Answers to Review Questions

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  = ω   f  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 and ,  x1 (t ) =  A sin 1t   x 2 (t ) =  A sin 2 t  =  A sin( 1t + δ  1t )  x(t ) =  x1 (t ) + x 2 (t ) = 2 A sin ( ω 1t +

1 2

⎛ δω 1t  ⎞ ⎟ 2 ⎝   ⎠

δω 1t ). cos ⎜

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   ⎞ ⎟⎟  X  ⎝  0  ⎠

dB = 20 log ⎜⎜

where  X 0 is a specified reference value of X  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.

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  = ω   f  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 and ,  x1 (t ) =  A sin 1t   x 2 (t ) =  A sin 2 t  =  A sin( 1t + δ  1t )  x(t ) =  x1 (t ) + x 2 (t ) = 2 A sin ( ω 1t +

1 2

⎛ δω 1t  ⎞ ⎟ 2 ⎝   ⎠

δω 1t ). cos ⎜

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   ⎞ ⎟⎟  X  ⎝  0  ⎠

dB = 20 log ⎜⎜

where  X 0 is a specified reference value of X  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 11. b 12. c 13. a 14. b 14. b 15. a 16. a Question 1.5: 1. 2. 3. 4. 5.

 ⎯  b  b  c  ⎯  c  e  ⎯  e  ⎯  d  d  a  ⎯  a

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

 c  ⎯  c  ⎯  e  e  a  ⎯  a  d  ⎯  d  ⎯  b  b

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

 b  ⎯  b  ⎯  c  c  ⎯  e  e  d  ⎯  d  ⎯  f  f  ⎯  h  h  g  ⎯  g  a  ⎯  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 δ  =

⎛   x  ⎞ ln⎜⎜ 1 ⎟⎟ m ⎝  x m +1  ⎠ 1

The damping ratio (ς  ) can be found as

ς  =

δ  (2π ) +δ  2

2

.

2.  No. 3. Mass moment of inertia, torsional damping constant, torsional stiffness, and angular displacement, respectively. 4. Since the natural frequency is given by ω n =

k  m

, a decrease in m will cause the

natural frequency to increase. 5. Since the natural period is given by τ  =

2π  ω n

= 2π 

m k 

, a decrease in k   will cause

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

 g  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.

20. Damping ratio(ς  ): ς  =

c cc

=

c 2 km

Logarithmic decrement (δ  ): π  c 2πς  = δ  = mω d  1 − ς  2 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 ) where k   = stiffness, i =

− 1 , h  = hysteresis damping constant, and  β  =

h k 

=  a

measure of damping. 23. Hysteresis damping constant (h) is the proportionality constant that relates the damping coefficient (c) and the frequency ( ) 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  x(t ) =  x0 e

−

c m

t 

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.  N  7. loss 8. rigid 9. critical 10. amplitude 11. natural 12. logarithmic 13. ω d  = 1 − ς  2 ω n 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 ) =  F 0 cos t , the steady-state vibration response will have the following characteristics:  x p (t ) =  X cos t  Amplitude = X  =

 F 0 k − m ω 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 cos t  is given by

⎛   F 0  ⎞ ⎟ cos ω t  2 − k  m ω  ⎝   ⎠

 x p (t ) = ⎜

= 0 and hence  x p (t ) =

 F 0

= δ st  = constant static k  deflection of the mass due to  F 0 . This amounts to “no effect” on steady-state For a constant force  F 0 ,

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 − ⎜⎜ ⎝ ω n  ⎠ 4. If

 X  δ  st 

< 1 , then

ω n

> 1  or

>

n

.

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   ⎞ 2 −1 ⎜ 2ς  1 − ς  ⎟ −1 ⎜ 2 1 − ς  ⎟ φ  = tan ⎜ = tan ⎟  with r  = 1 − ς  = tan 2 ⎜ 1 − (1 − ς  2 ) ⎟ ⎜ ⎟ ⎝ 1 − r   ⎠ ⎝   ⎠ ⎝  ς   ⎠

−1

For ς  < 1  (underdamped system), φ  = tan −1 ( w)  where w < 2 . Hence φ  < 90 0 . 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 ) =  F 0 cos ω t  Vector representation: mω 2 X  c  X  t   F 0

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 + iβ ) , 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) , both the inertia and damping force will be small, which result in a small phase angle φ . Then the magnitude of the applied force will be nearly equal to the spring force. For large 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 n

≠ 1.

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 eω 2 , due to the rotating unbalance. 20. Frequency of response will be . The response will be harmonic. 21. Peak amplitude ( X  p ) occurs when  X  is maximum. Resonance amplitude ( X r  ) 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  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 − . 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 ω 1 = . T  14. n th  frequency ( n ) is given by n = n. 1 ; 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 du ring 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 )]  is defined by [ Z (iω )] X  = F 0 where,  Z rs (iω ) = −ω 2 mrs + 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  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, a ij , is defined as the deflection at point i due to a unit load at point j. The stiffness influence coefficient, k ij , 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 energy): 1 T  V  =  x [ k ] x 2

Kinetic energy: 1

. T 

.

T  =  x [ m] x 2 4. The generalized mass matrix will have nonzero non-diagonal terms as: ⎡ m11 m12 L m1n ⎤

⎢m m 22 21 [ m] = ⎢ ⎢ M ⎢ ⎣mn1 mn 2

L

L

⎥ ⎥ ⎥ ⎥ mnn ⎦

m2 n

5. The mass matrix [m] is always positive definite because the kinetic energy, 1

. T 

.

.

