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Vibration Isolation of the Wind Tunnel Drive System
Available online at www.sciencedirect.com
ScienceDirect Materials Today: Proceedings 4 (2017) 7778–7792
www.materialstoday.com/proceedings
ICAAMM-2016
Vibration Isolation of the Wind Tunnel Drive System Gopala, Dr L Suresh Kumarb, V Jaipal Reddyc, M Uma Maheshwara Raod, K PavanKumare1 a Chinthalapudi Engg College, Ponnur, Guntur 522124, India C.B.I.T, Hyderabad, Telangana, India.(d&e) SMICH, Hyderabad, Telangana, India.
&cb,
Abstract The paper deals with the analysis of the wind tunnel drive to isolate the vibrations generated during testing. A wind tunnel is a tool used in aerodynamic research to study the effects of air moving past solid objects. Wind tunnels involve intricate study of various parameters by the addition of various accessories to the model. At the test section, the lift forces are predominant due to the vibrations. The aerofoil structure which is placed in the test section for study gets disturbed by the vibrations which effect the coefficient of lift parameter. Keeping in view all these effects, the study of vibrations is essential to minimize vibrational effects over mechanical components. The torsional natural frequency of the wind tunnel system is found out by both Analytical method and Finite Element (or) Eigen value methods. The mode shapes are drawn. Mathematical modelling of the physical system is done. Critical speeds are calculated. The amplitudes of vibration prior to the introduction of damping were measured. Suitable Dampers are selected and placed under the wind tunnel system for vibration isolation. Damping pads are selected as they are a perfect match for this wind tunnel. Isolation of the vibrations are confirmed by both analytical calculations and practical values measured by the vibration analyzers. © 2017 Elsevier Ltd. All rights reserved. Selection and Peer-review under responsibility ofthe Committee Members of International Conference on Advancements in Aeromechanical Materials for Manufacturing (ICAAMM-2016). Keywords:Wind tunnel; Torsional vibrations; Critical speeds; Transmissibility; Damping; Isolators;
1
* G Gopal. Tel.: +91 94416 32074. E-mail address:[email protected]
2214-7853© 2017 Elsevier Ltd. All rights reserved. Selection and Peer-review under responsibility ofthe Committee Members of International Conference on Advancements in Aeromechanical Materials for Manufacturing (ICAAMM-2016).
G Gopal / Materials Today: Proceedings 4 (2017) 7778–7792
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1. Introduction A wind tunnel is a tool used in aerodynamic research to study the effects of air moving past solid objects. It is used to study the characteristics and the effects of air moving past the miniature model of the body. Wind tunnels enables a stationary observer to study the aircraft in action, and measure the aerodynamic forces being imposed on the aircraft. A wind tunnel consists of a closed tubular passage with the object under test mounted in the middle. A powerful fan system moves air past the object; the fan must have straightening vanes to smooth the airflow. The test object is instrumented with a sensitive balance to measure the forces generated by airflow; or, the airflow may have smoke or other substances injected to make the flow lines around the object visible. It has a circular cross section which allows a smoother flow i.e. laminar flow. Other sections would constrict the air flow and may create turbulence. Multiple fans run by turbofan engines suck air from the atmosphere. The inside surfacing is made smooth to a very large extent to reduce turbulence. Even smooth walls induce drag on the test specimen. Hence the specimen is mounted in the centre of the wind tunnel cross section. In wind tunnels, due to the high speed of the rotors the support structures are subjected to vibrations. Faulty design and poor manufacturing create imbalance in the engines and give rise to excessive and unpleasant stresses in the rotating system because of vibrations. The vibrations cause rapid wear of machine parts such as bearings and pulleys. If the frequency of excitation coincides with one of the natural frequencies of the system, a condition of resonance is reached, dangerously large oscillations may occur which may result in mechanical failure of the system. At the test section the lift forces are predominant which is caused by the vibration and it is undesirable. The aerofoil structure which is placed in the test section for study has stream lines circumventing it in one direction as air is passed through it, due to vibrations this gets disturbed and it effects the coefficient of lift parameter. Hence vibrations are undesirable in the wind tunnel. Keeping in view all these effects, the study of vibrations is essential to minimize vibrational effects over mechanical components by designing them suitably. The undesirable vibrations can be eliminated or reduced up to certain extent by the following methods:• • • •
Removing external excitation, if possible. Using shock absorbers. Dynamic absorbers. Resting the system on proper vibration isolators.
The vibration isolation is done by resting the system on damping pads in the present experiment. The paper deals with the analysis of the wind tunnel drive system which is a multi-rotor system coupled with pulley. The mode shapes and torsional natural frequency of the system are estimated. Machine torsional frequencies and frequencies of any torsional load fluctuations should not coincide with the torsional natural frequencies as they lead to resonance and large amplitudes. This results in excessive shaking, component wear, and ultimately results in failure of the structure or component. The schematic diagram of a wind tunnel is shown in fig 1.
