Introduction

In the last stage of university level training, where the mechanical area is cultivated through careers such as Mechanical, Electromechanical, Mechatronics, or Industrial Engineering, the study of mechanical vibrations is encouraged as a specialty, either in mechanical design or diagnosis. This need is due to the widespread use of rotating machines at an industrial level, ranging from rotating motors weighing a few grams to turbines that can reach several tons; however, at an educational level, theoretical training does not necessarily reflect the industrial need.

Prototype development has been commonly used in the engineering field to attempt to measure how much a student has achieved in a particular competency, as well as their previously acquired knowledge; however, due to the high cost of obtaining a commercial prototype, functional prototypes are commonly made that might not be safe for continuous classroom or laboratory use. The maturity of rapid prototyping technology and its consequent cost decrease has led different universities and individuals to acquire 3D printing equipment for use in generating prototypes that help stimulate students and motivate them to see their initial designs when using this developing technology for educational purposes, as stated in [1]. The use of 3D stereolithography printing for the manufacture of a variety of mechanical equipment is presented in [2], where knowledge acquired in previous courses in mechanical and thermal stress analysis, CAD drawing tools, finite element simulation, plastic injection manufacturing processes, mechanical joints, and others are applied together. In [3], additive manufacturing is used for the creation of training robots to improve the teaching–learning process for master's students in robot design. Along the same lines, [4] presents how Nanyang Technology University adjusted its curriculum towards the precision manufacturing industry and emerging nanotechnology, as in 2004, this branch represented 28% of Singapore’s gross domestic product. In [5], Ahmad and Sosa present a case study in which full-scale prototypes were developed with students in their final semesters from the interior design, graphic design, visual arts, and multimedia areas, interacting with technological tools such as numerical control, welding, sheet bending, and 3D printing machines, allowing participants to develop practical skills that they will require in their careers, resulting in a high degree of commitment on the part of students to obtain 80% of the finished products. However, this is limited by the relatively small number of students who can be trained in this manner. In [6], the authors proposed starting the vibration and control laboratory equipment via projects developed by students who are taking the subject, using specific analysis systems studied during the course, and found an improvement in learning compared to theoretical teaching of the subject. The use of simulations and experimentation is presented in [7], mainly in the control area, as a tool to improve the teaching of historically theoretical courses, achieving an increase in retention of the field’s concepts, as well as in problem-solving ability. Likewise, a mechanical system was developed in [8] to analyze the critical speed and imbalance effects in a Jeffcott rotor coupled to an electric motor with a flexible coupling. In [9], the analytical, numerical, and experimental study of the concept of stress by mechanical deformation is illustrated, which will allow generating significant training in machine element design or related subjects, with application at the undergraduate and postgraduate levels. Likewise, an analysis of the Computer Science area in a Malaysian University is presented in [10], which proposes an online tool that allows increasing the number of passes in Introduction to Programming. Along the same line of work, [11] explains an analysis that allows knowing the learning styles applied in introductory programming courses, finding that the Soleman-Felder model is the most widely used in this area. In [12], the design, development, and implementation of a complementary tool for onsite classes is presented to improve the learning process at the Modibbo Adama Technological University in Yola, Nigera; this was applied to third-year Computer Science students, and the results show an increase in the student’s use of knowledge development in comparison to the traditional methodology (fully in person). Related to the original topic, another area of interest where practical training in students is high is the field of robotics, where the use of commercial computer packages is often chosen for instruction and learning. In [13] and [14], two methodologies for mapping the development and application of software tools commonly used in robotics are discussed, finding that the largest number of articles published focus on software design and construction and also presenting the databases where the relevant research articles can be found. In [15], an empirical study is presented through application in the industrial and academic sectors for the development of software applied to robotics, identifying the C +  + and Python languages as the most widely used. Additionally, [16] discusses a proposal for the design of the necessary libraries to be developed in the control software of a robotic system with industrial or training applications. Expansion of the development of graphical interfaces in other areas of mechanics which seek to improve undergraduate learning in the subject should be highlighted, such as mechanism analysis, as stated in [17], by developing a user interface with Matlab software to implement the kinematic equations that govern a four-bar crank-rocker mechanism. This type of mechanism is used in [18] at the training level to know the performance of a sliding mode movement control when using a direct current (DC) motor. Likewise, simulation of the application of a PID control in the dynamic equations of the crank-rocker mechanism is presented in [19], considering a DC motor with a gear speed reducer. In [20], the input speed variation in a crank-rocker mechanism using a CVT transmission is proposed, this being the link between an alternating current (AC) motor and the 4-bar mechanism, using the simulation to present the conclusions. [21] presents the design of a rehabilitation prototype using a 5-link planar mechanism made with additive manufacturing, while [22] uses the educational level mathematical tool called GeoGebra to determine the displacement, velocity, and acceleration parameters of a 4-bar mechanism. In [23], the SolidWorks tool is used to determine the speed, acceleration, and output force of two coupled crank-rocker mechanisms. The same type of mechanism is used in [24] when applying the Lagrange method to obtain the corresponding kinematic equations and their simulation using Matlab's Simulink without presenting some type of verification of the characteristic patterns obtained. In [25], a specific approach for the balancing of a crank-rocker mechanism is shown using the optimization process called Differential Evolution when using Cartesian coordinates, while in [26], the experimental balancing procedure in a crank-rocker is studied, considering an elastodynamic behavior in the links. A review of the state of the art for the dynamic balancing of mechanisms is presented in [27].

