VTU-NPTEL-NMEICT Project

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1 MODUE-X -CONTINUOUS SYSTEM : APPROXIMATE METHOD VIBRATION ENGINEERING 14 VTU-NPTE-NMEICT Project Progress Report The Project on Deveopment of Remaining Three Quadrants to NPTE Phase-I under grant in aid NMEICT, MHRD, New Dehi SME Name : Course Name: Type of the Course Modue Subject Matter Expert Detais Dr.MOHAMED HANEEF PRINCIPA, VTU SENATE MEMBER Vibration engineering web X VTU-NPTE-NMEICT Project DEPARTMENT OF MECHANICA ENGINEERING, GHOUSIA COEGE OF ENGINEERING, RAMANARAM Dr.MOHAMED HANEEF,PRINCIPA,GHOUSIA COEGE OF ENGINEERING,RAMANAGARAM Page 1 of 4

2 MODUE-X -CONTINUOUS SYSTEM : APPROXIMATE METHOD VIBRATION ENGINEERING 14 CONTENTS S. No. DISCRETION 1. ecture Notes (Continuous System: Approximate method).. Quadrant a. Animations. b. Videos. c. Iustrations. a. Wikis. b. Open Contents a. Probems. Quadrant -3 VTU-NPTE-NMEICT Project Quadrant -4 b. Sef Assigned Q & A.. Dr.MOHAMED HANEEF,PRINCIPA,GHOUSIA COEGE OF ENGINEERING,RAMANAGARAM Page of 4

3 MODUE-X -CONTINUOUS SYSTEM : APPROXIMATE METHOD VIBRATION ENGINEERING 14 Modue-X Continuous system : Approximate Soution ECTURE NOTES 1. Rayeigh-Ritz method Rayeigh-Ritz method is an extension of Rayeigh s method. It not ony provides a means of obtaining a more accurate vaue for the fundamenta frequency, but it aso gives approximations to the higher frequencies and mode shapes. In Rayeigh s method a singe shape function is used to approximate the natura mode of vibration but in Rayeigh-Ritz s method a series of shape functions mutipied by constant coefficients are used. The coefficients are adjusted by minimizing the frequency with respect to each of the coefficients, which resuts in n agebraic equations in ω. The soution of these equations then gives the natura frequencies and mode shapes of the system. The number frequencies cacuated is equa to the number arbitrary functions used. If n functions are seected for approximating the defection y(x) for the transverse vibration of a uniform beam, then. y(x) = c 1 y 1 (x) + c y (x) + c 3 y 3 (x) c n y n (x) (1) where y 1 (x), y (x), y 3 (x), y n (x) are the constant coefficients to be determined. We have, ω n = Maximum potentia energy (U max) Maximum kinetic energy (T max ) = EI d i y (x) dx dx = d EI i y dx dx ρa{y(x)} dx my dx () Substituting equation (1) in equation () and equating the first partia derivative of the resuting expression with respect to each of the constant coefficients C i to zero wi make the natura frequency stationary. i.e., ω n c 1 = ; ω n c VTU-NPTE-NMEICT Project = ; ω n c 3 = In genera ω n c i = where I = 1,, n (3) Equation 3 represents a set of n agebraic equation in the n coefficients c 1, c, c 3, ---- c n and ω n to be determined. Since the right hand side of equation is a function of c 1, c, c 3, ---- c i, its partia derivatives with respect to c 1, c, c 3, ---- c i wi a be zero for minimum vaue of ω n. Hence we have Dr.MOHAMED HANEEF,PRINCIPA,GHOUSIA COEGE OF ENGINEERING,RAMANAGARAM Page 3 of 4

4 MODUE-X -CONTINUOUS SYSTEM : APPROXIMATE METHOD VIBRATION ENGINEERING 14 my dx EI d y c i dx dx EI d y dx dx my c i dx = where i = 1,, 3, n (4) From equation () Therefore equation (4) becomes where A = EI d y EI d y dx EI d y c i dx dx dx dx dx = ω n my dx ω n my dx = (5) c i B i.e., A ω c n = (6) i c i and B = my dx Equation 6 are set of I homogeneous inear equations in unknowns c 1, c, c 3, ---- c n. It wi have a non-trivia soution if the determinant of its coefficients is equa to zero. It gives an i th degree equation in ω n, the owest root of which is equa to ω n1 and the next higher root ω n and so on.. Rayeigh s energy method: Using the energy method to find the natura frequencies of singe degree of freedom system, the principe of conservation of energy, in the context of an undamped vibrating system, can be restated as T 1 + U 1 = T + U (1) Where the subscripts 1 and denote two different instants of time; Specificay, we use the subscript 1 to denote the time when the mass is passing through its static equiibrium position and choose u 1 = as reference for the potentia energy. If we et the subscript indicate the time corresponding to the maximum dispacement of the mass, we have T =. Thus Eq. [1] Becomes. VTU-NPTE-NMEICT Project T 1 +O = O + U --- () If the system is undergoing harmonic motion, then T1 and U denote the maximum vaues of T and U, respectivey, and Eq. [] becomes. T max = U max --- (3) Dr.MOHAMED HANEEF,PRINCIPA,GHOUSIA COEGE OF ENGINEERING,RAMANAGARAM Page 4 of 4

5 MODUE-X -CONTINUOUS SYSTEM : APPROXIMATE METHOD VIBRATION ENGINEERING Ritz-Gaerkin method In the Ritz-Gaerkin method, an approximate soution of the probem is found by satisfying the governing noninear equation in the average. To see how the method works, et the noninear differentia equations be represented as E[x] = (1) An approximate soution of Eq. (1)is assumed as x (t) = a 1 1 (t) + a (t) + + a n n (t) () Where 1 (t), (t). n (t) are prescribed functions of time and a 1, a,. a n are weighting factors to be determined. If Eq. (), we get a function E x (t). since x (t) is not, in genera, the exact soution of Eq. (1), E (t) = E x (t)wi not be zero. However, the vaue of E (t)wi serve as a measure of the accuracy of the approximation; in fact, E (t) as x x. The weighting factor a i are determined by minimizing the integra τ E [t]dt (3) Where τ denoted as the period of the motion. The minimization of the function of Eq.(3) requires τ E τ E [t] [t]dt = E [t] dt = (4) a i a i Eq. (4) represents a system of n agebraic equations can be soved simutaneousy to find vaues of a 1, a. a n. VTU-NPTE-NMEICT Project Dr.MOHAMED HANEEF,PRINCIPA,GHOUSIA COEGE OF ENGINEERING,RAMANAGARAM Page 5 of 4

6 MODUE-X -CONTINUOUS SYSTEM : APPROXIMATE METHOD VIBRATION ENGINEERING 14 Animations QUADRANT- (Animation inks reated, Continuous System: Approximate method) gation.htm Videos (Animation inks reated, Continuous System: Approximate method) rayeigh-s-energy-method gaerkin-s-method-1d-finite-eement-method VTU-NPTE-NMEICT Project Dr.MOHAMED HANEEF,PRINCIPA,GHOUSIA COEGE OF ENGINEERING,RAMANAGARAM Page 6 of 4