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Fig. 1 Wind Tunnel 1.1.Terms associated with the Wind Tunnel Wind tunnels involve intricate study of various parameters by the addition of various accessories to the model which demands a clear and perfect recording of the readings. Angle of attack (AOA) :It is the angle between the body's reference line and the oncoming flow. Lift coefficient (CL):The lift coefficient is a dimensionless coefficient that relates the lift generated by a lifting body to the density of the fluid around the body, its velocity and an associated reference area. Mach number: Mach number is a dimensionless quantity representing the ratio of speed of an object moving through a fluid and the local speed of sound. Reynolds number: The Reynolds number is defined as the ratio of inertial forces to viscous forces. Turbulence: Turbulence or turbulent flow is a flow regime characterized by chaotic property changes. This includes low momentum diffusion, high momentum convection, and rapid variation of pressure and velocity in space and time. Laminar: Laminar flow (or streamline flow) occurs when a fluid flows in parallel layers, with no disruption between the layers. It is a flow regime characterized by high momentum diffusion and low momentum convection. Viscosity: The viscosity of a fluid is a measure of its resistance to gradual deformation by shear stress or tensile stress. The friction between neighbouring particles in a fluid that are moving at different velocities. Stall: A stall is a reduction in the lift coefficient generated by a foil as angle of attack increases. This occurs when the critical angle of attack of the foil is exceeded. 1.2.Classification of Wind Tunnels • • • • •
Spin Tunnels: Aircraft have a tendency to go to spin when they stall (flight). These tunnels are used to study that phenomenon. Subsonic Tunnels: Low-speed wind tunnels are used for operations at very low mach number, with speeds in the test section up to 480 km/h. Mach number is 0.4. Transonic Tunnels: These are used for operations in the range 0.75
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Air velocity and pressures are measured in several ways in wind tunnels. Aerodynamic forces on the test model are usually measured with beam balances, connected to the test model with beams, strings or cables. The following factors are to be clearly understood : Periodic Motion: It is a motion which repeats itself after equal intervals of time. Time Period (T): It is the time required for one complete cycle or to and fro motion. The unit is seconds. Frequency (f or ω) : It is the number of cycles per unit time. The unit are radians/sec or Hz. Amplitude (X or A) : It is the displacement of a vibrating body from its equilibrium position. It has units of length in general. Natural Frequency (f n): It is the frequency with which a body vibrates when subjected to an initial external disturbance and allowed to vibrate without external force being applied subsequently. Also known as resonant frequency, is the specific frequency at which a material will naturally vibrate. 1.3. Fundamental Mode of Vibration: A vibrating body may have more than one natural frequency and when it vibrates with the lowest natural frequency. It is the Fundamental mode of vibration. Degrees of Freedom: It is the minimum number of coordinates required to describe the motion of system. 1DOF system will have one mass, e.g., a spring attached with one mass, 2 DOF system will have two masses and likewise we have 3DOF system. Simple Harmonic Motion (SHM): It is a periodic motion with acceleration always directed towards the equilibrium position. Damped vibrations: It is the resistance offered to the motion of a vibrating body by absorbing the energy of vibrations. Such vibrations are termed as damped vibrations. Resonance: It is said to occur in the system when the amplitude of vibrations are excessive leading to failure. This occurs in forced vibrations when the frequency of externally applied force is same as that of natural frequency of the body. A vibratory system basically consists of three elements as shown in fig 2 namely:1. 2. 3.
Mass (M) Spring (K) and Damper (C)
Fig. 2 Vibratory System representation Mass: The mass is assumed to be rigid and concentrated at the centre of gravity. Spring: It is assumed that the elasticity is represented by a helical spring. When deformed it stores energy. The springs work as energy restoring element. They are treated as massless.
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Damper: In a vibratory system the damper is an element which is responsible for loss of energy in the system. It converts energy into heat due to friction which may be either sliding friction or viscous friction. A vibratory system stops vibration because of energy conversion by damper. There are two types of dampers: Viscous Damper: A viscous damper consists of viscous friction which converts energy into heat due to this. For this damper, force is proportional to the relative velocity.
Coulumb’s Damper: The dry sliding friction acts as a damper. It is almost a constant force but direction is always opposite to the sliding velocity. Therefore, direction of friction changes due to change in direction of velocity.
The equation of motion of such a vibratory system can be written as, mx2+cx1+kx=0 where, mx2 is the inertia force,
cx1 is the damping force and
kx is the spring force.
1.Types of vibrations Free vibration: It occurs when a mechanical system is set off with an initial input and then allowed to vibrate freely. Forced Vibrations: It is the vibration of a body when subjected to an external force which is periodic in nature and vibrations occur as long as external force is present . Linear and Non linear vibration: If in a vibratory system mass, spring, damper behaves in linear manner, the vibrations caused are known as linear vibrations. Linear vibrations are governed by differential equations. If any of the basic components of vibratory system behaves non-linearly the vibrations are called nonlinear. Longitudinal, Transverse and Torsional Vibrations: When the motion of mass of the system is : i) Parallel to the axis of the system – they are Longitudinal vibrations. ii) Perpendicular to the system axis – they are Transverse vibrations. iii) Twists and untwists about the axis – they are Torsional vibrations. Damped and Undamped Vibration: If the vibratory system has a damper, the motion of the system will be opposed by it and the energy of the system will be dissipated in friction. This type of vibration is called damped vibration. The system having no damper is known as undamped vibration. Transient Vibration: In ideal system the free vibration continue indefinitely as there is no damping. The amplitude of vibration decays continuously because of damping and vanishes ultimately. Such vibration in real system is called transient vibration. 2. Procedure 2.1. Calculation of Lift Force in wind Tunnel To calculate the lift coefficient, a small experiment has been performed using wind tunnel coupled with a DC motor, a pitot static tube with manometer and a symmetric aero foil.
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Procedure is as follows: • Start the motor and note down the initial readings of pitot static tube in still state condition. • Check if the angle of attack is zero i.e. α=0. • Now, hold on the digital meter on to the motor and note the displayed values of L. • Note the readings in the manometer. • The CL (Lift coefficient) is calculated from the formulae are shown in Table.1. Table 1 : Readings obtained from experiment to find the Lift Coefficient H Static H H2O H Air Speed Angle of H Total (cm) (cm) (cm) (m) Attack 0 29.8 29.6 0.0002 1.6260 Initial 0 29.2 33.6 4.4 35.77 Final Angle of Attack (∞) 0
Lift Force (L) 1.2*9.8 = 11.76 N
Fig. 3 Wind tunnel system model – Air Supply Side
V (m) 5.6481 26.478
q 431
Lift Coefficient 0.6960
Fig. 4 Wind tunnel system model – Power Drive Side
Fig. 5Drive System The fig 3 and fig 4 shows two different views of the wind tunnel system model. Consider a rotor system as shown Fig.5 • The shaft is considered as massless and it provides torsional stiffness only. • The disc is considered as rigid and has no flexibility. If an initial disturbance is given to the disc in the torsional mode and allow it to oscillate its own, it will execute free vibrations. It shows that rotor is spinning with a nominal speed of w and executing torsional vibrations, q(t), due to this it has actual speed of {w + q(t)}.
