturally, it possible and usually clear that there is the possibility of configuring and coming up with the physical sets of the coordinates and this often tend to yield overall decoupling as far as the motion equations are concerned (BuchAcz, P?Aczek and WróBel 2014).

Gyroscopic Effect Explanation

 The attainment of the physical coordinates in the systems tends not to possible in most cases.  In essence, the process extends decisively to the lateral rotating vibrations system. Moreover, the imposition of the kx =ky =k as well as cx=cy=c result in the isotropic model.  The result of the process is that the system resultant can eventually be rotated to the parametric orientation in line with the plane x-y coordinates. Conversely, the models for such systems mainly depicted to be in the x-y adjustments. Notably, isotropic models in line with the rotating vibration models show that the overall radial stiffness values tend to be same provided that the operations are in the same radial directions in comparison to the radial damping.  Preferably, various studies have indicated that most of the radial bearing types have following fluid dynamical elements. These features produce some amount of lateral rotating vibrations which have the potential of coupling both in the orthogonal directions as well as in the crossing the system. The integration of the system  to produce a significant amount of the lateral rotating vibration mechanism results in the generation of the definitive version mainly termed as a 2-of system.

Furthermore, matrix forms part and parcel of the analysis for the lateral vibration system in line with the seals and the bearings. Mostly, they are being viewed as fundamental inputs in the investigation. In essence, there is extensive research on increasing their quantification context in the parallel rotating vibration systems.  The diagonalization of the coefficient matrixes often difficult to be computed as individual data but have to be performed as multiples rather than just carrying them as single x-y coordinates.  Moreover, both the seal as well as the bearing stiffness and their overall coefficient often depicted as nonsymmetrical. Furthermore, the matrices associated with the damping coefficients also viewed and appraised as nonsymmetrical. The overall nonsymmetrical elements mainly described when the systems tend to have fluid dynamical and related factors such as the fluid inertia in a significant level.

This is also an essential model used in the examination and evaluation of the lateral vibrations. The model utilizes the slender flexible and the uniform shaft which is often mounted on the disk. This critical role of using the model is to help in obtaining the shafts vibrations at the lowest critical speed.  The figure below illustrations action of lateral vibrations in line with the systems in planar view as well as the overall effects which it has in the long run. Also, figure forms the extension of the first one aims at helping in giving in-depth analogy and understanding on the bearing flexibility (Mohammadi et al. 2016  p.2207).  The identifications regarding the stiffness coefficient in this case mainly regarded as symmetrical. The best method for adopting in computing the mechanism with such parameters includes mostly the 4-DOF models.  This model primarily appraised and denoted as undamped. It means that the system will have makeable coupling existing between the k and the m matrices. This analysis mainly illustrated using these equations of motions. Thus, the comparison below demonstrates mostly the matrices computations in line with the 4-DOF systems

The equation above, mainly used in the computation of the moment stiffness. It is evidential that the above matrix mainly multiplied by the overall velocity vector and these results in some of the elements is zero. There are 16 elements in the system; however, it is clear that 14 of the results in zero. Moreover, the two existing nonzero elements in the given equation account for the matrix embody. These matrixes embody mainly defined as the gyroscopic disk effect. The elements mainly represented as the overall skew components in line with the symmetric arrangement.

Moreover, it is important to note that the obtained gyroscopic effect in the system is conservative and this means that it will retain the inertia in it and thus, no energy will be dissipated in the long run. From the analogy view, the nonzero stiffness mainly represents the stiffness matrix and the overall elements, and these are described as flexible shafts. The shafts often have the isotropic force as well as a moment which are in a position to respond to the changes in the orientations of the disk.  The changes in the system mainly recorded concerning the overall four coordinates (Neubauer and Wallaschek 2013 p.36).

The method used in the evaluation of the vibrations impacts in the lateral rotating vibrations mainly utilized was the simulation approach.

The results sections mainly divided into various sections as discussed below  (Ebrahimi, Barati and Zenkour 2018 p.512).

Notably, this was a symmetrically unbalanced shaft and therefore, there was no essence of graphing the displacements in the x and y directions. For the moments, it is evidential from the results that station two had greater value compared station 1 and this account for the overall displacement depicted by station 2. The value obtained from station two mainly established at 16.388mm as compared to the one for station 1 depicted at 1.897mm.

