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.

14. .12f

==ωπτ

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 tAtx 11 sin)( ω= and )sin(sin)( 1122 ttAtAtx δωωω +== ,

⎟⎠⎞

⎜⎝⎛+=+=

2cos).

21(sin2)()()( 1

1121tttAtxtxtx δω

δωω

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:

⎟⎟⎠

⎞⎜⎜⎝

⎛=

0

log20XXdB

where 0X 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. )()( tftf −=− 17. half 18. harmonic 19. 104.72 rad/s 20. 0.01 s

Question 1.4: 1. b 2. a 3. c 4. a 5. c 6. b 7. c 8. b

9. a 10. a 11. b 12. c 13. a 14. b 15. a 16. a

Question 1.5:

1. ⎯ b 2. ⎯ c 3. ⎯ e 4. ⎯ d 5. ⎯ a

Question 1.6: 1. ⎯ c 2. ⎯ e 3. ⎯ a 4. ⎯ d 5. ⎯ b

Question 1.7:

1. ⎯ b 2. ⎯ c 3. ⎯ e 4. ⎯ d 5. ⎯ f 6. ⎯ h 7. ⎯ g 8. ⎯ 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

1ln1

mxx

mδ

The damping ratio (ς ) can be found as

22)2( δπ

δς+

= .

2. No. 3. Mass moment of inertia, torsional damping constant, torsional stiffness, and

angular displacement, respectively.

4. Since the natural frequency is given by mk

n =ω , a decrease in m will cause the

natural frequency to increase.

5. Since the natural period is given by km

n

πωπτ 22== , 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 stress-strain 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

example, in the case of hysteresis damping, ωhceq = , indicating that the

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.

18. st

ngδ

ω =

where stδ is the static deflection under self-weight and g is the acceleration due to gravity.

19. Mechanical clock, Wind turbine.

20. Damping ratio(ς ):kmc

cc

c 2==ς

Logarithmic decrement (δ ):

dmcωπ

ς

πςδ =−

=21

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 = ihk + = )1( βik +

where k = stiffness, 1−=i , h = hysteresis damping constant, and ==khβ a

measure of damping. 23. Hysteresis damping constant (h) is the proportionality constant that relates the

damping coefficient (c) and the frequency (ω ) as

ωhc = .

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

tmc

extx−

= 0)( 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. T 2. T 3. T 4. F 5. F 6. F 7. T 8. 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. Nμ 7. loss 8. rigid 9. critical 10. amplitude 11. natural 12. logarithmic 13. nd ωςω 21−= 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 ⎯ c 2 ⎯ a 3 ⎯ d 4 ⎯ e 5 ⎯ b

Question 3.1:

1. If the applied force is tFtF ωcos)( 0= , the steady-state vibration response will have the following characteristics:

tXtx p ωcos)( =

Amplitude = X = 20

ωmkF

−

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 tFtF ωcos)( 0= is given by

tmkF

tx p ωω

cos)( 20 ⎟

⎠⎞

⎜⎝⎛

−=

For a constant force 0F , 0=ω and hence === stp kF

tx δ0)( constant static

deflection of the mass due to 0F . 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,

forceconstant under massofdeflectionamplitude Maximumfactor ion Magnificat =

or 2

1

1

⎟⎟⎠

⎞⎜⎜⎝

⎛−

=

n

st

X

ωωδ

4. If 1<st

Xδ

, then 1>nωω or nωω > .

5. In the neighborhood of resonance, the amplitude (X) is given by

ςδ2

stX =

and the phase angle by

2)(tan 1 πφ =∞= − .

6. Phase corresponding to peak amplitude is given by

⎟⎠⎞

⎜⎝⎛−

= −2

1

12tan

rrςφ with

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ −=

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−−−

=−= −−

ςς

ςςς

ς2

12

212 12

tan)1(1

12tan1r

For 1<ς (underdamped system), )(tan 1 w−=φ where 2<w . Hence 090<φ .

7. Because it avoids the amplitude from reaching a value of infinity. 8. Forced equation of motion:

tFtFxkxcxm ωcos)( 0

...==++

Vector representation:

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.

kX

Xcω

0F

Xm 2ω

tω

φ

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

st

X

ωωδ

=

, is called

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, )1( βik + , 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 )1( <<rr , 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 )1( >>rr ,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 2<r . 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 1≠nωω .

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, 2ωem , due to the rotating unbalance.

