Investigation of Pressure Distribution on Slippers of a Monorail Sled with Vibration Damper
الموضوعات :
Mohammad Reza Najafi
1
(Department of Mechanical Engineering, Imam Hosein comprehensive University, Tehran, Iran)
Saied Mahjoub Moghadas
2
(Department of Mechanical Engineering, Imam Hossein University, Tehran, Iran)
الکلمات المفتاحية: Modal Analysis, Natural Frequency, Pressure Distribution, Sled Test, Slipper,
ملخص المقالة :
In this paper, the pressure distribution on the slippers of a mono-rail sled with vibration damping is investigated. Due to the many applications of sled testing in the aerospace industry, the study of system vibrations is highly noticeable. In this research, first, by mathematical modelling of the sled, the governing Equations are extracted and natural frequencies and vibration modes are obtained from the analytical method using the mass and stiffness matrix of the system. Then, using numerical simulation and validation methods with experimental results performed in wind tunnels, the modal analysis of the designed sled sample is performed. A difference of less than eight percent in both numerical and analytical methods proves the accuracy of the results. The results show that the role of the slipper in the vibrations created in the sled is very important due to the large torsional and transverse oscillations in different positions, and the highest static pressure occurs in the inner layer of the slipper.
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Int. J. Advanced Design and Manufacturing Technology, 2023, Vol. 16, No. 4, pp. 87-97
DOI: 10.30495/ADMT.2022.1949929.1334 ISSN: 2252-0406 https://admt.isfahan.iau.ir
Investigation of Pressure Distribution on Slippers of a Monorail Sled with Vibration Damper
Department of Mechanical Engineering,
University of Imam Hossein Comprehensive, Iran
E-mail: drmrnajafi@ihu.ac.ir
Saeed Mahjoub Moghadas *
Department of Mechanical Engineering,
University of Imam Hossein Comprehensive, Iran
E-mail: smahjoubmoghadas@ihu.ac.ir
*Corresponding author
Received: 15 January 2022, Revised: 3 April 2022, Accepted: 15 April 2022
Keywords: Modal Analysis, Natural Frequency, Pressure Distribution, Sled Test, Slipper
Biographical notes: Mohammad Reza Najafi received his PhD in Mechanical Engineering from Imam Hossein Comprehensive University in Tehran, Iran. His current research interests include vibration and dynamics. Saeed Mahjoub Moghadas is an Associate Professor of Mechanical engineering at the University of Imam Hossein Comprehensive, Iran, since 1986. He received his PhD with the thesis subject “Internal Combustion Engines Control and Diagnostics through Instantaneous Speed of Rotation Analysis” at L'ensam University, Paris, France, in 1985. He has authored 20 books and translated 15 others in the field of dynamic, vibration, and control.
1 Introduction
Sled testing technology is now limited to a few countries. Experimental facilities of this technology include long rails, chassis, carriage, propulsion engines, and sled or projectile body. Slippers are used to connect this system to the rails. The slippers slide on the rails to allow the sled to move. Fig. 1 shows how to connect rail and slipper.
Fig. 1 Sled and rail connection.
The sled technology is used to achieve supersonic speeds up to 8 Mach and to provide experimental conditions in the field of space equipment [1], projectile penetration [2], parachute [3], pilot seat [4], anti-penetration structures [5], propulsion tests [6], and ultrasonic aerodynamic tests [7]. Real examples of sleds are also shown in “Fig. 2”.
Fig. 2 Real example of mono-rail sled system [8].
Sled testing technology has attracted the attention of researchers in recent years [9–13]. Due to the high speed of this technology, the vibrations on the sled are one of the main bottlenecks of this system, which has been noticed by many researchers [14–17]. Minto discussed a sled capacity development program to achieve high supersonic speeds. The results of their research show that the generated vibrational loads due to the impact of the slipper on the rail at speeds more than 6000 ft/sec lead to the failure of the sled body [18]. Bosmajian et al. studied the magnetic rail system of the sled. The aim of this study was to increase the sled capability for reduction of the vibrational environment at a final speed of 3000 m/s [19], and in the continuation of this research, Gurol et al. investigated the condition of the magnetic rail track in sled test at a higher speed. They showed that using a magnetic suspension system could significantly reduce sled vibrations [20]. Chen et al. investigated the amplitude of rocket sled vibrations and the time and frequency measurements of irregularity characteristics using statistical methods [21]. Turnbull et al. investigated the dynamic analysis of a sled system using narrow-gauge rails. The aerodynamic parameters of drag and lift, and pitch and yaw torques have been studied as a function of the angle of attack and acceleration has been obtained at four points on the sled [22]. Dunshee has modeled the mono-rail sled system as a two-degree-of-freedom beam with the capability to twist slightly in addition to moving in a vertical direction, using dynamic and vibrational analysis, and also has assumed the dynamic movement and a harmonic stimulation of a slipper [23]. Researchers have used mathematical models such as beams to analyze the behavior of dynamic systems such as high-speed trains, bridges, and automobiles [24–27]. Afshar et al. modelled the train bridge as a beam and investigated the use of linear and nonlinear vibration dampers for nonlinear beams at transient loads. The results show that for normal forces and short beams, linear and nonlinear models have similar behaviors [28].