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It should be noted that the spinning speed remains same however angular velocity due to torsion have varying direction over a period. The oscillation will be simple harmonic motion with a unique frequency, which is called the torsional natural frequency of the rotor system. 1.
System with stepped shaft The wind tunnel which is under the study of the project has drive system with stepped shaft.
Fig. 7 Stepped Shaft
Fig. 6 Two Disc Stepped Shaft Fig. 8 Equivalent Stepped Shaft A two disc stepped shaft shown in fig6 has to be replaced by an un-stepped equivalent shaft shown in fig 7 and fig 8 for the purpose of the analysis. On applying equivalence to the stepped shaft, we get, De1=38mm
and Le1= 130.31 mm.
De2 = 48mm and Le2 =268.00 mm
Fig. 9 Equivalence of the drive system in the form of rotors The line diagram of the drive system in the form of motors is shown in fig 9. When there are several number of discs in the rotor system it becomes a multi-DOF system. With increasing number of lumped masses (rigid discs), the resulting mathematical formulation becomes cumbersome. 1.
The torsional natural frequency of the wind tunnel system is found out by: Analytical method and 2. Finite Element (or) Eigen value method.
1.Analytical method With the help of tachometer, the gear ratio has been found out by operating the wind tunnel. The readings are tabulated in table 2
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Table 2 : Tachometer readings Tachometer reading at Motor 2200
Tachometer reading at Shaft 1490
Gear Ratio (n) 1.5
In the analysis we need to convert the original system into an equivalent system. The basis for this conversion is that the kinetic and potential energies for the equivalent system should be the same as that for the original system. We have taken equivalence to the stepped shaft shown in fig 10,
Fig. 10 Equivalent of the Stepped Shaft Table 3 : Stiffness and Mass Moment of Inertia from calculations Parameter Pulley Pulley Counter Mass Equival 1 2 / Fan ent Stiffness 12036 14888 275090 6311 Mass Moment of Inertia
0.022
0.023
0.015
0.102
Equivalent Polar Moment of Inertia, J = 0.01195 Kgm2 J = 0.01195 Kgm2 The natural frequency of the system, f = 115 Hz. 2.
Finite Element Method This method can be regarded as a general case of Rayleigh-Ritz method. It involves the formulation of the eigen value problem by determining the stiffness and mass matrices using Lagrange’s equation or direct method for each of the elements and their finite assemblage to get the global stiffness and mass matrix respectively are shown in Table.3. If the inertial of the shafts connecting the rotors is neglected, then the finite element method reduces to representation of the equations of motion for rotor in the form of an eigen value problem. The eigen values and eigenvectors hence found are the fundamental frequencies and the mode shapes respectively. As there are 3 degrees of freedom, we end up with three natural frequencies. These are tabulated in table 4 and table 5. Table 4: Natural Frequencies Ang. Speed ω Wave length, λ λ1=|-600288| λ2=|70907| λ3=|-134618.76|
ω1=774 ω 2 = 266.3 ω 3 = 366.4
N, rpm N1=7391 N2=2542 N3=3503
Table 5: Mode Shape values Mode x 2 / x1 x 3 / x1 Shape no. 1 =-4.0024 =-20.13 2 =1.59 =-0.28 3 =-0.1218 =1.008
For calculating the mode shapes, [λ-D][X]=0, the mode shapes obtained are shown in fig 11.
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Mode Shape 1
Mode Shape 2
Mode Shape 3
Fig. 11 Mode Shapes The comparison of the frequencies obtained are tabulated in table 6. Table 6: Comparison of frequencies obtained by: Analytical ω =726
FEA / Eigen Value ω2 =266.3 ω1 =774 ω3 =366.4
2.1. Modelling Of The System Modelling refers to mathematical modelling of the physical system. The purpose of mathematical modelling is to represent all the important features of the system for the purpose of deriving the mathematical or analytical equations governing the systems behaviour.
Fig. 12: Physical model representation of the Wind Tunnel
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The physical model of the wind tunnel is represented as shown in fig 12, the lumped mass and the supporting structures of the wind tunnel is taken each as a stiffness element. 2.2. Calculation of Mass of the Wind tunnel The calculation of mass in the wind tunnel is done as shown in fig 13 and are tabulated in table 7. Table 7: Mass Calculations 1&2 3 4 5 6&7 Shaft Total
Mass (Kg) 361 135 52.44 69.86 188.4 17.1 824.00
``Fig. 13: Mass Calculation ``The natural frequency of the system is obtained as 463 Hz. 2.3. Critical speed of the shafts: As the speed of rotation approaches the object's natural frequency, the object begins to resonate which dramatically increases system vibration. The resulting resonance occurs regardless of orientation. When the rotational speed is equal to the numerical value of the natural vibration, then that speed is referred to as critical speed. All rotating shafts, even in the absence of external load, will deflect during rotation. The unbalanced mass of the rotating object causes deflection that will create resonant vibration at certain speeds, known as the critical speeds. The magnitude of deflection depends upon the following: • • • •
stiffness of the shaft and its support total mass of shaft and attached parts unbalance of the mass with respect to the axis of rotation the amount of damping in the system
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For the wind tunnel system, the speed is obtained as 4430 rpm for the frequency 73Hz. This is the speed at which the resonance will occur. 2.4. Secondary Critical Speed: Apart from the Main critical speed resulting from the centrifugal forces from the vibrational sources, a good amount of vibration has been observed at half the critical speed. This effect has been noticed in horizontal shafts only and has been found in this and usually absent in the vertical shafts, understanding that gravity must be one of the causes of it. The importance and severity of the critical speed known as Secondary critical speed which is much less than that of Main or Primary critical speed. 2.5. Vibrations in the Wind Tunnel System: As stated earlier, vibrations near test section is undesirable, hence needs to be reduced. Analysis of forced vibrations can be divided into following categories • Forced vibration with constant harmonic excitation. • Forced vibration with rotating and reciprocating unbalance. • Forced vibration due to excitation of the support. A: Absolute amplitude
Relative amplitude
• Force and motion transmissibility. 2.6. Amplitudes of vibration The amplitudes of vibration prior to the introduction of damping are in tabulated in table 8. The values have been noted using vibration analyzer shown in fig 14. Table 8: Amplitudes before damping Location at Test Section Top Bottom
Displacement, x (micron) 0.02*103 0.01*103
Veloctiy, v (mm/s) 0.1*10 0.1*10
Acceleration, a (mm/s2) 0.2*10 0.21*10
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Fig. 14 Amplitude measurement setup. Damping refers to the action of a substance or an element in a mechanical or electrical device, which gradually reduces the degree of oscillation, vibration, or signal intensity. This prevents it from increasing in quantity. The damping of a system can be described as being one of the following: •
Overdamped: The system returns (exponentially decays) to equilibrium without oscillating.