Therefore, for courses focused on mechanical vibrations in rotating systems at the undergraduate level, an area of opportunity is identified by designing a low-cost training rotor for balancing training as well as the development of the methodology via the influence coefficient method on a plane.

Low-Cost Experimental Rotor Design

In higher education, in final semester academic training courses, where the objective is to study the phenomenon of mechanical vibrations in rotating systems, not only is the theoretical part required, but also the corresponding laboratory, where practical study of the subject is developed and evaluated. However, due to the high costs involved in training prototypes, and since these are subjects at the end of professional training, it is difficult to obtain the necessary resources and equipment. For example, an RK-4 training rotor from Blenty Nevada was quoted at $21,176.00 USD (plus tax) in 1984, not considering any type of electronic instrumentation. This represented a remarkably high cost for many higher educational institutions at in Mexico for training in a very specific area.

This work proposes the development of a lower-cost training rotor than those commercially available. To do this, the elements shown in the following table will be used (Table 1).

Table 1 Experimental rotor elements
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As a fundamental element, a commercial grinder is used, as it is an easy to locate and acquire electromechanical device, as well as for its ease of starting and stopping. An illustrative image is presented below.

This commercially available device has the following advantages:

  • It is available from local hardware stores, commercial franchises (for example: Home Depot), or online retailers (Amazon, Mercado Libre).

  • It allows the user to start and stop the system at will.

  • It has two natural frequencies in its working range (0 to 3600 rpm) in its original form (see Fig. 1).

  • It can be trained for balancing on one and two planes.

  • Reducing speed due to friction during shutdown allows each of the two natural frequencies to be audibly detected; however, the system does not vibrate during fast startup when it crosses each of the natural frequencies.

  • It is low cost.

Fig. 1
figure 1

3,600 rpm commercial grinder

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Drawing of the balancing planes, which are placed where the whetstones are currently located, are carried out in parallel with those of the commercial grinder. Care must be taken to match the mass of the grinding stone to that of the balancing plane to be machined to avoid significantly affecting the rotor’s two natural frequencies in its speed range. Thus, the following figure shows the standardized schematic based on [28] (Fig. 2).

Fig. 2
figure 2

Schematic of a balancing plane

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As can be seen in the figure above, the balancing plane contains 28 holes that will allow (correction or imbalance) weights to be placed at a distance of 12.85°, which generates a low-cost rotor by coupling the balancing planes to the commercial grinding wheel to provide students with specialized training in their final round of univeristy courses. The final rotor is shown in the figure below:

The rotor shown in Fig. 3 allows performing balancing exercises (modal or influence coefficient) and vibration fault diagnosis (imbalance level or of initial problem detection in the various bearing components), as well as the identification of natural frequencies during rotor stop.

Fig. 3
figure 3

Training balancing rotor

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The proposed device generated a total cost of $6,830 USD, which represents 1.6% (considering an exchange rate of $20 MXN to the dollar) of the cost of the commercial training rotor available for balance training. Note that the commercial training rotor has the capacity to simulate misalignment faults, imbalance, faulty bearings, natural frequency identification, and hydrodynamic instability.

