Version 05 June 16 2017 Page 1 School of Engineering and Technology Higher Education Division ENEM14015 – Dynamic System Modelling and Control Laboratory Instruction Sheet Torsional Vibration ___________________________________________________________________________ Compulsory Personal Protection Equipment (PPE) required to access laboratories. Closed footwear Long or short sleeve shirts. No tank tops allowed. Lecturer: Mitchell McClanachan Lab OIC: Anwar Mohammad Learning outcomes (LOs): LO1: Design mathematical models that analyse and evaluate mechanical systems LO4: Relate theory to problems of introducing, operating and maintaining mechanical systems in the industrial context Introduction Torsional vibrations can occur in shafts due to the shaft acting as a torsional spring and attached masses such as gears acting as inertial masses. Excitation can be produced from changes in input or output power and defects in equipment attached to the shaft such as gears and bearings. This lab sheet provides the experimental procedure for the laboratory. Some theory is also presented along with prompts on possible discussions/reflections. You are encouraged to expand on these suggested reflections as this will help to improve your portfolio. Objectives – Observations of shaft-disc system vibrations and determination of the rigidity modulus of the shaft material; – Observation of two disc-shaft system vibrations, determination of the natural frequency and calculation of the node position; – Comparison of theoretical and experimental results.Version 05 June 16 2017 Page 2 Equipment The laboratory equipment consists of two discs connected by a solid round shaft and supporting vertically. Both discs are supported on bearings with full freedom of rotation; thus the system can rotate and vibrate simultaneously. The upper disc can be fixed rigidly to the bracket and in such case the system is converted into a one disc vibrating system. An indicating arrow can be positioned along the shaft to show the shaft rotation at any chosen point. When placed at a node position, the arrow will have no movement. Theoretical Background One disc system Analysing a system with a shaft and disc is shown in Figure 1. A metal shaft has a rotational spring characteristic (k) and the disc has a rotational inertia (I). Assuming there is little or no damping in the system the modelling equation is that of a 1DOF undamped free vibration system, shown in Figure 1 and associated equations. Modelling equation: 𝐵𝐵𝘅̈ + 𝘅 = 0 where: 𝐠= 𝐵𝔎𝐵𝐍 𝐍 and 𝐍 𝔎𝐵𝐠= 𝐵4 32 𝐵𝐵𝐠= 𝐠𝐲 8 G = modulus of rigidity (Pa), m = disc mass (kg), D = disc diameter (m), l = shaft length (m), d = shaft diameter (m) Figure 1: Single disc system From the general solution for the natural period of vibration: 𝐠= 2𝟽𝐵𝐠𝐵𝐠Eqn. 1 Expressing natural period of vibration in shaft and disc dimensions: 𝐠= 2𝟽𝐴 𝐠𝐵42𝐵 (seconds) D l d m φVersion 05 June 16 2017 Page 3 Two disc system For a two disc system as shown in Figure 2 the natural period of vibration and the node point can be derived by using the conservation of momentum in the system, as shown in Equation 2. The conservation of motion highlights that the rotational speeds of the discs ω1 and ω2 are in opposite directions. So there is a nodal point between the discs where ω is zero. It also indicates that both discs have the same period of vibration since when ω1= 0 then ω2 = 0. Figure 2: Two disc system 𝐱𝐱 + 𝐲𝐲 = 0 Eqn. 2 Since T1 = T2 and using the single disc relationship for T of Equation 1 gives: 𝐱 𝐲 = 𝐱 𝐲 And using the relationship for k gives: 𝐱 𝐲 = 𝐠𝐍 , and we know a + b = l thus the node position can be calculated as: 𝐠= 𝐠𝐲 𝐱+𝐲 and 𝐠= 𝐠𝐱 𝐱+𝐲 and the period of natural vibration is: 𝐠= 2 𝟽𝐱𝐱+𝐲𝐲 ∙ 𝐳2 𝐴𝐵 D2 b d D1 a l nodeVersion 05 June 16 2017 Page 4 Experimental Procedures Experiment 1: Determining shaft modulus of rigidity 1) Fix or hold the upper disc to create a one disc (bottom) vibrating system. 2) Measure the shaft length (within 1 mm) and shaft diameter (within 0.1 mm) three times. Record the measurements and calculate the average values. 3) Measure the mass and diameter of the bottom disc and calculate the mass-moment of inertia. 4) Rotate the disc by a small angle about the vertical axis and release. 5) Observe the oscillation and measure the time of 10 full oscillations using a stopwatch. Repeat this three times. Record the measurements and calculate the average period. 6) Calculate the shaft modulus of rigidity for the period of vibration measured and compare this with values in literature. 7) Perform an error analysis and draw out the conclusions. Experiment 2: Torsional vibrations of a two disc system 1) Loosen the upper disc and check if the system rotates free so it acts like a two disc system. 2) Measure the mass and diameter of the upper disc and calculate the mass-moment of inertia. 3) Using the shaft modulus of rigidity determined in Experiment 1, calculate the theoretical period of natural vibration of the two discs system. 4) Rotate two discs in opposite directions by a small angle about vertical axis and release. Observe the oscillations. Check if both discs have the same period of vibrations. 5) Measure the time of 10 full oscillations using a stopwatch. Repeat this three times. Record the measurement and calculate the average period. Compare the period with the theoretical calculation in Step 3. 6) Calculate the theoretical position of the node. 7) Place the indicating arrow at the theoretical node position and start the system oscillating. Observe the indication of the arrow. If the arrow is not at the node determine a more accurate location of the node by adjusting the arrow up/down and restarting the free oscillation. 8) Perform an error analysis and draw out the conclusions.Version 05 June 16 2017 Page 5 Results Record your results, see the following tables for a possible format. Table 1: Torsional Vibration Measurements Measurement # Average 1 2 3 Shaft length (mm) Shaft Diameter (mm) Mass of bottom disc (kg) Diameter of bottom disc (mm) Mass of top disc (kg) Diameter of top disc (mm) Table 2: Single Disc Vibration Results Measurement # Average Calculated Shaft modulus 1 2 3 (GPa) Single Disc System Time for 10 oscillations (s) Table 3: Two Disc Vibration Results Measurement # Average Theoretical 1 2 3 Value Two disc System Time for 10 oscillations (s) Distance of node from bottom disc (mm) Discussion 1) Discuss the results of the experiments including any sources of inaccuracies that could exist in the procedures used. 2) Discuss the comparison of the measured results from the experimental values. 3) Provide a discussion on the listed laboratory objectives. 4) Discuss how your learning from this laboratory could be applied in engineering practice.