20. Frequency of response will be ω . The response will be harmonic. 21. Peak amplitude ( pX ) occurs when X is maximum. Resonance amplitude ( rX )

occurs when 1=r . For underdamped systems, rp XX > . 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 ωi 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 0=t , the initial conditions can be

taken as 0)0( ==tx ,m

Ftx ~)0( ==

⋅

5. Equation of motion of a system subjected to base excitation )(ty is given by .....ymkzzczm −=++ where yxz −= .

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. ∫∞

−==0

)()()( dttxetxLsx st

10. Generalized impedance ( )(sZ ): kcsmssZ ++= 2)(

Admittance ( )(sY ):

kcsmssZsY

++== 2

1)(

1)(

11. Step function and linear function. 12. If the forcing function is neither periodic nor harmonic, there will be no

resonance conditions.

13. If the period is T, the first harmonic frequency is given by Tπω 2

1 = .

14. thn frequency ( nω ) is given by L,3,2;. 1 == nnn ωω 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.

18. (i) ⎩⎨⎧

≠=∞

=000

)(xatxat

xδ

(ii) 1)( =∫∞

∞−

dxxδ

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 ix to cause a unit displacement along jx is same as the force along jx to cause a unit displacement along ix .

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 ix , it causes displacement along

jx as well. Dynamic coupling: If a dynamic force is applied along ix , it causes displacement along jx 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 )]([ ωiZ is defined by 0)]([ FXiZ =ω where,

rsrsrsrs kcimiZ ++−= ωωω 2)( 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 ωis = 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. a 2. b 3. c 4. a 5. c 6. a 7. a 8. 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, ija , is defined as the deflection at point i due to a unit load at point j. The stiffness influence coefficient, ijk , 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

1][][ −= ak and 1][][ −= ka . 2. Equations of motion:

Fxkxcxm =++ ][][][...

or

Fxaxcxm =++ −1...

][][][ 3. Elastic potential energy (strain energy):

xkxVT

][21

=

Kinetic energy: ..

][21 xmxT

T

=

4. The generalized mass matrix will have nonzero non-diagonal terms as:

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

nnnn

n

n

mmm

mmmmmm

m

L

M

L

L

21

22221

11211

][

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

][21 xmxT

T

= , cannot be negative or zero for nonzero velocity vector .

x .

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:

njQqV

qT

q

Tdtd n

jjj

j

,,2,1;)(. L==

∂∂

+∂∂

−⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

∂

∂

where t

qq j

j ∂

∂=

.= generalized velocity, )(n

jQ = nonconservative generalized

coordinate jq , T = kinetic energy , V = strain energy, and t = time. 9. Matrix eigenvalue problem:

XkXm ][][2 =ω

where 2ω is the eigenvalue and X is the eigenvector. 10. Mode shape is same as the eigenvector X in the eigenvalue problem,

XkXm ][][2 =ω (E.1)

The eigenvector )(i

X corresponding to the eigenvalue 2iω can be computed by

substituting 2iω in Eq. (E.1) and solving the resulting linear algebraic equations to

find )(i

X : 0]][][[

)(2 =−i

i Xkmω 11. n distinct natural frequencies.

12. Dynamic matrix = ][][][ 1 mkD −= . 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:

XXIXD λλ == ][][ 13. Frequency equation:

0][][2 =+− kmω or

0][][ =− DIλ

where 2ωλ = and ][][][ 1 mkD −= .

14. Orthogonality of normal modes )(i

X implies

0][)()(=

iTjXmX for ji ≠

and 0][)()(=

iTjXkX for ji ≠

Orthogonal modal vectors implies

niXmXiTi

,,2,1;1][)()(

L== ; and

niXkX i

iTi,,2,1;][ 2)()(

L== ω 15. Any set of n linearly independent vectors in an n-dimensional space is called a

basis in that space.

16. If the eigenvectors ,,,2,1,)(

niXi

L= are used as the basis, any vector x in the

n-dimensional space can be expressed as a linear combination of )(i

X as

∑=

=n

i

i

i Xcx1

)( (E.1)

where the constants ic can be determined as

xmXcTi

i ][)(

= ; ni ,,2,1 L= (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 ). The rigid body mode,

)0(X , can be found by solving the equations:

0][)0(=Xk

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

.;).()(1

)(nrXtqtx

r

i

i

i <= ∑=

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

][21 xcxR

T

=

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: ][][][ kmc βα +=

where α and β are constants. Modal damping ratio ( iς ): is defined by

iii ωςβωα 22 =+ where iω is the thi natural frequency of the system. Modal participation factor ( iq ): is the thi generalized coordinate used in the expansion theorem:

∑=

=n

i

i

i Xtqtx1

)().()(

23. When the system is damped, and damping is not proportional damping, that is, when ][][][ kmc βα +≠

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. ji, 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 ):

XmX

XkXR T

T

][

][2 == ω

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 1X 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

eigenvector and the constant used in the normalization process denotes 21

1ω

where

1ω is the fundamental eigenvalue. 8. Yes, provided we use 1][ −D 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 ][ iD 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

cossin

sincos

010001

][ 1

OM

O

O

L

L

θθ

θθnn

R

( E.1 ) where

⎟⎟⎠

⎞⎜⎜⎝

⎛

−=

jjii

ij

ddd2

2tan θ ( E.2 )

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

column i

column j

row i

row j

By carrying out the computations as ][][][][ 11 RDRD T= ( E.3 )

the off-diagonal components ijd and jid of ][D will be reduced to zero. By carrying out the computations according to Eq. ( E.3 ) using different rotation matrices L],[],[ 32 RR , 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 L]][[ 21 RR denote the eigenvectors of the matrix ][D .

12. Standard eigenvalue problem: [ ] [ ][ ] 0=− XID λ

where λ is the eigenvalue and X is the eigenvector. 13. The general eigenvalue problem

XkXk ][][ 2ω= ( E.1 ) can be converted to a standard eigenvalue problem as

XXD λ=][ ( E.2 )

where ⎟⎠⎞

⎜⎝⎛= 2

1ω

λ and ].[][][ 1 mkD −=

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

][][][ UUk T= ( E.3 ) where ][U is an upper triangular matrix, and Eq. ( E.1 ) can be converted to a standard eigenvalue problem as

YYD λ=][ ( E.4 ) where 11 ][][)][(][ −−= UmUD T ( E.5 )

and XUY ][= or YUX 1][ −= . The matrix ][D in Eq. ( E.4 ) will be symmetric.

14. If ][][ Uuij = is an upper triangular matrix, its inverse, 1][][ −= Uaij , can be determined as follows:

][][][ 1 IUU =− Equating the corresponding elements on both sides of Eq. ( E.1 ), we can determine the elements of 1][ −U .

Question 7.2:

1. F 2. T 3. F 4. T 5. T 6. T 7. 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. a 2. b 3. b 4. a 5. c 6. 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

2

2

22

tw

xwc

∂∂

=∂∂

where ρPc = .

Traveling – wave solution: )()(),( 21 ctxwctxwtxw ++−=

where 1w and 2w are arbitrary functions of tcx − and tcx + , respectively, which are determined from the initial conditions.

5. Wave velocity gives the velocity with which the waves )(1 tcxw − and )(2 tcxw + 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):

0,0 2

2

=∂∂

=xwEIw

(ii) Timoshenko beam theory (w = transverse displacement, Φ = bending

slope): 0,0 =∂Φ∂

=x

EIw

7. Possible boundary conditions at the ends of a string: (i) Fixed end: 0=w (ii) String connected to a pin that can move in a perpendicular direction:

0=∂∂

xwP ( P = tension in string )

(iii) Free end : 0=∂∂

xw

(iv) Elastically supported by a spring of stiffness ~k :

wkxwP

~−=

∂∂

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, criP , 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 2

1ω , attains minimum at the exact fundamental mode. Hence any other mode, used as an approximation, yields a larger value of 2

1ω 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 shape.

Rayleigh’s quotient for a beam, ( ) ⎪

⎪

⎭

⎪⎪

⎬

⎫

⎪⎪

⎩

⎪⎪

⎨

⎧⎟⎟⎠

⎞⎜⎜⎝

⎛

==

∫

∫l

l

dxxWA

dxdx

xWdEIR

0

2

0

2

2

2

2

)(

)(

)(ρ

ωω

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. b 2. a 3. c 4. a 5. c 6. b 7. 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 LW 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 ω .

(ii) Remove LW and add a known weight RW in the right plane at a known angular position and measure the resulting vibration while the rotor is running at speed ω . (iii) Using relevant vector equations, find the unbalance vectors LU and RU in the left and right planes, respectively. (iv) Balance the rotor by adding equal and opposite balancing weight LB and RB as LL UB −= and RR UB −= .

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

mk

n =ω

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 ( cm ) can be made zero

by counter-balancing the crank. However, the equivalent reciprocating mass ( pm ) cannot be balanced. For a multi-cylinder engine, the axial displacements ( il ) and angular orientations ( iα ) of cylinder i from those of the first cylinder ( Ni ,,3,2 L= ) 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 ( 2ω ) decreases

with a soft spring. The force transmissibility becomes smaller with a reduced value of 2ω .

