Answers to review questions and solutions with detailed information Chapter 1 (Edition 6 2020-07-27) PDF

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Answers to Review QuestionsMechanical Vibrations , Fifth Edition in SI UnitsSingiresu S. RaoQuestion 1: Bad effects: (a) Blade and disk failure in turbines (b) Poor surface finish in metal cutting Good effects: (a) vibratory conveyors and hoppers (b) Pile driving and vibratory finishing processes Me...

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Answers to Review Questions Mechanical Vibrations, Fifth Edition in SI Units Singiresu S. Rao

Question 1.1: 1. Bad effects: (a) Blade and disk failure in turbines (b) Poor surface finish in metal cutting Good effects: (a) vibratory conveyors and hoppers (b) Pile driving and vibratory finishing processes 2. Means to store potential energy: spring Means to store kinetic energy: mass Means by which energy is lost: damper 3. Degree of freedom is the minimum numbers of independent coordinates required to determine completely the positions of all parts of a system at any instant of time. 4. A discrete system is one that has a finite number of degrees of freedom. A continuous system is one that has an infinite number of degrees of freedom. Any continuous system can be approximated as a discrete system. 5. It may not be possible to disregard damping always, especially if the system is excited near resonance. 6. Yes. If the differential equation is nonlinear, the corresponding system will be nonlinear. 7. If the system parameters are completely known and the magnitude of excitation acting on the vibratory system is known at any given time, the resulting vibration is known as deterministic vibration. Examples are (i) simple pendulum, and (ii) vibration of a cantilever beam subjected to harmonic base motion. If the system parameters and/or excitation of a system are random or nondeterministic, the resulting vibration is called random vibration. Examples are (i) vibration of an automobile due to road roughness, and (ii) vibration of a multistory building subjected to an earthquake. 8. Standard methods of solving differential equations, Laplace transform methods, matrix methods, and numerical methods. 9. In parallel. 10. Spring stiffness is the force necessary to deform the spring by a unit amount. Damping constant is the force necessary to cause a unit velocity across the damper. 11. Viscous damping, Coulomb (dry-friction) damping, and solid(hysteretic) damping. 12. Fourier series in terms of trigonometric functions, complex Fourier series, and frequency spectrum.

13. Cycle: The movement of vibratory body from its equilibrium position to its extreme position in one direction, then to the equilibrium position, then to its extreme position in other direction, and back to equilibrium position is called a cycle of vibration. Amplitude: The maximum displacement of a vibrating body from its equilibrium position is called the amplitude of vibration. Phase angle: The angular difference between the occurrence of the maxima of two harmonic motions having the same frequency is called the phase difference. Linear frequency: The number of cycles per unit time. Period: The time taken to complete one cycle of motion is called the period. Natural frequency: If a system, after an initial disturbance, is left to vibrate on its own, the frequency with which it oscillates without external forces, is known as its natural frequency. 2π 1 14. τ = = . f ω 15. Frequency: Angular velocity of the rotating vector (ω). Phase: If the vertical projection of the rotating vector is nonzero at time t = 0, the angular difference from the occurrence of zero vertical projection to t = 0 is called the phase. Amplitude: maximum projection of the rotating vector on the vertical axis. 16. If x1 (t ) = A sin ω1t , and x 2 (t ) = A sin ω 2 t = A sin(ω1t + δω1t ) 1 ⎛ δω t ⎞ x (t ) = x1 (t ) + x 2 (t ) = 2 A sin ( ω1t + δω1 t ). cos ⎜ 1 ⎟ 2 ⎝ 2 ⎠ 17. When two harmonic motions, with frequencies close to one another, are added, the resulting motion exhibits a phenomenon known as beats. In beat phenomenon, the amplitude builds up and dies down at a frequency known as beat frequency. 18. Decibel (dB) is defined as: ⎛ X ⎞ ⎟ dB = 20 log ⎜⎜ ⎟ ⎝ X0 ⎠ where X 0 is a specified reference value of X. Octave: The frequency range in which the maximum value is twice the minimum value is called an octave band. 19. When a periodic function is approximated by n terms of the Fourier series, the approximation improves everywhere except in the vicinity of the discontinuity as the value of n increases. This phenomenon is called the Gibbs phenomenon. 20. If a function, defined only in the interval 0 to τ , is extended arbitrarily to include the interval − τ to 0 for the purpose of Fourier series expansion, the resulting expansion is known as the half-range expansion.

