Pavan Kumar Vodnala in partial fulfillment of the requirements for the award of a Master of Technology degree in Mechanical Engineering with specialization in Manufacturing Engineering during session 2009-2011 at the National Institute of Technology, Rourkela (Deemed University) is an authentic work done by him under my supervision and guidance. Bhagat Singh (Ph.D. Fellow) and well wishers directly or indirectly involved in successful completion of this work. A Cross-sectional area of the beam Ao Cross-sectional area of the rivet A′ Area under a connecting rivet head d Diameter of the connecting rivet.
R Any radius within the zone of influence RB Radius of the connecting section RM Limiting radius of zone of influence t time coordinate. All of the above vital parameters are greatly influenced by the thickness ratio of the beam and thereby affect the damping ability of the constructions. In addition to this, the number of layers, beam length and diameter of the joint also play key roles in the damping performance of the joined structures quantitatively.
It is determined that the damping capacity can be significantly improved by using larger beam length and rivet diameters as well as lower thickness ratio of the beams. Extensive experiments were carried out on a number of mild steel specimens under different initial conditions of excitation to establish the validity of the developed theory.
Damping 2
- Classification of damping 3
- Damping due to sandwich construction 6
- Damping capacity of riveted joint 8
- Damping ratio of beam 9
With the development of jointed beams, fabricated structures can be used as a substitute for rigid structures with increased damping. Damping due to the distribution mechanism operating within the material of the part is called material damping. An important step towards the analysis and representation of the machine's vibration mode is the analysis of the spatial vibration behavior of the structure when it vibrates at its resonance frequencies.
The following basic characteristics of the assembly must be considered when replacing machine tool assemblies with mathematical model:. In ordinary slides, the dynamic characteristics are mainly determined by damping caused by friction, the mass of the carriage and the stiffness and damping of the drive. The stiffness and damping of the practical manufactured structures are not always linear due to the existence of some prestress.
Core thickness equal to or greater than the thickness of the metal confining layer can provide high overall attenuation. The damping capacity of a riveted joint depends mainly on the following factors: i) Coefficient of friction between joint surfaces. ii) Microslip between joint surfaces. iii) Reaction force of a base applied under vibration. iv) Joint material, processing method of joint surfaces, interface layer, beam thickness, vibration amplitude have great effect on damping and frequency.
Objective 10
Interface pressure distribution 20
A layered and joined construction is made by means of rivets holding the parts together at the interfaces. In such circumstances, the pressure distribution profile of the interface takes on an important role, especially in the presence of slip, to dissipate the vibration energy. Consequently, it is necessary to examine the exact nature of the interface pressure profile and its magnitude along a beam layer for the accurate assessment of the damping capacity of a jointed structure.
This pressure distribution at the interfaces is due to the clamping of the rivets on the contact elements. When two or more elements are pressed together by riveting, a contact circle will be formed around the rivet with separation occurring at a certain distance from the rivet hole as shown in fig. 1 (a) Plates fastened with a rivet (b) ) Free-body diagram of a riveted joint showing the zone of influence The contact between the connecting elements develops an interfacial pressure, the exact nature and magnitude of which across the beam layer is very important for the correct assessment of the damping capacity of the a joined construction.
As defined, the contact pressure at the joint interface is non-uniform in nature being maximum at the rivet hole and decreasing with distance from the rivet. 35] have found that the interfacial pressure distribution due to this contact is parabolic with a circular impact zone surrounding the rivet of diameter equal to 4.125,.
Determination of Pressure Distribution at the Interfaces 22
Evaluation of relative dynamic slip 24
Dynamic equations of free transverse vibration of cantilever and fixed-fixed
Cantilever beam 25
Fixed-Fixed beam 27
B=0 (16b) Inserting the values of A and B into equation (14), we obtain the transverse deflection of the cantilever beam at the free end as;.
Analysis of energy dissipated 29
Cantilever beam 29
Assuming that the dynamic slip ratio, α, is independent of the distance from the fixed end of the cantilever beam and time, the above equation (20) was modified using equation (19) as;. If we replace 3EI/l3=k, i.e. static bending stiffness of a laminated and riveted cantilever beam, the above equation (26) reduces to.
Fixed-Fixed beam 31
Following the above procedure for a fixed-fixed beam, the energy ratio is evaluated to;. If we replace 192EI/l3=kf, i.e. static bending stiffness of a laminated and riveted cantilever beam, the above equation (30) reduces to.