7 MODUE-X -CONTINUOUS SYSTEM : APPROXIMATE METHOD VIBRATION ENGINEERING 14 IUSTRATIONS 1. Expain Rayeigh Energy Method. So: Rayeigh-Ritz method is an extension of Rayeigh s method. It not ony provides a means of obtaining a more accurate vaue for the fundamenta frequency, but it aso gives approximations to the higher frequencies and mode shapes. In Rayeigh s method a singe shape function is used to approximate the natura mode of vibration but in Rayeigh-Ritz s method a series of shape functions mutipied by constant coefficients are used. The coefficients are adjusted by minimizing the frequency with respect to each of the coefficients, which resuts in n agebraic equations in ω. The soution of these equations then gives the natura frequencies and mode shapes of the system. The number frequencies cacuated is equa to the number arbitrary functions used. If n functions are seected for approximating the defection y(x) for the transverse vibration of a uniform beam, then. y(x) = c 1 y 1 (x) + c y (x) + c 3 y 3 (x) c n y n (x) (1) where y 1 (x), y (x), y 3 (x), y n (x) are the constant coefficients to be determined. We have, ω n = Maximum potentia energy (U max) Maximum kinetic energy (T max ) = EI d i y (x) dx dx = d EI i y dx dx ρa{y(x)} dx my dx () Substituting equation (1) in equation () and equating the first partia derivative of the resuting expression with respect to each of the constant coefficients C i to zero wi make the natura frequency stationary. i.e., ω n c 1 = ; ω n c = ; ω n = c 3 In genera ω n c i = where I = 1,, n (3) Equation 3 represents a set of n agebraic equation in the n coefficients c 1, c, c 3, ---- c n and ω n to be determined. Since the right hand side of equation is a function of c 1, c, c 3, ---- c i, its partia derivatives with respect to c 1, c, c 3, ---- c i wi a be zero for minimum vaue of ω n. Hence we have my dx EI d y c i dx dx EI d y dx dx my c i dx = where i = 1,, 3, n (4) From equation () Therefore equation (4) becomes VTU-NPTE-NMEICT Project EI d y dx dx = ω n my dx Dr.MOHAMED HANEEF,PRINCIPA,GHOUSIA COEGE OF ENGINEERING,RAMANAGARAM Page 7 of 4

8 MODUE-X -CONTINUOUS SYSTEM : APPROXIMATE METHOD VIBRATION ENGINEERING 14 where A = EI d y EI d y c i dx dx dx dx ω n my dx = (5) c i i.e., A ω B c n = (6) i c i and B = my dx Equation 6 are set of I homogeneous inear equations in unknowns c 1, c, c 3, ---- c n. It wi have a non-trivia soution if the determinant of its coefficients is equa to zero. It gives an i th degree equation in ω n, the owest root of which is equa to ω n1 and the next higher root ω n and so on.. Expian the Concept of Rayeigh Energy Method. Soution) Using the energy method to find the natura frequencies of singe degree of freedom system, the principe of conservation of energy, in the context of an undamped vibrating system, can be restated as T 1 + U 1 = T + U (1) Where the subscripts 1 and denote two different instants of time; Specificay, we use the subscript 1 to denote the time when the mass is passing through its static equiibrium position and choose u 1 = as reference for the potentia energy. If we et the subscript indicate the time corresponding to the maximum dispacement of the mass, we have T =. Thus Eq. [1] Becomes. T 1 +O = O + U --- () If the system is undergoing harmonic motion, then T1 and U denote the maximum vaues of T and U, respectivey, and Eq. [] becomes. T max = U max --- (3) VTU-NPTE-NMEICT Project Dr.MOHAMED HANEEF,PRINCIPA,GHOUSIA COEGE OF ENGINEERING,RAMANAGARAM Page 8 of 4

9 MODUE-X -CONTINUOUS SYSTEM : APPROXIMATE METHOD VIBRATION ENGINEERING Expain the concept of Ritz Gaerkins Method. Soution) In the Ritz-Gaerkin method, an approximate soution of the probem is found by satisfying the governing noninear equation in the average. To see how the method works, et the noninear differentia equations be represented as E[x] = (1) An approximate soution of Eq. (1)is assumed as x (t) = a 1 1 (t) + a (t) + + a n n (t) () Where 1 (t), (t). n (t) are prescribed functions of time and a 1, a,. a n are weighting factors to be determined. If Eq. (), we get a function E x (t). since x (t) is not, in genera, the exact soution of Eq. (1), E (t) = E x (t)wi not be zero. However, the vaue of E (t)wi serve as a measure of the accuracy of the approximation; in fact, E (t) as x x. The weighting factor a i are determined by minimizing the integra τ E [t]dt (3) Where τ denoted as the period of the motion. The minimization of the function of Eq.(3) requires τ E τ E [t] [t]dt = E [t] dt = (4) a i a i VTU-NPTE-NMEICT Project Eq. (4) represents a system of n agebraic equations can be soved simutaneousy to find vaues of a 1, a. a n. 4. Determine the first two natura frequencies of fixed-fixed uniform string of mass density P per unit ength stretched between x = and x = with an initia tension T. The defection functions are given by y1(x) = x( x) and y(x) = x ( x) Soution : We have, y (x) = c 1 y 1 (x) + c y (x) = c 1 x( x) + c x ( x) = c 1 x c 1 x +c x + c x 4 c x 3 dy = c 1 c 1 x + c x + 4c x 3 6c x dx Dr.MOHAMED HANEEF,PRINCIPA,GHOUSIA COEGE OF ENGINEERING,RAMANAGARAM Page 9 of 4

10 c c MODUE-X -CONTINUOUS SYSTEM : APPROXIMATE METHOD VIBRATION ENGINEERING 14 For a string, Max. PE = U max = 1 Max. KE = T max = ω n T dy et A = T c 1 3 dx Tρy 3 B = ρ c 1 5 Equating the equations (1) and () we get Considering, ω n c 1 = ω n A = Bω n ω n = A B c = A A B = A c 1 B c 1 A A B = A c B c Now, A dx = c dx = c c 7 + c 1c c 9 + c 1c c 7 + c 1c 5 T c 9 + c 1c 7 ρω n (1) () (3) (4) ω B c n = (5) 1 c 1 ω B c n = (6) c = T c 1 c c B = ρ c c c A = T c 4c 5 + c B = ρ c c 9 + c Therefore A ω B c n = gives 1 c 1 T c c 5 3 i.e., c 1 T3 ρω n i.e., c 1 T5 VTU-NPTE-NMEICT Project 15 ω n ρ c R 15 + c 7 7 = T5 ρω n 7 15 Simiary, A c ω n B c = gives T 4c 5 + c 1 5 ω n ρ c 9 + c 1 7 = ρω n R T7 ρω n = (7) 63 = (8) Equations (7) and (8) are homogeneous inear equations in unknowns c 1 and c. For a non trivia soution of c 1 and c, the determinant of the coefficients of c 1 and c must be zero. Hence Dr.MOHAMED HANEEF,PRINCIPA,GHOUSIA COEGE OF ENGINEERING,RAMANAGARAM Page 1 of 4