Question 1.2: 1. T 2. F 3. T 4. T 5. T 6. T 7. T 8. T 9. T 10. F Question 1.3: 1. resonance 2. energy 3. mass 4. periodic 5. simple 6. period 7. frequency 8. synchronous 9. phase difference 10. infinite 11. discrete 12. coordinates 13. free 14. forced 15. natural 16. f (− t ) = − f (t ) 17. half 18. harmonic 19. 104.72 rad/s 20. 0.01 s Question 1.4: 1. 2. 3. 4. 5. 6. 7. 8.

b a c a c b c b

9. a 10. a 11. b 12. c 13. a 14. b 15. a 16. a Question 1.5: 1. 2. 3. 4. 5.

⎯b ⎯c ⎯e ⎯d ⎯a

Question 1.6: 1. 2. 3. 4. 5.

⎯c ⎯e ⎯a ⎯d ⎯b

Question 1.7: 1. 2. 3. 4. 5. 6. 7. 8.

⎯b ⎯c ⎯e ⎯d ⎯f ⎯h ⎯g ⎯a

Question 2.1: 1. Assume that the system is underdamped. Then by measuring the amplitudes of vibration m cycles apart, the logarithmic decrement (δ) can be computed as 1 ⎛ x ⎞ δ = ln⎜⎜ 1 ⎟⎟ m ⎝ x m+1 ⎠

The damping ratio ( ς ) can be found as

ς=

δ (2π )2 +δ 2

.

2. No. 3. Mass moment of inertia, torsional damping constant, torsional stiffness, and angular displacement, respectively. k , a decrease in m will cause the 4. Since the natural frequency is given by ω n = m natural frequency to increase. m 2π = 2π , a decrease in k will cause 5. Since the natural period is given by τ = k ωn the natural period to increase. 6. Due to the damping present in the surroundings. 7. To avoid resonance. 8. Two. Constants are determined using two initial conditions ( usually, using the initial values of the variable and its derivative). 9. Energy method cannot be used for damped systems. 10. No dissipation of energy due to damping. 11. If the system is underdamped or critically damped, the frequency of damped vibration will be smaller than the natural frequency of the system. 12. Logarithmic decrement can be used to determine the damping constant of a system by experimentally measuring any two consecutive displacement amplitudes. 13. Since hysteresis damping depends on the area of the hysteresis loop (in the stressstrain diagram), the maximum stress influences hysteresis damping. 14. Critical damping corresponds to a damping ratio of one. It is important because the motion will be aperiodic (non-oscillatory) with critical damping. 15. It is mostly dissipated as heat. 16. Equivalent viscous damping is defined such that the energy dissipated per cycle during harmonic motion will be same in both the actual and the equivalent viscous dampers. Equivalent viscous damping factor need not be a constant. For h example, in the case of hysteresis damping, ceq = , indicating that the

ω

equivalent viscous damping depends on the frequency ( ω ). 17. Several mechanical and structural systems can be approximated, reasonably well, as single degree of freedom systems. g 18. ω n =

δ st

where δ st is the static deflection under self-weight and g is the acceleration due to gravity. 19. Mechanical clock, Wind turbine.

c c = cc 2 km Logarithmic decrement ( δ ): 2πς πc δ= = mω d 1− ς 2

20. Damping ratio( ς ): ς =

Loss coefficient: It is the ratio of energy dissipated per radian and the total strain energy. Specific damping capacity: It is the ratio of energy dissipated per cycle and the total strain energy. 21. (i) Damping force is independent of the displacement and velocity. (ii) Damping force depends only on the normal force (weight of the mass) between the sliding surfaces. (iii) Governing equation is nonlinear. 22. Complex stiffness = k + ih = k(1 + iβ ) h where k = stiffness, i = − 1 , h = hysteresis damping constant, and β = = a k measure of damping. 23. Hysteresis damping constant (h) is the proportionality constant that relates the damping coefficient (c) and the frequency ( ω ) as h c= .

