A Numerical Systematic Review and Meta-Analysis of Diagnosing the Vibration Modes of the Cylindrical Shell in the MRI Machine

Abstract

Magnetic Resonance Imaging (MRI) is a non-invasive imaging method that utilizes radio waves and magnetic fields. This study focuses on reducing the acoustic noise produced inside the cylindrical shell of the scanner, where the patient is located. Vibration modes are generated by eddy currents in the cylindrical shell induced by gradient magnetic fields. Additionally, the scanner wall is typically joined to the gradient spiral cylinder, causing vibrations to be transmitted to the wall and thereby producing extra sound waves. The present study investigates methods for mitigating noise from the scanner wall and reducing the transmission noise from the spiral gradient cylinder. Numerical methods and practical solutions for lowering acoustic noise in MRI gradient coils are explored. A 20 mm uniform absorber is demonstrated as an effective design for significantly reducing acoustic noise in the frequency range 0 to 3 kHz. Finally, numerical analysis of gradient cycles yields solutions that lower both vibration and noise levels.

Keywords

MRI numerical modeling vibrational modes cylindrical shell (CS)nonlinear

Introduction

In the medical field, magnetic resonance imaging (MRI) is one of the essential methods used to diagnose diseases and disorders within the human body. In this method, magnetic receptors and radio frequencies are employed to obtain accurate, 3-dimensional images of the body’s tissues and structures. However, the operation of the MRI machine can also induce significant vibrations on the surface of its cylindrical shell (CS). It is therefore necessary to model and analyze these vibrations in order to improve and control them within the MRI system.

For this purpose, the study aims to theoretically and practically analyze the cylindrical shell model of the MRI machine using various analytical methods, including the Finite Element Method (FEM). In addition, the research will investigate how different factors affect the vibrations of the cylindrical shell and explore methods to control them. Given that the MRI machine is one of the most widely used imaging devices, many manufacturers are seeking ways to reduce and manage these vibrations. Thus, the findings of this research can serve as a valuable guide for MRI machine manufacturers aiming to enhance their performance and efficiency.

This study will also explore the theoretical foundations and research background behind the analytical calculation of vibration modes in the MRI machine’s cylindrical shell. It discusses the fundamental concepts of structural mechanics, the analysis of cylindrical shells, numerical and experimental solution methods, and the operational principles of MRI. By reviewing a range of sources, the study examines previous research on the analytical calculation of vibration modes in MRI cylindrical shells. Cylindrical shells have attracted the attention of many engineers and scientists due to their crucial industrial applications, and numerous studies have been conducted in this field. These investigations can be classified in various ways. In the following sections, research on cylindrical shells will be reviewed in 4 segments, and the strengths and weaknesses of each approach will be discussed. Figure 1 illustrates the main components and operational algorithm of an MRI machine.

Figure 1.

Main components and operation algorithm of an MRI machine.

In modern medicine, Magnetic Resonance Imaging (MRI) is a crucial method for diagnosing diseases and disorders in the human body. However, the operation of an MRI machine can also induce severe vibrations on the surface of its cylindrical shell. Therefore, it is necessary to model and analyze these vibrations to enhance and control them effectively. This study aims to examine both the theoretical and practical models of the MRI machine’s cylindrical shell using various analytical methods. In addition, the influence of different parameters on the shell’s vibrations—and methods for mitigating them—will be investigated. Since MRI machines are among the most widely used imaging devices, many manufacturers are seeking ways to reduce these vibrations. The results of this research can thus serve as a valuable guide for improving the performance and efficiency of MRI systems.

Free Vibrations of CSs with Simple Support Conditions

Livanov studied CSs with axial symmetry and simple support boundary conditions. In the equations presented for the shell’s movement, the assumption of axial symmetry meant that changes in the peripheral location were not considered, resulting in a system of only 2 motion equations. Assuming harmonic motion, he derived 4 relations for the natural frequency of the shell, corresponding to two-to-two correlations.¹

The assumption of axial symmetry reduces the equations of motion from 3 to 2, thereby simplifying the process of solving for the natural frequencies. However, this simplification compromises accuracy and fails to provide a complete description of the shell’s vibration behavior.