X-Displacement (mm) vs. Y-Displacement (mm)

The results indicated that the value will be depicted at the designated two perpendicular axes in line with symmetrical system. This is because the shaft used essentially used was symmetrically unbalanced.

From the figure obtained, the greatest displacement mainly obtained at critical speed

X-Displacement (mm) vs. Time (s)

 The graph below indicates the representation of the movements recorded in line with the x-plane with imminent time. The often displacements as indicated in figure mainly recorded at critical speed in line with the displacement amount. Also, the overall speed tends to increase as a makeable wave tends to be distributed outwards.

For the asymmetrically unbalanced data for station 1 and station 2, the following graphs were obtained in line with the analysis (Cremer and Heckl 2013). 

The shaft mainly used in the process mainly depicted as symmetrical unbalanced and thus, the curves obtained mainly demarcated as oval shaped. These curves were estimated not to be perfect circles.  Different colors mainly used and this helped in developing comparison in line with station 1 and station 2 data (Ebrahimi, Barati and Zenkour 2018 p.512).

Also, the curves given below mainly indicates the analysis regarding the overall analysis for the

A graph of X-Displacement vs. Time (s)

Also, the graph, depicting the results for the Y-displacement verses time in seconds mainly recorded as follows

Preliminary Conclusions

In summary, it is important to state that this research covered various sections. These sections included introduction, literature reviews, theoretical modeling and backgrounds, computer program calculations as well as results and experiment.   Additionally, the introduction mainly covered the vibration concept as the various aspects which it has in line with the unbalance and equilibrium elements in line with the shaft vibration. The research for this mainly was to give an in-depth understanding on the dynamic forces. Also, the analogy helped in give analysis on the machine life, safety hazards as well as the related costs which these effects have on the organization as a whole. On the other hand, the literature reviews mainly focused on the academic research conducted on the various shaft vibration and unbalance areas. In the process, the past studies mainly reviewed and appraised accordingly.

Furthermore, analysis on both the asymmetrical unbalanced as well as symmetrical balanced mainly evaluated and this were expressed with respect to the below conditions

Subsequently, the experimental analysis was also performed. The experiment was vital and conducted in line with the electronic balancing.  The mainly objective of carried out the task was  to established the balance recorded when the masses  are added to the rotating disk .these masses mainly added at various weights and at different locations. In the process, considerations were incorporated in the design. In essence, the vibration amplitude mainly reduced to the parametric 4.8mm using the 10g mas and this was mainly designated at a distance of 25 radians.

Conversely, the computer program utilized for the study mainly incorporated the symmetric system. The program was set and the input data keyed in as well as the output program. The output mainly set in line with the response positions alongside phase and amplitude.

The results for the experiment mainly expressed in graphical forms. The graphs utilized mainly included the speed against time, and displacement verse time. These graphs were plotted as those of symmetrical system and asymmetric system. These results obtained from the graphics indicated that the most circles depicted were concentric in the shape form as a result of the symmetrical unbalance nature of the system. The vibration conducted at the lowest velocity set, demarcated to be the overall least in line with others. The same results were obtained for the graph of displacement against time.

MONTHS	1STMONTH NOV,2018	2NDMONTH DEC,2018	3RDMONHT JAN,2019	4THMONTH FEB,2019	5THMONTH MAR,2019	APRIL, JUNE,2019
Project initiation
Project system analysis
Project system design
Project implementation
Documentation and reporting

References

Ansari, R., Gholami, R., and Norouzzadeh, A., 2016. Size-dependent thermo-mechanical vibration and instability of conveying fluid functionally graded nanoshells based on Mindlin’s strain gradient theory. Thin-Walled Structures, 105, pp.172-184.

Ashworth, S., Rongong, J., Wilson, P. and Meredith, J., 2016. Mechanical and damping properties of resin transfer moulded jute-carbon hybrid composites. Composites Part B: Engineering, 105, pp.60-66.

Avallone, E. and Baumeister, T., 2017. Mark’s standard handbook for mechanical engineers. McGraw-Hill.

Barati, M.R., Shahverdi, H. and Zenkour, A.M., 2017. Electro-mechanical vibration of smart piezoelectric FG plates with porosities according to a refined four-variable theory. Mechanics of Advanced Materials and Structures, 24(12), pp.987-998.

Benaroya, H., Nagurka, M. and Han, S., 2017. Mechanical vibration: analysis, uncertainties, and control. CRC Press.

Bies, D.A., Hansen, C. and Howard, C., 2017. Engineering noise control. CRC press.