11 MODUE-X -CONTINUOUS SYSTEM : APPROXIMATE METHOD VIBRATION ENGINEERING 14 1 μ 1 μ μ μ = (9) where µ = ρ ω n T Expanding (9) we get, µ - 11µ + 18 = ω n1 = 3.14 T rad ρ sec From equation (1), µ = +11± µ 1 = 9.87 and µ = 1.13 and ω n = 1.11 T rad ρ sec 5. Find the fundamenta natura frequency of transverse vibration of a tapered bar fixed at its base as shown in figure beow. Consider the width of this bar as unity. Soution : Thickness of bar at distance x from base t = T 1 x VTU-NPTE-NMEICT Project Moment of inertia at distance x from base I(x) = 1 bt3 Mass per unit ength at distance x from base. = 1 1 T3 1 x 3 (since b = 1) m(x) = ρt = ρt1 x we have y(x) = c 1 y 1 (x) + c y (x) c n y n (x) x Assume y = c 1 + c x 3 + c x in order that the foowing 4 boundary conditions are satisfied. Dr.MOHAMED HANEEF,PRINCIPA,GHOUSIA COEGE OF ENGINEERING,RAMANAGARAM Page 11 of 4

12 MODUE-X -CONTINUOUS SYSTEM : APPROXIMATE METHOD VIBRATION ENGINEERING 14 Tria : 1 We know, y(o) = ; dy () dx Consider y = c 1 x Soving the equation (1) we get, Tria : We know, dy dx = c 1x and d y dx = c 1 ω n = EI(x)dy ω n1 = 1.58 T rad = ; EI () d y () dx = ; EI () d3 y () dx 3 = dx dx = 1 E 1 T3 1 x 3 c 1 m(x)y dx ρt1 x x c 1 dx sec To get more accurate resuts consider first two terms. i.e., y = c 1 x + c dy = c 1x + 3c x dx 3 d y = c 1 + 6c x dx 3 EI(x) d y dx c i dx i.e., E 1 c i 1 T 1 x 3 c 1 + 6c x 3 x 3 3 ω n m(x)y dx = c i dx ω n ρt 1 x c i x x c c Integrating first and then differentiating with the respect to c 1 and c separatey, we get, (.8µ)c 1 + (1..57µ)c = (1..57µ) + (1..43µ)c = VTU-NPTE-NMEICT Project where + µ = ρ4 ωn T E (1) 3 dx = Equations () and (3) are homogeneous inear equations in unknowns c 1 and c. For a non trivia soution of c 1 and c, the determinant of the coefficients of c 1 and c must be zero. Hence.8µ 1..57µ 1..57µ 1..43µ = Dr.MOHAMED HANEEF,PRINCIPA,GHOUSIA COEGE OF ENGINEERING,RAMANAGARAM Page 1 of 4

13 MODUE-X -CONTINUOUS SYSTEM : APPROXIMATE METHOD VIBRATION ENGINEERING 14 i.e., (.8µ) (1..43µ) - (1..57µ) (1..57µ) = i.e.,.4.86µ -.96µ +.344µ µ µ = i.e.,.191µ.45µ +.96 = i.e., µ 3.665µ = For the fundamenta natura frequency, i.e., µ = 3.665± µ 1 =.36 and µ = 1.31 µ 1 =.36 = ρ4 ωn T E ω n1 = T rad/sec It is more accurate than the tria (1) vaue. 6. Use the Rayeigh-Ritz method to approximate the two owest natura frequencies of the torsiona system of figure beow. Use cubic poynomias as tria functions. J = m 4 G = N m The Rayeigh-Ritz method can be appied using tria functions satisfying ony the geometric boundary conditions (i.e., boundary conditions deveoped soey from geometric considerations). Poynomias satisfying the boundary conditions for a fixed-free shaft are φ i (x) = x 3 3 x, φ (x) = x x The coefficients used in Equation (7.) are α ij = JG d i d j dx + k dx dx t i () j () VTU-NPTE-NMEICT Project β ij = ρj i (x) j (x)dx Kt = N m r Evauation of these coefficients eads to α 11 = JG(3x 3 ) dx + k t ( ) = α 1 = JG(3x 3 )(x ) dx + k t ( )( ) = α 13 = JG(x ) dx + t ( ) = β 11 = ρj (x ) dx =.589 β 1 = ρj (x 3 3 x)(x x)dx =.93 Dr.MOHAMED HANEEF,PRINCIPA,GHOUSIA COEGE OF ENGINEERING,RAMANAGARAM Page 13 of 4

14 MODUE-X -CONTINUOUS SYSTEM : APPROXIMATE METHOD VIBRATION ENGINEERING 14 β = ρj (x x) dx =.3159 Equation (7.) become ( ω )C 1 + ( ω )C = ( ω )C 1 + ( ω )C = A nontrivia soution is obtained if and ony if the determinant of the coefficient matrix is set to zero, eading to.1369ω = whose soutions are ω 1 = 886 rad/s, ω = 6643 rad/s, 7. Use the Rayeigh-Ritz method to approximate the two owest natura frequencies of a uniform fixed-fixed beam. Use the foowing tria functions, which satisfy a boundary conditions: = x 4 x 3 + x, φ (x) = x 5 3 x x The Rayeigh-Ritz approximation to the mode shape is w(x) = C 1 φ 1 (x) + C φ (x) The appropriate form of the coefficients for equation (7.) is α ij = EI d i d j dx dx dx β ij = ρa i (x) j (x)dx Using the suggested tria functions, the coefficients are cacuated as α 11 = EI(1x 1x + ) dx =.8EI 5 α 1 = EI(1x 1x + )(x 3 18 x ) dx = EI 6 α = EI(x 3 18 x ) dx = 5.148EI 7 VTU-NPTE-NMEICT Project β 11 = ρa (x 4 x 3 + x ) dx =.1587ρA 9 β 1 = ρa (x 4 x 3 + x )(x 5 3 x + 3 x )dx =.3968ρA 1 β = ρa (x 5 3 x x ) dx =.99567ρA 11 Substitution into equation (7.) and rearrangement yieds ( φ)C 1 + (.3968φ)C = (.3968φ)C 1 + ( φ)C = where φ = ω ρa4 EI Dr.MOHAMED HANEEF,PRINCIPA,GHOUSIA COEGE OF ENGINEERING,RAMANAGARAM Page 14 of 4