Rinehart and Wang studied the vibrations of CSs with simple support boundary conditions, assuming that the shell was equipped with longitudinal stiffeners. They employed the energy method to analyze the vibrations and represented displacements in the radial, circumferential, and axial directions using a Fourier series that satisfied the boundary conditions. Subsequently, by employing Lagrange’s equations and incorporating beam functions for the longitudinal wavenumber, they derived a relation for the natural frequency.² The resulting frequency relation specifies only the radial frequency, which is typically the lowest natural frequency; consequently, only the radial mode shapes can be investigated.

Reinforcements can also be installed around the perimeter of the shell in the form of ring-stiffened reinforcements. Beskos and Oates³ conducted a dynamic analysis of CSs with ring reinforcements.

Mustafa and Ali⁴ continued previous work by investigating the vibrations of CSs with the amplifier, incorporating the rotational inertia effect in the equations related to the amplifier. In the presented theory of CSs, the influences of shear deformations and rotational inertia are neglected; in other words, Euler-Bernoulli beam theory is used to derive the equations.

Soedel, one of the renowned scientists in the field of plate and shell vibrations, generalized Law’s theory to include the influences of shear deformation and rotational inertia in his calculations, presenting relations equivalent to those of Timoshenko’s beam theory. His research provided a straightforward formulation of the equations of motion for CSs by generalizing Lowe’s theory.⁵ Although this valuable work explains the method of obtaining the equations of motion, it does not address the determination of natural frequencies.

Suzuki and Leissa studied the free vibrations of CSs with variable thickness and obtained expressions for the natural frequencies and mode shapes of these shells. The governing equations for CSs with variable thickness consist of 3 differential equations with variable coefficients, and their solution method is analogous to that for CSs with constant thickness and curvature. Their results indicate that while the mode shapes are similar, the natural frequencies of the 2 types of shells differ significantly.⁶ In this study, beam functions served as substitutes for the boundary condition equations.

Sivadas and Ganesan⁷ further investigated CSs with variable thickness by considering 4 cases: asymmetric linear variations, symmetric linear variations, asymmetric parabolic variations, and symmetric parabolic variations.

In much of the research investigating the vibration behavior of CSs with variable thickness, only longitudinal thickness variations have been considered. For this reason, Zhang and Xiang studied the vibrations of stepped CSs. In this case, the shell is divided into several segments along its length, with each segment having a distinct thickness. A differential equation is written for each segment, and boundary conditions must be applied not only at the ends of the shell but also at the points where the thickness changes.⁸

Bhangale and Ganesan investigated the free vibrations of functionally graded material CSs. In their work, the variations in the circumferential and radial displacements of the shell were represented by trigonometric harmonic functions. By employing beam functions appropriate to simple support boundary conditions, they obtained natural frequencies for different modes and examined the effects of the length-to-radius and radius-to-thickness ratios on the natural frequency. The basic shell equations associated with the boundary conditions were not used directly; instead, corresponding beam functions were employed. Although the study explored the influence of geometric parameters on the frequency behavior, their effects on the mode shapes were not investigated.⁹

Kadoli and Ganesan further investigated the free vibrations of functionally graded material CSs with 2-end boundary conditions. In their work, the material properties were considered not only as varying slowly along the thickness but also as temperature dependent. They assumed that the outer surface of the shell was maintained at ambient temperature while the inner surface was at a higher temperature, formulated the 1-dimensional heat transfer equation along the thickness, and determined the temperature distribution in the shell. Then, using the finite element method, they calculated the critical temperature for thermal buckling.¹⁰

Several studies have also examined the vibration behavior of CSs on an elastic bed. Among these, Paliwal et al Chen et al, and Pellicano are notable. In these articles, the elastic bed is modeled using the Winkler/Pasternak models.^11
-13