ω

24. Hammer, baseball bat, pendulum used in Izod impact testing of materials. 25. One. 26. Time constant is the value of time which makes the exponent in the solution −

c

t

x( t) = x0 e m equal to -1. 27. A graph that shows how changes in one of the parameters of the system will change the roots of the characteristic equation of the system is known as the root locus plot. 28. Negative damping corresponds to an unstable system. 29. A system whose characteristics do not change with time is called a time invariant system.

Question 2.2: 1. 2. 3. 4. 5. 6. 7. 8.

T T T F F F T T

9. T 10. F 11. T 12. T 13. F 14. T 15. T 16. T 17. T 18. T 19. T 20. T 21. T 22. F Question 2.3: 1. kinetic, potential 2. harmonic 3. torsional 4. percussion 5. continues 6. μ N 7. loss 8. rigid 9. critical 10. amplitude 11. natural 12. logarithmic 13. ωd = 1 − ς 2 ωn 14. 63.2% 15. faster 16. damped

Question 2.4: 1. b 2. c 3. c 4. b 5. a 6. a 7. b 8. b 9. a 10. c

11. b 12. b 13. a 14. b 15. b 16. c 17. b 18. a 19. a Question 2.5: 1 ⎯g 2 ⎯d 3 ⎯f 4 ⎯a 5 ⎯b 6 ⎯e 7 ⎯c Question 2.6: 1 2 3 4 5

⎯c ⎯a ⎯d ⎯e ⎯b

Question 3.1: 1. If the applied force is F (t ) = F0 cos ωt , the steady-state vibration response will have the following characteristics: x p (t ) = X cosω t Amplitude = X =

F0 k − mω 2

Frequency = ω Phase = 0 (no phase difference between applied force and response). 2. For simplicity, consider an underdamped system. The steady-state response under a harmonic force F (t ) = F 0 cosω t is given by F0 ⎛ ⎞ x p (t ) = ⎜ ⎟ cosω t 2 ⎝ k − mω ⎠

F0 = δ st = constant static k deflection of the mass due to F0 . This amounts to “no effect” on steady-state response since the vibration due to additional time-dependent forces can be considered to be about the new static equilibrium position of the mass. 3. For an underdamped system, Maximum amplitude Magnification factor = deflection of mass under constant force X 1 = or 2 δ st ⎛ω ⎞ 1 − ⎜⎜ ⎟⎟ ⎝ωn ⎠

For a constant force F0 , ω = 0 and hence x p ( t) =

X

4. If

< 1 , then

ω > 1 or ω > ω n . ωn

δ st 5. In the neighborhood of resonance, the amplitude (X) is given by δ X = st 2ς and the phase angle by π φ = tan − 1 (∞ ) = . 2 6. Phase corresponding to peak amplitude is given by 2 ⎞ 2 ⎛ ⎛ ⎛ 2ς r ⎞ − 1⎜ 2ς 1 − ς 2 ⎟ = tan −1 ⎜ 2 1 − ς = − = r 1 tan with ς ⎟ 2 ⎜ 1 − (1 − ς 2 ) ⎟ ⎜ ς ⎝1 − r ⎠ ⎝ ⎠ ⎝

φ = tan −1 ⎜

⎞ ⎟ ⎟ ⎠

For ς < 1 (underdamped system), φ = tan − 1 (w) where w < 2 . Hence φ < 900 . 7. Because it avoids the amplitude from reaching a value of infinity. 8. Forced equation of motion: ..

.

m x+ c x+ k x = F ( t) = F0 cos ω t Vector representation:

mω 2 X cω X

ωt F0

φ

kX

9. Response becomes infinity. 10. Beating: This is a phenomenon that occurs when the forcing frequency is close to, but not exactly equal to, the natural frequency of the system.

