VTU-NPTEL-NMEICT Project

Transcription

1 VTU-NPTEL-NMEICT Project Progress Report The Project on Development of Remaining Three Quadrants to NPTEL Phase-I under grant in aid NMEICT, MHRD, New Delhi SME Name : Course Name: Type of the Course Module Subject Matter Expert Details Dr.MOHAMED HANEEF PRINCIPAL, VTU SENATE MEMBER Vibration engineering web V DEPARTMENT OF MECHANICAL ENGINEERING, GHOUSIA COLLEGE OF ENGINEERING, RAMANARAM Page of 25

2 Sl. No a. Animations. b. Videos. c. Illustrations. a. Wikis. b. Open Contents CONTENTS DISCRETION Lecture Notes (Vibration absorber). Quadrant -2 Quadrant -3 Quadrant -4 a. Self Answered Question & Answer b. Assignment Page 2 of 25

3 Module-V VIBRATION ABSORBER LECTURE NOTES. Tuned absorber, Determination of mass ratio. Un-damped Dynamic Vibration Absorber Consider a two degree of freedom system with a forcing function F F o sint as shown in figure- (a). Consider original system of mass m which is under excitation. In this system suppose the excitation frequency is equal to the resonant frequency n k m the amplitude becomes very large. If we coupled a spring mass system to the original system as shown in the fig (a), this spring mass system acts as an absorber and reduces the amplitude of vibration of mass m to zero if its natural frequency is equal to the excitation frequency k 2 m 2 Hence we have of original system of modified system k 2 m m 2 This relation is called as tuned absorber, which satisfied the absorber condition. Figure- (a) k Figure (b) Figure- () Two degree of freedom system with forcing functionf on mass Page 3 of 25

4 2. Derivation on tuned absorber Let x and x 2 be the displacement of the masses m and m 2 respectively at any given instant of time measured from the equilibrium position with x 2 > x. Then the spring forces acting on the masses are as shown in free body diagram in Figure (b) Based on Newton s second law of motion f mx For mass m we have [m x + (k + k 2 )x ] (k 2 x 2 ) F 0 sint () [m 2 x 2 + (x 2 x )k 2 ] 0 For the principal modes of vibration the solution to above equation become Substituting equation (2) in () we get. Rearranging equation (3) we get x X sin t > x 2 X sin t x 2 X 2 sin t > x 2 2 (2) X 2 sin t [k + k 2 m 2 ]X (k 2 X 2 ) F (k 2 X 2 ) + [k 2 m 2 2 (3) ]X 2 0 [k 2 m 2 2 ]F X m m 2 4 [m k 2 +m 2 (k k 2 ) 2 + k k 2 ] (4) k F X 2 m m 2 4 [m k 2 +m 2 (k k 2 ) 2 + k k 2 ] To bring these equations into dimensionless forms, let divided by their numerator and denominator by k k 2 and introduce the following notations are, X St F 0 k Frequency deflection of first mass k m natural frequency of main system alone 2 k 2 m 2 natural frequency of the absorber system alone μ m 2 m ratio of absorber mass to the main mass Equation (4) becomes. Page 4 of 25

5 X X ( + μ) ( + μ) (5) Equation (5) clearly shows that X 0 when ; that is when the excitation frequency of the absorber, the main system amplitude becomes zero even it is excited by harmonic force. This is the principle of an un-damped dynamic vibration absorber, since, if a main system has undesirable vibrations at the operating frequency, a secondary spring mass system (absorber system) having its natural frequency equal to the operating frequency can be coupled to the main system to reduce its amplitude to zero. Substituting in equation (5) X 2 amplitude become X 2 (μ) 2 2 m k 2 m 2 2 m k m i. e. X 2 F 0 k2 (6) F 0 X 2 k m 2 m k 2 m m 2 k Above equation (6) indicates that, the main mass vibrations have been reduced to zero and the vibrations have been taken up by the absorber system. Hence the name called as vibration absorber. The vibration absorber to main system is not much effective unless the main system is operating at least nearest to it, for this condition. But for the absorber to be effective, we have 2. Therefore for the effectiveness of the absorber at the operating frequency corresponding to the natural frequency of the main system alone, we have 2 or k 2 m 2 k m (7) When the above condition is fulfilled, the absorber is known as tuned absorber. For tuned absorber equation (5) becomes Page 5 of 25