T  =  x [ m] x , cannot be negative or zero for nonzero velocity vector  x . 2 6.  No. The stiffness matrix [k ]  is positive definite only if the system is constrained and stable. For a semi-definite system, the matrix [k ]  will be singular and is said to be just positive (not positive definite). 7. The generalized coordinates are a set of n independent coordinates that describe the motion of an n  degree of freedom system uniquely. They may be lengths, angles or other set of numbers. On the other hand, if Cartesian coordinates are used to describe an n  degree of freedom system, we may require more than n coordinates along with certain constraints to describe the system uniquely. 8. Lagrange’s equations:

⎛ 

 ⎞

d  ⎜ ∂T  ⎟

∂T  ∂V  − + = Q j( n) ;  j = 1,2,L, n ⎜ ⎟ . dt ⎜ ⎟ ∂q j ∂q j ⎝ ∂ q  j  ⎠ . ∂q j where q  j = = generalized velocity, Q j( n ) = nonconservative generalized ∂t  coordinate q j , T  = kinetic energy , V  = strain energy, and t = time. 9. Matrix eigenvalue problem: ω 2 [ m] X  = [k ] X  where ω 2  is the eigenvalue and  X  is the eigenvector. 10. Mode shape is same as the eigenvector  X  in the eigenvalue problem, ω 2 [m] X  = [ k ] X   

(E.1) (i )

The eigenvector  X    corresponding to the eigenvalue ω i2   can be computed by substituting ω i2  in Eq. (E.1) and solving the resulting linear algebraic equations to (i )

find  X  : (i )

[ ω i2 [ m] − [ k ] ] X 

=0

11. n distinct natural frequencies.

12. Dynamic matrix = [ D] = [ k ]−1 [ m ] . It is useful because it leads to a special eigenvalue problem, instead of general eigenvalue problem, that needs to be solved to find the eigenvalues and eigenvectors of a system: [ D]  X  = λ  [ I ]  X  = λ  X  13. Frequency equation:

− ω 2 [m] + [k ] = 0 or λ [ I ] − [ D]

=0

where λ  = ω 2 and [ D] = [ k ]−1 [ m ] . (i )

14. Orthogonality of normal modes  X   implies ( j ) T 

 X 

(i )

= 0 

[ m]  X  ( j ) T 

for i  ≠  j

(i )

and  X  [ k ]  X  = 0   for i  ≠  j Orthogonal modal vectors implies ( i ) T 

(i )

 X  [m]  X  and ( i ) T 

 X 

(i)

[k ]  X 

= 1; i = 1, 2, L, n ; = ω i2 ; i = 1, 2, L, n

15. Any set of n  linearly independent vectors in an n-dimensional space is called a  basis in that space. (i )

16. If the eigenvectors  X 

, i = 1, 2, L , n,  are used as the basis, any vector  x  in the (i )

n-dimensional space can be expressed as a linear combination of  X   as  x =

n

∑

(i)

ci X 

 

(E.1)

i =1

where the constants ci  can be determined as ( i ) T 

ci =  X 

[ m] x ; i = 1, 2, L , n  

(E.2)

Equations (E.1) and (E.2) denote the expansion theorem. The expansion theorem is very useful in finding the response of multidegree of freedom systems subject to arbitrary forcing conditions according to a procedure known as modal analysis. 17. Modal analysis procedure: (i) Solve the eigenvalue problem and find eigenvalues and eigenvectors of the system. (ii) Express the solution vector in terms of normal modes (or eigenvectors) using the expansion theorem. The constants used are known as generalized coordinates.

(iii)

Uncouple the equations of motion and solve the resulting system of n second order ordinary differential equations. (iv) Apply the known initial conditions and find the generalized coordinates( or generalized displacements ). (v) Using the known generalized displacements, find the physical displacements of the system. 18. A rigid body mode is one in which the system moves as a rigid body ( either in translatory or rotary motion ). (0)

The rigid body mode,  X  , can be found by solving the equations: (0)

[k ]  X  = 0 The frequency corresponding to the rigid body mode will be zero. 19. A degenerate system is an unrestrained system for which at least one eigenvalue is zero ( corresponding to a rigid body motion or mode ). 20. Use only r  modes ( r  < n ) in the modal analysis so that the displacement vector of the n degree of freedom system,  x , is expressed as  x(t ) =

r 

∑

(i)

q i (t ). X  ;

r  < n.

i =1

21. Rayleigh’s dissipation function ( R ) is defined as 1

. T 

.

 R =  x [c]  x 2 where [c ]  is called the damping matrix. 22. Proportional damping: is one in which the damping matrix [c ] is assumed to be a linear combination of the mass and stiffness matrices as: [c] = α [m] + β [k ] where α   and  β   are constants. Modal damping ratio ( ς i ): is defined by α  + ω i2 β  = 2ς i ω i where

i

 is the i th  natural frequency of the system.

Modal participation factor ( qi ): is the i th generalized coordinate used in the expansion theorem:  x(t ) =

n

(i)

∑ q (t ).X  i

i =1

23. When the system is damped, and damping is not proportional damping, that is, when [c] ≠ α [ m] + β [ k ] 24. Routh-Hurwitz criterion can be used to investigate the stability of a multidegree of freedom system.

Question 6.2: 1. T 2. F 3. T 4. T 5. T 6. T 7. T 8. T 9. F 10. T 11. F 12. T 13. T 14. F 15. T 16. T 17. T Question 6.3: 1. force 2. i,  j 3. stiffness 4. orthogonal 5. influence 6. generalized 7. 0 8. singular 9. six 10. modal 11. basis 12. expansion 13. modal 14. uncoupled 15. basis 16. energy 17. characteristic 18. Maxwell’s 19. symmetric 20. stable 21. synchronous 22. stiffness, mass

Question 6.4: 1. c 2. a 3. c 4. a 5.  b 6.  b 7.  b 8. a 9. c 10. b 11. a 12. b 13. a Question 6.5: 1  ⎯  c 2  ⎯  f 3  ⎯  d 4  ⎯  h 5  ⎯  b 6  ⎯  g 7  ⎯  a 8  ⎯  e Question 7.1: 1. Dunkerley’s formula, Rayleigh’s method, Holzer’s method, matrix iteration method, and Jacobi’s method. 2. Higher natural frequencies of a system are large compared to its fundamental frequency. 3. The frequency of vibration of a conservative system vibrating about an equilibrium position has a stationary value in the neighborhood of a natural mode. This stationary value, in fact, is a minimum value in the neighborhood of the fundamental natural mode. 4. The fundamental frequency given by Dunkerley’s formula will always be smaller than the exact value The fundamental frequency given by Rayleigh’s method will always be larger than the exact value. 5. Rayleigh’s quotient ( R ): T 

 R = ω 2 =

 X  [k ] X  T 

 X  [ m] X 

6. Holzer’s method is a trial and error method. In this method, first a trial frequency of the system is assumed and a solution is found when the assumed frequency satisfies the constraints of the system. 7. In matrix iteration method, a trial vector  X 1  is assumed for the mode shape, and is premultiplied by the dynamical matrix [ D]. The resulting column vector is normalized, usually by making one of its components to unity. The normalized column vector premultiplied by [ D] to obtain a third column vector, which is normalized in the same way as before, and becomes still another trial column vector. The process is repeated until the successive normalized column vectors converge to a common vector. The converged vector represents the fundamental 1 eigenvector and the constant used in the normalization process denotes 2  where ω 1 1

 is the fundamental eigenvalue.