Quality factor: The value of the amplitude ratio at resonance,

X

δ st

, is called ω =ω n

the quality factor of the system. Transmissibility: When a system is subjected to harmonic base motion, the ratio of the amplitude of the response to that of the base motion is called the displacement transmissibility. Complex stiffness: The term, k (1 + i β ) , in the equation of motion of a hysteretically damped system is called complex stiffness. Quadratic damping: When the damping force is proportional to the square of the velocity of the mass, the corresponding damping is said to be quadratic damping. 11. For small values of r ( r > 1) , φ will be nearly π , and all the applied force will be overcoming the large inertia force. Hence the response will be small. 12. Addition of damping reduces the force transmitted to the base only when r < 2 . 13. For small values of damping, the force transmitted to the base due to rotating unbalance increases from zero to a peak value, then decreases for a while, and then increases as the speed of the machine increases. 14. Yes. 15. Yes, theoretically possible. 16. Harmonic response is assumed. 17. Yes, under the following conditions: (a) small damping values (b) away from resonance. 18. Yes, only for

ω ≠ 1. ωn

19. Using mass of the system equal to the total mass of the machine, and magnitude of the applied harmonic force equal to the centrifugal force, m eω2 , due to the rotating unbalance. 20. Frequency of response will be ω . The response will be harmonic. 21. Peak amplitude ( X p ) occurs when X is maximum. Resonance amplitude ( Xr ) occurs when r = 1 . For underdamped systems, X p > X r . 22. It is simple to handle mathematically. Governing differential equation will be linear. 23. Self-excited vibration is one that results when the external force is a function of the motion parameters of the system (such as displacement, velocity or acceleration).

24. Transfer function is defined as the ratio of the Laplace transform of the output (or response function) to the Laplace transform of the input (or forcing function), assuming zero initial conditions. 25. By substituting i ω for s. 26. Graphs of logarithm of the magnitude of the frequency transfer function versus logarithm of the frequency and phase angle versus logarithm of the frequency are known as Bode diagrams. 27. A decibel is defined as 10 times the logarithm to base 10 of the ratio of two power quantities. Question 3.2: 1. T 2. T 3. T 4. F 5. T 6. T 7. T 8. F 9. F 10. T 11. T 12. T 13. T 14. T 15. T 16. T Question 3.3: 1. harmonic 2. harmonic 3. transient 4. resonance 5. magnification 6. beating 7. transmissibility 8. impedance 9. bandwidth 10. quality 11. Coulomb 12. large 13. complex 14. turbulent 15. motion

16. self-excited 17. diverges 18. Laplace 19. transfer function 20. F(s) 21. algebraic Question 3.4: 1. b 2. a 3. a 4. a 5. a 6. b 7. c 8. b 9. a 10. b 11. a Question 3.5: 1⎯d 2⎯a 3⎯f 4⎯e 5⎯c 6⎯b Question 3.6: 1⎯c 2⎯e 3⎯a 4⎯d 5⎯b Question 4.1: 1. Any periodic function can be expressed as a sum of harmonic functions using Fourier series. 2. a. Representing the excitation by a Fourier integral. b. Using the method of convolution integral c. Using the method of Laplace transfor d. Numerical integration of equations of motion

3. The equation denoting the response of an underdamped single degree of freedom system to an arbitrary excitation is called Duhamel integral. 4. When an impulse of magnitude F is applied at t = 0 , the initial conditions can be ~

⋅

taken as x(t = 0) = 0 , x (t = 0) =

F ~

m 5. Equation of motion of a system subjected to base excitation y (t ) is given by ..

.

..

m z + c z + kz = −m y where z = x − y . 6. Response spectrum is a graph showing the variation of the maximum response, such as maximum displacement , with the natural frequency of a single degree of freedom system to a specified forcing function. 7. It can treat discontinuous functions without any particular difficulty. It automatically takes into account the initial conditions. 8. The response spectrum associated with the fictitious velocity associated with the apparent harmonic motion is called pseudo spectrum. ∞

9.

x (s ) = L x (t ) = ∫ e− st x (t ) dt 0

10. Generalized impedance ( Z ( s ) ):

Z (s ) = ms 2 + cs + k Admittance ( Y ( s ) ): 1 1 Y ( s) = = 2 Z( s) ms + cs + k 11. Step function and linear function. 12. If the forcing function is neither periodic nor harmonic, there will be no resonance conditions. 2π 13. If the period is T, the first harmonic frequency is given by ω1 = . T 14. n th frequency ( ωn ) is given by ω n = n.ω1 ; n = 2,3, L 15. Transient response is due to initial conditions. Steady state response is due to the applied force. 16. First order system is one whose governing differential equation is of order one. 17. A large force acting over a short period is called an impulse. ⎧ ∞ at x = 0 18. (i) δ ( x ) = ⎨ ⎩0 at x ≠ 0 ∞