Balancing Method by Influence Coefficient on a Plane

In general, influence coefficient and modal balancing methods are used in the rotor balancing process, as well as a method that blends the two. The influence coefficient method allows calculating the correction weights from the cause-and-effect phenomenon in a mechanical system, that is, by placing a known test weight and observing the reaction in vibration amplitude to that weight, while the modal balancing method requires knowing the modal shapes and modal parameters to perform the same process. Therefore, the modal method requires that the analyst have a deep knowledge of the dynamics of rotating systems, while influence coefficient does not; however, the latter method is widely used today for balancing in the field and employed in commercial tools for the balancing process. Part of the mathematical development for balancing is presented in [29,30,31,32] for rotors with two or more planes; however, the information presented omits information that only a balancing specialist would know, which complicates training undergraduate students.

This method is based on consideration of a rotor’s linear behavior when known test weights are added, generating a proportional vibration amplitude, as shown in the figure below.

Figures 4 and 5 shows that placing a known initial weight (\(W_{0}\)) generates an initial vibration amplitude (\(V_{0}\)). Subsequently, a new test weight (\(W_{1}\)) is placed when removing the previous weight, which generates a vibration with a magnitude of (\(V_{1}\)); a relationship between these parameters—known as the influence coefficient—can be found, as expressed in the following equation:

Fig. 4
figure 4

Basic concept of the influence coefficient method

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Fig. 5
figure 5

Systems diagrams of a plane

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$$\alpha = \frac{\Delta V}{\Delta W}= \frac{{V}_{1}- {V}_{0}}{{W}_{1}-{W}_{0}}$$
(1)

A methodology for balancing the rotor on a plane at working speed was developed in a university course on Mechanical Vibrations through the research work of a graduate student, as presented in [33]. This is done by starting from the image below, which allows locating the placement of the vibration measurement sensors and the balancing plane.

Prior to the balancing process, the equipment and the rotor bearings must be confirmed to be in good condition. To do this, the instructor must perform a start and stop run with the necessary safety elements. With the rotor turned off, rotor instrumentation is carried out by placing the vibration sensors (displacement, speed, or acceleration) as well as the optical tachometer connected to the acquisition cards. Likewise, it is essential that the phase angle (equal to zero at low speed) coincide with the position of the reflective tape.

The rotor is started at working speed, and the speeds in amplitude and phase angle of the sensor S1 and S2 are recorded, identifying the signal that generates data with the highest value. It will be assumed that the sensor S1 is used, which measures the vibration amplitude \({V}_{a1}\) with its corresponding phase angle \({\varphi }_{a1}\). The original vibration amplitude is then resolved into its real and imaginary components, as shown in the following equation:

$$\begin{array}{l}{V}_{a1{\text{Real}}}={V}_{a1}\mathit{cos}{\varphi }_{a1}\\ {V}_{a1{\text{Imag}}}=i{V}_{a1}sen{\varphi }_{a1}\end{array}$$
(2)

where \({V}_{a1{\text{Real}}}\) Y \({V}_{a1{\text{Imag}}}\) are the real and imaginary components of the real vibration amplitude. Then, the rotor is stopped, and a test weight is placed on plane 1, recording its magnitude \({W}_{1}\) and its angular placement position \(\beta\). Breaking this down into its real and imaginary components returns

$$\begin{array}{l}{W}_{1{\text{Real}}}={W}_{1}\mathit{cos}\beta \\ {W}_{1{\text{Imag}}}=i{W}_{1}sen\beta \end{array}$$
(3)

With the test weight placed on plane 1, the rotor is brought to its working speed and the vibration amplitude is remeasured to determine the degree of impact on the dynamic response; then, the variable \({V}_{d1}\) and its phase angle \({\varphi }_{d1}\) are recorded to subsequently obtain their corresponding components again, as shown below.

$$\begin{array}{l}{V}_{d1\mathit{Re}al}={V}_{d1}\mathit{cos}{\varphi }_{d1}\\ {V}_{d1{\text{Imag}}}=i{V}_{d1}sen{\varphi }_{d1}\end{array}$$
(4)

where \({V}_{d1\mathit{Re}al}\) and \({V}_{d1{\text{Imag}}}\) are the real and imaginary components, respectively, of the vibration amplitude after placement of the test weight.