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 ( )( ) ν21

//

+≈ΔΔ

=LLRRK

where R = initial resistance, RΔ = change in resistance, L = initial length of wire (strain gage) , LΔ = 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 53 ≤≤nω

ω where nω 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 phase-shift 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 )( ωiH versus frequency ω . 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:

∫∞

∞−

−= dxxfxxk )()(1 44σ

where )(xf is the probability density function of the instantaneous amplitude, )(tx , at time t, x is the mean value, and σ is the standard deviation of )(tx .

Cepstrum, )(τc , is defined as the inverse Fourier transform of the logarithm of the power spectrum:

)}(log{)( 1 ωτ XSFc −= where the power spectrum, )(ωXS , of the time signal )(tx is given by [ ]2)}({)( txFS X =ω with { }F denoting the Fourier transform of { }:

{ } dtetxT

txFT

T

ti∫−

=2/

2/

)(1)( ω

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. transducer 2. charge 3. spring-mass-damper 4. accelerometer 5. accelerometers 6. velometer 7. resonance 8. cantilever 9. contact

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 1+ix and 1−ix can be expressed about the grid point i as

L&&&&&& ++++=+ iiiii xhxhxhxx62

32

1 (E.1)

L&&&&&& +−+−=− iiiii xhxhxhxx62

32

1 (E.2)

where )( ii ttxx == and .1 ttth ii Δ=−= + By taking two terms only and substracting Eq.(E.2) from (E.1) , we get the central difference formula for the first derivative as

)(21

11 −+ −== iiit

i xxhdt

dxx& (E.3)

By taking three terms only and adding Eqs.(E.1) and (E.2), we get the central difference formula for the second derivative as

)2(11122

2

−+ +−== iii

iti xxx

hdtxdx&& (E.4)

3. A conditionally stable method is a numerical method that requires the use of the time step ( tΔ ) smaller than a critical time step ( critΔ ). If tΔ is chosen to be larger than critΔ , 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

0=dxdφ at a boundary (grid point i), the use of central difference approximation

gives

02

12 =Φ−Φ

= −

hdxd

i

φ

where )( 1111 hxxxx −=Δ−=Φ=Φ −− 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, it and θ+it .

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

)()( 11

iiii

ii xx

tttt

xtx &&&&&&&& −⎟⎟⎠

⎞⎜⎜⎝

⎛−−

+= ++

( E.1 )

where )( ii txx &&&& = and 1+≤≤ ii ttt . By integrating Eq. ( E.1 ) once and twice, we 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 1+it to find the solution 1+ix , it is called an implicit integration method. If a numerical integration method requires the values of the response at previous time steps it and 1−it , including the use of the equilibrium equation at time it , to find the value of the response at 1+it , the method is known as an explicit integration method.

10. No, not directly. Question 11.2:

1. F 2. T 3. T 4. F 5. T 6. T 7. F 8. T

Question 11.3:

1. closed 2. derivatives 3. three 4. mesh/grid 5. Taylor’s 6. conditionally 7. unstable 8. 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, ),( yxNi , is a polynomial in x and y defined such that its value is 1 at node i (with coordinates ix and iy ) and 0 at all other nodes j (with coordinates jx and jy ) 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. a 2. a 3. b 4. b 5. a 6. b 7. c 8. 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 ( nω ) are related to

the forcing frequency (ω ) as

L,4,3,2; == nnnωω

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

dxdy by giving it a constant value 1c and solve the equation, 1),( cyx

dxdy

== φ for

the trajectory. The curve 0),( 1 =− cyxφ thus represents an isocline in the phase plane. We plot several isoclines L,, 21 hh by giving different values L,, 21 cc to

the slope ),( yxdxdy φ= . These isoclines are then used to plot the trajectory passing

through any specific point in the phase plane. 10. For a nonlinear spring with deformation x , the restoring force can be expressed

as )(xf . If =dxdy constant, the spring is linear. If

dxdy is a strictly increasing

function of x , the spring is called a hard spring. If dxdy is a strictly decreasing

function of x , the spring is called a soft spring. 11. Subharmonic oscillations are oscillations whose frequencies ( nω ) are related to

the forcing frequency (ω ) as

L,4,3,2; == nnnωω

Superharmonic oscillations are oscillations whose frequencies ( nω ) are related to the forcing frequency (ω ) as

L,4,3,2; == nnn ωω 12. In the solution of the pendulum equation of the form

0320 =++ xxx αω , ( E.1 )

If a two-term solution is assumed as )()()( 10 txtxtx α+= ( E.2 )

the resulting expressions of )(0 tx and )(1 tx make )(tx 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

)(1 tx 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 , )(xpX , of a random variable X is defined by =dxxpX ).( Probability of realizing the value of X in the interval x and x + dx.