8. Yes, provided we use [ D ]−1   in place of [ D]   for premultiplication in the matrix iteration method. 9. A procedure known as matrix deflation is used to find a deflated matrix [ Di ]  to  be used in place of the dynamical matrix [ D]  for premultiplication in the matrix iteration method. 10. The matrix iteration method finds one eigenvalue and the corresponding eigenvector at a time while the Jacobi’s method finds all the eigenvalues and eigenvectors simultaneously. 11. Rotation matrix, [ R ] , is defined as

⎡1 0 ⎢0 1 ⎢ ⎢ ⎢ ⋅ [ R1 ] = ⎢ ⎢ n× n ⎢ ⎢ ⎢ ⎢ ⎣⎢

⋅ ⋅

⋅ ⋅

⋅ ⋅

L L

O

cos θ 

sin θ  O

sin θ 

cos θ 

M

O

0⎤

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ 1⎦⎥ 0

row

i

row

j

( E.1 ) column i

column  j

where

⎛  2 d ij  ⎞ ⎟ ⎜ d ii − d  jj ⎟ ⎝   ⎠

tan 2θ  = ⎜

( E.2 )

and [ D ]  is the matrix whose eigenvalues and eigenvectors are to be found.

By carrying out the computations as [ D ] = [ R1 ]T [ D ][ R1 ]

( E.3 )

the off-diagonal components d ij  and d  ji  of [ D] will be reduced to zero. By carrying out the computations according to Eq. ( E.3 ) using different rotation matrices [ R2 ], [ R3 ],L , the final matrix [ D ]  will be reduced to a diagonal matrix. The diagonal elements will then represent the eigenvalues and the columns of the  product of rotation matrices [ R1 ][ R2 ]L denote the eigenvectors of the matrix [ D ] . 12. Standard eigenvalue problem:

[ [ D] − λ [ I ] ] X  = 0 where λ  is the eigenvalue and  X  is the eigenvector. 13. The general eigenvalue problem [k ]  X  = ω 2 [ k ]  X  can be converted to a standard eigenvalue problem as

( E.1 )

[ D]  X  = λ  X 

( E.2 )

⎛  1  ⎞ 1 ⎟ and [ D] = [k ]− [m]. 2 ⎝ ω   ⎠

where λ  = ⎜

However, [ D ] will be nonsymmetric in Eq. ( E.2 ) although [k ] and [m] are symmetric. Choleski decomposition method can be used to express [k ] as [k ] = [U ]T [U ] ( E.3 ) where [U ] is an upper triangular matrix, and Eq. ( E.1 ) can be converted to a standard eigenvalue problem as [ D] Y  = λ  Y 

( E.4 )

T  − − where [ D ] = ( [U ] ) 1 [ m ] [ U  ] 1

( E.5 )

and Y  = [U ]  X   or  X  = [U ]−1 Y  . The matrix [ D] in Eq. ( E.4 ) will be symmetric. 14. If [u ij ] = [U ] is an upper triangular matrix, its inverse, [a ij ] = [U ]−1 , can be determined as follows: [U ] [U ] −1 = [ I ] Equating the corresponding elements on both sides of Eq. ( E.1 ), we can determine the elements of [U ] −1 . Question 7.2: 1. 2. 3. 4. 5. 6. 7.

F T F T T T T

8. T 9. F 10. T 11. T Question 7.3: 1. upper triangular 2. Choleski 3. zero 4. expansion 5. largest 6. upper, lower 7. eigenvector 8. static 9. trial and error 10. Holzer’s 11. deflation Question 7.4: 1. 2. 3. 4. 5. 6.

a  b  b a c a

Question 7.5: 1  ⎯  d 2  ⎯  e 3  ⎯  a 4  ⎯  b 5  ⎯  c Question 8.1: 1. Equations of motion will be partial differential equations for continuous systems and ordinary differential equations for discrete systems. 2. Infinity 3.  No. Because they are taken care, in an indirect way, in generating the influence coefficients.

4. Wave equation: 2 ∂2w 2 ∂ w c = 2 ∂ 2 ∂t  where c  =

 P   ρ 

.

Traveling – wave solution: w( x, t ) = w1 ( x − ct ) + w2 ( x + ct ) where w1  and w2 are arbitrary functions of − c t   and + c t  , respectively, which are determined from the initial conditions. 5. Wave velocity gives the velocity with which the waves w1 ( x − c t )  and w2 ( x + c t ) propagate in the positive and negative directions of the  x-axis, respectively. 6. Boundary conditions for a simply supported end of a beam: (i) Thin beam theory (w = transverse displacement): ∂2w w = 0,  EI  2 = 0 ∂ x (ii) Timoshenko beam theory (w = transverse displacement, Φ = bending slope): w = 0, EI 

∂Φ =0 ∂ x

7. Possible boundary conditions at the ends of a string: (i) Fixed end: w = 0 (ii) String connected to a pin that can move in a perpendicular direction: ∂w  P  = 0  ( P  = tension in string ) ∂ x ∂w =0 (iii) Free end : ∂ x (iv) Elastically supported by a spring of stiffness k : ~

∂w = − k w ~ ∂ x

 P 

8. Frequency equations: For discrete systems: Polynomial equation. For continuous systems: Transcendental equation. 9. For tensile force, the natural frequencies of the beam increase. 10. As the axial force (compressive) approaches the Euler buckling load,  P cri , the natural frequency of the beam approaches zero. 11. The beam becomes less stiff when the effects of shear deformation and rotary inertia are considered. 12. Drumhead, cover of a cylindrical soda can. 13. The maximum potential (strain) energy is equal to the maximum kinetic energy. 14. The Rayleigh’s quotient, which gives ω 12 , attains minimum at the exact fundamental mode. Hence any other mode, used as an approximation, yields a larger value of ω 12  than the exact value.