•
Critically damped: The system returns to equilibrium as quickly as possible without oscillating.
•
Underdamped: The system oscillates (at reduced frequency compared to the undamped case) with the amplitude gradually decreasing to zero.
•
Undamped: The system oscillates at its natural resonant frequency (ωo). Vibration isolation is the process of isolating an object, such as a piece of equipment, from the source of vibrations. They are Passive Isolation and Negative Isolation systems. Passive isolation - "Passive vibration isolation" refers to vibration isolation or mitigation of vibrations by passive techniques such as rubber pads or mechanical springs. The different passive vibration isolation systems are shown in fig 15.
Pneumatic Isolator
Mechanical Isolator
Elastomeric Isolators
Fig. 15: Different Passive Isolation Systems
Flexible materials Isolators
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Common negative isolation systems – 1. Wire rope isolators 2. Base isolators 3. Tuned mass dampers Table 9: The damping values for various visco-elastic materials System Viscous damping ratio ξ Metals (in elastic range) < 0.01 Continuous Metal Structures 0.02 to 0.04 Metal Structure with Joints 0.03 to 0.07 Aluminum Steel Transmission Lines ~ 0.0004 Small Diameter Piping Systems 0.01 to 0.02 Large Diameter Piping Systems 0.02 to 0.03 Auto Shock Absorbers ~0.30 Rubber ~0.05 Large Buildings During Earthquakes 0.01 to 0.05 Prestressed Concrete Structures 0.02 to 0.05 Reinforced Concrete Structures 0.04 to 0.07 The material used for the isolation of the wind tunnel is the compound rubber, from the table 9 the damping ratio is 0.15. They are made of knitted stainless steel mesh. They form a perfect match for wind tunnel.
Fig. 16: Isolator used for damping – at different places The damping pads are introduced in the wind tunnel as shown in fig 16. On the introduction of damping, the wind tunnel has been operated and the amplitudes have been noted further by using vibration analyzer. The amplitude with dampers are tabulated in table 10. Table 10: Measured Amplitudes after keeping Dampers
Location at Test Section Base Top
Displacement, x (micron) 0.1*102 0.6*10
Veloctiy, v (mm/s) 0.5*10 0.44*1
Acceleration, a (mm/s2) 0.12*102 0.08*102
Transmissibility Transmissibility is a ratio of the vibrational force being measured in a system to the vibrational force entering a system. The amplitude ratio or the transmissibility versus the frequency ratio has been plotted for different damping ratios by using MATLAB. The plot obtained from MATLAB is as follows;
G Gopal / Materials Today: Proceedings 4 (2017) 7778–7792
Fig. 17: MATLAB Plot – Transmissibilty
Fig. 18 Reduction in Amplitudes
3. Results and Discussions From the plot of transmissibility shown in fig 17 and fig 18, we infer that all curves start from1and transmissibility Tr is always desired to be less than1, as it ensures that transmitted force to the is minimum and better isolation is achieved. The operating values of frequency ratio to achieve this effect should be greater than √2 and the region beyond this value of frequency ratio is called mass control zone where isolation is most effective. In the plot, the frequency ratio values up to 0.6 are spring control zone and from 0.6 to √2 is damping control zone and beyond that is mass control zone. Reduction in Amplitude Reduction in amplitude (X) = (1.8-1.45)Y =0.35Y = 0.35 * 0.02 mm = 0.007 mm Table 11: Practical values observed (Using a Dial gauge)
X (mm) (Test Section) Y (mm) (Base)
Before Damping 0.1 0.02
After Damping 0.08 0.01
Table11 shows the practical values observed using adial guage Reduction at test section =0.1-0.08 =0.02mm Reduction in amplitude both calculated analytically and practically are tabulated in table 12, Table 12: Comparison of reduction in amplitude values Analytical (X) mm 0.007
Practical (X) mm 0.02
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References [1] Justin D Periera, “Wind tunnels – aerodynamics, models and experiments”, Nova Science Publishers. [2] Design Rules For Small Low Speed Wind Tunnels”, Journal of Royal Aeronautical Society 1979, Vol. 73. [3] ac 2012-3461: undergraduate research on conceptual design of a wind tunnel for instructional purposes peter john arslanian, nasa/computer sciences corporation [4] https://www.grc.nasa.gov/www/k-12/airplane/tunnel1.html [5] https:///V3I10-IJERTV3IS100995.pdf [6] Peter John Arslanian,: Undergraduate research on conceptual design of a wind tunnel for instructional purposes, nasa/computer sciences corporation, ac 2012-3461. [7] Singiresu S Rao , Mechanical Vibrations, 4thedition, Pearson Education [8] G K Groover, Mechanical Vibrations [9] John J Uicker, Theory of Machines and Mechanisms. [10] http://ethesis.nitrkl.ac.in/3454/ [11] http://www.rogerscorp.com/hpf/vibration/best_solution.aspx
ScienceDirect Materials Today: Proceedings 4 (2017) 7778–7792
www.materialstoday.com/proceedings
ICAAMM-2016
Vibration Isolation of the Wind Tunnel Drive System Gopala, Dr L Suresh Kumarb, V Jaipal Reddyc, M Uma Maheshwara Raod, K PavanKumare1 a Chinthalapudi Engg College, Ponnur, Guntur 522124, India C.B.I.T, Hyderabad, Telangana, India.(d&e) SMICH, Hyderabad, Telangana, India.