6 X X (2 + μ) (2 + μ) (8) Equation (7) is holds good at 2, at which point X Determination of mass ratio. μ is the mass ratio m 2 /m. where k m and 2 k 2 m 2, However, the natural or resonant frequencies of the 2-degree of freedom system formed after attaching the auxiliary system are given by different expressions in terms of m, m 2, k, and k 2. Plot of the dimensionless responses of each of the two masses is shown above when a harmonic force acts on m when excitation frequency is close to resonant frequency of main system,, but much away from resonant frequency of auxiliary system, 2, Mass ratio between μm 2 /m 0.5 to 2.5. In denominators of equations (8) are identical. At a value of when these denominators are zero, the two masses have infinite amplitudes of vibration. The expression for the denominators is a quadratic in 2 and therefore there are two values of for which these expressions vanish. From equation (8) Solving for 2 we have (2 + μ) + 0 (9) μ 2 2 ± μ + μ2 4 (0) The relationship of equation (0) is plotted in figure below from this plot we see that greater the mass ratio μ, greater is the spread between the two resonant frequencies. Page 6 of 25

7 2 2 μ Figure: effect of mass ratio on natural (resonant) frequency. The dimensionless response curves for the main system and the absorber system given by the equation (0) are shown in figures below for a value of μ 0.2. the dotted curves shown actually mean that the amplitude is negative or its phase difference with respect to the exiting force is These portions of the curves, however, are shown on positive side of the ordinate. It can be seen from hese curves that when <, the phase difference between them is For the system alone without absorber we have only one resonant frequency at 2. Imagine for the main system alone, the exciting frequency is very close to its natural frequency. To overcome this resonant condition we attach a vibration absorber ( 2 ) to the main system, thereby reducing its vibration to zero. When the amplitudes are equal 2 + μ X Xst 2 Page 7 of 25

8 Fig:(a) 2 μ 0.20 Fig: (b) 2 Figure: frequency-response curves for (a) main system, (b) absorber. Page 8 of 25

9 4. Tuned and damped absorber, Unturned Dry friction damper: This type of damper is very useful in reducing the amplitude of torsional vibrations near resonance conditions figure shown below a dry friction type of damper known as Lanchester damper. Figure: Lanchester damper. It consists of two flywheels which can rotate freely on the shaft bearings. There is a hub in between rigidly fixed to the shaft. The hub carries friction lining on its faces against which the flywheels can be pressed by screwing down the coil springs. If the engine I.e. hub executes torsional vibrations, the motion of the flywheels depend upon the amount of friction between them and the hub. If the friction torque is zero i.e. pressure between the frictional plates and flywheel become zero, the relative velocity is maximum. Since friction torque is zero, there is no energy dissipation. On the other hand, if friction torque is very large i.e. pressure between the friction plates and flywheels is excessive, the flywheels become rigid with the shaft and have same oscillation at that shaft. There is thus, no relative slip and hence no energy dissipation. However, between these two extremes, there is both friction torque and slip, so that energy is dissipated. This reduces the amplitude of torsional oscillations. Greater is the amount of energy dissipation, greater will be reduction in amplitude of vibration. The energy dissipation V/a frictional torque plot is shown in figure below. Page 9 of 25

10 Figure: energy dissipation V/a frictional torque. 5. Unturned viscous dampers: This type of damper is useful for damping out torsional oscillations. Shown in Fig: (a) a unturned viscous damper commonly known as Haudaillc Damper. It is similar in principle to the Lanchester Damper Except that instead of dry friction damping is employed in this case. It consists of a flywheel which can rotate freely about the hub. The hub is splined inside so that it can be attached easily to the shaft end. The flywheel is enclosed in housing, welded to the hub. The clearance between the housing and flywheel is filled with silicone fluid. Silicone oil is used because of its high viscosity index i.e., its viscosity changes relatively little with temperature. The flywheel rotates at the shaft speed owing to viscous drag of the fluid. Figure: Unturned viscous dampers When the damping is zero in the damper, it is ineffective and the system reduces to one degree of freedom system. If the damping is infinite, the damper mass becomes integral with the main Page 0 of 25

11 mass. It still remains a single degree of feedom. The frquency response curve is of same nature except that the peak shifts to the right as shown by curve (b) Figure: response curve for untuned viscous damper for (a) zero damping, (b) infinte damping, (c) optimal damping. The point of intersection of the above two curve (for ζ 0 and ζ ) is the point through which the response curve of different damping values pass. It is impossible to reduce the peak amplitude below this point. So, a system having optimal damping has its response curve with P as the highest point. We start by assuming optimal damping, calculate the system. Find the dangerous frequency and then specify the damping, c for the damping the flywheel and housing, affect the value of C. Page of 25

