STUDENT NAME STUDENT ID COURSE LECTURER SUBMISSION DATE SUBJECT MD ATIQUR RAHMAN FAISAL SCM-012154 BACHELOR IN MECHANICAL ENGINEERING DR. CHIA 19th April 2013 ENGINEERING MECHANICS (EAT 227) Torsion Vibration Experiment Title: Introduction : Torsion Vibration. Torsion is the twisting of a metallic rod shaped object, when a torque is applied on two sides’ perpendicular to the radius of a uniform cross-sectional bar. To study the response of materials under a torsional force, the torsion test is performed by mounting the specimen onto a torsion testing machine, then applying the twisting moment till failure.
The torque and degree of rotation are measured and plotted. It can be seen higher torsion force is required at higher degree of rotation. Normally the test specimens used are of a cylindrical rod type since the stress distribution across the section of the rod is the simplest geometry, which is easy for the calculation of the stresses. Both ends of the cylindrical specimen are tightened to hexagonal socket in which one is fitted to a torque shaft and another is fitted to an input shaft. The twisting moment is applied by turning the input hand wheel to produce torque until specimen fails.
When the twisting moment is applied, the torque is reacted by a torque shaft, which moves in relation to the deflection arm. The movement of the deflection arm is measured by a linear potentiometer, which is connected to a calibrated TQ digital torque meter to give readout of the torque in unit of Nm or lb. in. The move we turned the input hand wheel clockwise to increase the degree of rotation, the more torque is produced. At the initial stage, the graphical relationship of the torque and degree of rotation measured is linear as demonstrated.
The specimen is elastically deformed and the recovery of the specimen to its original shape is possible if the specimen is unloaded. However, if a high degree of rotation is applied passing a proportional limit, the specimen starts to deform plastically and will not return to its original shape when the input hand wheel is turned anti-clockwise. Determining the natural frequency of a system undergoing torsional vibration. Using Newton’s second law of torsional system. o Mechanism : Objective Theory ? : : ? ? T ? I ?? …………………. (Equation 1) Where Io = mass moment of inertia of the disk Hence, ? k? ? I o?? …….. ……… (Equation 2) Where k = torsional stiffness of the shaft Rearrange Equation 2 ? ? ?? ? ? n ? ? 0 . ……….. ……… (Equation 3) 2 Where natural frequency of the system, MD. Atiqur Rahman Faisal Page 2 Torsion Vibration ?n ? k ….. ……. ….. ……… (Equation 4) Io From Simple Theory of Torsion, T ? G? ? ? J R L J = Polar second moment of area R = Radius of shaft Where T = Applied torque ? = Shear stress G = Shear modulus L = Length of shaft As torsional stiffness k ? Apparatus ? ? ? : ? = Angle of twist T ? , it can be determined through k ? GJ ………….. Equation 5) L One solid circular disk with mass = 4. 536kg, diameter = 150mm and thickness = 30mm. One annular circular disk with mass 1. 89kg, outer diameter 150mm, inner diameter = 110mm and thickness = 30mm. Two chucks; one steel rod; one stopwatch. MD. Atiqur Rahman Faisal Page 3 Torsion Vibration Procedure : 1. The diameter of the provided rod is measured at three different locations to get the average diameter of the rod. 2. The anchor is chucked tightly to the solid circular disk. 3. The length of the rod or the distance between the two chucks is initially kept 30cm. 4.
The disk is displaced slightly, so that the rod can be twisted. 5. The disk is released and the stopwatch is switched on simultaneously. 6. The time taken is recorded according 10, 20, 30, 40 and 50 cycles of the disk. 7. From step 3 to step 6 is repeated by increasing the length between the two chucks from 35 cm to 40 cm. 8. The whole procedure is repeated by attaching the annular circular disk on top of the solid disk. Results : Time, (s) 0. 3m(With Annular disk) Length Number of cycles 10 20 30 40 50 0. 3m 0. 4m 0. 5m 0. 4m(With Annular disk) 0. 5m(With Annular disk) 2. 63 6. 6 9. 25 11. 84 14. 67 3. 30 6. 12 8. 94 13. 31 16. 38 3. 41 6. 50 9. 61 12. 56 16. 56 3. 75 7. 55 10. 93 14. 3 18. 0 3. 95 7. 90 12. 13 15. 90 20. 15 4. 15 8. 52 12. 59 16. 7 21. 15 Given Information Diameter Of the shaft : D1 = 40. 6 mm, D2 = 42 mm, D3 = 45. 8 mm. Davg = 42. 8 mm. Mass of the circular disk = 4. 536kg. Diameter of the circular disk = 150mm. Thickness of the circular disk = 20mm. Mass of the annular disk = 1. 86kg. Outer diameter of the annular disk = 150mm. Inner diameter of the annular disk = 110mm. Thickness of the annular disk = 30mm. G (Shear module) = 80MPa.
