154
FREE VIBRATIONS OF MULTI DEGREEOFFREEDOM SYSTEMS
whose displacement is described by
body diagrams of Fig. 518 leads to
X,.
[CHAP. 5
Application of the equations of equilibrium to the free
2: F = 0 = 1 
2: Mo = 0 = k(a" 
k(a"  ra,)
ra , )r  k(2
VIBRATION CONTROL
244
8.S
~f+
ii.
Mathcad
[CHAP. 8
A 100kg turbine operates at 2000 r/min. What percent isolation is achieved if the turbine
is mounted on four identical springs in parallel, each of stiffness 3 X 10' N/m?
The equivalent stiffness of the
(IIAP. 4)
FORCE D RESPONSE OF IDEGREEOFFREED OM
121
Ass uming the sys te m is at res t in eq uilibrium at ( = 0 and lak in g th e Laplace transform o f the
differential equation leads to
O(s)
= 36Fo
1
7m L (s + a)(s + w"')
Partial fraction decompositio
CHAP. 7]
Since the end at x
VIBRATIONS OF CONTINUOUS SYSTEMS
=0
211
is fixed,
u(O, I) = 0
The resultant of the normal stress at the right e nd of the bar must equal the force in the spring at
any insta nt; thus
EA ~ax (L, I) = ku(L, I)
= X(x)e'"' to the
268
FINITE ELEMENT METHOD
Element J:
U,
V
= 0, U , = V, .
[CHAP. 9
Hence in te rms of the global displacement vector,
lEA
l EAf
=2f V, , =2f V, V,
V,
I[~ ~ ~ ~][ ~]
26
'
26
(;.
.
,
I
0
.
VJ
0
VJ
0000
T=!pAf2U,=~pAf[U
VJ
0
0
V,
J[~ ~ ~ ~][~]
U
V'OOOO
J
U,
280
FIN ITE ELEMENT METHOD
[CHAP. 9
A threeelement model of the fixedfree beam of Fig. 915 leads to a 4degreeoffreedom
system. The global coordinates are illustrated in Fig. 915. The rela tions between the local and
global coordinates for each elem
292
[CHAP. 10
NONLINEA R SYSTEMS
(20 kg) (9.8 1
tl=m g
Also,
~)
1.96 x 10' m
I X ](t~
k,
m
Define
The chain rule yields
d
d dr
d
==w dt dr dt
dt
0
Equation (10.6) is rewritten using nondimensional variables as
2
2
d x
dx
l3.
W
mwo tldr+ cwo tldr +k, l
MATHe AD SAMPLES
346
Undamped Absorber Design
(Schaum's Mechanical Vibrations, Solved Problems 8.27 and 8.28, p. 254)
Statement
A machine of mass m is attached to a spring of stiffness k 1 . During operation the
machine is subject to a harmonic excitation
Chapter 11
Computer Applications
Vibration analysis often requires much mathematical analysis and co mputation. Digital
computa tion can be used in lieu of manual computation for many of the tedious tasks
performed in vibration analysis. Computer algebra
184
FORCED VIBRATIONS OF MULTI DEGREEOFFREEDOM SYSTEMS
[CHAP. 6
The magnitude o f the harmonic excitation provided by the rotating unbalance is
cycle 2;c rad )'
F;, = m oew' = (0.45 kgm) ( 200      = 7.11 X 10' N
s
cycl e
Th e force vector is
F=
CHAP. 3)
HARMONIC EXC ITATION OF ID EG REEOFFREE DOM SYSTEMS
75
Fo r r < I, requi ring M < 3.74 leads to
r
<
)1 3.~4
= 0.856
For r > I, requiring M < 3.74 leads to
r
1
+~=)1 +3~4 = 1.I 26
Thus the allowable ra nges of freq ue ncies are
w < O. 856w. =
298
[CHAP. 10
NONLINEAR SYSTEMS
y
E
= 1.000  01
.
~
= 0.10
A = 1.00
(c)
r= 1.05
Fig. 109 (Continued)
If the system develops a limit cycle, the total work done by the nonconservative forces over each
cycle is zero. Assume the system is nondimensionalized
322
COMPUTER APPLICATIONS
[CHAP. 11
200
150
50
4 000
4500
5000
5500
6000
lambda
Fig. 1120
Supplementary Problems
11.17 Use VIBES or another dedicated vibrations software package to develop the force spectrum for a
system with a damping ratio of 0.2 subj
196
6.16
sa.
~ (+
Mattlcad
FORCED VIBRATION S O F MULTIDEGREEOFFREEDOM SYSTEMS
ICHAP. 6
D e te rmine the timedepe nd e nt response of the system of Fig. 66 if th e system is at rest in
equilibrium a t 1=0 whe n an impulse of magnitud e 0.5 Ns is app
CHAP. 6] FORCED VIBRATIONS OF MULTIDEGREEOFFREEDOM SYSTEMS
x,(t)
1
1
193
1
= &p,(t)  V2;,p ,(t)  w;mp, (t)
I 2m .~ I.ff)
I
(3m tSIO m6m 3 u(t)
m
= 
6.14

+  S IO
Repeat Problem 6.13 if each coupli ng is modeled as a spring of stiffness k in para
Chapter 2
Free Vibrations of 1DegreeofFreedom Systems
2.1
DERIVATION OF DIFFERENTIAL EQUATIONS
All linear Idegreeoffreedom systems can be modeled using either the system of Fig. 21 or
the system of Fig. 22. The equivalent system method, or the ene
12
MECHANICAL SYSTEM ANALYSIS
[CH A P. 1
Using the resuits of Proble m 1.9, it is observed that the beam and the spring act as two springs in
parall el.