&cb,
Abstract The paper deals with the analysis of the wind tunnel drive to isolate the vibrations generated during testing. A wind tunnel is a tool used in aerodynamic research to study the effects of air moving past solid objects. Wind tunnels involve intricate study of various parameters by the addition of various accessories to the model. At the test section, the lift forces are predominant due to the vibrations. The aerofoil structure which is placed in the test section for study gets disturbed by the vibrations which effect the coefficient of lift parameter. Keeping in view all these effects, the study of vibrations is essential to minimize vibrational effects over mechanical components. The torsional natural frequency of the wind tunnel system is found out by both Analytical method and Finite Element (or) Eigen value methods. The mode shapes are drawn. Mathematical modelling of the physical system is done. Critical speeds are calculated. The amplitudes of vibration prior to the introduction of damping were measured. Suitable Dampers are selected and placed under the wind tunnel system for vibration isolation. Damping pads are selected as they are a perfect match for this wind tunnel. Isolation of the vibrations are confirmed by both analytical calculations and practical values measured by the vibration analyzers. © 2017 Elsevier Ltd. All rights reserved. Selection and Peer-review under responsibility ofthe Committee Members of International Conference on Advancements in Aeromechanical Materials for Manufacturing (ICAAMM-2016). Keywords:Wind tunnel; Torsional vibrations; Critical speeds; Transmissibility; Damping; Isolators;
1
* G Gopal. Tel.: +91 94416 32074. E-mail address:[email protected]
2214-7853© 2017 Elsevier Ltd. All rights reserved. Selection and Peer-review under responsibility ofthe Committee Members of International Conference on Advancements in Aeromechanical Materials for Manufacturing (ICAAMM-2016).
G Gopal / Materials Today: Proceedings 4 (2017) 7778–7792
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1. Introduction A wind tunnel is a tool used in aerodynamic research to study the effects of air moving past solid objects. It is used to study the characteristics and the effects of air moving past the miniature model of the body. Wind tunnels enables a stationary observer to study the aircraft in action, and measure the aerodynamic forces being imposed on the aircraft. A wind tunnel consists of a closed tubular passage with the object under test mounted in the middle. A powerful fan system moves air past the object; the fan must have straightening vanes to smooth the airflow. The test object is instrumented with a sensitive balance to measure the forces generated by airflow; or, the airflow may have smoke or other substances injected to make the flow lines around the object visible. It has a circular cross section which allows a smoother flow i.e. laminar flow. Other sections would constrict the air flow and may create turbulence. Multiple fans run by turbofan engines suck air from the atmosphere. The inside surfacing is made smooth to a very large extent to reduce turbulence. Even smooth walls induce drag on the test specimen. Hence the specimen is mounted in the centre of the wind tunnel cross section. In wind tunnels, due to the high speed of the rotors the support structures are subjected to vibrations. Faulty design and poor manufacturing create imbalance in the engines and give rise to excessive and unpleasant stresses in the rotating system because of vibrations. The vibrations cause rapid wear of machine parts such as bearings and pulleys. If the frequency of excitation coincides with one of the natural frequencies of the system, a condition of resonance is reached, dangerously large oscillations may occur which may result in mechanical failure of the system. At the test section the lift forces are predominant which is caused by the vibration and it is undesirable. The aerofoil structure which is placed in the test section for study has stream lines circumventing it in one direction as air is passed through it, due to vibrations this gets disturbed and it effects the coefficient of lift parameter. Hence vibrations are undesirable in the wind tunnel. Keeping in view all these effects, the study of vibrations is essential to minimize vibrational effects over mechanical components by designing them suitably. The undesirable vibrations can be eliminated or reduced up to certain extent by the following methods:• • • •
Removing external excitation, if possible. Using shock absorbers. Dynamic absorbers. Resting the system on proper vibration isolators.
The vibration isolation is done by resting the system on damping pads in the present experiment. The paper deals with the analysis of the wind tunnel drive system which is a multi-rotor system coupled with pulley. The mode shapes and torsional natural frequency of the system are estimated. Machine torsional frequencies and frequencies of any torsional load fluctuations should not coincide with the torsional natural frequencies as they lead to resonance and large amplitudes. This results in excessive shaking, component wear, and ultimately results in failure of the structure or component. The schematic diagram of a wind tunnel is shown in fig 1.