In most articles, the longitudinal, radial, and circumferential displacement functions of the shell are modeled as harmonic functions. Pellicano presented a new proposal, he assumed that the displacement field is expressed as a combination of harmonic functions and Chebyshev polynomials. He tested this assumption on a CS with simply supported and 2-supported boundary conditions and compared the theoretical predictions with experimental and numerical data, he used ninth-degree polynomials in the calculations, yielding results that closely matched laboratory data. It is noteworthy that although the theory is highly accurate, achieving such precision necessitates the use of high-degree polynomials, which increases the computational effort.¹²

Another approach to modeling the shell’s displacement involves the wave propagation method, where the longitudinal, radial, and circumferential displacements are expressed as power functions. Xuebin used this method to study the free vibrations of CSs, presenting results for 3 different support boundary conditions: simply supported at both ends, single-end support, and 1 jointed end. In a continuation of this research, he employed beam functions in place of longitudinal wavenumbers for all 3 boundary conditions.¹⁴

Free Vibrations of CSs with Different Boundary Conditions

Lin and Bell used the solution for a CS with simple support conditions when 2 moments were applied at the boundaries as constraints, and the free vibrations of 2-headed CSs were verified with this approach. Also, the natural frequencies of a CS with simple support conditions served as a guide to determine those of the 2-headed CS.¹⁵

Tottenham and Shimizu applied the matrix expansion method to analyze the free vibrations of truncated shells. In this method, after incorporating the variations in the longitudinal, radial, and circumferential displacements into the differential equations of motion and nondimensionalizing the resulting expressions, a first-order differential equation in matrix form is obtained and solved using the described procedure.¹⁶

Askari and Daneshmand¹⁷ employed the integral equations technique to determine the natural frequencies of CSs.

Other methods for investigating the free vibrations of shells have also been explored. For instance, Lee et al¹⁸ used the Ritz method to obtain the natural frequencies and mode shapes of a CS. Later, Leissa et al¹⁹ applied the same method to study CSs with 2 different curvatures.

Ganesan and Sivadas, as described in the previous section, have conducted several studies on CSs with variable thickness. In addition to examining CSs with simple support conditions, they have also investigated conical shells. In another study, they analyzed the vibration behavior of this type of shell using the semi-analytical finite element method (FEM).²⁰

Annigeri et al²¹ also employed the semi-analytical FEM to study the effects of electric and magnetic fields on the vibration behavior of CSs with 2-end boundary conditions.

The asymptotic expansion method is another approach to studying the vibrations of CSs. In this method, Wong and Bush expanded the variations in the longitudinal, circumferential, and radial displacements, as well as the shell frequency, in terms of the thickness-to-radius ratio parameter. This method yields acceptable solutions for long, thin shells and for frequencies lower than the ring frequency.²²

The preceding sections provide a historical overview of work conducted in the field of free vibrations of CSs under various boundary conditions. Among the valuable references are the books Vibration of Plates²³ and Vibration of Shells,²⁴ which are widely consulted by researchers and engineers in pursuit of their scientific and research goals.

Nonlinear Vibrations of CSs

The equation of motion for cylindrical shells (CSs) consists of 3 coupled differential equations, which are typically treated as linear. When a nonlinear theory is applied to derive the equations of motion, even formulating the governing equations becomes extremely challenging. In the field of nonlinear vibrations of CSs, most work dates from the 1990s, whereas research on linear vibrations flourished in the 1970s and 1980s. One reason for this disparity is that solving nonlinear problems requires high-performance computing resources and advanced engineering software, in addition to a solid understanding of linear vibrations.

Among the early contributions in this field is the research by Dowell and Ventres,²⁵ who attempted to derive nonlinear equations for the bending vibrations of CSs. Atluri²⁶ employed the multiple time scale method to solve these nonlinear equations. Initially, he converted the equations with relative derivatives into ordinary differential equations using the Galerkin technique, thereby obtaining results for 3 modes. Birman and Bert²⁷ investigated the vibrations of long CSs subjected to large displacements. They approximated the vibrations of a CS as analogous to those of a beam, derived the nonlinear motion equation in the lateral direction, and calculated the transverse motion frequency using elliptic integrals. Chiba²⁸ conducted experiments to study the nonlinear vibrations of a cylindrical tank and examined the influence of various parameters on its behavior.