(ii)

∫ δ (x ) dx = 1

−∞

Question 4.2: 1. T 2. T 3. T 4. F 5. T 6. T 7. T 8. T 9. T 10. F 11. T 12. T 13. T Question 4.3: 1. superposing 2. Fourier 3. short 4. impulse 5. convolution 6. response 7. convolution 8. steady 9. algebraic 10. reciprocal 11. momentum 12. impulse 13. undamped 14. pseudo 15. Fourier 16. initial 17. impulse 18. steady state 19. X(s) 20. F(s) 21. Second 22. 1

Question 4.4: 1. b 2. b 3. c 4. c 5. b 6. b 7. a 8. b 9. b 10. a 11. a 12. c 13. c 14. a 15. b 16. a 17. b 18. a

Question 4.5: 1⎯c 2⎯e 3⎯a 4⎯f 5—b 6⎯d Question 4.6: a—2 b—5 c—1 d—3 e—4

Question 5.1: 1. Number of degrees of freedom = (number of masses in the system)×(number of possible types of motion of each mass) 2. If the mass matrix is not diagonal, the system is said to have mass coupling. If the damping matrix is not diagonal, the system is said to have velocity coupling. If the stiffness matrix is not diagonal, the system is said to have elastic coupling. 3. Yes. 4. (a) Six: for a rigid body (b) Infinity: for an elastic body. 5. The coordinates that lead to equations of motion that are both statically and dynamically uncoupled, are known as principal coordinates. They are useful since the resulting equations of motion can be solved independently of one another. 6. Due to symmetry of influence coefficients; that is, the force along xi to cause a unit displacement along x j is same as the force along x j to cause a unit displacement along xi . 7. Node is a point in the system which does not move during vibration in a particular mode. 8. Static coupling: If a static force is applied along xi , it causes displacement along

x j as well. Dynamic coupling: If a dynamic force is applied along xi , it causes displacement along x j as well. Coupling of the equations of motion can be eliminated by using a special system of coordinates known as principal coordinates. 9. Impedance matrix [ Z ( i ω)] is defined by [Z (iω )] X = F 0 where, Z rs ( iω ) = −ω 2 m rs + iω c rs + k rs 10. By giving initial conditions that simulate the displacement pattern of the particular mode shape. 11. Degenerate system is one for which at least one of the natural frequencies is zero ( that is, the stiffness matrix is singular ) . Examples: Two railway cars connected by a spring. Two rotors connected by an elastic shaft. 12. At the most, six, corresponding to three translational and three rigid body rotational motions. 13. The frequency transfer function can be obtained by substituting s = i ω in the general transfer function. 14. One.

Question 5.2: 1. T 2. F 3. T 4. F 5. T 6. T 7. T 8. T 9. T 10. T 11. F 12. F 13. F 14. F 15. T 16. T 17. T 18. T 19. T 20. T Question 5.3: 1. natural/principal/normal 2. independent 3. resonance 4. initial 5. mass moments of inertia, torsional springs 6. coupling 7. rigid 8. static 9. dynamic 10. velocity 11. uncoupled 12. stability 13. physically 14. free 15. forced 16. characteristic 17. elastic

Question 5.4: 1. 2. 3. 4. 5. 6. 7. 8.

a b c a c a a b

Question 5.5: 1⎯c 2⎯a 3⎯d 4⎯b Question 5.6: 1⎯b 2⎯d 3⎯e 4⎯c 5⎯a Question 6.1: 1. The flexibility influence coefficient, aij , is defined as the deflection at point i due to a unit load at point j. The stiffness influence coefficient, kij , is defined as the force at point i due to a unit displacement at point j when all the points other than the point j are fixed. If [ a ] and [ k ] denote the flexibility and stiffness matrices, respectively, then [ k ] = [ a]−1 and [ a] = [ k ]− 1 . 2. Equations of motion: ..

.

[ m] x + [ c] x + [ k ] x = F or ..

.

[ m] x + [ c] x+ [ a] −1 x = F 3. Elastic potential energy (strain ...