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Fig. 1 Wind Tunnel 1.1.Terms associated with the Wind Tunnel Wind tunnels involve intricate study of various parameters by the addition of various accessories to the model which demands a clear and perfect recording of the readings. Angle of attack (AOA) :It is the angle between the body's reference line and the oncoming flow. Lift coefficient (CL):The lift coefficient is a dimensionless coefficient that relates the lift generated by a lifting body to the density of the fluid around the body, its velocity and an associated reference area. Mach number: Mach number is a dimensionless quantity representing the ratio of speed of an object moving through a fluid and the local speed of sound. Reynolds number: The Reynolds number is defined as the ratio of inertial forces to viscous forces. Turbulence: Turbulence or turbulent flow is a flow regime characterized by chaotic property changes. This includes low momentum diffusion, high momentum convection, and rapid variation of pressure and velocity in space and time. Laminar: Laminar flow (or streamline flow) occurs when a fluid flows in parallel layers, with no disruption between the layers. It is a flow regime characterized by high momentum diffusion and low momentum convection. Viscosity: The viscosity of a fluid is a measure of its resistance to gradual deformation by shear stress or tensile stress. The friction between neighbouring particles in a fluid that are moving at different velocities. Stall: A stall is a reduction in the lift coefficient generated by a foil as angle of attack increases. This occurs when the critical angle of attack of the foil is exceeded. 1.2.Classification of Wind Tunnels • • • • •
Spin Tunnels: Aircraft have a tendency to go to spin when they stall (flight). These tunnels are used to study that phenomenon. Subsonic Tunnels: Low-speed wind tunnels are used for operations at very low mach number, with speeds in the test section up to 480 km/h. Mach number is 0.4. Transonic Tunnels: These are used for operations in the range 0.75
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Air velocity and pressures are measured in several ways in wind tunnels. Aerodynamic forces on the test model are usually measured with beam balances, connected to the test model with beams, strings or cables. The following factors are to be clearly understood : Periodic Motion: It is a motion which repeats itself after equal intervals of time. Time Period (T): It is the time required for one complete cycle or to and fro motion. The unit is seconds. Frequency (f or ω) : It is the number of cycles per unit time. The unit are radians/sec or Hz. Amplitude (X or A) : It is the displacement of a vibrating body from its equilibrium position. It has units of length in general. Natural Frequency (f n): It is the frequency with which a body vibrates when subjected to an initial external disturbance and allowed to vibrate without external force being applied subsequently. Also known as resonant frequency, is the specific frequency at which a material will naturally vibrate. 1.3. Fundamental Mode of Vibration: A vibrating body may have more than one natural frequency and when it vibrates with the lowest natural frequency. It is the Fundamental mode of vibration. Degrees of Freedom: It is the minimum number of coordinates required to describe the motion of system. 1DOF system will have one mass, e.g., a spring attached with one mass, 2 DOF system will have two masses and likewise we have 3DOF system. Simple Harmonic Motion (SHM): It is a periodic motion with acceleration always directed towards the equilibrium position. Damped vibrations: It is the resistance offered to the motion of a vibrating body by absorbing the energy of vibrations. Such vibrations are termed as damped vibrations. Resonance: It is said to occur in the system when the amplitude of vibrations are excessive leading to failure. This occurs in forced vibrations when the frequency of externally applied force is same as that of natural frequency of the body. A vibratory system basically consists of three elements as shown in fig 2 namely:1. 2. 3.
Mass (M) Spring (K) and Damper (C)
Fig. 2 Vibratory System representation Mass: The mass is assumed to be rigid and concentrated at the centre of gravity. Spring: It is assumed that the elasticity is represented by a helical spring. When deformed it stores energy. The springs work as energy restoring element. They are treated as massless.
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Damper: In a vibratory system the damper is an element which is responsible for loss of energy in the system. It converts energy into heat due to friction which may be either sliding friction or viscous friction. A vibratory system stops vibration because of energy conversion by damper. There are two types of dampers: Viscous Damper: A viscous damper consists of viscous friction which converts energy into heat due to this. For this damper, force is proportional to the relative velocity.
Coulumb’s Damper: The dry sliding friction acts as a damper. It is almost a constant force but direction is always opposite to the sliding velocity. Therefore, direction of friction changes due to change in direction of velocity.
The equation of motion of such a vibratory system can be written as, mx2+cx1+kx=0 where, mx2 is the inertia force,
cx1 is the damping force and
kx is the spring force.
1.Types of vibrations Free vibration: It occurs when a mechanical system is set off with an initial input and then allowed to vibrate freely. Forced Vibrations: It is the vibration of a body when subjected to an external force which is periodic in nature and vibrations occur as long as external force is present . Linear and Non linear vibration: If in a vibratory system mass, spring, damper behaves in linear manner, the vibrations caused are known as linear vibrations. Linear vibrations are governed by differential equations. If any of the basic components of vibratory system behaves non-linearly the vibrations are called nonlinear. Longitudinal, Transverse and Torsional Vibrations: When the motion of mass of the system is : i) Parallel to the axis of the system – they are Longitudinal vibrations. ii) Perpendicular to the system axis – they are Transverse vibrations. iii) Twists and untwists about the axis – they are Torsional vibrations. Damped and Undamped Vibration: If the vibratory system has a damper, the motion of the system will be opposed by it and the energy of the system will be dissipated in friction. This type of vibration is called damped vibration. The system having no damper is known as undamped vibration. Transient Vibration: In ideal system the free vibration continue indefinitely as there is no damping. The amplitude of vibration decays continuously because of damping and vanishes ultimately. Such vibration in real system is called transient vibration. 2. Procedure 2.1. Calculation of Lift Force in wind Tunnel To calculate the lift coefficient, a small experiment has been performed using wind tunnel coupled with a DC motor, a pitot static tube with manometer and a symmetric aero foil.
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Procedure is as follows: • Start the motor and note down the initial readings of pitot static tube in still state condition. • Check if the angle of attack is zero i.e. α=0. • Now, hold on the digital meter on to the motor and note the displayed values of L. • Note the readings in the manometer. • The CL (Lift coefficient) is calculated from the formulae are shown in Table.1. Table 1 : Readings obtained from experiment to find the Lift Coefficient H Static H H2O H Air Speed Angle of H Total (cm) (cm) (cm) (m) Attack 0 29.8 29.6 0.0002 1.6260 Initial 0 29.2 33.6 4.4 35.77 Final Angle of Attack (∞) 0
Lift Force (L) 1.2*9.8 = 11.76 N
Fig. 3 Wind tunnel system model – Air Supply Side
V (m) 5.6481 26.478
q 431
Lift Coefficient 0.6960
Fig. 4 Wind tunnel system model – Power Drive Side
Fig. 5Drive System The fig 3 and fig 4 shows two different views of the wind tunnel system model. Consider a rotor system as shown Fig.5 • The shaft is considered as massless and it provides torsional stiffness only. • The disc is considered as rigid and has no flexibility. If an initial disturbance is given to the disc in the torsional mode and allow it to oscillate its own, it will execute free vibrations. It shows that rotor is spinning with a nominal speed of w and executing torsional vibrations, q(t), due to this it has actual speed of {w + q(t)}.