It was found that the degree of nonlinearity in oscillations depends on both the vibration mode and the tank’s length. In other words, the ambient and longitudinal wavenumbers affect the nonlinearity of the motion. Moreover, if the tank is filled with fluid, the degree of nonlinearity also depends on the fluid’s weight, Amabili et al²⁹ investigated the forced and free vibrations of CSs in contact with static fluid. Amabili et al, like Atluri,²⁶ employed the Galerkin method to reduce the governing equations to ordinary differential equations.³⁰ To further this research, they published additional studies using the same approach.^31
-33

Sun and Liu³⁴ investigated the effects of geometrical parameters—such as length, radius, and thickness—on the nonlinear vibrations of CSs. Similarly, Gonçalves et al³⁵ and Amabili et al³⁶ used this method to examine the vibration behavior of nonlinear CSs. Amabili³⁷ compared the nonlinear vibrations of CSs for well-known theories. Kurylov and Amabili³⁸ also explored, for the first time, the nonlinear vibrations of CSs under captive and free-end boundary conditions.

Jansen employed both analytical³⁹ and semi-analytical numerical techniques⁴⁰ to study the nonlinear vibrations of CSs. In these methods, an analytical technique such as the perturbation method is applied to solve the nonlinear equations, with numerical methods subsequently used to obtain the solution. Another approach is the averaging technique introduced by Popov for CSs; these methods are widely applicable to many nonlinear problems.⁴¹

In the previous section, works on the linear vibrations of CSs on an elastic bed were reviewed. The nonlinear case of this issue can also be investigated, as Bakhtiari-Nejad and Mousavi Bideleh used the disturbance method in their study. Owing to the substantial complexity and computational intensity of the perturbation method, most research simplifies the equations as much as possible or employs mathematical software to obtain a solution.⁴²

Furthermore, alternative methods and theories have been developed for investigating CSs, yielding more accurate solutions. For example, Leizerovich and Seregin⁴³ extended the shear deformation theory to higher orders to facilitate solving nonlinear equations. This approach can be applied to both the linear⁴⁴ and nonlinear⁴⁵ vibrations of CSs.

Among the contributions in the field of nonlinear vibrations of CSs, some works have been particularly influential and have served as the foundation for many subsequent studies. Notable examples include the comprehensive review by Amabili and Paı··doussis and Amabili’s book, Nonlinear Vibrations and Stability of Shells and Plates.^46,47

CSs with Composite Structure

Composite shells have been widely used in industry, and considerable research has addressed various aspects of using composite materials. Soldatos discussed the dynamic analysis of orthogonal composite multilayer cylindrical panels, expressing the free vibrations of these panels using thin shell theory in most formulas.⁴⁸ In his research, he presented numerical results and compared 4 well-known theories.

A few years later, Chandrashekhara and Kumar⁴⁹ applied a similar comparison to static response analysis. Messina and Soldatos investigated the influence of various boundary conditions on cylindrical plates and panels.^50,51

Case Study and Numerical Model Comparison

Accurate numerical modeling of the MRI scanner cylinder is essential for producing realistic predictions of vibration levels. The scanner cylinder can be analyzed using a comprehensive numerical modeling approach. The vibration levels generated by the body and head sections can be compared, and it is possible to investigate how the main field strength influences acoustic vibrations and overall vibration levels. Once these effects are understood, efforts can be focused on reducing the vibration levels.

All numerical simulations are performed in ANSYS using the finite element method. This design serves as a base model for high-performance human brain imaging. Figure 2 presents the dimensions of the single-layer MRI scanner cylinder structure, which is a circular shell.

Figure 2.

Dimensions of the single-layer gradient MRI scanner cylinder structure.

The finite element mesh of the cylindrical shell structure (Figure 3) is selected after checking the grid independence according to the parameters in Table 1. Also, the validation of the results is confirmed according to Table 2.

Figure 3.

Finite element mesh of gradient coil structure A single-layer gradient coil structure for validation with Li et al.⁵²

Table 1.

Specifications and settings for vibration analysis of the structural model.