1.14
What is the equiyalent stiffness of the system o f Fig. 119 u sing the displacement of the
block
9.2
265
FINITE ELEMENT M ETHO D
CHA P. 9]
FORCED VIBRATIONS
If F (x, t) represents the timedependent extern al force applied to a continuo us syste m, then
the virtual wo rk done by the external force due to variations in the glo bal displacements is
c'J
114
[CHAP. 4
FORCED RESPONSE OF IDEGR EEOFFREEDOM
Substituting into the convolution integra l solution, Eq. (4.5), leads to
0(1)
= 7
j ~ LF.,e "
1
4il mL ' w
n
si n Wn(1  r) dr
o
36 mLw
F., J' e ' Sin
. Wn(1 ="7
n
r) dr
o
Pe rforming the integrat
VIBRATIONS OF CONTINUOUS SYSTEMS
214
[CHAP. 7
from which it is determined that C, = C, = O. Application of the bound ary conditions at x = L
Eq. (7. 11) leads to
to
X(L) = O > C,sinAL+ C,sinhAL = O
d' X
dx' (L) = O> A'C,sinAL+A' C,sinhAL=O
Nontrivial s
MECHANICAL SYSTEM ANALYSIS
CHAP. II
21
The kinetic energy of th e system at an arbitrary instant is
T
= ~IABe2 + ~mABiJA/ + UCD4:i + ~m cDiJcD2
Hh.mL')iI' + lmcfw_lLiI)' + Hh.mL')(lil)' + lmClLj il)'
= Hbm LZ) 8Z
=
The potential energy of the system at an
232
7.46
VIBRATIONS OF CONTINUOUS SYSTEMS
[CHAP. 7
A torque To is statically applied to the midspan of the shaft of Fig. 719. Determine the
mathematical form of the timedependent torsional oscillations when the torque is suddenl y
removed:
Ans.
8(x, f) =
84
H A RM ON IC EXCITATION OF 1DEG R EEO FFR EEDO M SYSTEM S
[C H A P. 3
In o r d~r for r < 0.788 over the entire freque ncy range, r = 0.788 should corres pond to a freq ue ncy
less than.600 r/ min . T hus
~)(2Jr
min
(600 mm rad)( l 60 s )
r
w
w., >r
142
FREE VIBRATIONS OF MULTI DEGREEOFFREEDOM SYSTEMS
[CHAP. 5
Rearranging and rewriting in matrix form leads to
[
5.5
m
o
O][i] + [2k
1 i!
I kL
IkL
ftkL'
][X] [ 0 ]
e = M(t)
Use Lagrange's equations to derive the differentia l equations governing the m
166
FREE VIBRATIONS OF MULTI DEGREEOF FREEDOM SYSTEMS
[CHAP. 5
H ence the itera ti on has converged to A ; 31.564> and
X, ; [0.7071
1
w, ; V3 1.564> ;
1 0.707W which leads 10
1
cfw_El
31.56( mL' ) = 9866 \j ;Li
30721
5.41
~f+
sa.
Matncad
Use matrix it
HARMONIC EXCITATION OF I DEGREEOFFREEDOM SYSTEMS
96
3.33
[CHAP. 3
D etermine the steadystate ampli tude for the machine in the system of Fig. 327.
r
y(l)
= 0 .005 sin
351 m
E=2 10X IO'!,
Ct1.8
1=4 .1 X 10.6 m"
m
Fig. 327
The system is modeled as a
54
FREE VIBRATIONS OF JDEGREEOFFREEDOM SYSTEMS
2.26
[CHAP. 2
List three differences between the free vibration response of a system with Coulomb
damping and the free vibration response for a system with viscous damping whose free
vibrations are underda
FREE VIBRATIONS OF MULTIDEGREEOFFREEDOM SYSTEMS
178
5.79
[CHAP. 5
Determine the natural freq uencies for the system of Problem 5.68 if the machines both have a mass
m.
Ans.
5.692
5.80
l
,.
mL
22.03
(EI
\j;JJ
Determine the natural frequencies for the
This is my 2nd time to Da Gu Di and I took 14 years old years girls with Junyin. There were 8
of them and they were very pleasant and friendly :) All of them were attending the camp for
the first time. 23 of the girls are staying in thus attends church r
Mission trip 2014
Only in Da gu di , I feel the sense of urgency and hear the cry
for hope and pleas for salvation to be upon the land.
Although it is my 2nd trip to Da gu di, I still end up having the
same takeaway I should have more to offer, I could ha
Junpei
It's been yet another year since the last JubileeDa Gu Di camp. There were many
new faces this year, but yet there were some familiar faces. It has been 5 years
since I have come for this annual mission trip, and each year's camp gives me new
lear
Jamie
Reflection
Before I went to Da Gu Di, I didn't know what to expect as it was my first mission trip. I was worried
about what was expected of me, whether I could meet others' expectations and if I could deliver the
lessons well, after all, the main p
This is my very first mission trip with the Church. I have really learnt many things in this trip. It has
enabled me to see how fortunate we are to live our lives with abundance. But for the youths in
DaGuDi they are less fortunate than us. Yet despite th