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It should be noted that the spinning speed remains same however angular velocity due to torsion have varying direction over a period. The oscillation will be simple harmonic motion with a unique frequency, which is called the torsional natural frequency of the rotor system. 1.
System with stepped shaft The wind tunnel which is under the study of the project has drive system with stepped shaft.
Fig. 7 Stepped Shaft
Fig. 6 Two Disc Stepped Shaft Fig. 8 Equivalent Stepped Shaft A two disc stepped shaft shown in fig6 has to be replaced by an un-stepped equivalent shaft shown in fig 7 and fig 8 for the purpose of the analysis. On applying equivalence to the stepped shaft, we get, De1=38mm
and Le1= 130.31 mm.
De2 = 48mm and Le2 =268.00 mm
Fig. 9 Equivalence of the drive system in the form of rotors The line diagram of the drive system in the form of motors is shown in fig 9. When there are several number of discs in the rotor system it becomes a multi-DOF system. With increasing number of lumped masses (rigid discs), the resulting mathematical formulation becomes cumbersome. 1.
The torsional natural frequency of the wind tunnel system is found out by: Analytical method and 2. Finite Element (or) Eigen value method.
1.Analytical method With the help of tachometer, the gear ratio has been found out by operating the wind tunnel. The readings are tabulated in table 2
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Table 2 : Tachometer readings Tachometer reading at Motor 2200
Tachometer reading at Shaft 1490
Gear Ratio (n) 1.5
In the analysis we need to convert the original system into an equivalent system. The basis for this conversion is that the kinetic and potential energies for the equivalent system should be the same as that for the original system. We have taken equivalence to the stepped shaft shown in fig 10,
Fig. 10 Equivalent of the Stepped Shaft Table 3 : Stiffness and Mass Moment of Inertia from calculations Parameter Pulley Pulley Counter Mass Equival 1 2 / Fan ent Stiffness 12036 14888 275090 6311 Mass Moment of Inertia
0.022
0.023
0.015
0.102
Equivalent Polar Moment of Inertia, J = 0.01195 Kgm2 J = 0.01195 Kgm2 The natural frequency of the system, f = 115 Hz. 2.
Finite Element Method This method can be regarded as a general case of Rayleigh-Ritz method. It involves the formulation of the eigen value problem by determining the stiffness and mass matrices using Lagrange’s equation or direct method for each of the elements and their finite assemblage to get the global stiffness and mass matrix respectively are shown in Table.3. If the inertial of the shafts connecting the rotors is neglected, then the finite element method reduces to representation of the equations of motion for rotor in the form of an eigen value problem. The eigen values and eigenvectors hence found are the fundamental frequencies and the mode shapes respectively. As there are 3 degrees of freedom, we end up with three natural frequencies. These are tabulated in table 4 and table 5. Table 4: Natural Frequencies Ang. Speed ω Wave length, λ λ1=|-600288| λ2=|70907| λ3=|-134618.76|
ω1=774 ω 2 = 266.3 ω 3 = 366.4
N, rpm N1=7391 N2=2542 N3=3503
Table 5: Mode Shape values Mode x 2 / x1 x 3 / x1 Shape no. 1 =-4.0024 =-20.13 2 =1.59 =-0.28 3 =-0.1218 =1.008
For calculating the mode shapes, [λ-D][X]=0, the mode shapes obtained are shown in fig 11.
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Mode Shape 1
Mode Shape 2
Mode Shape 3
Fig. 11 Mode Shapes The comparison of the frequencies obtained are tabulated in table 6. Table 6: Comparison of frequencies obtained by: Analytical ω =726
FEA / Eigen Value ω2 =266.3 ω1 =774 ω3 =366.4
2.1. Modelling Of The System Modelling refers to mathematical modelling of the physical system. The purpose of mathematical modelling is to represent all the important features of the system for the purpose of deriving the mathematical or analytical equations governing the systems behaviour.
Fig. 12: Physical model representation of the Wind Tunnel
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The physical model of the wind tunnel is represented as shown in fig 12, the lumped mass and the supporting structures of the wind tunnel is taken each as a stiffness element. 2.2. Calculation of Mass of the Wind tunnel The calculation of mass in the wind tunnel is done as shown in fig 13 and are tabulated in table 7. Table 7: Mass Calculations 1&2 3 4 5 6&7 Shaft Total
Mass (Kg) 361 135 52.44 69.86 188.4 17.1 824.00
``Fig. 13: Mass Calculation ``The natural frequency of the system is obtained as 463 Hz. 2.3. Critical speed of the shafts: As the speed of rotation approaches the object's natural frequency, the object begins to resonate which dramatically increases system vibration. The resulting resonance occurs regardless of orientation. When the rotational speed is equal to the numerical value of the natural vibration, then that speed is referred to as critical speed. All rotating shafts, even in the absence of external load, will deflect during rotation. The unbalanced mass of the rotating object causes deflection that will create resonant vibration at certain speeds, known as the critical speeds. The magnitude of deflection depends upon the following: • • • •
stiffness of the shaft and its support total mass of shaft and attached parts unbalance of the mass with respect to the axis of rotation the amount of damping in the system
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For the wind tunnel system, the speed is obtained as 4430 rpm for the frequency 73Hz. This is the speed at which the resonance will occur. 2.4. Secondary Critical Speed: Apart from the Main critical speed resulting from the centrifugal forces from the vibrational sources, a good amount of vibration has been observed at half the critical speed. This effect has been noticed in horizontal shafts only and has been found in this and usually absent in the vertical shafts, understanding that gravity must be one of the causes of it. The importance and severity of the critical speed known as Secondary critical speed which is much less than that of Main or Primary critical speed. 2.5. Vibrations in the Wind Tunnel System: As stated earlier, vibrations near test section is undesirable, hence needs to be reduced. Analysis of forced vibrations can be divided into following categories • Forced vibration with constant harmonic excitation. • Forced vibration with rotating and reciprocating unbalance. • Forced vibration due to excitation of the support. A: Absolute amplitude
Relative amplitude
• Force and motion transmissibility. 2.6. Amplitudes of vibration The amplitudes of vibration prior to the introduction of damping are in tabulated in table 8. The values have been noted using vibration analyzer shown in fig 14. Table 8: Amplitudes before damping Location at Test Section Top Bottom
Displacement, x (micron) 0.02*103 0.01*103
Veloctiy, v (mm/s) 0.1*10 0.1*10
Acceleration, a (mm/s2) 0.2*10 0.21*10
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Fig. 14 Amplitude measurement setup. Damping refers to the action of a substance or an element in a mechanical or electrical device, which gradually reduces the degree of oscillation, vibration, or signal intensity. This prevents it from increasing in quantity. The damping of a system can be described as being one of the following: •
Overdamped: The system returns (exponentially decays) to equilibrium without oscillating.