15. In Rayleigh’s method, a one-term solution is assumed for the mode shape. In the Rayleigh-Ritz method, a multi-term solution is assumed for the mode shap e.

⎧ l  ⎛ d 2W ( x) ⎞ 2 ⎫ ⎟⎟ dx ⎪ ⎪ ∫ EI ⎜⎜ 2 dx ⎪ ⎪ ⎝   ⎠ Rayleigh’s quotient for a beam,  R (ω ) = ω 2 = ⎨ 0 l  ⎬ 2 ⎪  ρ  A(W ( x) ) dx ⎪ ⎪ ∫0 ⎪ ⎩ ⎭ Question 8.2: 1. T 2. T 3. F 4. T 5. T 6. T 7. T 8. T 9. T 10. F Question 8.3: 1. wave 2. characteristic 3.  product 4. initial 5.  positive 6. flexural, torsional 7. Euler- Bernoulli 8. fourth 9. increases 10. thick 11. membrane 12. plate 13. fundamental/first 14. bending moment 15. ordinary 16. stiffness 17. kinetic 18. strain

Question 8.4: 1. 2. 3. 4. 5. 6. 7.

 b a c a c  b a

Question 8.5: 1  ⎯  d 2  ⎯  c 3  ⎯  b 4  ⎯  a Question 8.6: 1  ⎯  b 2  ⎯  d 3  ⎯  a 4  ⎯  c Question 8.7: 1  ⎯  c 2  ⎯  a 3  ⎯  b Question 9.1: 1. Impact processes, such as pile driving and blasting; Rotating and reciprocating machinery such as engines, compressors, and motors; Transportation vehicles such as trucks, trains, and aircraft; Flow of fluids in pipes. 2. Balancing of machines; Control of natural frequencies; Introduction of damping; Use of vibration isolation; Use of vibration absorbers. 3. Static balancing; the unbalance can be corrected by removing or adding material in a single plane. 4. (i) First, add a known weight W  L   in the left plane at a known angular position and measure the displacement and phase of vibration at the two bearings, while the rotor is rotating at speed .

(ii) Remove W  L   and add a known weight W  R in the right plane at a known angular position and measure the resulting vibration while the rotor is running at speed . (iii) Using relevant vector equations, find the unbalance vectors U  L  and U  R  in the left and right planes, respectively. (iv) Balance the rotor by adding equal and opposite balancing weight  B L  and  B R as  B L = −U  L  and  B R = −U  R . 5. Whirling is defined as the rotation of the plane made by the line of centers of the  bearings and the bent shaft. 6. Any rotating system responds in two different ways to damping, depending upon whether the forces rotate with the shaft or not. When the positions at which the forces act remain fixed in space, the damping is called stationary damping. On the other hand, if the positions at which they act rotate with the shaft in space, the damping is called rotary damping. 7. A critical speed is one at which the frequency of rotation of a shaft equals one of the natural frequencies of the shaft. For an undamped system, the critical speed is given by ω n =

k 

m where k  and m denote the stiffness and mass of the shaft. 8. Instability in a flexible rotor system can occur due to reasons such as internal friction, eccentricity of the rotor, and oil whip in the bearings. 9. For a single-cylinder engine, the equivalent rotating mass ( mc  ) can be made zero  by counter-balancing the crank. However, the equivalent reciprocating mass ( m p ) cannot be balanced. For a multi-cylinder engine, the axial displacements ( l i ) and angular orientations ( α i ) of cylinder i from those of the first cylinder ( i = 2, 3, L, N  ) can be selected to balance the inertia forces in  x  and  y  directions (vertical and horizontal directions) and the moments about the z and x- axes. 10. Force transmitted to the base of a vibrating system can be reduced. The vibrating mass can be protected from the base vibration. 11. A vibration absorber is a spring-mass system that is added to a vibrating system so that the natural frequencies of the resulting system are away from the excitation frequency. 12. Vibration isolator involves the design of spring and/or damper to reduce the vibration (transmissibility). Vibration absorber involves the design of a new spring-mass system to be added to the original vibrating system so that the natural frequencies of the resulting system are away from the excitation frequency. 13. Yes. 14. Yes. The frequency of the machine-isolator-supporting system ( 2 ) decreases with a soft spring. The force transmissibility becomes smaller with a reduced value of 2 .

15. Shaking force ( unbalanced force ) in an unbalanced machine is proportional to the square of the speed ( angular frequency ) of the machine. The force transmitted to the foundation increases with the speed of the machine. 16. Static balancing involves balancing an unbalanced mass in a single plane. Dynamic balancing involves balancing the unbalanced masses in two different  planes. Since any unbalanced mass in a single plane can be replaced by two unbalanced masses in two different planes, dynamic balancing implies static  balancing. 17. Dynamic balancing involves balancing an unbalanced mass of an elongated rotor. The amount of unbalance and the plane in which it occurs are difficult to determine using a static test. 18. Because of the flexibility of the shaft. Source of shaking force is the inertia force due to the rotor mass rotating off-center with an eccentricity. 19. Yes. It reduces large amplitudes of vibration as the machine passes through the first peak during startup and stopping. 20. A vibration isolator that uses external power to perform the function of isolation is called an active vibration isolator. 21. Passive isolation consists of a spring and/or a damper to reduce the vibration transmitted to the mass from base motion or the force transmitted to the base from the vibrating mass. There is no external power used in passive isolation. On the other hand, active isolation involves the use of a servomechanism with a sensor, signal processor and an actuator for isolation. External power is used in active isolation. Question 9.2: 1. T 2. T 3. T 4. T 5. T 6. T 7. F 8. T 9. T 10. F 11. F 12. T 13. F 14. T

