Machine Design Databook Episode 2 part 10 ppt

I mass moment of inertia of rotating disk or rotor, N s2m lbf s2in J polar second moment of inertia, m4or cm4in4 k spring stiffness or constant, kN/m lbf/in ke equivalent spring constant,

12 Lingaiah, K and B R Narayana Iyengar, Machine Design Data Handbook, Vol I (SI and Customary MetricUnits), Suma Publishers, Bangalore, India, 1973.

13 Lingaiah, K., Machine Design Data Handbook, Vol II (SI and Customary Metric Units), Suma Publishers,Bangalore, India, 1986

14 Bureau of Indian Standards

15 Albert, C D., Machine Design Drawing Room Problems, John Wiley and Sons, New York, 1949

16 V-Belts and Pulleys, SAE J 636C, SAE Handbook, Part I, Society of Automotive Engineers, Inc., 1997

17 SI Synchronous Belts and Pulleys, SAE J 1278 Oct.80, SAE Handbook, Part I, Society of AutomotiveEngineers, Inc., 1997

18 Synchronous Belts and Pulleys, SAE J 1313 Oct.80, SAE Handbook, Part I, Society of Automotive Engineers,Inc., 1997

19 Wolfram Funk, ‘Belt Drives,’ J E Shigley and C R Mischke, Standard Handbook of Machine Design, 2ndedition, McGraw-Hill Publishing Company, New York, 1996

Cc critical viscous damping, N s/m (lbf s/in)

Ct coefficient of torsional viscous damping, N m s/rad

G modulus of rigidity, GPa (Mpsi)

h thickness of plate, m (in)

i integer (¼ 0, 1, 2, 3, )

I mass moment of inertia of rotating disk or rotor, N s2m

(lbf s2in)

J polar second moment of inertia, m4or cm4(in4)

k spring stiffness or constant, kN/m (lbf/in)

ke equivalent spring constant, kN/m (lbf/in)

kt torsional or spring stiffness of shaft, J/rad or N m/rad (lbf in/rad)

K kinetic energy, J (lbf/in)

me equivalent mass, kg (lb)

Mt torque, N m (lbf ft)

p circular frequency, rad/s

q damped circular frequencyð¼pffiffiffiffiffiffiffiffiffiffiffiffiffi1
2

Þ

R¼ 1
TR percent reduction in transmissibility

R2¼ D2=2 radius of the coil, m (in)

U vibrational energy, J or N m (lbf in)

potential energy, J (lbf in)

x1, x2 successive amplitudes, m (in)

xo maximum displacement, m (in)

_xx linear velocity, m/s (ft/min)

€xx linear acceleration, m/s2(ft/s2)

Xst static deflection of the system, m (in)

y deflection of the disk center from its rotational axis, m or mm (in)

 weight density, kN/m3(lbf/in3)

 ¼ C

Cc damping factor

deflection, m (in)

st static deflection, m (in)

 mass density, kg/m3(lb/in3)

 normal stress, MPa (psi)

shear stress, MPa (psi)

period, s

angular deflections, rad (deg)

angular velocity, rad/s

€ angular acceleration, rad/s2

! forced circular frequency, rad/s

SIMPLE HARMONIC MOTION (Fig 22-1)

The displacement of point P on diameter RS (Fig 22-1)

The wavelength

The periodic time

The frequency

The maximum velocity of point Q

The maximum acceleration of point Q

Single-degree-of-freedom system without

damping and without external force (Fig 22-2)

Linear system

The equation of motion

The general solution for displacement

The equation for displacement of mass for the initial

condition x¼ xoand _xx ¼ 0 at t ¼ 0

The natural circular frequency

The natural frequency of the vibration

The natural frequency in terms of static deflectionst

FIGURE 22-1 Simple harmonic motion.

r

¼

ffiffiffiffiffig

st

1=2

 0:5

1

st

1=2

ð22-13aÞwherestin m and fnin Hz

FIGURE 22-3 Static deflection (  st ) vs natural frequency.

(Courtesy of P H Black and O E Adams, Jr., Machine

Design, McGraw-Hill, New York, 1955.)

The plot of natural frequency vs static deflection

st

1=2

 15:76

1

st

1=2

SI ð22-13bÞwherestin mm and fnin Hz

fn¼52:67

1

st

1=2

 0:9

1

st

1=2USCS ð22-13cÞwherestin ft and fnin Hz

fn¼19:67

2

1

r

p¼

ffiffiffigl

Potential energy

Maximum kinetic energy is equal to maximum

poten-tial energy according to conservation of energy

FIGURE 22-4 Simple pendulum.