•
Critically damped: The system returns to equilibrium as quickly as possible without oscillating.
•
Underdamped: The system oscillates (at reduced frequency compared to the undamped case) with the amplitude gradually decreasing to zero.
•
Undamped: The system oscillates at its natural resonant frequency (ωo). Vibration isolation is the process of isolating an object, such as a piece of equipment, from the source of vibrations. They are Passive Isolation and Negative Isolation systems. Passive isolation - "Passive vibration isolation" refers to vibration isolation or mitigation of vibrations by passive techniques such as rubber pads or mechanical springs. The different passive vibration isolation systems are shown in fig 15.
Pneumatic Isolator
Mechanical Isolator
Elastomeric Isolators
Fig. 15: Different Passive Isolation Systems
Flexible materials Isolators
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Common negative isolation systems – 1. Wire rope isolators 2. Base isolators 3. Tuned mass dampers Table 9: The damping values for various visco-elastic materials System Viscous damping ratio ξ Metals (in elastic range) < 0.01 Continuous Metal Structures 0.02 to 0.04 Metal Structure with Joints 0.03 to 0.07 Aluminum Steel Transmission Lines ~ 0.0004 Small Diameter Piping Systems 0.01 to 0.02 Large Diameter Piping Systems 0.02 to 0.03 Auto Shock Absorbers ~0.30 Rubber ~0.05 Large Buildings During Earthquakes 0.01 to 0.05 Prestressed Concrete Structures 0.02 to 0.05 Reinforced Concrete Structures 0.04 to 0.07 The material used for the isolation of the wind tunnel is the compound rubber, from the table 9 the damping ratio is 0.15. They are made of knitted stainless steel mesh. They form a perfect match for wind tunnel.
Fig. 16: Isolator used for damping – at different places The damping pads are introduced in the wind tunnel as shown in fig 16. On the introduction of damping, the wind tunnel has been operated and the amplitudes have been noted further by using vibration analyzer. The amplitude with dampers are tabulated in table 10. Table 10: Measured Amplitudes after keeping Dampers
Location at Test Section Base Top
Displacement, x (micron) 0.1*102 0.6*10
Veloctiy, v (mm/s) 0.5*10 0.44*1
Acceleration, a (mm/s2) 0.12*102 0.08*102
Transmissibility Transmissibility is a ratio of the vibrational force being measured in a system to the vibrational force entering a system. The amplitude ratio or the transmissibility versus the frequency ratio has been plotted for different damping ratios by using MATLAB. The plot obtained from MATLAB is as follows;
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Fig. 17: MATLAB Plot – Transmissibilty
Fig. 18 Reduction in Amplitudes
3. Results and Discussions From the plot of transmissibility shown in fig 17 and fig 18, we infer that all curves start from1and transmissibility Tr is always desired to be less than1, as it ensures that transmitted force to the is minimum and better isolation is achieved. The operating values of frequency ratio to achieve this effect should be greater than √2 and the region beyond this value of frequency ratio is called mass control zone where isolation is most effective. In the plot, the frequency ratio values up to 0.6 are spring control zone and from 0.6 to √2 is damping control zone and beyond that is mass control zone. Reduction in Amplitude Reduction in amplitude (X) = (1.8-1.45)Y =0.35Y = 0.35 * 0.02 mm = 0.007 mm Table 11: Practical values observed (Using a Dial gauge)
X (mm) (Test Section) Y (mm) (Base)
Before Damping 0.1 0.02
After Damping 0.08 0.01
Table11 shows the practical values observed using adial guage Reduction at test section =0.1-0.08 =0.02mm Reduction in amplitude both calculated analytically and practically are tabulated in table 12, Table 12: Comparison of reduction in amplitude values Analytical (X) mm 0.007
Practical (X) mm 0.02
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References [1] Justin D Periera, “Wind tunnels – aerodynamics, models and experiments”, Nova Science Publishers. [2] Design Rules For Small Low Speed Wind Tunnels”, Journal of Royal Aeronautical Society 1979, Vol. 73. [3] ac 2012-3461: undergraduate research on conceptual design of a wind tunnel for instructional purposes peter john arslanian, nasa/computer sciences corporation [4] https://www.grc.nasa.gov/www/k-12/airplane/tunnel1.html [5] https:///V3I10-IJERTV3IS100995.pdf [6] Peter John Arslanian,: Undergraduate research on conceptual design of a wind tunnel for instructional purposes, nasa/computer sciences corporation, ac 2012-3461. [7] Singiresu S Rao , Mechanical Vibrations, 4thedition, Pearson Education [8] G K Groover, Mechanical Vibrations [9] John J Uicker, Theory of Machines and Mechanisms. [10] http://ethesis.nitrkl.ac.in/3454/ [11] http://www.rogerscorp.com/hpf/vibration/best_solution.aspx