Question 9.3: 1. resonance 2. less 3. vibration 4. critical 5.  piston 6. secondary 7. high 8. small 9. source 10. passive 11. actuator 12. absorber 13. two 14. static 15. single 16. unbalance 17. vibration 18. critical 19. instability Question 9.4: 1. a 2.  b 3. c 4. a 5. a 6. c 7. a 8.  b 9. c 10. a 11. b Question 9.5: 1  ⎯  d 2  ⎯  a 3  ⎯  b 4  ⎯  c

Question 10.1: 1. (i) To ensure safety of machinery and structures. (ii) To find natural frequencies of a machine or structure to select the operational speeds of nearby machines to avoid resonance. (iii) To find the discrepancy between theorectical and actual vibration characteristics of systems. (iv) For system identification. 2. A vibrometer is an instrument that measures the displacement of a vibrating body. If the instrument that measures the displacement of a vibrating body also records the measured displacement, it is called a vibrograph. 3. A transducer is a device that transforms changes in mechanical quantities (such as displacement, velocity, acceleration or force) into changes in electrical quantities (such as current or voltage). 4. A strain gage consists of a fine wire whose resistance changes when it is subjected to mechanical deformation. When the strain gage is bonded to a structure, it experiences the same strain as the structure and hence its resistance change gives the strain applied to the structure. 5. The gage factors of a strain gage ( K  ) is defined as (Δ R / R )  K  = ≈ 1 + 2ν  (Δ L / L ) where R = initial resistance, Δ R = change in resistance, L = initial length of wire (strain gage) , Δ L = change in length of wire, and ν  = Poisson’s ratio of the wire. 6. A transducer is a device that transforms values of physical variables into equivalent electrical signals. When a transducer is used in connection with other components that permit the processing and transmission of the signal, the device is called a pickup. 7. Piezoelectric material is one that generates electrical charge when subjected to a deformation or mechanical stress. Examples: quartz and Rochelle salt. 8. When an electrical conductor, in the form of a coil, moves in a magnetic field, a voltage, proportional to the relative velocity of the coil, will be generated. This is the working principle of an electrodynamic transducer. 9. An LVDT (linear variable differential transformer) transducer consists of a long magnetic core, a primary coil wrapped around the center of the core, and two secondary coils wrapped around at the two ends of the core. The magnetic core can move freely inside the coils in the axial direction. When an a.c. input voltage is applied to the primary coil, an output voltage, depending on the amount of axial displacement of the core, will be induced in the secondary coils. 10. A seismic instrument is a vibration pickup which can be used to measure the displacement of a mass relative to the base on which it is mounted. 11. The frequency range of a seismometer is given by 3 ≤

ω n

≤ 5 where

n

 is the

natural frequency of the mass. 12. An accelerometer is an instrument that measure the acceleration of a vibrating  body.

13. The distortion in the wave form of a recorded signal due to different phase (time) lags affecting different harmonic components of the signal is called the phaseshift error. It becomes important when the vibration signal consists of a sum of two or more harmonic components. 14. Scotch yoke mechanism, and a device consisting of two identical unbalanced masses rotating at the same speed in opposite directions. 15. An electromagnetic shaker is a device that is based on the following principle. When current passes through a coil placed in a magnetic field, a force  proportional to the current and the magnetic flux density is produced. This force accelerates the component placed on the shaker table. 16. If a particular part or location of a machine or structure is found to have excessive deflection through the operational deflection shape measurement, that part or location can be stiffened subsequently to increase the natural frequency of the machine or structure beyond its operational frequency range. 17. Experimental modal analysis deals with the determination of natural frequencies, damping ratios and mode shapes through vibration testing. 18. Since the response of a system exhibits a sharp peak at resonance when the forcing frequency is equal to its natural frequency, the natural frequencies can be determined from the frequency response function. 19. Single-reed instrument (Fullarton tachometer), multi-reed instrument (Frahm tachometer), and stroboscope. 20. Plot of  H (i ω ) versus frequency . Real and imaginary components of response versus frequency, and vector diagram of the real component versus the imaginary component of the response. 21. The graphs showing the variations of the magnitude of the response and its phase angle of a single degree of freedom system in the frequency domain are called Bode diagrams. These diagrams can be used to find the natural frequency and the damping ratio of the system. 22. Nyquist diagram is constructed by plotting the real and imaginary parts of the frequency response function of a single degree of freedom system along the horizontal and vertical axes of a graph for a range of frequencies. The Nyquist diagram will be in the form of a circle. 23. The mode superposition principle states that the dynamic response is given by a linear superposition of the normal modes of vibration of the system. In experimental modal analysis, the response of the system to vibration is measured and the results are used to identify the modal (natural) frequencies, mode shapes and the system parameters, namely, the equivalent mass, stiffness and damping ratio. 24. Breakdown maintenance, preventive maintenance, and condition-based maintenance. 25. Any change in the pattern of the orbits can be used to identify faults such as misalignment in shafts, unbalance in shafts, shaft rub, wear in journal bearings, and hydrodynamic instability in lubricated bearings.

26. Kurtosis ( k  ) is defined as the fourth order moment: k  =

1 σ 

4

∞

∫ ( x − x)

4

 f ( x) dx

−∞

where  f ( x )   is the probability density function of the instantaneous amplitude,  x(t ) , at time t ,  x   is the mean value, and σ    is the standard deviation of  x(t ) . Cepstrum, c(τ ) , is defined as the inverse Fourier transform of the logarithm of the power spectrum: c (τ ) =  F −1 log{S  X  (ω )} where the power spectrum, S  X  ( ) , of the time signal  x(t )  is given by S  X  (ω ) = [ F { x(t )}]

2

with  F { }  denoting the Fourier transform of  F { x(t )} =

1

T  / 2

∫ x(t ) e

T  −T  / 2

Question 10.2: 1. T 2. T 3. T 4. T 5. T 6. T 7. F 8. T 9. F 10. T 11. T 12. T 13. T 14. T 15. T Question 10.3: 1. 2. 3. 4. 5. 6. 7. 8. 9.

transducer charge spring-mass-damper accelerometer accelerometers velometer resonance cantilever contact

i ω t 

dt 

{ }:

10. frequency 11. health 12. octave 13. deformation 14. unconstrained/free-free 15. load 16. accelerometers 17. spectrum 18. vibration 19. bathtub 20. changes 21. power spectrum Question 10.4: 1.  b 2. a 3. a 4. a 5. c 6. a 7.  b 8. c 9. a 10. c 11. b Question 10.5: 1  ⎯  d 2  ⎯  c 3  ⎯  b 4  ⎯  e 5  ⎯  a Question 11.1: 1. The governing differential equation and the associated boundary conditions are replaced by finite difference equations. For this, each derivative is replaced by its finite difference equivalence. This leads to a system of linear algebraic equations instead of a differential equation.