Torsional system (Fig 22-5)

The equation of motion of torsional system (Fig 22-5)

with torsional damping under external torque Mtsin pt

The equation of motion of torsional system without

considering the damping and external force on the

rotor

The equation for angular displacement

The angular displacement for o and _

t¼ 0

The natural circular frequency

The natural circular frequency taking into account

the shaft mass

The natural frequency

The expression for torsional stiffness

Single-degree-freedom system with damping

and without external force (Fig 22-6)

The equation of motion

The general solution for displacement

FIGURE 22-6 Single-degree-of-freedom

spring-mass-dashpot system.

For the damped oscillation of the

single-degree-freedom system with time for damping factor < 1

ffiffiffiffiffiffiffiffi

 2
1

pÞpnt ð22-30Þ

where C1, C2, and A are arbitrary constants ofintegration (They can be found from initialconditions.)

s1;2¼
C2m

C2m

2

km



¼ ffiffiffiffiffiffiffiffiffiffiffiffiffi1
2q

pn¼

k

m
c24m2

1=2

ð22-33aÞ

the exciting forceRefer to Figs 22-7 and 22-8

FIGURE 22-7 Damped motion  < 1:0 FIGURE 22-8 Logarithmic decrement (Reproduced from

Marks’ Standard Handbook for Mechanical Engineers, 8th edition, McGraw-Hill, New York, 1978.)

LOGARITHMIC DECREMENT (Fig 22-8)

The equation for logarithmic decrement

EQUIVALENT SPRING CONSTANTS

For spring constants of different types of springs,

beams, and plates

FIGURE 22-9 Springs in series and parallel.

Single-degree-of-freedom system with

damping and external force (Fig 22-10)

The equation of motion

Formula for spring

Linear Spring Stiffness or Constants [Load per mm (in) Deflection]

Helical spring subjected to tension with i number of turns

k ¼ Gd

4

64iR 3

(22-41)

Cantilever beam subjected to transverse load at the free

Simply supported beam with concentrated load at the

Simply supported beam subjected to a concentrated load

a 2 b 2

(22-46)

Beam fixed at both ends subjected to a concentrated load

l 3

(22-47)

Beam fixed at one end and simply supported at another

end subjected to concentrated load at the center k¼768EI7l3

(22-48)

Circular plate clamped along the circumferential edge

subjected to concentrated load at the center whose

flexural rigidity is D ¼ Eh 3 =12ð1

2 Þ, thickness h

and Poisson ratio

k ¼16D

R 2

(22-49)

Circular plate simply supported along the

circumferential edge with concentrated load at the

String fixed at both ends subjected to tension T k¼4Tl String tension T (22-51)

Torsional or Rotational Spring Stiffness or Constants (Load per Radian Rotation) Spiral spring whose total length is l and moment of

Helical spring with i turns subjected to twist whose wire diameter is d, the coil diameter is D

The complete solution for the displacement

The steady-state solution for amplitude of vibration

The phase angle

The magnification factor

The plot of magnification factor ðXo=XstÞ vs

kt¼64iDEd4

(22-53)TABLE 22-1

Spring constants or spring stiffness of various springs, beams, and plates (Cont.)

Formula for spring

Bending of helical spring of i number of turns

Twisting of a hollow circular shaft with length l,

whose outside diameter is Do, and inside

FIGURE 22-11 Phase angle nÞ.

The amplitude at resonance (i.e for!=pn¼ 1)

UNBALANCE DUE TO ROTATING MASS

(Fig 22-13)

The equation of motion

The steady-state solution for displacement

FIGURE 22-12 Magnification factor ðX o =X st Þ vs quency ratio ð!=p n Þ.

The complete solution for the displacement

Nondimensional form of expression for Eq (22-65b)

The phase angle

For a schematic representation of Eqs (22-67) and

(22-68) or harmonically disturbing force due to

FIGURE 22-14 MX =me vs frequency ratio ð!=p n Þ.

WHIPPING OF ROTATING SHAFT

(Fig 22-16)

The equation of motion of shaft due to unbalanced

mass

The solution

The displacement of the center of the disk from the

line joining the centers of bearings

The phase angle

FIGURE 22-16 Whipping of shaft (Reproduced from

Marks’ Standard Handbook for Mechanical Engineers, 8th

edition, McGraw-Hill Book Company, New York, 1978.)

EXCITATION OF A SYSTEM BY MOTION

OF SUPPORT (Fig 22-17)

The equation of motion

The absolute value of the amplitude ratio of x and y

m€xxcþ c _xxcþ kxc¼ me!2cos!t ð22-69aÞ

m€yycþ c€yycþ kyc¼ me!2sin!t ð22-69bÞwhere xcand ycare coordinates of position of center

of shaft with respect to x and y coordinates

The phase angle

The plot of Eq (22-55) for motion due to support

INSTRUMENT FOR VIBRATION

MEASURING (Fig 22-18)

The equation of motion

The steady-state solution for relative displacement Z

The phase angle

The plot of absolute value ofjZ=Yj vs frequency ratio

ð!=pn

ð!=pnÞ

FIGURE 22-18 Instrument for vibration measuring.