12 Animations (Animation links related to Vibration Absorber) QUADRANT Videos (video links related,to Vibration Absorber) le-2-lecture-2-tuned-vibration-absorber ILLUSTRATION Page 2 of 25

13 . A section of pipe pertaining to a certain machine vibrates with large amplitude at a compressor speed of 220 r.p.m. for analyzing this system a spring-mass system was suspended form the pipe to act an absorber. A kg absorber mass tuned to 220 c.p.m. what must be the mass and the spring stiffness of the absorber if the resonant frequencies are to lie outside the range of 50 to 30 c.p.m.? Solution: 220 2π 23.0 rad/sec rad/sec M kg k 2 M N/m n 88 2π rad/sec n π rad/sec For 2, we apply equation to find µ by taking first n, and then n2, When n 9.7, μ 0.00 n2 27.0, μ 0.04 The average value of μ therefore is Also μ 0.02 M 2 M M M kg 0.02 k k N/m 0.02 After finding the weight and the stiffness of the main system it is required first to find μ so that new resonant frequencies will not be in specified range. n Page 3 of 25

14 n W.K.T two corresponding values of μ are 0.62 and choosing the higher the value, we get μ 0.62 Therefore, M kg 2. Derive an expression for tuned Absorber. Solution: k N/m Let x and x 2 be the displacement of the masses m and m 2 respectively at any given instant of time measured from the equilibrium position with x 2 > x. Then the spring forces acting on the masses are as shown in free body diagram in Figure (b) Based on Newton s second law of motion f mx For mass m we have [m x + (k + k 2 )x ] (k 2 x 2 ) F 0 sint () [m 2 x 2 + (x 2 x )k 2 ] 0 For the principal modes of vibration the solution to above equation become Substituting equation (2) in () we get. Rearranging equation (3) we get x X sin t > x 2 X sin t x 2 X 2 sin t > x 2 2 (2) X 2 sin t [k + k 2 m 2 ]X (k 2 X 2 ) F (k 2 X 2 ) + [k 2 m 2 2 (3) ]X 2 0 [k 2 m 2 2 ]F X m m 2 4 [m k 2 +m 2 (k k 2 ) 2 + k k 2 ] (4) k F X 2 m m 2 4 [m k 2 +m 2 (k k 2 ) 2 + k k 2 ] To bring these equations into dimensionless forms, let divided by their numerator and denominator by k k 2 and introduce the following notations are, X St F 0 k Frequency deflection of first mass Page 4 of 25

15 k m natural frequency of main system alone 2 k 2 m 2 natural frequency of the absorber system alone μ m 2 m ratio of absorber mass to the main mass Equation (4) becomes. X ( + μ) X (5) ( + μ) Equation (5) clearly shows that X 0 when ; that is when the excitation frequency of the absorber, the main system amplitude becomes zero even it is excited by harmonic force. This is the principle of an un-damped dynamic vibration absorber, since, if a main system has undesirable vibrations at the operating frequency, a secondary spring mass system (absorber system) having its natural frequency equal to the operating frequency can be coupled to the main system to reduce its amplitude to zero. Substituting in equation (5) X 2 amplitude become X 2 (μ) 2 2 m k 2 2 m m 2 m k 2 m m 2 k 2 2 m k 2 m i. e. X 2 F 0 k2 (6) F 0 X 2 k 2 Above equation (6) indicates that, the main mass vibrations have been reduced to zero and the vibrations have been taken up by the absorber system. Hence the name called as vibration absorber. The vibration absorber to main system is not much effective unless the main system is operating at least nearest to it, for this condition. But for the absorber to be effective, we Page 5 of 25

16 have 2. Therefore for the effectiveness of the absorber at the operating frequency corresponding to the natural frequency of the main system alone, we have 2 or k 2 m 2 k m (7) When the above condition is fulfilled, the absorber is known as tuned absorber. For tuned absorber equation (5) becomes X X (2 + μ) (8) 2 (2 + μ) Equation (7) is holds good at 2, at which point X Explain with neat sketche Lanchester Damper. Solution : This type of damper is very useful in reducing the amplitude of torsional vibrations near resonance conditions figure shown below a dry friction type of damper known as Lanchester damper. Figure: Lanchester damper. Page 6 of 25