2. The Taylor’s series expansions for  xi +1  and  xi −1  can be expressed about the grid  point i as  xi +1 =  xi + h x& i +  xi −1 =  xi − h x& i +

h2 2 h2

&&i +  x

h3 6 h3

&x&&i

+ L   (E.1)

&&i − &x&&i + L   (E.2)  x 2 6 where  xi =  x(t  = t i ) and h = t i +1 − t i = Δt . By taking two terms only and

substracting Eq.(E.2) from (E.1) , we get the central difference formula for the first derivative as dx 1  x& i = = ( xi +1 − xi −1 )   (E.3) dt  t i 2h By taking three terms only and adding Eqs.(E.1) and (E.2), we get the central difference formula for the second derivative as &&i =  x

d 2 x

=

2

dt 

t i

1 h2

( xi +1 − 2 xi + xi −1 )   (E.4)

3. A conditionally stable method is a numerical method that requires the use of the time step ( Δt ) smaller than a critical time step ( Δt cri ). If Δt  is chosen to be larger than Δt cri , the method becomes unstable. 4. The Runge-Kutta method requires the function value at a single previous point to find the function value at the current point. The central-difference method requires function values at two previous points to find the function value at the current point. 5. If a derivative is to be approximated at a boundary point using the finite difference method, it may require the use of the function value at a grid point outside the material (which is called a fictitious grid point). For example, if d φ  = 0  at a boundary (grid point i), the use of central difference approximation dx gives Φ − Φ −1 d φ  = 2 =0 dx i 2h where Φ −1 = Φ ( x −1 =  x1 − Δ x =  x1 − h)  with –1 denoting a fictitious grid point. 6. Tridiagonal matrix is a square matrix that has non-zero elements only along the main diagonal and each diagonal that lies on either side of the main diagonal. 7. The acceleration of the system is assumed to vary linearly between two instants of time, t i  and t i +θ  .

8. Linear acceleration method is one in which the acceleration is assumed to vary linearly between two time stations t i  and t i +1  so that

⎛  t − t i  ⎞ &&(t ) =  x &&i + ⎜  x ⎜ t  − t  ⎟⎟( x&&i +1 − &x&i ) ⎝  i +1 i  ⎠

( E.1 )

&&i =  x && ( t i )  and t i ≤ t  ≤ t i +1 . By integrating Eq. ( E.1 ) once and twice, we where  x

can find expressions for the velocity  x&  and displacement  x , respectively. 9. If a numerical integration method requires the use of the equilibrium equation at time t i +1  to find the solution  xi +1 , it is called an implicit integration method. If a numerical integration method requires the values of the response at previous time steps t i  and t i −1  , including the use of the equilibrium equation at time t i , to find the value of the response at t i +1 , the method is known as an explicit integration method. 10. No, not directly. Question 11.2: 1. 2. 3. 4. 5. 6. 7. 8.

F T T F T T F T

Question 11.3: 1. 2. 3. 4. 5. 6. 7. 8.

closed derivatives three mesh/grid Taylor’s conditionally unstable recurrence

Question 11.4: 1. c 2. a 3. a 4. c 5.  b 6. a 7. c 8. a 9.  b 10. a Question 11.5: 1  ⎯  e 2  ⎯  a 3  ⎯  b 4  ⎯  c 5  ⎯  f 6  ⎯  d

Question 12.1: 1. The basic idea behind the finite element method is to replace the actual structure  by several pieces called finite elements. The finite elements are assumed to be interconnected at certain points known as nodes. A simple solution is assumed within each element and equilibrium of forces at the nodes and compatibility of displacements between the elements are enforced. This leads to a system of equations valid for the overall structure (assembly of elements) whose solution yields an approximate solution of the problem. 2. The shape function,  N i ( x, y ) , is a polynomial in and  y  defined such that its value is 1 at node i (with coordinates  xi  and  y i ) and 0 at all other nodes  j (with coordinates  x j  and  y j ) of the finite element. 3. A transformation matrix, [λ ] , relates the nodal displacements of an element  between local and global coordinate systems. It permits conversion of element matrices (such as stiffness matrices) derived in a local coordinate system to those valid in a global coordinate system so that the element matrices can be assembled to derive a system or overall matrix. 4. Transformation of nodal displacements of one coordinate system to those of another coordinate system. 5. The rows and columns of the system matrices and system load vectors corresponding to zero degree of freedom are deleted.

6. By using symmetry conditions. The system must remain symmetric even after deformation. Hence suitable displacement conditions are to be incorporated along axes of symmetry. 7. Because the variation of displacement within an element is expressed as a  polynomial in terms of its nodal displacement values (unknowns). 8. A consistent mass matrix is a mass matrix derived using the same displacement model that is used for deriving the element stiffness matrix. 9. A lumped mass matrix is a mass matrix derived by distributing the mass of the element to its various nodes. 10. In the finite element method, an approximate solution is assumed within each finite element. In the Rayleigh-Ritz method, an approximate solution is assumed for the whole system. In the case of structural problems, the potential energy of the complete system is minimized in both the finite element and Rayleigh-Ritz methods to derive the system equilibrium equations. 11. The distributed load is used to compute the virtual work of the element. By equating this virtual work with the virtual work associated with the equivalent  joint forces and the associated virtual joint displacements, the expressions for the equivalent joint forces are derived. Question 12.2: 1. T 2. T 3. F 4. T 5. F 6. T 7. F 8. T 9. T 10. T Question 12.3: 1. finite elements 2. nodes/joints 3. approximate 4. shape 5. two 6. three 7. displacement 8. consistent 9. lumped 10. dynamic 11. transformation