With the expressions in equations (1) to (4), the influence coefficient \(\alpha\) is calculated using the following expression:

$$\alpha =\frac{\Delta V}{\Delta W}=\frac{\left({V}_{d1\mathit{Re}al}+{V}_{d1{\text{Imag}}}\right)-\left({V}_{a1\mathit{Re}al}+{V}_{a1\mathit{Im}ag}\right)}{\left({W}_{1{\text{Real}}}+{W}_{1{\text{Imag}}}\right)}$$
(5)

The expression in equation (5) is known as the influence coefficient in its real and imaginary components, and this parameter has a permanent relationship with this rotor for subsequent balancing processes. This relationship represents the rotor response in terms of vibration amplitude given a known weight.

From this step, it is possible to determine the correction weight to place on plane one, by considering the previous expression, where it is redefined as

$${P}_{1}=-\frac{\left({V}_{a1\mathit{Re}al}+{V}_{a1\mathit{Im}ag}\right)}{\alpha }$$
(6)

where \({P}_{1}\) is the correction weight on plane one in its real and imaginary parts, the term \(\left({V}_{a1\mathit{Re}al}+{V}_{a1\mathit{Im}ag}\right)\) is the original vibration amplitude without any trial weight, and \(\alpha\) is the calculated influence coefficient, while the negative sign means that the correction weight will be placed at 180° from the imbalance weight.

However, the calculated weight is on the Real-Imaginary plane, which has no meaning for physical placement of the correction mass on plane 1. To determine the Real-Real plane, the correction weight uses the expression

$${P}_{Total{P}_{1}}=\sqrt{{\left({P}_{1{\text{Real}}}\right)}^{2}+{\left({P}_{1{\text{Imag}}}\right)}^{2}}$$
(7)

where \({P}_{Total{P}_{1}}\) is the correction weight to place on plane 1, thus \({P}_{1{\text{Real}}}\) and \({P}_{1{\text{Imag}}}\) are the real and imaginary components of the variable \({P}_{1}\),while the correction weight will be placed at an angle given by

$$\gamma =t{g}^{-1}\left(\frac{{P}_{1{\text{Imag}}}}{{P}_{1{\text{Real}}}}\right)$$
(8)

Consider that, for placement of the correction mass, it is necessary to start from the zero generated by the reflective tape and in the direction opposite to the rotation of the rotor.

In the event that the calculated correction weight reduces the vibration level, then the calculated influence coefficient can be used permanently for the balancing process with only the original measurement. However, when the rotor undergoes a significant change during the maintenance process or a change of supports or foundations, a new influence coefficient must be determined.

If, when balancing the first run, the vibration levels are not within the standard used, then the same procedure must be applied again with the calculated influence coefficient until the levels are within the range determined by the specific company or standard.

The proposed methodology was validated using a commercial device given an installed instrumentation, as shown in the figure below (Fig. 6).

Fig. 6
figure 6

Instructional rotor instrumentation

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CTC Industrial sensors with a sensitivity of 200 mV/g were used for this case study, as well as an optical tachometer to determine rotor speed and the phase angle of the vibration vector. The equipment used was the DSP Logger MX 300 model, whose technical characteristics are shown in [34].

The data presented in the table below were acquired by following the procedure presented above (equations (2)–(8)) (Table 2).

Table 2 Vibration amplitude parameters at running speed
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Using a programming tool to implement the one-plane balancing methodology, the correction weight is found to be \({P}_{Total{P}_{1}}=2.3281 grams\) placed at \(\mathrm{\angle }-47.2507^\circ\).

Meanwhile, the MX 300 DSP Logger device returned the information presented in the following figure.

The correction weight in the figure above is \(2.3131grams\) placed at \(-47^\circ\); the difference in mass between the proposed methodology and that used for commercial equipment is 0.64%, while it is 0.53% at the placement angle. This difference may be due to the DSP Logger MX 300 device using integer values in its display.

The commercial equipment provides the size of the weights and the corresponding hole into which the mass must be placed, in such a way that its vector sum is equal to the original. The balancing weights presented in Fig. 7 were difficult to apply, so a 1.1-g correction weight was placed for hole 25, and an 0.8-g weight for hole 26. When remeasuring with these weights, the final vibration amplitude is \(0.9413\frac{mm}{seg}a\angle 57^\circ\), which represents a 64.45% decrease in comparison to the original vibration amplitude (\({V}_{a1}=2.6479\frac{mm}{seg}\)). The behavior is shown graphically in Fig. 8.

