Structural Dynamics Theory and Computation Sixth Edition Mario Paz - Complete Solution Manual

Solution Manual for Structural Dynamics Theory and Comput
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ation Sixth Edition by Paz
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1

,1.1


Ifotheoweightowoisodisplacedobyoamount,oy,otheobeamoandotheospringsowilloexertoaototaloforceoo
notheomassoof
3𝐸𝐼o
𝑃o=o# +o2𝑘,o𝑢
𝐿(

Theobeamoandospringsoactoinoparallel.oTheoequivalentostiffnessois:
𝑃o 3𝐸𝐼o
𝑘o o =o =o # +o2𝑘,
.
𝑢 𝐿(

Naturalofrequency:

𝑘 𝑔 3𝐸𝐼
𝜔o=o0
o o
=o0 # +o2𝑘,
𝑚 𝑊 𝐿(

Naturaloperiod:

2𝜋o 𝑊o 𝐿
𝑇 o =o =o 2𝜋𝐿0 # ,
𝜔 𝑔 3𝐸𝐼o+o2𝑘𝐿 (




1.2


Stiffness:

3𝐸𝐼 3o×o109
𝑘.o=o#o (o +o2𝑘,o +o2o×o1000o =o 4,300o𝑙𝑏/𝑖𝑛.
𝐿 100(
=


Naturalofrequency:
0o𝑘 4300o×o386
𝜔o = =o0 =o 23.52o𝑟𝑎𝑑/𝑠𝑒𝑐
𝑚o 3000

Freeovibrationoresponseoofoundampedooscillator:
𝑢Oo
𝑢(𝑡)o =o 𝑢O 𝑐𝑜𝑠𝜔𝑡o+o 𝑠𝑖𝑛𝜔𝑡
𝜔


2

, 𝑢(𝑡)o =o −𝑢O𝜔𝑠𝑖𝑛𝜔𝑡o +o𝑢O𝑐𝑜𝑠𝜔𝑡

Displacementoandovelocityoato𝑡o =o 1𝑠𝑒𝑐owithotheoinitialovalues

𝑢Oo =o 1o𝑖𝑛.o,o𝑢Oo =o 20o𝑖𝑛./𝑠𝑒𝑐:
20
𝑢(1)o =o 1o∙o𝑐𝑜𝑠(23.5o∙o1)o+o 𝑠𝑖𝑛(23.5o∙o1)o =o −0.89o𝑖𝑛.
23.5
𝑢(1)o =o −1o∙o23.5𝑠𝑖𝑛(23.5o∙o1)o+o20𝑐𝑜𝑠(23.5o∙o1)o =o 22.66o𝑖𝑛./𝑠𝑒𝑐




1.3

Theostiffnessoofotheobeamois


12𝐸𝐼W 3𝐸(2𝐼X) 12o∙o(30o∙o10Z)o∙o170.9 3o∙o(30o∙o10Z)o∙o82.5
𝑘o = o V o o ( + Yo= + =o 25,577o𝑙𝑏/𝑖𝑛.
𝐿 𝐿( 144( 144(


Naturalofrequency:

1o 𝑘 1 25,577o×o386
0o
𝑓o = o =o 0 =o2.24o𝑐𝑝𝑠
2𝜋 𝑚o 2𝜋o 50,000




1.4

a) Infinitelyorigidohorizontalomember

Stiffness:
o




12𝐸𝐼 12o∙o(30o∙o10Z)o∙o171
𝑘o =o 2o#o (o ,o= =o 21,100o𝑙𝑏/𝑖𝑛.
𝐿 (12o∙o15)(


Naturalofrequency:



3

, 0o𝑘 21,100o×o386
𝜔o = =o0 =o 18.05o𝑟𝑎𝑑/𝑠𝑒𝑐
𝑚o 25,000
𝜔
𝑓o =o =o 2.87o𝑐𝑝𝑠
2𝜋

b) FlexibleohorizontalomemberoconsistingoofoW18X30

Computeotheostiffnessobyomomentodistributionomethod.oDisplaceotheoframeohorizontallyobyo
oneoinchoandodetermineotheostiffnessoofotheoframeoasotheosumoofotheoshearoforcesoinobothocol
umns.oTakeoadvantageoofotheosymmetryobyomodifyingotheostiffnessoofohorizontalomemberobyo
factoro3/2.


Distributionofactors
4𝐸𝐼o 4𝐸o
𝑘 =o =o ∙o171o →o 171o →o 0.1244
^_
𝐿 𝐿
4𝐸𝐼o3o 4𝐸o o 3
𝑘 =o =o ∙ ∙o802o →o 1203o →o 0.8756
^a
𝐿o 2 𝐿o 2o


Fixedoendomoments:
0.1244 0.8756
C
950 B
6𝐸𝐼o 6o∙o(30o∙o10Z)o∙o170.9
𝑀 =o 𝑀 =o = -118 -832
^_ _^ 𝐿X (12o∙o15)X o o -832o

832
=o 950o(𝑘o−o𝑖𝑛.o)

Shearoforce:
832o+o891o
𝑘o =o ∙o2o =o 19.14o𝑘𝑖𝑝/𝑖𝑛.
180 950 A D
-59
Naturalofrequency:
891

1o
𝑘 1 19,140o×o386
0o =o 0
𝑓o = o =o 2.74o𝑐𝑝𝑠
2𝜋 𝑚o 2𝜋o 25,000



Note:
1. Assumingoaoflexibleogirderodecreasesotheonaturalofrequencyobyoonly


4