BuchAcz, A., P?Aczek, M. and WróBel, A., 2014. Modelling of passive vibration damping using piezoelectric transducers–the mathematical model. Eksploatacja i Niezawodno??, 16.

Crandall, S.H. and Mark, W.D., 2014. Random vibration in mechanical systems. Academic Press.

Cremer, L. and Heckl, M., 2013. Structure-borne sound: structural vibrations and sound radiation at audio frequencies. Springer Science & Business Media.

Ebrahimi, F. and Barati, M.R., 2017. Hygrothermal effects on vibration characteristics of viscoelastic FG nanobeams based on nonlocal strain gradient theory. Composite Structures, 159, pp.433-444.

Ebrahimi, F. and Barati, M.R., 2017. Small-scale effects on hygro-thermo-mechanical vibration of temperature-dependent nonhomogeneous nanoscale beams. Mechanics of Advanced Materials and Structures, 24(11), pp.924-936.

Ebrahimi, F. and Jafari, A., 2018. A four-variable refined shear-deformation beam theory for thermo-mechanical vibration analysis of temperature-dependent FGM beams with porosities. Mechanics of Advanced Materials and Structures, 25(3), pp.212-224.

Ebrahimi, F., Barati, M.R. and Zenkour, A.M., 2018. A new nonlocal elasticity theory with graded nonlocality for thermo-mechanical vibration of FG nanobeams via a nonlocal third-order shear deformation theory. Mechanics of Advanced Materials and Structures, 25(6), pp.512-522.

Gatti, P.L., 2014. Applied Structural and Mechanical Vibrations: Theory and Methods. CRC Press.

Inman, D.J., 2017. Vibration with control. John Wiley & Sons.

Jing, D., Yan, Z., Cai, J., Tong, S., Li, X., Guo, Z. and Luo, E., 2018. Low-1 level mechanical vibration improves bone microstructure, tissue mechanical properties and porous titanium implant osseointegration by promoting anabolic response in type 1 diabetic rabbits. Bone, 106, pp.11-21.

Kheibari, F. and Beni, Y.T., 2017. Size dependent electro-mechanical vibration of single-walled piezoelectric nanotubes using thin shell model. Materials & Design, 114, pp.572-583.

Lallart, M., Yan, L., Richard, C. and Guyomar, D., 2016. Damping of periodic bending structures featuring nonlinearly interfaced piezoelectric elements. Journal of Vibration and Control, 22(18), pp.3930-3941.

Le Guen, M.J., Newman, R.H., Fernyhough, A., Emms, G.W. and Staiger, M.P., 2016. The damping–modulus relationship in flax–carbon fibre hybrid composites. Composites Part B: Engineering, 89, pp.27-33.

Mohammadi, M., Safarabadi, M., Rastgoo, A. and Farajpour, A., 2016. Hygro-mechanical vibration analysis of a rotating viscoelastic nanobeam embedded in a visco-Pasternak elastic medium and in a nonlinear thermal environment. Acta Mechanica, 227(8), pp.2207-2232.

Naudascher, E., 2017. Flow-induced Vibrations: an Engineering Guide: IAHR Hydraulic Structures Design Manuals 7. Routledge.

Neubauer, M. and Wallaschek, J., 2013. Vibration damping with shunted piezoceramics: Fundamentals and technical applications. Mechanical Systems and Signal Processing, 36(1), pp.36-52.

Newland, D.E., 2013. Mechanical vibration analysis and computation. Courier Corporation.

Piersol, A.G. and Harris, C.M., 2017. Harri’s Shock and Vibration Handbook Fifth Edition. Mcgraw-hill.

Scarpa, F., Smith, C.W., Miller, W., Evans, K. and Rajasekaran, R., Rolls-Royce PLC, 2016. Vibration damping structures. U.S. Patent 9,382,962.

Sonoda, N. and Toyozawa, Y., FANUC Corp, 2018. Drive apparatus comprising mechanical vibration suppression function, and system comprising mechanical vibration suppression function. U.S. Patent Application 15/711,466.

Wang, C.Y. and Wang, C.M., 2016. Structural vibration: exact solutions for strings, membranes, beams, and plates. CRC Press.

Zhou, X.Q., Yu, D.Y., Shao, X.Y., Zhang, S.Q. and Wang, S., 2016. Research and applications of viscoelastic vibration damping materials: a review. Composite Structures, 136, pp.460-480.