15 MODUE-X -CONTINUOUS SYSTEM : APPROXIMATE METHOD VIBRATION ENGINEERING 14 A nontrivia soution of the above system exists if and ony if the determinant of the coefficient matrix is set to zero. To this end ( φ)( φ) (.3968φ) = φ φ = The soutions of the above equations are 54.9 and 48.3 eading to ω 1 =.4 EI ρa 4, ω = EI ρa 4 8. The exhaust from a singe-cyinder four-stroke diese engine is to be connected to a siencer, and the pressure therein is to be measured with a simpe U-tube manometer [see fig. beow]. Cacuate the minimum ength of the manometer tube so that the natura frequency of osciation of the mercury coumn. Wi be 3.5 times sower than the frequency of the pressure fuctuation in the siencer at an engine speed of 6 rpm. The frequency of pressure fuctuation in the siencer is equa to. Number of cyinders x speed of the engine x Figure: U-tube manometer VTU-NPTE-NMEICT Project x SOUTION 1. Natura frequency of osciation of the iquid coumn: et the datum in fug..18 be taken as the equiibrium position of the iquid. If the dispacement of the iquid coumn from the equiibrium position is denoted by x, the change in potentia energy is given by potentia energy of raised potentia energy of depressed U = + iquid coumn iquid coumn Dr.MOHAMED HANEEF,PRINCIPA,GHOUSIA COEGE OF ENGINEERING,RAMANAGARAM Page 15 of 4

16 MODUE-X -CONTINUOUS SYSTEM : APPROXIMATE METHOD VIBRATION ENGINEERING 14 U = weight mercury raised x dispacement of the C. G of the segment weight of mercury depressed + x dispacement of the C. G. of the segment = [Axγ] x + [Axγ] = [Ax γ] (1) Where A is the cross-sectiona area of the mercury coumn and y is the specific weight of mercury. The change in kinetic energy is given by T = 1 [mass of mercury] [veocity] = 1 Aγ g x () Where is the ength of the mercury coumn. By assuming harmonic motion, we can write where x is the maximum dispacement and ω n is the natura frequency. x(t) = Xcosω n t (3) By substituting Eq. [3] into Eqs. [1] and [], we obtain U = U max cos ω n t T = T max sin ω n t U max = AγX (4) (5) (6) T max = 1 Aγω n X (7) g By equating U max to T max we obtain the norma frequency: VTU-NPTE-NMEICT Project ω n = g (8). ength of mercury coumn: the frequency of pressure fuctuation in the siencer = 1 6 = 3 rpm 3 π = = 1π 6 rad/sec Thus the frequency of osciation of iquid coumn in the manometer is 1 π /3.5 = 9. rad/sec. by using Eq. [8], we obtain Dr.MOHAMED HANEEF,PRINCIPA,GHOUSIA COEGE OF ENGINEERING,RAMANAGARAM Page 16 of 4

17 MODUE-X -CONTINUOUS SYSTEM : APPROXIMATE METHOD VIBRATION ENGINEERING 14 ω n = g = 9. = =. 43 m VTU-NPTE-NMEICT Project Dr.MOHAMED HANEEF,PRINCIPA,GHOUSIA COEGE OF ENGINEERING,RAMANAGARAM Page 17 of 4

18 MODUE-X -CONTINUOUS SYSTEM : APPROXIMATE METHOD VIBRATION ENGINEERING 14 QUADRANT-3 Wikis: (This incudes wikis reated to Continuous System: Approximate Method) Open Contents: (This incudes wikis reated to Continuous System: Approximate Method) Mechanica Vibrations, S. S. Rao, Pearson Education Inc, 4 th edition, 3. Mechanica Vibrations, V. P. Singh, Dhanpat Rai & Company, 3 rd edition, 6. Mechanica Vibrations, G. K.Grover, Nem Chand and Bros, 6 th edition, 1996 Theory of vibration with appications,w.t.thomson,m.d.daheh and C Padmanabhan,Pearson Education inc,5 th Edition,8 Theory and practice of Mechanica Vibration : J.S.Rao&K,Gupta,New Age Internationa Pubications,New Dehi,1 VTU-NPTE-NMEICT Project Dr.MOHAMED HANEEF,PRINCIPA,GHOUSIA COEGE OF ENGINEERING,RAMANAGARAM Page 18 of 4

19 MODUE-X -CONTINUOUS SYSTEM : APPROXIMATE METHOD VIBRATION ENGINEERING 14 QUADRANT-4 Probems 1. Determine the first two natura frequencies of fixed-fixed uniform string of mass density P per unit ength stretched between x = and x = with an initia tension T. The defection functions are given by y1(x) = x( x) and y(x) = x ( x) Soution : We have, y (x) = c 1 y 1 (x) + c y (x) = c 1 x( x) + c x ( x) For a string, Max. PE = U max = 1 Max. KE = T max = ω n = c 1 x c 1 x +c x + c x 4 c x 3 dy = c 1 c 1 x + c x + 4c x 3 6c x dx T dy dx dx = c 1 3 Tρy dx = c 1 5 et A = T c B = ρ c 1 5 Equating the equations (1) and () we get Considering, ω n c 1 = ω n A = Bω n ω n = A B c = A A B = A c 1 B c 1 A A B = A c B c + c 7 + c 1c c 9 + c 1c c 7 + c 1c 5 T c 9 + c 1c 7 ρω n VTU-NPTE-NMEICT Project Now, A (1) () (3) (4) ω B c n = (5) 1 c 1 ω B c n = (6) c = T c 1 c c 5 15 B = ρ c c c A = T c 4c 5 + c B = ρ c c 9 + c Dr.MOHAMED HANEEF,PRINCIPA,GHOUSIA COEGE OF ENGINEERING,RAMANAGARAM Page 19 of 4

20 c c MODUE-X -CONTINUOUS SYSTEM : APPROXIMATE METHOD VIBRATION ENGINEERING 14 T c i.e., c 1 T3 i.e., c 1 T5 Therefore A c 1 ω n B c 1 = gives + c 5 15 ω n ρ c c 7 ρω n R 7 = T5 ρω n 7 15 Simiary, A c ω n B c = gives T 4c 5 + c 1 5 ω n ρ c 9 + c 1 7 = ρω n R T7 ρω n = (7) 63 = (8) Equations (7) and (8) are homogeneous inear equations in unknowns c 1 and c. For a non trivia soution of c 1 and c, the determinant of the coefficients of c 1 and c must be zero. Hence 1 μ 1 μ μ = (9) μ where µ = ρ ω n Expanding (9) we get, µ - 11µ + 18 = ω n1 = 3.14 T rad ρ sec T From equation (1), µ = +11± and ω n = 1.11 T 1 µ 1 = 9.87 and µ = 1.13 rad ρ sec. Find the fundamenta natura frequency of transverse vibration of a tapered bar fixed at its base as shown in figure beow. Consider the width of this bar as unity. VTU-NPTE-NMEICT Project Soution : Dr.MOHAMED HANEEF,PRINCIPA,GHOUSIA COEGE OF ENGINEERING,RAMANAGARAM Page of 4