MD. Atiqur Rahman Faisal Page 4 Torsion Vibration Sample calculation : = = We know, J (polar second moment of area) = We know, We know, (moment of inertia) = mr2 = (4. 54) (0. 075)2 = 0. 013 (circular disk) (moment of inertia) for annular disk = =0. 28728. = We know, K (torsion stiffness of the shaft) = We know, (angular speed) = = = = = 82. 2 rad/s = 87. 84 We know, ? (shear stress) = = 0. 076 Comparing experimental and theoretical value of = 2??? f, where f is collected from graph. = 2??? 3. 3387 = 20. 98rad/s Percentage errors ? 100%, : , = 291. 8% MD. Atiqur Rahman Faisal Page 5
Torsion Vibration Plotted Graph : N vs T (without annular disk) L=30cm 60 y = 3. 3387x + 0. 3193 50 40 30 20 10 0 0 5 10 15 20 N vs T (without annular disk) Linear (N vs T (without annular disk)) N vs T (without annular disk) L=35cm 60 y = 2. 9785x + 1. 3763 50 40 30 20 10 0 0 5 10 15 20 N vs T (without annular disk) L=35cm Linear (N vs T (without annular disk) L=35cm) MD. Atiqur Rahman Faisal Page 6 Torsion Vibration N vs T (without annular disk) L=40 cm 60 50 40 30 20 10 0 0 5 10 15 20 10 20 30 40 N vs T (without annular disk) L=40 cm Linear (N vs T (without annular disk) L=40 cm) y = 3. 805x + 0. 0328 50 N vs T (with annular disk) L=30 cm 60 50 40 30 20 10 0 0 5 10 15 20 N vs T (with annular disk) L=30 cm Linear (N vs T (with annular disk) L=30 cm) y = 2. 8355x – 0. 9242 MD. Atiqur Rahman Faisal Page 7 Torsion Vibration N vs T (with annular disk) L=35 cm 60 50 40 30 20 10 0 0 5 10 15 20 25 N vs T (with annular disk) L=35 cm Linear (N vs T (with annular disk) L=35 cm) y = 2. 4746x + 0. 2905 N vs T (with annular disk) L=40cm 60 50 40 30 20 10 0 0 5 10 15 20 25 N vs T (with annular disk) Linear (N vs T (with annular disk)) y = 2. 3702x + 0. 0832 MD.
Atiqur Rahman Faisal Page 8 Torsion Vibration Graph Comparison : 60 50 y = 3. 3387x + 0. 3193 40 30 20 10 0 0 5 10 15 20 y = 2. 8355x – 0. 9242 N vs T (With annular disk) L=30cm N vs T (Without annular disk) L=30cm Linear (N vs T (With annular disk) L=30cm) Linear (N vs T (Without annular disk) L=30cm) 60 50 y = 2. 9785x + 1. 3763 40 30 20 10 0 0 5 10 15 20 25 y = 2. 4746x + 0. 2905 N vs T (with Annular disk) L=35 cm N vs T (without Annular disk) L=35cm Linear (N vs T (with Annular disk) L=35 cm) Linear (N vs T (without Annular disk) L=35cm) MD. Atiqur Rahman Faisal
Page 9 Torsion Vibration 60 y = 2. 3702x + 0. 0832 50 y = 3. 0805x + 0. 0328 40 N vs T (with annular disk) L=40cm N vs T(without annular disk) L=40cm Linear (N vs T (with annular disk) L=40cm) Linear (N vs T(without annular disk) L=40cm) 30 20 10 0 0 5 10 15 20 25 Discussion : From the graph it can be observed that, the specimen at first deformed elastically and then deformed plastically. The initial stage of the elastic behavior describes a linear relationship between torque and degree of rotation with the slope representing the shear modulus of rigidity, G.
Beyond the proportional limit, specimen deformed in a plastic manner. The stresses vary from section to section of the specimen. To get an average value, the diameter of the specimen is measured from 3 different places and then calculated the average value. Conclusion : Comparing results, it is observed that the error percentage is quiet high; this is due to the high sensitive values that have been collected during the experiment. Possible reasons for the errors are, when the disk was rotating, it was quiet difficult to take the exact turn, or rotation of the disk.
So there was also some human reflex and visual errors. Even though, when the experiment was performing, there was some unwanted air resistance application running beside the experiment. Some of the measurement was so sensitive, and it was hard to take the measurement. The apparatus tools, and measurement machines was also not so accurate. The experiment could be repeated for some more time to get some more result to find it more accurate. Considering all the facts, the experiment can be accepted with errors. MD. Atiqur Rahman Faisal Page 10