17 It consists of two flywheels which can rotate freely on the shaft bearings. There is a hub in between rigidly fixed to the shaft. The hub carries friction lining on its faces against which the flywheels can be pressed by screwing down the coil springs. If the engine I.e. hub executes torsional vibrations, the motion of the flywheels depend upon the amount of friction between them and the hub. If the friction torque is zero i.e. pressure between the frictional plates and flywheel become zero, the relative velocity is maximum. Since friction torque is zero, there is no energy dissipation. On the other hand, if friction torque is very large i.e. pressure between the friction plates and flywheels is excessive, the flywheels become rigid with the shaft and have same oscillation at that shaft. There is thus, no relative slip and hence no energy dissipation. However, between these two extremes, there is both friction torque and slip, so that energy is dissipated. This reduces the amplitude of torsional oscillations. Greater is the amount of energy dissipation, greater will be reduction in amplitude of vibration. The energy dissipation V/a frictional torque plot is shown in figure below. 4. Explain with a neat sketch Viscous Damper. Solution) Figure: energy dissipation V/a frictional torque. This type of damper is useful for damping out torsional oscillations. Shown in Fig: (a) a unturned viscous damper commonly known as Haudaillc Damper. It is similar in principle to the Lanchester Damper Except that instead of dry friction damping is employed in this case. Page 7 of 25

18 It consists of a flywheel which can rotate freely about the hub. The hub is splined inside so that it can be attached easily to the shaft end. The flywheel is enclosed in housing, welded to the hub. The clearance between the housing and flywheel is filled with silicone fluid. Silicone oil is used because of its high viscosity index i.e., its viscosity changes relatively little with temperature. The flywheel rotates at the shaft speed owing to viscous drag of the fluid. Figure(a): Unturned viscous dampers When the damping is zero in the damper, it is ineffective and the system reduces to one degree of freedom system. If the damping is infinite, the damper mass becomes integral with the main mass. It still remains a single degree of feedom. The frquency response curve is of same nature except that the peak shifts to the right as shown by curve (b) Figure: response curve for untuned viscous damper for (a) zero damping, (b) infinte damping, (c) optimal damping. Page 8 of 25

19 The point of intersection of the above two curve (for ζ 0 and ζ ) is the point through which the response curve of different damping values pass. It is impossible to reduce the peak amplitude below this point. So, a system having optimal damping has its response curve with P as the highest point. We start by assuming optimal damping, calculate the system. Find the dangerous frequency and then specify the damping, c for the damping the flywheel and housing, affect the value of C. QUADRANT-3 Wikis: (This includes wikis related to Vibration Absorber) Open Contents: (This includes wikis related to Continuous System : Closed form solution)) Mechanical Vibrations, S. S. Rao, Pearson Education Inc, 4 th edition, Mechanical Vibrations, V. P. Singh, Dhanpat Rai & Company, 3 rd edition, Mechanical Vibrations, G. K.Grover, Nem Chand and Bros, 6 th edition, 996 Theory of vibration with applications,w.t.thomson,m.d.dahleh and C Padmanabhan,Pearson Education inc,5 th Edition,2008 Theory and practice of Mechanical Vibration : J.S.Rao&K,Gupta,New Age International Publications,New Delhi,200 Page 9 of 25

20 Self Answered Question & Answer QUADRANT-4. Derive an expression for tuned Absorber. Solution: Let x and x 2 be the displacement of the masses m and m 2 respectively at any given instant of time measured from the equilibrium position with x 2 > x. Then the spring forces acting on the masses are as shown in free body diagram in Figure (b) Based on Newton s second law of motion f mx For mass m we have [m x + (k + k 2 )x ] (k 2 x 2 ) F 0 sint () [m 2 x 2 + (x 2 x )k 2 ] 0 For the principal modes of vibration the solution to above equation become Substituting equation (2) in () we get. Rearranging equation (3) we get x X sin t > x 2 X sin t x 2 X 2 sin t > x 2 2 (2) X 2 sin t [k + k 2 m 2 ]X (k 2 X 2 ) F (k 2 X 2 ) + [k 2 m 2 2 (3) ]X 2 0 [k 2 m 2 2 ]F X m m 2 4 [m k 2 +m 2 (k k 2 ) 2 + k k 2 ] (4) k F X 2 m m 2 4 [m k 2 +m 2 (k k 2 ) 2 + k k 2 ] To bring these equations into dimensionless forms, let divided by their numerator and denominator by k k 2 and introduce the following notations are, X St F 0 k Frequency deflection of first mass k m natural frequency of main system alone 2 k 2 m 2 natural frequency of the absorber system alone Page 20 of 25