21 MODUE-X -CONTINUOUS SYSTEM : APPROXIMATE METHOD VIBRATION ENGINEERING 14 Thickness of bar at distance x from base t = T 1 x Moment of inertia at distance x from base I(x) = 1 bt3 Mass per unit ength at distance x from base. = 1 1 T3 1 x 3 (since b = 1) m(x) = ρt = ρt1 x we have y(x) = c 1 y 1 (x) + c y (x) c n y n (x) x Assume y = c 1 + c x 3 + c x in order that the foowing boundary conditions are satisfied. Tria : 1 We know, y(o) = ; dy () Consider y = c 1 x Soving the equation (1) we get, Tria : We know, dy = c 1x dx dx and d y ω n = EI(x)dy ω n1 = 1.58 T rad = ; EI () d y () dx = ; EI () d3 y () dx 3 = = c 1 dx dx dx = 1 E 1 T3 1 x 3 c 1 m(x)y dx ρt1 x x c 1 dx sec To get more accurate resuts consider first two terms. VTU-NPTE-NMEICT Project x i.e., y = c 1 + c dy dx = c 1x + 3c x 3 d y = c 1 + 6c x dx 3 EI(x) d y dx c i dx x 3 3 ω n m(x)y dx = c i (1) Dr.MOHAMED HANEEF,PRINCIPA,GHOUSIA COEGE OF ENGINEERING,RAMANAGARAM Page 1 of 4

22 MODUE-X -CONTINUOUS SYSTEM : APPROXIMATE METHOD VIBRATION ENGINEERING 14 i.e., E 1 c i 1 T 1 x 3 c 1 + 6c x 3 dx ω n ρt 1 x c i x x c c Integrating first and then differentiating with the respect to c 1 and c separatey, we get, (.8µ)c 1 + (1..57µ)c = (1..57µ) + (1..43µ)c = where + µ = ρ4 ωn T E 3 dx = Equations () and (3) are homogeneous inear equations in unknowns c 1 and c. For a non trivia soution of c 1 and c, the determinant of the coefficients of c 1 and c must be zero. Hence.8µ 1..57µ 1..57µ 1..43µ = i.e., (.8µ) (1..43µ) - (1..57µ) (1..57µ) = i.e.,.4.86µ -.96µ +.344µ µ µ = i.e.,.191µ.45µ +.96 = i.e., µ 3.665µ = For the fundamenta natura frequency, i.e., µ = 3.665± µ 1 =.36 and µ = 1.31 µ 1 =.36 = ρ4 ωn T E ω n1 = T rad/sec It is more accurate than the tria (1) vaue. VTU-NPTE-NMEICT Project 3. Use the Rayeigh-Ritz method to approximate the two owest natura frequencies of the torsiona system of figure beow. Use cubic poynomias as tria functions. J = m 4 G = N m Kt = N m r The Rayeigh-Ritz method can be appied using tria functions satisfying ony the geometric boundary conditions (i.e., boundary conditions deveoped soey from geometric considerations). Poynomias satisfying the boundary conditions for a fixed-free shaft are φ i (x) = x 3 3 x, φ (x) = x x The coefficients used in Equation (7.) are Dr.MOHAMED HANEEF,PRINCIPA,GHOUSIA COEGE OF ENGINEERING,RAMANAGARAM Page of 4

23 MODUE-X -CONTINUOUS SYSTEM : APPROXIMATE METHOD VIBRATION ENGINEERING 14 α ij = JG d i dx β ij = ρj i (x) j (x)dx d j dx dx + k t i () j () Evauation of these coefficients eads to α 11 = JG(3x 3 ) dx + k t ( ) = α 1 = JG(3x 3 )(x ) dx + k t ( )( ) = α 13 = JG(x ) dx + t ( ) = β 11 = ρj (x ) dx =.589 β 1 = ρj (x 3 3 x)(x x)dx =.93 β = ρj (x x) dx =.3159 Equation (7.) become ( ω )C 1 + ( ω )C = ( ω )C 1 + ( ω )C = A nontrivia soution is obtained if and ony if the determinant of the coefficient matrix is set to zero, eading to.1369ω = whose soutions are ω 1 = 886 rad/s, ω = 6643 rad/s, 4. Use the Rayeigh-Ritz method to approximate the two owest natura frequencies of a uniform fixed-fixed beam. Use the foowing tria functions, which satisfy a boundary conditions: = x 4 x 3 + x, φ (x) = x 5 3 x x The Rayeigh-Ritz approximation to the mode shape is w(x) = C 1 φ 1 (x) + C φ (x) The appropriate form of the coefficients for equation (7.) is VTU-NPTE-NMEICT Project α ij = EI d i dx d j dx dx β ij = ρa i (x) j (x)dx Using the suggested tria functions, the coefficients are cacuated as α 11 = EI(1x 1x + ) dx =.8EI 5 α 1 = EI(1x 1x + )(x 3 18 x ) dx = EI 6 Dr.MOHAMED HANEEF,PRINCIPA,GHOUSIA COEGE OF ENGINEERING,RAMANAGARAM Page 3 of 4

24 MODUE-X -CONTINUOUS SYSTEM : APPROXIMATE METHOD VIBRATION ENGINEERING 14 α = EI(x 3 18 x ) dx = 5.148EI 7 β 11 = ρa (x 4 x 3 + x ) dx =.1587ρA 9 β 1 = ρa (x 4 x 3 + x )(x 5 3 x + 3 x )dx =.3968ρA 1 β = ρa (x 5 3 x x ) dx =.99567ρA 11 Substitution into equation (7.) and rearrangement yieds where ( φ)C 1 + (.3968φ)C = (.3968φ)C 1 + ( φ)C = φ = ω ρa4 EI A nontrivia soution of the above system exists if and ony if the determinant of the coefficient matrix is set to zero. To this end ( φ)( φ) (.3968φ) = φ φ = The soutions of the above equations are 54.9 and 48.3 eading to ω 1 =.4 EI ρa 4, ω = EI ρa 4 5. The exhaust from a singe-cyinder four-stroke diese engine is to be connected to a siencer, and the pressure therein is to be measured with a simpe U-tube manometer [see fig. beow]. Cacuate the minimum ength of the manometer tube so that the natura frequency of osciation of the mercury coumn. Wi be 3.5 times sower than the frequency of the pressure fuctuation in the siencer at an engine speed of 6 rpm. The frequency of pressure fuctuation in the siencer is equa to. Number of cyinders x speed of the engine VTU-NPTE-NMEICT Project Dr.MOHAMED HANEEF,PRINCIPA,GHOUSIA COEGE OF ENGINEERING,RAMANAGARAM Page 4 of 4