Question 12.4: 1. 2. 3. 4. 5. 6. 7. 8.

a a  b  b a  b c c

Question 12.5: 1  ⎯  d 2  ⎯  c 3  ⎯  a 4  ⎯  b Question 13.1: 1. If the governing differential equation is nonlinear, the vibrating system is nonlinear. 2.  Nonlinearity of the system may be due to the nonlinearity associated with the mass, damper or spring. 3. Spring stiffness. 4. The frequency of the response increases with the amplitude for a hardening spring and decreases for a softening spring. 5. Subharmonic oscillations are oscillations whose frequencies ( n ) are related to the forcing frequency ( ω n =

) as

; n = 2,3,4, L n 6. Jump phenomenon is one where the amplitude of vibration increases or decreases suddenly as the excitation frequency is increased or decreased. Thus there exist two amplitudes of vibration for certain values of the forcing frequency. 7. In the Ritz-Galerkin method, an approximate solution of the problem is found by satisfying the governing nonlinear equation in the average. 8. Phase plane: It is a graph with displacement and velocity denoting the two coordinate axes. Trajectory: As time changes, the state of a system or solution changes. The graph showing the variation of the solution with time in the phase-plane is called the trajectory. Singular point: A singular point is a point in the phase plane where the velocity and the force are zero. It corresponds to a state of equilibrium of the system. Phase velocity: The velocity with which a representative point moves along a trajectory in the phase-plane is called the phase velocity.

9. An isocline is defined as the locus of points at which the trajectories passing through them have a constant slope, c. In the method of isoclines, we fix the slope dy dy  by giving it a constant value c1  and solve the equation, = φ ( x, y ) = c1  for dx dx the trajectory. The curve φ ( x, y ) − c1 = 0  thus represents an isocline in the phase  plane. We plot several isoclines h1 , h2 , L   by giving different values c1 , c2 , L  to dy the slope = φ ( x, y) . These isoclines are then used to plot the trajectory passing dx through any specific point in the phase plane. 10. For a nonlinear spring with deformation  x , the restoring force can be expressed dy dy as  f  ( x ) . If   is a strictly increasing = constant, the spring is linear. If dx dx dy function of  x , the spring is called a hard spring. If  is a strictly decreasing dx function of , the spring is called a soft spring. 11. Subharmonic oscillations are oscillations whose frequencies ( n  ) are related to the forcing frequency (

) as

ω n =

; n = 2,3,4, L n Superharmonic oscillations are oscillations whose frequencies ( the forcing frequency ( ; n = 2,3,4,L n = n

n

 ) are related to

) as

12. In the solution of the pendulum equation of the form ( E.1 )  x + ω 02  x + α  x 3 = 0 , If a two-term solution is assumed as  x(t ) =  x0 (t ) + α  x1 (t ) ( E.2 ) the resulting expressions of  x0 (t ) and  x1 (t ) make  x(t ) given by Eq. ( E.2 ) approach infinity as t   tends to infinity although the exact solution of Eq. ( E.1 ) is known to be bounded for all values of t . One of the terms in the expression of  x1 (t )  is called a secular term. 13. A simple pendulum whose pivot point is subjected to harmonic motion in the vertical direction. 14. Stable node: It is a point in the phase plane towards which all trajectories converge as t  → ∞ . Unstable node: It is a point in the phase plane from which all trajectories move away as t  → ∞ . Saddle point: If one solution tends to the origin while the other tends to infinity in the phase plane, the origin is called a saddle point and it corresponds to unstable equilibrium. Focus: If the trajectory is in the form of a logarithmic spiral, the equilibrium point or origin is called a focus.

Center: If the trajectory is a circle with origin as the equilibrium point, the motion will be periodic and hence stable. The equilibrium point ( origin ), in this case, is called the center. 15. In certain vibration problems involving nonlinear damping, the trajectories, starting either very close to the origin or far away from the origin, tend to a single closed curve, which corresponds to a steady-state periodic ( not harmonic ) solution of the system. This means that every solution of the system tends to a  periodic solution as t  → ∞ . The closed curve to which all the solutions approach is called a limit cycle. 16. Certain electrical feedback circuits controlled by valves where there is a source of  power that increases with the amplitude of vibration. Question 13.2: 1. T 2. T 3. F 4. T 5. F 6. T 7. T 8. T 9. T 10. T 11. T 12. T 13. T 14. T 15. T Question 13.3: 1. nonlinear 2. superposition 3. Mathew 4. Mathew 5.  phase 6. trajectory 7.  phase 8.  jump 9. two 10. algebraic 11. self-excited 12. autonomous 13. isoclines 14. limit

Question 13.4: 1. a 2.  b 3.  b 4. a 5. a 6. a 7. c 8.  b 9.  b 10. a 11. b Question 13.5: 1  ⎯  c 2  ⎯  e 3  ⎯  a 4  ⎯  b 5  ⎯  d Question 13.6: 1  ⎯  b 2  ⎯  d 3  ⎯  a 4  ⎯  c Question 14.1: 1. Each outcome of an experiment , in the case of a random variable, is called a sample point. If n experiments are conducted, all the n possible outcomes of the random variable constitute what is known as the sample space of the random variable. Each outcome of an experiment , in the case of a random process, is called a sample function. If n experiments are conducted, all the n  possible outcomes of the random process constitute what is known as the ensemble of the process. 2. The probability density function ,  p X  ( x ) , of a random variable X  is defined by  p X  ( x).dx = Probability of realizing the value of X  in the interval x and x + dx. The probability distribution function,  P   X  ( x ) , of a random variable X  is defined as  P   X  ( x) = Probability of realizing the value of X less than or equal to x. The probability density and distribution functions are related as dP X  ( x)  p X  ( x) = dx

3. Mean value of a random variable X : ∞

∫

μ  X  =  X  =  x ⋅  p X  ( x) dx −∞

Variance of a random variable X : ∞

∫ ( x − μ 

σ  = 2

 X 

) 2 ⋅ p X  ( x) dx

−∞

4. The distribution involving two random variables is called a bivariate distribution. 5. Covariance between the random variables X   and Y , denoted σ  XY  , is defined as follows: σ  XY  =  E [( x − μ  X  )( y − μ Y  )] ∞ ∞