(Reproduced from Marks’ Standard Handbook for

Mechan-ical Engineers, 8th edition, McGraw-Hill, New York, 1978.)

ISOLATION OF VIBRATION (Fig 22-19)

The force transmitted through the springs and

damper

1 2ð!=pnÞ3f1
ð!=pnÞ2g2þ ð2!=pnÞ2

ð22-75ÞRefer to Fig 22-20 forjX=Yj vs !=pn

The curves are similar

m€zz þ c _zz þ kz ¼
m€yy ¼ mY!2sin!t ð22-76Þ

Refer to Figs 22-14 and 22-15

The curves forjZ=Yj vs !=pnand nare tical

iden-FIGURE 22-19 External force transmitted to foundation through damper and springs (Reproduced from Marks’ Standard Handbook for Mechanical Engineers, 8th edition, McGraw-Hill, New York, 1978.)

Comparison of Eqs (22-81) and (22-85) indicates that

the plot of F =Fois identical tojX=Yj

Transmissibility when damping is negligible

The transmissibility in terms of static deflectionst

The frequency from Eq (22-83)

FIGURE 22-20 Transmissibility (TR) vs frequency ratio

½f1
ð!=pnÞ2g2þ ð2!=pnÞ21=2 ð22-81ÞRefer to Fig 22-20 for TRandjX=Yj

TRþ 1



¼ 0:5 1st

The percent reduction in the transmissibility is defined

The amplitude ratio

DYNAMIC VIBRATION ABSORBER

(Fig 22-23)

Equations of motion

The solution of the forced vibration of the absorber

will be of the form

FIGURE 22-21 Static deflection (  st ) vs disturbing frequency for various

percent reduction in transmissibility (TR) for  ¼ 0 (Courtesy of F S Tes,

I E Morse, and R T Hinkle, Mechanical Vibration—Theory and Applications,

CBS Publishers and Distributors, New Delhi, India, 1983.)

FIGURE 22-22 Undamped two-degree-of freedom system.

The ratio of amplitudes a1and a2to the static

deflec-tion of the main system xst

If the main system is in resonance, then considering

The natural frequencies

The mass equivalent for the absorber

TORSIONAL VIBRATING SYSTEMS

Two-rotor system (Fig 22-24)

The torque on rotor A

The total torque on two rotors

The angular displacement or angle of twist of rotor B

p2



1þk

K
!2

p2 m



kK



kK

ð22-90bÞ

where

xst¼ Fo=K ¼ static deflection of main system

p2¼ K=m ¼ natural circular frequency of absorber

p2m¼ k=M ¼ natural circular frequency of main system

x2

½1
ð!=paÞ2½1 þ Rm
ð!=paÞ2
Rm

sin!tð22-91bÞ







RmþR2m4

The frequency equation

The natural circular frequency

The natural frequency

The amplitude ratio

The relation between Ia, Ib, la, and lb

The distance of node point from left end of rotor A

Two rotors connected by shaft of varying

diameters

The length of torsionally equivalent shaft of diameter

dwhose varying diameters are d1, d2, and d3

Three-rotor torsional system (Fig 22-25)

The algebraic sum of the inertia torques of rotors A,

FIGURE 22-25 Three-rotor system.

The frequency equation

The amplitude ratio

The relation between Ia, Ic, la, and lc

The relation between Ia, Ib, la, and lc

Frequency can also be found from Eqs (22-108) and

(22-109)

For collection of mechanical vibration formulas to

calculate natural frequencies

For analogy between different wave phenomena

For analogy between mechanical and electrical

Refer to Table 22-3

Refer to Table 22-4

TABLE 22-2

A collection of formulas

Natural Frequencies of Simple Systems End mass M, spring mass m, spring

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k

M þ 0:23m

r

(22-113)

Simply supported beam central mass M;

beam mass m; stiffness by formula (22-95) pn ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k

u t

Longitudinal vibration of beam clamped

or free at both ends; n ¼ number of half waves along length

pn¼ n

ffiffiffiffiffiffiffiffiffi AE

1 l 2

s

n ¼ 1; 2; 3;

(22-121)

Particular Formula Equation

Water column in rigid pipe closed (or

Epipe¼ elastic modulus of pipe, MPa

D, t ¼ pipe diameter and wall thickness, same units For water columns in nonrigid pipes fnonrigid

frigid ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

1 þ300;000D

tE pipe

Epipe¼ elastic modulus of pipe, psi

D, t ¼ pipe diameter and wall thickness, same units Torsional vibration of beams Same as (22-117) and (22-118); replace

tensional stiffness AE by torsional stiffness GIp; replace 1 by the moment

of inertia per unit length i1¼ I bar =l Uniform Beams (Transverse or Bending Vibrations)

The same general formula holds for all the following cases,

pn¼ a n

ffiffiffiffiffiffiffiffiffi EI

TABLE 22-2

A collection of formulas (Cont.)