Despite the research that has been done on sled testing, vibration analysis and modal analysis of sleds considering study of the role and the pressure distribution on slipper, which are the main outstanding of the present paper, have not been carried out by researchers so far. In the present paper, using the analytical method, first the vibrational Equations of the system are extracted and the natural frequencies are obtained. Then, the numerical simulation method is used to validate the obtained results and the modal analysis of a mono-rail sled with dampers is performed and the shape of its modes is clarified. Finally, the pressure and velocity distributions in the computational domain and the body of the sled and the slippers are also examined.
2 Mathematical modeling
Most of the sled vibrations are caused by the impact of the slippers on the rails. Since the materials used in sleds and slippers are stiff, considerable vibrations are transmitted from the slippers and the rails to the body due to the impact of the rails [29], most of which occur in the upper parts of the slipper [30]. Thus, the use of dampers at the junction of the sled body to the slipper can greatly prevent the transmission of vertical vibrations from the rail surface to the sled. In this case, the sled can be modeled as a system that is excited transversely from the slipper. An overview of this type of sled modeling is shown in “Fig. 3”.
The parameters of present modeling describe mb as sled body mass, mw1 as front slipper mass, mw2 as rear slipper mass, zb as vertical sled body displacement, zb1 as rear slipper displacement, zb2 as rear slipper displacement, θb as a rotation of sled body around Y-axis (pitch motion) and Iyy as inertia moment.
3 EXTRACTIONS of governing Equations
Considering the displacement of the slippers due to the transverse motion of the center of mass as well as the twisting motion around the Y-axis (θb), there is:
(1) |
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(2) |
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(3) |
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(4) |
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So, the vibrational Equations of the various components of the sled are now being extracted. To move the front slipper, there is mw1 in the Z direction as obtained in relation (5):
For the rear slipper motions, there is mw2 in the Z direction as discussed in the relation (6):
For the sled body motion (mb) in the Z direction, there is:
(7) |
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For the sled body rotation around the Y-axis, there is:
By multiplying the above values as indicated in relation (8), and simplifying them, the vibrational Equations of the four-degree-of-freedom system are obtained.
(9) |
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(10) |
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(11) |
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(12) |
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In the above Equations, the parameters are introduced as follows:
(13) |
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(14) |
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(15) |
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(16) |
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(17) |
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(18) |
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(19) |
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(20) |
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(21) |
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(22) |
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(23) |
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(24) |
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(25) |
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(26) |
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(27) |
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(28) |
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That Equation of system motions is obtained as relation (29):
Thus, the Equations of vibrational motion of the system are obtained as relation (30):
According to the modeling of a mono-rail sled, mass values and geometry of different components were extracted. The amount of elastomer stiffness used on top of the slipper was also obtained experimentally. “Table 1” shows the values of the various parameters of the designed sled.
Table 1 The values of the various parameters of the designed sled [31]
Parameter | Values | Unit | |
| 14.69 | kg | |
| 0.79 | kg | |
| 370 | kN/m | |
| 29.94 | N.s/m | |
| 450 | mm | |
| 393 | mm | |
| 285 | mm | |
| 195 | mm |
(31) |
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(32) |
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For the nose profile relations, the following values are placed and the y parameter is calculated for different x parameters.
(33) |
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(34) |
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Fig. 4 Measurement with parameters of Ogive nose.
Considering the obtained values, “Fig. 5” is represented.
Fig. 5 Measurement of the sled nose section.
Then, the Ogive nose is designed, which is shown in “Fig. 6”.