=

∫ ∫ ( x − μ 

 X 

)( y − μ Y  ) p X ,Y  ( x, y ) dxdy

−∞−∞

where  p X ,Y  ( x, y )  is the joint density function of X  and Y . 6. Correlation coefficient of X  and Y  ( ρ  XY  ):  ρ  XY  =

σ  XY  σ  X σ Y 

where σ  X   and σ Y   are the standard deviations of X   and Y , respectively, and σ  XY  is the covariance of  X  and Y . 7. − 1 ≤  ρ XY  ≤ 1 8. Marginal density function is the density function of one random variable obtained from a joint density function. For example, the marginal density function,  p X  ( x ) , can be obtained from  p XY  ( x, y )  as ∞

∫

 p X  ( x) =  p XY  ( x, y) dy −∞

9. The mathematical expectation of  x1 x2 is called the autocorrelation function, denoted as  R(t 1 , t 2 ) : ∞ ∞

 R (t 1 , t 2 ) =  E [ x1 x 2 ] =

∫ ∫ x  x  p( x , x 1

2

1

2

) dx1 dx 2

−∞− ∞

where  x1  and  x2 denote the values of the random process  x(t )  at t 1  and t 2 , respectively, and  p( x1 , x 2 )  is the joint density function of  x1  and  x 2 . 10. A stationary random process is one for which the probability distributions remain invariant under a shift of the time scale. On the other hand, a nonstationary random process is one for which the probability distributions are functions of time. 11. The autocorrelation function of a stationary random process,  R(τ ) , is bounded as follows: − σ  2 + μ 2 ≤ R(τ ) ≤ σ  2 + μ 2 where and σ   are the mean and standard deviation of the process.

12. Ergodic process is a stationary random process for which we can obtain all the  probability information from a single sample function and assume that it is applicable to the entire ensemble. 13. If  x(t ) is an ergodic random process, and  x i (t )  denotes a typical sample function of duration T , the averages of  x(t )  can be computed by averages with respect to time along  x i (t ) . Such averages are called temporal averages. For example, the temporal averages of  x(t ) , denoted as  x(t ) , is defined as  x(t ) =  E [ x ] = lim

T → ∞

1

T  / 2

∫ x

T  −T  / 2

(i )

(t )dt 

14. A Gaussian random process ( x(t ) ) is one for which the probability density function is given by 1 ⎛  x − x ⎞

1

 p( x) =

2π σ  x

e

2

⎜ ⎟ 2 ⎜⎝  σ x  ⎠⎟

where  x  and σ  x  denote the mean and standard deviation of x. The Gaussian random process is used frequently in vibration analysis because of the following reasons: ( i ) It is simple to use. ( ii ) Most physical random processes can be very well modeled as Gaussian  processes. 15. The Parseval’s formula for a periodic function,  x(t ) , states that the mean square value of  x(t ) is equal to the sum of the squares of the absolute values of the Fourier coefficients. Thus if  x(t ) =

∞

∑c e

inω 0t 

n

n = −∞

The Parseval’s formula states that  x (t ) = 2

∞

∑c

2 n

n = −∞

16. Power spectral density function, S ( ) : For a stationary random process, S ( )   is defined as the Fourier transform of  R (τ ) where  R(τ )  is the autocorrelation function. 2π  White noise: A random process whose power spectral density is constant, with respect to frequency, is called white noise. Band-limited white noise: It is random process for which the power spectral density is constant over a frequency band. Wide-band process: It is a stationary random process whose spectral density function S ( )  has significant values over a range or band of frequencies which is approximately the same order of magnitude as the center frequency of the band.

 Narrow-band process: It is a stationary random process whose spectral density function S ( ) has significant values only in a range or band of frequencies whose width is small compared to the magnitude of the center frequency of the  process. ∞

∫ S (ω ) d ω 

17.  E [ x ] =  R (τ  = 0) = 2

−∞

If the mean is zero, the variance of the stationary random process,  x(t ) , is given  by ∞

∫ S (ω ) d ω 

σ  =  R (0) = 2  x

−∞

18. The response of a single degree of freedom system to a unit impulse is called the impulse response function. 19. Response of a single degree of freedom system,  x(t ) , using Duhamel integral: t 

1

∫

 x(t ) =  F (τ ) g (t − τ ) d τ  =

mω d 

0

t 

∫ F (τ ) e

−ςω n ( t −τ  )

sin ω d  (t − τ ) d τ 

0

where  F (τ ) = force applied at time τ  and  g (t − τ ) is the impulse response function due to a unit impulse applied at τ  . 20. If the forcing frequency on a single degree of freedom system is assumed as  x(t ) = e i ω t  , the response can be expressed as ~

 y (t ) =  H (ω ) e

i ω t 

~

where  H ( )  is called the complex frequency response function. It is given by 1

 H (ω ) =

{(1 − r  ) 2

where r  =

ω n

2

+ (2ς  r )

1

2

}

2

 and ς   is the viscous damping ratio.

21. The power spectral densities of the input, S  x ( ) , and the output, S  y ( ) , are related by the complex frequency response function,  H ( ) , as 2

S  y (ω ) =  H (ω ) S (ω ) 22. Wiener-Khintchine relations: S (ω ) =

1

∞

 R (τ ) e 2π  ∫

−i ω τ 

−∞

∞

 R (τ ) =

∫ S (ω ) e

−∞

i ω τ 

d ω 

d τ 

Question 14.2: 1. T 2. F 3. T 4. T 5. T 6. T 7. T 8. T 9. T 10. T 11. F Question 14.3: 1. deterministic 2. random 3. random 4.  parameter 5. variance 6.  joint 7. single 8.  bivariate 9. multivariate 10. t 11. engodic 12. bell 13. zero,one 14. infinite 15. power 16. wide-band 17. narrow-band 18. Fourier 19. band-limited Question 14.4: 1. 2. 3. 4. 5. 6. 7. 8.

a c a c a  b a  b