Evaluation of damping ratio 32
Calculation of logarithmic damping decrement 33
Determination of the product of dynamic slip ratio and kinematics coefficient
6 Variation of α.μ with frequency with a beam thickness ratio of 1.0 for a cantilever beam at different initial excitation amplitudes (y). 7 Variation of α.μ with frequency with a beam thickness ratio of 1.5 for a cantilever beam at different initial excitation amplitudes (y). 8 Variation of α.μ with frequency with a beam thickness ratio of 2.0 for a cantilever beam at different initial excitation amplitudes (y).
9 Variation of α.μ with frequency with a beam thickness ratio of 1.0 for a fixed-fixed beam at different initial excitation amplitudes (y). 10 Variation of α.μ with vibration frequency with a beam thickness ratio of 1.5 for a fixed-fixed beam at different initial excitation amplitudes (y). 11 Variation of α.μ with oscillation frequency with a beam thickness ratio of 2.0 for a fixed-fixed beam at different initial excitation amplitudes (y).
Experimental set-up 39
Test specimen 43
Instrumentation 45
The trace left behind can be used to measure the input signal voltage (off the Y-axis) and the duration or frequency can be read from the X-axis. A grid with a 1 cm grid allows you to get voltage and time measurements from the display. The graph is a trace and is drawn by a bunch of electrons hitting the phosphor coating of the screen causing it to emit light, usually green or blue.
Oscilloscopes contain a vacuum tube with a cathode (negative electrode) at one end to emit electrons and an anode (positive electrode) to accelerate them so that they move quickly down the tube to the screen. It's quite easy to 'lose' track of the screen if controls are misaligned. There is some variation in the arrangement and labeling of the many controls, so the following instructions may need to be adapted for your instrument.
Adjust Y SHIFT (up/down) and X SHIFT (left/right) to create a trail across the middle of the screen like a picture. The center wire carries the signal and the screen is connected to ground (0 V) to protect the signal from electrical interference (commonly called noise). The oscilloscope is connected like a voltmeter, but you should be aware that the shield (black) connection of the input cable is connected to ground on the oscilloscope.
Choose a setting so that the trace covers at least half the screen height, but does not disappear from the screen. The TIMEBASE (TIME/CM) control determines the speed at which the dot moves across the screen. Choose a setting so that the trace shows at least one cycle of the signal across the screen.
The time period (often just called period) is the time for one cycle of the signal. When a coil of wire is moved through a magnetic field, a voltage is induced across the end wires of the coil. The induced voltage is caused by the transfer of energy from the flux field of the.
Experimental techniques 56
The static bending stiffness (k) of the joined specimens was determined and was always found to be lower than that of a corresponding solid specimen. The logarithmic attenuation reduction was then evaluated from the measured values of the amplitudes of the first cycle (a1), last cycle (an+1) and the number of cycles (n) of the stable signal using the equation;. 26 Variation of logarithmic decrement with the diameter of rivet with beam length 240 mm and thickness (2+3) mm at different excitation amplitudes (y).
27 Variation of logarithmic decrease with the diameter of a rivet with a radius length of 243 mm and a thickness (2+4) mm at different excitation amplitudes (y). 29 Variation of logarithmic decrease with the diameter of the rivet with fixed-solid beam length 240 mm and thickness (2+3) mm at different excitation amplitudes (y). The increase in length results in a larger interfacial length, creating a larger area for energy dissipation from the structures.
From the above discussions, it is found that the damping of laminated and riveted structures can be improved by the following influencing parameters: (a) excitation amplitude, (b) excitation frequency, (c) length of specimens (d) end condition of the beam specimen. The damping of riveted structures can be studied under vacuum conditions to properly assess the damping. Cochardt, A.W., 1954, A method for determining the internal damping of machine parts, ASME, Journal of Applied Mechanics, Vol.
Ungar, E.E., 1973, The status of engineering knowledge concerning the damping of built-up structures, Journal of Sound and Vibration, Vol. Den Hartog, J.P., 1931, Forced vibrations with combined coulomb and viscous friction, Transactions of the ASME, Vol. Sidorov, O.T., 1983, Change of damping of vibrations during the operation in dependence of the parameters of bolted joints, Strength of Materials, Vol.
El-Zahry, R.M., 1985, Investigation of the vibration behavior of preloaded bolted connections, Dirasat-Engineering Technology, Vol. 2007, Further results on static analysis of slip damping with clamped laminated beams, European Journal of Scientific Research, Vol. Beards, C.F., 1983, The damping of structural vibration by controlled interface slip in joints, ASME, Journal of Vibration, Acoustics, Stress and Reliability and in Design, Vol.