25 MODUE-X -CONTINUOUS SYSTEM : APPROXIMATE METHOD VIBRATION ENGINEERING 14 x x SOUTION Natura frequency of osciation of the iquid coumn: et the datum in fug..18 be taken as the equiibrium position of the iquid. If the dispacement of the iquid coumn from the equiibrium position is denoted by x, the change in potentia energy is given by potentia energy of raised potentia energy of depressed U = + iquid coumn iquid coumn weight mercury raised U = x dispacement of the C. G of the segment Figure: U-tube manometer weight of mercury depressed + x dispacement of the C. G. of the segment = [Axγ] x + [Axγ] = [Ax γ] (1) Where A is the cross-sectiona area of the mercury coumn and y is the specific weight of mercury. The change in kinetic energy is given by VTU-NPTE-NMEICT Project T = 1 [mass of mercury] [veocity] = 1 Aγ g x () Where is the ength of the mercury coumn. By assuming harmonic motion, we can write where x is the maximum dispacement and ω n is the natura frequency. x(t) = Xcosω n t (3) By substituting Eq. [3] into Eqs. [1] and [], we obtain Dr.MOHAMED HANEEF,PRINCIPA,GHOUSIA COEGE OF ENGINEERING,RAMANAGARAM Page 5 of 4

26 MODUE-X -CONTINUOUS SYSTEM : APPROXIMATE METHOD VIBRATION ENGINEERING 14 U = U max cos ω n t T = T max sin ω n t U max = AγX (4) (5) (6) T max = 1 Aγω n X (7) g By equating U max to T max we obtain the norma frequency: ω n = g (8) ength of mercury coumn: the frequency of pressure fuctuation in the siencer = 1 6 = 3 rpm 3 π = = 1π 6 rad/sec Thus the frequency of osciation of iquid coumn in the manometer is 1 π /3.5 = 9. rad/sec. by using Eq. [8], we obtain ω n = g = = 9. =. 43 m VTU-NPTE-NMEICT Project Dr.MOHAMED HANEEF,PRINCIPA,GHOUSIA COEGE OF ENGINEERING,RAMANAGARAM Page 6 of 4

27 MODUE-X -CONTINUOUS SYSTEM : APPROXIMATE METHOD VIBRATION ENGINEERING 14 Frequenty asked Questions. 1. Determine the first two natura frequencies of fixed-fixed uniform string of mass density P per unit ength stretched between x = and x = with an initia tension T. The defection functions are given by y 1 (x) = x( x) and y (x) = x ( x). Find the fundamenta natura frequency of transverse vibration of a tapered bar fixed at its base as shown in figure beow. Consider the width of this bar as unity. 3. Use the Rayeigh-Ritz method to approximate the two owest natura frequencies of the torsiona system of figure beow. Use cubic poynomias as tria functions. J = m 4 G = N m VTU-NPTE-NMEICT Project Kt = N m r 4. Use the Rayeigh-Ritz method to approximate the two owest natura frequencies of a uniform fixed-fixed beam. Use the foowing tria functions, which satisfy a boundary conditions: = x 4 x 3 + x, φ (x) = x 5 3 x x Dr.MOHAMED HANEEF,PRINCIPA,GHOUSIA COEGE OF ENGINEERING,RAMANAGARAM Page 7 of 4

28 MODUE-X -CONTINUOUS SYSTEM : APPROXIMATE METHOD VIBRATION ENGINEERING Use the Rayeigh-Ritz method to approximate the two owest natura frequencies of the system of figure beow. Use the two owest mode shapes for a uniform fixed-free bar as tria functions. 6. Write a short Notes on i) Rayeigh Ritz method ii) iii) Rayeigh Energy Method Ritz Gaerkin Method Sef Question & Answer 1. Determine the first two natura frequencies of fixed-fixed uniform string of mass density P per unit ength stretched between x = and x = with an initia tension T. The defection functions are given by y1(x) = x( x) and y(x) = x ( x) Soution : We have, y (x) = c 1 y 1 (x) + c y (x) = c 1 x( x) + c x ( x) = c 1 x c 1 x +c x + c x 4 c x 3 dy = c 1 c 1 x + c x + 4c x 3 6c x dx For a string, Max. PE = U max = 1 Max. KE = T max = ω n T dy dx dx = c 1 3 Tρy dx = c 1 5 et A = T c B = ρ c 1 5 Equating the equations (1) and () we get A = Bω n + c 7 + c 1c c 9 + c 1c c 7 + c 1c 5 T c 9 + c 1c 7 ρω n VTU-NPTE-NMEICT Project (1) () (3) (4) Considering, ω n c 1 ω n = A B = ω n c = A A B = A c 1 B c 1 A A B = A c B c ω B c n = (5) 1 c 1 ω B c n = (6) c Dr.MOHAMED HANEEF,PRINCIPA,GHOUSIA COEGE OF ENGINEERING,RAMANAGARAM Page 8 of 4

29 c c MODUE-X -CONTINUOUS SYSTEM : APPROXIMATE METHOD VIBRATION ENGINEERING 14 T c i.e., c 1 T3 i.e., c 1 T5 Now, A = T c 1 c c 5 15 B = ρ c c c A = T c 4c 5 + c B = ρ c c 9 + c Therefore A c 1 ω n B c 1 = gives + c 5 15 ω n ρ c c 7 ρω n R 7 = T5 ρω n 7 15 Simiary, A ω B c n = gives c T 4c 5 + c ρω n ω n ρ c R 14 = (7) c 1 7 T7 ρω n = 63 = (8) Equations (7) and (8) are homogeneous inear equations in unknowns c 1 and c. For a non trivia soution of c 1 and c, the determinant of the coefficients of c 1 and c must be zero. Hence 1 μ 1 μ μ = (9) μ where µ = ρ ω n Expanding (9) we get, µ - 11µ + 18 = T From equation (1), µ = +11± VTU-NPTE-NMEICT Project 1 µ 1 = 9.87 and µ = 1.13 ω n1 = 3.14 T rad ρ sec and ω n = 1.11 T rad ρ sec. Find the fundamenta natura frequency of transverse vibration of a tapered bar fixed at its base as shown in figure beow. Consider the width of this bar as unity. Dr.MOHAMED HANEEF,PRINCIPA,GHOUSIA COEGE OF ENGINEERING,RAMANAGARAM Page 9 of 4

30 MODUE-X -CONTINUOUS SYSTEM : APPROXIMATE METHOD VIBRATION ENGINEERING 14 Soution : Thickness of bar at distance x from base t = T 1 x Moment of inertia at distance x from base I(x) = 1 bt3 Mass per unit ength at distance x from base. = 1 1 T3 1 x 3 (since b = 1) m(x) = ρt = ρt1 x we have y(x) = c 1 y 1 (x) + c y (x) c n y n (x) x Assume y = c 1 + c x 3 + c x in order that the foowing boundary conditions are satisfied. Tria : 1 We know, y(o) = ; dy () Consider y = c 1 x dx dy dx = c 1x and d y dx = c 1 = ; EI () d y () dx = ; EI () d3 y () dx 3 = VTU-NPTE-NMEICT Project Soving the equation (1) we get, dx dx = 1 E 1 T3 1 x 3 c 1 m(x)y dx ρt1 x x c 1 dx ω n = EI(x)dy (1) ω n1 = 1.58 T rad sec Tria : Dr.MOHAMED HANEEF,PRINCIPA,GHOUSIA COEGE OF ENGINEERING,RAMANAGARAM Page 3 of 4