21 μ m 2 m ratio of absorber mass to the main mass Equation (4) becomes. X X ( + μ) ( + μ) (5) Equation (5) clearly shows that X 0 when ; that is when the excitation frequency of the absorber, the main system amplitude becomes zero even it is excited by harmonic force. This is the principle of an un-damped dynamic vibration absorber, since, if a main system has undesirable vibrations at the operating frequency, a secondary spring mass system (absorber system) having its natural frequency equal to the operating frequency can be coupled to the main system to reduce its amplitude to zero. Substituting in equation (5) X 2 amplitude become X 2 (μ) 2 2 m k 2 m 2 2 m k m i. e. X 2 F 0 k2 (6) F 0 X 2 k m 2 m k 2 m m 2 k Above equation (6) indicates that, the main mass vibrations have been reduced to zero and the vibrations have been taken up by the absorber system. Hence the name called as vibration absorber. The vibration absorber to main system is not much effective unless the main system is operating at least nearest to it, for this condition. But for the absorber to be effective, we have 2. Therefore for the effectiveness of the absorber at the operating frequency corresponding to the natural frequency of the main system alone, we have 2 or k 2 m 2 k m (7) Page 2 of 25

22 When the above condition is fulfilled, the absorber is known as tuned absorber. For tuned absorber equation (5) becomes X X (2 + μ) (2 + μ) (8) Equation (7) is holds good at 2, at which point X Explain with neat sketch Lanchester Damper. Solution : This type of damper is very useful in reducing the amplitude of torsional vibrations near resonance conditions figure shown below a dry friction type of damper known as Lanchester damper. Figure: Lanchester damper. Page 22 of 25

23 It consists of two flywheels which can rotate freely on the shaft bearings. There is a hub in between rigidly fixed to the shaft. The hub carries friction lining on its faces against which the flywheels can be pressed by screwing down the coil springs. If the engine I.e. hub executes torsional vibrations, the motion of the flywheels depend upon the amount of friction between them and the hub. If the friction torque is zero i.e. pressure between the frictional plates and flywheel become zero, the relative velocity is maximum. Since friction torque is zero, there is no energy dissipation. On the other hand, if friction torque is very large i.e. pressure between the friction plates and flywheels is excessive, the flywheels become rigid with the shaft and have same oscillation at that shaft. There is thus, no relative slip and hence no energy dissipation. However, between these two extremes, there is both friction torque and slip, so that energy is dissipated. This reduces the amplitude of torsional oscillations. Greater is the amount of energy dissipation, greater will be reduction in amplitude of vibration. The energy dissipation V/a frictional torque plot is shown in figure below. 3. Explain with a neat sketch Viscous Damper. Solution) Figure: energy dissipation V/a frictional torque. This type of damper is useful for damping out torsional oscillations. Shown in Fig: (a) a unturned viscous damper commonly known as Haudaillc Damper. It is similar in principle to the Lanchester Damper Except that instead of dry friction damping is employed in this case. Page 23 of 25

24 It consists of a flywheel which can rotate freely about the hub. The hub is splined inside so that it can be attached easily to the shaft end. The flywheel is enclosed in housing, welded to the hub. The clearance between the housing and flywheel is filled with silicone fluid. Silicone oil is used because of its high viscosity index i.e., its viscosity changes relatively little with temperature. The flywheel rotates at the shaft speed owing to viscous drag of the fluid. Figure(a): Unturned viscous dampers When the damping is zero in the damper, it is ineffective and the system reduces to one degree of freedom system. If the damping is infinite, the damper mass becomes integral with the main mass. It still remains a single degree of feedom. The frquency response curve is of same nature except that the peak shifts to the right as shown by curve (b) Figure: response curve for untuned viscous damper for (b) zero damping, (b) infinte damping, (c) optimal damping. Page 24 of 25

25 The point of intersection of the above two curve (for ζ 0 and ζ ) is the point through which the response curve of different damping values pass. It is impossible to reduce the peak amplitude below this point. So, a system having optimal damping has its response curve with P as the highest point. We start by assuming optimal damping, calculate the system. Find the dangerous frequency and then specify the damping, c for the damping the flywheel and housing, affect the value of C. Assignment:. Derive an expression for tuned Absorber. 2. Explain with a neat sketch Viscous Damper. 3. Explain with neat sketche Lanchester Damper. 4. Derive an expression for tuned Absorber. 5. A section of pipe pertaining to a certain machine vibrates with large amplitude at a compressor speed of 220 r.p.m. for analyzing this system a spring-mass system was suspended form the pipe to act an absorber. A kg absorber mass tuned to 220 c.p.m. what must be the mass and the spring stiffness of the absorber if the resonant frequencies are to lie outside the range of 50 to 30 c.p.m.? Page 25 of 25