The probability distribution function, )(xPX , of a random variable X is defined as )(xPX = Probability of realizing the value of X less than or equal to x.

The probability density and distribution functions are related as

dxxdPxp X

X)()( =

3. Mean value of a random variable X:

∫∞

∞−

⋅== dxxpxX XX )(μ

Variance of a random variable X:

∫∞

∞−

⋅−= dxxpx XX )()( 22 μσ

4. The distribution involving two random variables is called a bivariate distribution. 5. Covariance between the random variables X and Y, denoted XYσ , is defined as

follows: [ ]

∫ ∫∞

∞−

∞

∞−

−−=

−−=

dxdyyxpyx

yxE

YXYX

YXXY

),())((

))((

,μμ

μμσ

where ),(, yxp YX is the joint density function of X and Y. 6. Correlation coefficient of X and Y ( XYρ ):

YX

XYXY σσ

σρ =

where Xσ and Yσ are the standard deviations of X and Y, respectively, and XYσ is the covariance of X and Y .

7. 11 ≤≤− XYρ 8. Marginal density function is the density function of one random variable obtained

from a joint density function. For example, the marginal density function, )(xpX , can be obtained from ),( yxpXY as

∫∞

∞−

= dyyxpxp XYX ),()(

9. The mathematical expectation of 21xx is called the autocorrelation function, denoted as :),( 21 ttR

∫ ∫∞

∞−

∞

∞−

== 2121212121 ),(][),( dxdxxxpxxxxEttR

where 1x and 2x denote the values of the random process )(tx at 1t and 2t , respectively, and ),( 21 xxp is the joint density function of 1x and 2x .

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, )(τR , is bounded as follows:

2222 )( μστμσ +≤≤+− R where μ 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 )(tx is an ergodic random process, and )(txi denotes a typical sample function of duration T, the averages of )(tx can be computed by averages with respect to time along )(txi . Such averages are called temporal averages. For example, the temporal averages of )(tx , denoted as )(tx , is defined as

[ ] ∫−

∞→==

2/

2/

)( )(1lim)(T

T

i

Tdttx

TxEtx

14. A Gaussian random process ( ))(tx is one for which the probability density function is given by

2

21

21)(

⎟⎟⎠

⎞⎜⎜⎝

⎛ −

= x

xx

x

exp σ

σπ

where x and xσ 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, )(tx , states that the mean square value of )(tx is equal to the sum of the squares of the absolute values of the Fourier coefficients. Thus if

∑∞

−∞=

=n

tinnectx 0)( ω

The Parseval’s formula states that

∑∞

−∞=

=n

nctx 22 )(

16. Power spectral density function, )(ωS : For a stationary random process, )(ωS is defined as the Fourier transform of

πτ

2)(R where )(τR is the autocorrelation function.

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 )(ωS 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 )(ωS 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.

17. ωωτ dSRxE ∫∞

∞−

=== )()0(][ 2

If the mean is zero, the variance of the stationary random process, )(tx , is given by

ωωσ dSRx ∫∞

∞−

== )()0(2

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, )(tx , using Duhamel integral:

ττωτω

τττ τςω dteFm

dtgFtx dt

t

d

tn )(sin)(1)()()( )(

00

−=−= −−∫∫

where )(τF = force applied at time τ and )( τ−tg 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 tietx ω=)(

~, the response can be expressed as

tieHty ωω)()(~

=

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

{ }21

222 )2()1(

1)(rr

Hς

ω+−

=

where n

rωω

= and ς is the viscous damping ratio.

21. The power spectral densities of the input, )(ωxS , and the output, )(ωyS , are related by the complex frequency response function, )(ωH , as

)()()( 2 ωωω SHS y = 22. Wiener-Khintchine relations:

ττπ

ω τω deRS i−∞

∞−∫= )(

21)(

ωωτ τω deSR i∫∞

∞−

= )()(

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. a 2. c 3. a 4. c 5. a 6. b 7. a 8. b

9. b 10. b 11. c 12. c

Question 14.5:

1 ⎯ c 2 ⎯ f 3 ⎯ a 4 ⎯ e 5 ⎯ b 6 ⎯ d