31 MODUE-X -CONTINUOUS SYSTEM : APPROXIMATE METHOD VIBRATION ENGINEERING 14 We know, To get more accurate resuts consider first two terms. i.e., y = c 1 x + c dy dx = c 1x + 3c x 3 d y = c 1 + 6c x dx 3 EI(x) d y dx c i dx i.e., E 1 c i 1 T 1 x 3 c 1 + 6c x 3 x 3 3 ω n m(x)y dx = c i dx ω n ρt 1 x c i x x c c Integrating first and then differentiating with the respect to c 1 and c separatey, we get, (.8µ)c 1 + (1..57µ)c = (1..57µ) + (1..43µ)c = where + µ = ρ4 ωn T E 3 dx = Equations () and (3) are homogeneous inear equations in unknowns c 1 and c. For a non trivia soution of c 1 and c, the determinant of the coefficients of c 1 and c must be zero. Hence.8µ 1..57µ 1..57µ 1..43µ = i.e., (.8µ) (1..43µ) - (1..57µ) (1..57µ) = i.e.,.4.86µ -.96µ +.344µ µ µ = i.e.,.191µ.45µ +.96 = i.e., µ 3.665µ = VTU-NPTE-NMEICT Project For the fundamenta natura frequency, i.e., µ = 3.665± µ 1 =.36 and µ = 1.31 µ 1 =.36 = ρ4 ωn T E ω n1 = T rad/sec It is more accurate than the tria (1) vaue. Dr.MOHAMED HANEEF,PRINCIPA,GHOUSIA COEGE OF ENGINEERING,RAMANAGARAM Page 31 of 4

32 MODUE-X -CONTINUOUS SYSTEM : APPROXIMATE METHOD VIBRATION ENGINEERING Use the Rayeigh-Ritz method to approximate the two owest natura frequencies of the torsiona system of figure beow. Use cubic poynomias as tria functions. J = m 4 G = N m The Rayeigh-Ritz method can be appied using tria functions satisfying ony the geometric boundary conditions (i.e., boundary conditions deveoped soey from geometric considerations). Poynomias satisfying the boundary conditions for a fixed-free shaft are φ i (x) = x 3 3 x, φ (x) = x x The coefficients used in Equation (7.) are α ij = JG d i d j dx + k dx dx t i () j () β ij = ρj i (x) j (x)dx Kt = N m r Evauation of these coefficients eads to α 11 = JG(3x 3 ) dx + k t ( ) = α 1 = JG(3x 3 )(x ) dx + k t ( )( ) = α 13 = JG(x ) dx + t ( ) = β 11 = ρj (x ) dx =.589 β 1 = ρj (x 3 3 x)(x x)dx =.93 β = ρj (x x) dx =.3159 Equation (7.) become ( ω )C 1 + ( ω )C = ( ω )C 1 + ( ω )C = A nontrivia soution is obtained if and ony if the determinant of the coefficient matrix is set to zero, eading to.1369ω = whose soutions are ω 1 = 886 rad/s, ω = 6643 rad/s, VTU-NPTE-NMEICT Project 4. Use the Rayeigh-Ritz method to approximate the two owest natura frequencies of a uniform fixed-fixed beam. Use the foowing tria functions, which satisfy a boundary conditions: = x 4 x 3 + x, φ (x) = x 5 3 x x Dr.MOHAMED HANEEF,PRINCIPA,GHOUSIA COEGE OF ENGINEERING,RAMANAGARAM Page 3 of 4

33 MODUE-X -CONTINUOUS SYSTEM : APPROXIMATE METHOD VIBRATION ENGINEERING 14 The Rayeigh-Ritz approximation to the mode shape is w(x) = C 1 φ 1 (x) + C φ (x) The appropriate form of the coefficients for equation (7.) is α ij = EI d i dx d j dx dx β ij = ρa i (x) j (x)dx Using the suggested tria functions, the coefficients are cacuated as α 11 = EI(1x 1x + ) dx =.8EI 5 α 1 = EI(1x 1x + )(x 3 18 x ) dx = EI 6 α = EI(x 3 18 x ) dx = 5.148EI 7 β 11 = ρa (x 4 x 3 + x ) dx =.1587ρA 9 β 1 = ρa (x 4 x 3 + x )(x 5 3 x + 3 x )dx =.3968ρA 1 β = ρa (x 5 3 x x ) dx =.99567ρA 11 Substitution into equation (7.) and rearrangement yieds where ( φ)C 1 + (.3968φ)C = (.3968φ)C 1 + ( φ)C = φ = ω ρa4 EI A nontrivia soution of the above system exists if and ony if the determinant of the coefficient matrix is set to zero. To this end ( φ)( φ) (.3968φ) = φ φ = The soutions of the above equations are 54.9 and 48.3 eading to ω 1 =.4 EI VTU-NPTE-NMEICT Project ρa 4, ω = EI ρa 4 5. The exhaust from a singe-cyinder four-stroke diese engine is to be connected to a siencer, and the pressure therein is to be measured with a simpe U-tube manometer [see fig. beow]. Cacuate the minimum ength of the manometer tube so that the natura frequency of osciation of the mercury coumn. Wi be 3.5 times sower than the frequency of the pressure fuctuation in the siencer at an engine speed of 6 rpm. The frequency of pressure fuctuation in the siencer is equa to. Dr.MOHAMED HANEEF,PRINCIPA,GHOUSIA COEGE OF ENGINEERING,RAMANAGARAM Page 33 of 4

34 MODUE-X -CONTINUOUS SYSTEM : APPROXIMATE METHOD VIBRATION ENGINEERING 14 Number of cyinders x speed of the engine x x SOUTION Natura frequency of osciation of the iquid coumn: et the datum in fug..18 be taken as the equiibrium position of the iquid. If the dispacement of the iquid coumn from the equiibrium position is denoted by x, the change in potentia energy is given by potentia energy of raised potentia energy of depressed U = + iquid coumn iquid coumn weight mercury raised U = x dispacement of the C. G of the segment Figure: U-tube manometer weight of mercury depressed + x dispacement of the C. G. of the segment = [Axγ] x + [Axγ] = [Ax γ] (1) VTU-NPTE-NMEICT Project Where A is the cross-sectiona area of the mercury coumn and y is the specific weight of mercury. The change in kinetic energy is given by T = 1 [mass of mercury] [veocity] = 1 Aγ g x () Where is the ength of the mercury coumn. By assuming harmonic motion, we can write where x is the maximum dispacement and ω n is the natura frequency. x(t) = Xcosω n t (3) Dr.MOHAMED HANEEF,PRINCIPA,GHOUSIA COEGE OF ENGINEERING,RAMANAGARAM Page 34 of 4