Element size (m)	0.02
Maximum edge length (m)	1.88
Number of nodes	5856
Number of elements	5760
Element type	181 Shell
Maximum modes	1000
Minimum value	0 Hz
Maximum value	3000 Hz
Solver	Sysnoise

Table 2.

Validation of case study results.

Label	n = 10			n = 1
Circumferential mode m	Li et al⁵²	Present	%Difference	Li et al⁵²	Present	%Difference
1	30.45	28.6	6.08	29.44	29.6	−0.54
2	10.66	10.1	5.25	10.15	10.5	−3.45
3	6.1	6.2	−1.64	6.09	6.4	−5.09
4	7.92	7.8	1.52	7.41	8.2	−10.66
5	12.2	12	1.64	11.67	12.5	−7.11
6	17.77	17.4	2.08	16.75	18.2	−8.66
7	24.36	23.9	1.89	23.04	25	−8.51
8	32.5	31.4	3.38	30.45	32.9	−8.05
9	40.6	40	1.48	39	41.8	−7.18
10	50.77	49.5	2.5	48.73	51.8	−6.3

Sample Numerical Model Report

The properties of the linear elastic material were modeled using a Poisson’s ratio of 0.4, a density of 1600 $\frac{kg}{m^{3}}$ , and a Young’s modulus of 13 GPa. The air inside and outside the chamber was modeled as a pressure acoustic fluid domain with a density of 1.2 $\frac{kg}{m^{3}}$ and a sound speed of 343 m/s (see Figure 4 and Table 3). The coupling between vibration and noise was implemented in the simulation model. At both ends of the ducts, a hemispherical air volume with a radius of 1 m was added to simulate sound wave propagation outside the chamber. A perfectly matched layer with a length of 20 cm was incorporated into the model to create an infinite simulation domain. An alternating current harmonic excitation with a magnitude of 50 A was used to drive the gradient coil, and the frequency range from 0 to 3000 Hz was chosen to cover most sequences used in MRI scanners.

Figure 4.

Finite element mesh of the MRI scanner cavity structure.

Table 3.

Specifications and settings for vibration analysis of the structural model.

Element size (m)	0.015
Number of nodes	18 330
Multiplicity of elements	17 825
Element type shell	181
Load conditions velocity	0.0001 m/s
Minimum value	100 Hz
Maximum value	3000 Hz
Frequency steps	100 Hz
Solver	Sysnoise

Figures 5 and 6 compare the vibration spectra of the head and body using 2 approaches independent analysis and full analysis, respectively. In the independent analysis (Figure 5), the body’s vibration spectrum appears higher, likely due to the excitation of more modes. However, in the full analysis (Figure 6), which incorporates realistic factors and applies a coupling boundary condition between the cylindrical shell and the structural vibrations, the acoustic vibration pressure levels for the head and body are similar. The average vibration levels were measured at 97.6 dB for the head and 90.5 dB for the body coil, indicating that the head is more vulnerable to damage from vibration exposure.

Figure 5.

Comparison of head and body vibration level spectra in independent analysis.

Figure 6.

Comparison of head and body vibration level spectra in the full analysis.

Vibration levels can be reduced by placing an absorber layer on the inner wall of the scanner housing (see Figure 7). The averaged spectrum of simulated vibrations shown in Figure 8 indicates that the overall noise reduction is approximately 12.5 dB. Furthermore, using a microporous panel absorber with an absorber layer thickness of 20 mm reduces transmission losses in the frequency range of 125 Hz to 3 kHz by 15 to 37 dB.

Figure 7.

Coil gradient model and absorber with support under environmental support.

Figure 8.

Sound reduction with and without absorber.

In most MRI scanners, some acoustic noise is transmitted directly from the gradient coil cylinder through the tunnel wall into the scanner cavity, while additional noise emanates from the open end of the wall. Moreover, since the scanner tunnel wall is typically connected to the gradient coil cylinder, vibrations are transmitted to the wall, causing it to vibrate and generate sound waves. A possible solution is to design an additional panel absorber between the gradient coil and the scanner tunnel wall.