Fig. 6 Design of the sled nose section.
The body geometry is designed using the model specifications shown in “Fig. 7”.
Fig. 7 Geometry of model regarding [32].
The computational field and the network around the model are taken into consideration using the software and the range of fluid flow is analyzed. The dimensions of the domain should be considered large enough that it does not affect the results. (“Fig. 8”).
Fig. 8 Boundary condition and computational domain.
Fig. 9 Networking model of reference [32].
In this study, the length of the calculated range is 28 times more than the length of the body, so the distance from the beginning of the domain range to the nose portion is 7 times more than the length, and the distance from the end of the body to the end of the domain is 20 times more than it. The width and height of the domain are also 12 times more than the length. In all cases, for all boundaries around the domain, a freestream boundary condition was applied. The mesh networking of the model is produced using the trimmer method, which has four rows of boundary layers, the number of cells is 2707757 and the total thickness of the boundary layer network is 0.0072 m. Fig. 9 shows the research networking of the aforementioned reference. The side view and the front view of the meshing in the whole computational range are also shown in “Fig. 10”.
Fig. 10 Side view and front view of meshing in the whole computational range.
In all simulations, the k-e model is selected, which is suitable for engineering problems with high Reynolds numbers. In the k-e model, the Y+ boundary layer network must be greater than 30 and less than 300, as “Fig. 11” confirms.
Fig. 11 Y+ values.
The amount of drag force in the present numerical method in comparison with the amount of drag force obtained from the experimental test at two different cross-sectional levels and pressures is given in “Table 2”.
Table 2 Comparison of numerical and experimental simulation results
P0 [psi] | Experimental results [N] | Present working [N] | Error percent |
7.5 | 64.76 | 59.23 | 8.54 |
4.5 | 40.39 | 36.8 | 8.88 |
The results show that the error of the numerical method is less than 10% of the experimental test results, so the results of the numerical simulation method in the present study are confirmed.
4.2. Numerical Simulation
Considering the accuracy of the numerical solution methods, the numerical analysis of the sled is carried out. The studied sled in the present study is integrated and due to the mono-rail option, it lacks chassis and carriage. Dimensions and computational domain meshing, coordinate system and boundary conditions were carried out regarding the validation model of [32]. The computational domain and meshing network around the body are shown in “Fig. 12” from the front view. The mesh number in the computational domain is 2846662.
Fig. 12 Front view of the complete computational domain meshing.
The simulation settings were considered to be the same as the validation. The properties of the working fluid are shown in “Table 3”.
Table 3 Working fluid properties and environmental conditions [33].
Parameter | Values | Unit |
| 101325 |
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| 288.15 |
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c | 340.3 |
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| 1.225 |
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Also, there is:
Considering Equation (36), to investigate the motion of free vibrations of the sled system, there is relation (37):
Considering orthogonality of modes, there is:
(38) |
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In the above relations, mr and kr are related to the mass and stiffness of the modal or the generalized mass and stiffness of the rth mode. [Kr] and [Mr] are considered as the mass and stiffness matrices for the rth mode respectively, so the natural frequency and the mode shape matrices can be written as (39) and (40):
The Equation of motion of a multi-degree-of-freedom system with disproportionate structural damping is expressed as relation (41):
The damping structure matrix can be considered as an imaginary part of a complex stiffness matrix, defined as relation (42):
To solve the above Equation, relation (43) is obtained:
In this relation, λ is a complex frequency and includes both vibration and damping terms, and {X} indicates the complex vector of displacement domains. By substituting the value, the following problem of complex eigenvalue is obtained as the following relation.
The solution of Equation (44) includes the diametric matrix of the eigenvalue [λr] and the eigenvector matrix [Ψ]. The relation of the eigenvalue λr2 with the natural frequency ωr and the damping dissipation coefficient of the system ηr is obtained as relation (45):
λr is known as the complex natural frequency of the system. [λr2] is the natural frequency matrix of the system. The corresponding eigenvector, {Ψ}r, is a complex vector. All eigenvectors, corresponding eigenvalues of ωr in ascending order, are placed next to each other and these values result in the mixed matrix as the mode shape matrix of [Ψ]. Considering the orthogonal properties of the system, there is:
In this case, the modal mass of mr and the modality stiffness of kr are complex values. To obtain natural frequencies, given the four degrees of freedom of the system, [λr2] value for the four modes of the system is considered equal to the matrix of [B].
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