Fig. 7
figure 7

Correction weight of commercial balancing equipment

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Fig. 8
figure 8

Final vibration amplitude in the first balancing run

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Therefore, the calculated influence coefficient is considered a significant constant and is used for further reduction of the vibration level, as well as for further balancing of the same rotor.

The results of the proposed methodology and those obtained using a commercial tool validate using procedure proposed in this research work in the classroom for rotor balancing training considering a balancing plane and using the influence coefficient method in undergraduate Mechanical Vibrations lectures.

Survey Analysis

During the pandemic, in the January-May 2021 semester, the subject denominated "Mechanical Vibrations" was taught online to a group of 15 students, with 13.3%, answering a survey, while the class was taught in person during the January-May semester to a group of 13 students, 61.5% of which responded to this instrument.

The methodology presented in equations (1)–(8) of the previous section was developed specifically in the online course: Studying balancing using influence coefficients on a plane. Validation was carried out by solving balancing problems with data stored from runs from previous courses, such as that shown in Fig. 7.

Meanwhile, the January-May 2022 group used the equipment presented in Fig. 9 below.

Fig. 9
figure 9

a Undergraduate student receiving training on the DSP Logger MX 300 equipment, b) data acquisition window for single-plane balancing

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Figure 9 (a) shows the training rotor in operation during the training process, with an imbalance weight being placed arbitrarily, while Fig. 9 (b) shows the system's acquisition interface with the test weight.

After this second analysis group’s normal semesters, a survey was developed and the two study groups were asked to respond, with the interest group which had most recently received the training showing greater participation in filling out the survey.

Therefore, the answers from the first analysis group respondents are presented in the table below.

The above table presents the survey results from an Electromechanical Engineering group studying the eighth semester specialization course entitled "Mechanical Vibrations" online during the August-December 2021 semester. Access to the institutional laboratory was not allowed at that time, so it was not possible to carry out the hands-on balancing training. 100% of the students surveyed consider hands-on training and the application of an international standard necessary for mechanical vibration control. The same instrument was applied during the January-May 2022 semester, obtaining results similar to those presented in Table 3.

Table 3 Group-cunducted survey in online class
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The above table shows that student perception is that training with balancing equipment which they may see applied to rotating systems diagnosis or balancing in the regional industry is fundamental, as is making the theoretical result converge with that obtained in industrial equipment. However, this way of working in subjects which are to be promptly applied to solving problems in the industry, due to the instructor’s high training cost and low resources for the renewal of industrial balancing equipment, means that it has been in use for 7 years as of today. Likewise, the latter is an area of opportunity for researching and developing low-cost equipment at the same University.

Conclusions

It is common to find the subject of Mechanical Vibrations in terminal degrees related to mechanics, and one approach offered in the subject is the balancing of rotating systems. However, to develop the technical skills needed in this critical process (a poorly placed correction weight can increase vibrations to levels so high that equipment may fail at working speed or during startup and shutdown), it is necessary to complete 3 stages: obtain a training balancing rotor, plan a clear balancing methodology, and acquire the necessary instrumentation (vibration sensors, optical tachometer, acquisition and processing cards, and interpretation software). However, the high cost of completing the necessary stages can barely be covered by public universities. Another additional problem is the reduced number of students who take the subject, since it is commonly offered as an elective. A 2014 quote for a training rotor was priced at $21,176.00 USD, demonstrating how valuable this specialized training equipment is. However, as long as rotating equipment which requires maintenance is used in the industry, professionals specialized in the area of mechanical balancing will still be needed. Therefore, the subject begins with the design of a low-cost experimental rotor, whose approximate value is $6,830 MXN, with applications in training in balancing and identification of specific faults when analyzing mechanical vibrations. This reduction in cost is 98.4%, considering an exchange rate of $20 MXN to the dollar. In addition to the high cost of specialized training equipment, the use of bids is a federal requirement in national public universities, so the price can increase from 20 to 30% when an intermediary enters the acquisition process. The second necessary element has to do with the balancing methodology to be used, since this is difficult to find in specialized material in such a way that allows the proposed expressions to be applied practically in the balancing process; therefore, it is necessary to resort to commercial equipment with balancing programming, such as the equipment used in this article (DSP Logger MX 300), which generates little understanding from the point of view of engineering training. This article presented a plane balancing methodology for using influence coefficient at the system’s working speed. Therefore, there is a significant advance that allows supporting training in specific skills in the area of mechanical vibrations, such as balancing rotating systems.