Numerical analyses of gradient coils have shown that, through acoustic analysis, vibration and noise levels can be effectively reduced, ensuring the safe operation of the gradient coil. Numerous methods for reducing acoustic noise have been reported in the literature. From an engineering perspective, the optimal solution is to redesign the gradient coil structure to prevent unwanted noise generation for example, by balancing the Lorentz forces produced by moving currents, which can reduce both noise and vibrations. However, retrofitting existing MRI systems with such redesigned gradient coil structures may be more expensive than alternative noise reduction methods.

Various software tools have been employed for numerical modeling, including COMSOL,⁵³ HISYS,⁵⁴ and ANSYS,^55,56 each offering its own capabilities and advantages. Numerous studies have also investigated the impact of several parameters on power and vibration amplitudes, considering both the absence and presence of delay.⁵⁷ Additionally, locally resonant acoustic metamaterials have proven effective at reducing low-frequency noise because their resonant systems act as spatial frequency filters,⁵⁸ whereas traditional sound absorbers, such as porous media are ineffective in attenuating low-frequency sound waves.⁵⁹ These insights suggest promising avenues for future research.

Conclusion

MRI is a non-invasive medical imaging method that employs radio waves and magnetic fields. This study focuses on reducing the acoustic noise inside the cylindrical shell (CS) of the scanner where the patient is positioned. These noise modes are generated by eddy currents in the CS induced by gradient magnetic fields. The study addresses methods for managing noise originating from the scanner wall as well as transmission noise emanating from the spiral gradient cylinder through the wall.

This study presents numerical methods and practical solutions for reducing acoustic noise in MRI gradient coils, with a focus on passive noise reduction techniques. Analysis of the dependence of the acoustic gradient on the main magnetic field strength indicates that both the acoustic noise and vibrations can be controlled by adjusting the field strength. It was demonstrated that a 20 mm uniform thickness absorber is a practical design that significantly reduces acoustic noise in the frequency range of 0 to 3 kHz. Given that MRI is one of the most popular and widely used imaging modalities, many manufacturers are seeking to improve and control these vibrations. Therefore, the results of this study can serve as a valuable guide for MRI device manufacturers aiming to enhance performance and efficiency.

Footnotes

ORCID iD

Hamidreza Mortazavy Beni

Ethics and Patient Consent

Human participants were not involved in the present study.

Author Contributions

All authors have contributed equally, including conceptualization, formal analyses, investigation, methodology, software, validation, writing original drafts, and review and editing.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Data Availability Statement

Data will be available upon reasonable request.

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Izadneshan

Mortazavy Beni

Study the effect of temperature variation and intrinsic layer thickness on the linear response of a PIN photodetector: a finite element method approach. Results in Engineering. 2023;17:100810.

54.

Mortazavi

Beni

Nadooshan

Islam

Ghalambaz

4E analysis and triple objective NSGA-II optimization of a novel solar-driven combined ejector-enhanced power and two-stage cooling (EORC-TCRC) system. Energy. 2024;294:130803.

55.

Islam

Fang

Oldfield

, et al. Heat wave and bushfire meteorology in New South Wales, Australia: air quality and health impacts. Int J Environ Res Public Health. 2022;19(16):10388.

56.

Islam

Rahman

Arsalanloo

, et al. How SARS-CoV-2 omicron droplets transport and deposit in realistic extrathoracic airways. Phys Fluids. 2022;34(11):1-17. doi:10.1063/5.0123213

57.

Firoozy

Ebrahimi-Nejad

Broadband energy harvesting from time-delayed nonlinear oscillations of magnetic levitation. J Intell Mater Syst Struct. 2020;31(5):737-755.

58.

Ravanbod

Ebrahimi-Nejad

Innovative lightweight re-entrant cross-like beam phononic crystal with perforated host for broadband vibration attenuation. Appl Phys A. 2023;129(2):102.

59.

Ghoudjani

Ravanbod

Ebrahimi-Nejad

Genetic-algorithm-assisted design of chiral honeycomb membrane acoustic metamaterials for broadband noise suppression. J Intell Mater Syst Struct. 2024;35(18):1426-1443.