‘‘Free-free’’ beam or floating ship a1¼ 22:0

pn¼1r

ffiffiffiffiffiffi Eg

1 þ n 2

p

ffiffiffiffiffiffiffiffiffi EI

Membrane of any shape of area A roughly of equal dimensions in all directions, fundamental mode:

pn¼ const

ffiffiffiffiffiffiffiffiffi T

1 r 4

For free edges, 2 perpendicular nodal diameters a ¼ 5:25

For free edges, one nodal circle, no diameters a ¼ 9:07

Free edges, clamped at center, umbrella mode a ¼ 3:75

Rectangular plate, all edges simply supported, dimensions l1and l2:

p n ¼ 2

m2

l 2 þnl22

 ffiffiffiffiffiD 1

Source: Formulas (Eqs.) (7-110) to (7-133) extracted from J P Den Hartog, Mechanical Vibrations, McGraw-Hill Book Company, New York, 1962.

TABLE 22-3

Analogy between different wave phenomena

Phenomenon

Mass per unit

G p

E p

1 LC r

Ratio of force to

ffiffiffiffiffiffi A pT

energy per sec

ffiffiffiffiffiffi pT A

Source: Courtesy G W van Santen, Introduction to Study of Mechanical Vibration, 3rd edition, Philips Technical Library, 1961.

Key: c ¼ capacitance; e ¼ voltage; i ¼ current, A; I ¼ intensity, W/m 2 ; J ¼ polar moment of inertia, m 4 or cm 4 ; k ¼ c p =c v ¼ ratio of specific heats; L ¼ inductance, H; n ¼ any integer ¼ 1, 2, 3, 4, ; p ¼ pressure of gas, sound pressure, MPa; p n ¼ average pressure of gas, MPa; R ¼ resistance, ; T ¼ tension; T ? ¼ component of tension T which returns the string to the position of equilibrium, kN;  ¼ specific mass of the material of string, density of air, kg/m 3 ;  n ¼ average density of gas, kg/m 3 ;  ¼ normal stress, MPa; ¼ shear stress, MPa;  ¼ wavelength, m.

The meaning of other symbols in Table 7-3 are given under symbols at the beginning of this chapter.

Electrical system

Rectilinear system Torsional system Electrical network Electrical network

F ¼ m _

¼ m€xx,

Kinetic energy ¼ 1 m
2

i ¼ C _cc, q ¼ C€cc, Energy ¼ 1 C2 e¼ Ldidt¼ L€qq

F ¼ kx ¼ kð_xx dt

Potential energy ¼12Fk2

i ¼1L

ð

e dt ; q ¼eL Energy ¼ 1 Li2

e ¼1

Cq¼1C

ð

i dt Energy ¼ 1 Ce2

(b) Parallel connected electrical elements

(c) Series connected electrical elements Differential equation of motion Differential equation for

1 Den Hartog, J P., Mechanical Vibrations, McGraw-Hill Book Company, New York, 1962

2 Thomson, W T., Theory of Vibration with Applications, Prentice-Hall, Englewood Cliffs, New Jersey, 1981

3 Baumeister, T., ed., Marks’ Standard Handbook for Mechanical Engineers, 8th ed., McGraw-Hill BookCompany, New York, 1978

4 Black, P H., and O E Adams, Jr., Machine Design, McGraw-Hill Book Company, New York, 1955

5 Lingaiah, K., and B R Narayana Iyengar, Machine Design Data Handbook, Engineering College Operative Society, Bangalore, India, 1962

Co-6 Myklestad, N O., Fundamentals of Vibration Analysis, McGraw-Hill Book Company, New York, 195Co-6

7 Tse, F S., I E Morse, and R T Hinkle, Mechanical Vibration—Theory and Applications, CBS Publishers andDistributors, New Delhi, India, 1983

TABLE 22 -2< /p>

A collection of formulas (Cont.)

‘‘Free-free’’ beam or floating ship a1¼ 22 :0... l 2< /small>

s

n ¼ 1; 2; 3;

(22 - 121 )

p n ẳ 2< /small>

m2< /small>

l ỵnl2< /small>2< /sup>