35 MODUE-X -CONTINUOUS SYSTEM : APPROXIMATE METHOD VIBRATION ENGINEERING 14 By substituting Eq. [3] into Eqs. [1] and [], we obtain U = U max cos ω n t T = T max sin ω n t U max = AγX (4) (5) (6) T max = 1 Aγω n X (7) g By equating U max to T max we obtain the norma frequency: ω n = g (8) ength of mercury coumn: the frequency of pressure fuctuation in the siencer = 1 6 = 3 rpm 3 π = = 1π 6 rad/sec Thus the frequency of osciation of iquid coumn in the manometer is 1 π /3.5 = 9. rad/sec. by using Eq. [8], we obtain = ω n = g = 9. VTU-NPTE-NMEICT Project =. 43 m Dr.MOHAMED HANEEF,PRINCIPA,GHOUSIA COEGE OF ENGINEERING,RAMANAGARAM Page 35 of 4

36 MODUE-X -CONTINUOUS SYSTEM : APPROXIMATE METHOD VIBRATION ENGINEERING 14 Probems 1. Use the Rayeigh-Ritz method to approximate the two owest natura frequencies of the torsiona system of figure beow. Use cubic poynomias as tria functions. J = m 4 G = N m The Rayeigh-Ritz method can be appied using tria functions satisfying ony the geometric boundary conditions (i.e., boundary conditions deveoped soey from geometric considerations). Poynomias satisfying the boundary conditions for a fixed-free shaft are φ i (x) = x 3 3 x, φ (x) = x x The coefficients used in Equation (7.) are α ij = JG d i d j dx + k dx dx t i () j () β ij = ρj i (x) j (x)dx Kt = N m r Evauation of these coefficients eads to α 11 = JG(3x 3 ) dx + k t ( ) = α 1 = JG(3x 3 )(x ) dx + k t ( )( ) = α 13 = JG(x ) dx + t ( ) = β 11 = ρj (x ) dx =.589 β 1 = ρj (x 3 3 x)(x x)dx =.93 β = ρj (x x) dx =.3159 Equation (7.) become ( ω )C 1 + ( ω )C = ( ω )C 1 + ( ω )C = A nontrivia soution is obtained if and ony if the determinant of the coefficient matrix is set to zero, eading to.1369ω = whose soutions are ω 1 = 886 rad/s, ω = 6643 rad/s, VTU-NPTE-NMEICT Project Dr.MOHAMED HANEEF,PRINCIPA,GHOUSIA COEGE OF ENGINEERING,RAMANAGARAM Page 36 of 4

37 MODUE-X -CONTINUOUS SYSTEM : APPROXIMATE METHOD VIBRATION ENGINEERING 14. Use the Rayeigh-Ritz method to approximate the two owest natura frequencies of a uniform fixed-fixed beam. Use the foowing tria functions, which satisfy a boundary conditions: = x 4 x 3 + x, φ (x) = x 5 3 x x The Rayeigh-Ritz approximation to the mode shape is w(x) = C 1 φ 1 (x) + C φ (x) The appropriate form of the coefficients for equation (7.) is α ij = EI d i dx d j dx dx β ij = ρa i (x) j (x)dx Using the suggested tria functions, the coefficients are cacuated as α 11 = EI(1x 1x + ) dx =.8EI 5 α 1 = EI(1x 1x + )(x 3 18 x ) dx = EI 6 α = EI(x 3 18 x ) dx = 5.148EI 7 β 11 = ρa (x 4 x 3 + x ) dx =.1587ρA 9 β 1 = ρa (x 4 x 3 + x )(x 5 3 x + 3 x )dx =.3968ρA 1 β = ρa (x 5 3 x x ) dx =.99567ρA 11 Substitution into equation (7.) and rearrangement yieds where ( φ)C 1 + (.3968φ)C = (.3968φ)C 1 + ( φ)C = φ = ω ρa4 EI A nontrivia soution of the above system exists if and ony if the determinant of the coefficient matrix is set to zero. To this end ( φ)( φ) (.3968φ) = φ φ = The soutions of the above equations are 54.9 and 48.3 eading to ω 1 =.4 EI ρa 4, VTU-NPTE-NMEICT Project ω = EI ρa 4 Dr.MOHAMED HANEEF,PRINCIPA,GHOUSIA COEGE OF ENGINEERING,RAMANAGARAM Page 37 of 4

38 MODUE-X -CONTINUOUS SYSTEM : APPROXIMATE METHOD VIBRATION ENGINEERING The exhaust from a singe-cyinder four-stroke diese engine is to be connected to a siencer, and the pressure therein is to be measured with a simpe U-tube manometer [see fig. beow]. Cacuate the minimum ength of the manometer tube so that the natura frequency of osciation of the mercury coumn. Wi be 3.5 times sower than the frequency of the pressure fuctuation in the siencer at an engine speed of 6 rpm. The frequency of pressure fuctuation in the siencer is equa to. Number of cyinders x speed of the engine SOUTION Natura frequency of osciation of the iquid coumn: et the datum in fug..18 be taken as the equiibrium position of the iquid. If the dispacement of the iquid coumn from the equiibrium position is denoted by x, the change in potentia energy is given by potentia energy of raised U = + iquid coumn x Figure: U-tube manometer VTU-NPTE-NMEICT Project x potentia energy of depressed iquid coumn U = weight mercury raised x dispacement of the C. G of the segment weight of mercury depressed + x dispacement of the C. G. of the segment = [Axγ] x + [Axγ] = [Ax γ] (1) Where A is the cross-sectiona area of the mercury coumn and y is the specific weight of mercury. The change in kinetic energy is given by Dr.MOHAMED HANEEF,PRINCIPA,GHOUSIA COEGE OF ENGINEERING,RAMANAGARAM Page 38 of 4

39 MODUE-X -CONTINUOUS SYSTEM : APPROXIMATE METHOD VIBRATION ENGINEERING 14 T = 1 [mass of mercury] [veocity] = 1 Aγ g x () Where is the ength of the mercury coumn. By assuming harmonic motion, we can write where x is the maximum dispacement and ω n is the natura frequency. x(t) = Xcosω n t (3) By substituting Eq. [3] into Eqs. [1] and [], we obtain U = U max cos ω n t T = T max sin ω n t U max = AγX (4) (5) (6) T max = 1 Aγω n X (7) g By equating U max to T max we obtain the norma frequency: ω n = g (8) ength of mercury coumn: the frequency of pressure fuctuation in the siencer = 1 6 = 3 rpm 3 π = = 1π 6 rad/sec VTU-NPTE-NMEICT Project Thus the frequency of osciation of iquid coumn in the manometer is 1 π /3.5 = 9. rad/sec. by using Eq. [8], we obtain ω n = g = 9. = =. 43 m Dr.MOHAMED HANEEF,PRINCIPA,GHOUSIA COEGE OF ENGINEERING,RAMANAGARAM Page 39 of 4

40 MODUE-X -CONTINUOUS SYSTEM : APPROXIMATE METHOD VIBRATION ENGINEERING 14 VTU-NPTE-NMEICT Project Dr.MOHAMED HANEEF,PRINCIPA,GHOUSIA COEGE OF ENGINEERING,RAMANAGARAM Page 4 of 4