Solution Manual for Structural Dynamics Theory and Computation 6th Edition by Mario Paz – Complete Step-by-Step Solutions for All Chapters

Solution Manual for Structural Dynamics Theory and Compu
M M M M M M M



tation Sixth Edition by Paz
M M M M




1

,1.1


IfMtheMweightMwMisMdisplacedMbyMamount,My,MtheMbeamMandMtheMspringsMwillMexertMaMtotalMf
orceMonMtheMmassMof
3𝐸𝐼M
𝑃M=M# +M2𝑘,M𝑢
𝐿(

TheMbeamMandMspringsMactMinMparallel.MTheMequivalentMstiffnessMis:
𝑃M 3𝐸𝐼M
𝑘M M =M =M # +M2𝑘,
.
𝑢 𝐿(

NaturalMfrequency:

M𝑔
M 3𝐸𝐼
𝑘 =M0 # +M2𝑘,
𝜔M=M0
𝑚 𝑊 𝐿(

NaturalMperiod:

2𝜋M 𝑊M 𝐿
𝑇 M =M =M 2𝜋𝐿0 # ,
𝜔 𝑔 3𝐸𝐼M+M2𝑘 (
𝐿

1.2


Stiffness:

3𝐸𝐼 3M×M109
𝑘.M=M#M (M +M2𝑘, +M2M×M1000M =M 4,300M𝑙𝑏/𝑖𝑛.
𝐿 100(
M=




NaturalMfrequency:
0M𝑘 4300M×M386
𝜔M = =M =M 23.52M𝑟𝑎𝑑/𝑠𝑒𝑐
𝑚M 3000
0

FreeMvibrationMresponseMofMundampedMoscillator:
𝑢OM
𝑢(𝑡)M =M 𝑢O𝑐𝑜𝑠𝜔𝑡M+M 𝑠𝑖𝑛𝜔𝑡
𝜔
2

, 𝑢(𝑡)M =M −𝑢O𝜔𝑠𝑖𝑛𝜔𝑡M +M𝑢O𝑐𝑜𝑠𝜔𝑡

DisplacementMandMvelocityMatM𝑡M =M 1𝑠𝑒𝑐MwithMtheMinitialMvalues

𝑢OM =M 1M𝑖𝑛.M,M𝑢OM =M 20M𝑖𝑛./𝑠𝑒𝑐:

𝑢(1)M =M 1M∙M𝑐𝑜𝑠(23.5M∙M1)M+M 𝑠𝑖𝑛(23.5M∙M1)M =M −0.89M𝑖𝑛.
20
23.5
𝑢(1)M =M −1M∙M23.5𝑠𝑖𝑛(23.5M∙M1)M+M20𝑐𝑜𝑠(23.5M∙M1)M =M 22.66M𝑖𝑛./𝑠𝑒𝑐




1.3

TheMstiffnessMofMtheMbeamMis


12𝐸𝐼W 3𝐸(2𝐼X) 12M∙M(30M∙M10Z)M∙M170 3M∙M(30M∙M10Z)M∙M82.5
.9
𝑘M = M V M M + YM= + =M 25,577M𝑙𝑏/𝑖𝑛.
𝐿( 𝐿( 144( 144(


NaturalMfrequency:

1M 𝑘 1 25,577M×M386
0M
𝑓M = M =M 0 =M2.24M𝑐𝑝𝑠
2𝜋 𝑚M 2𝜋M 50,000




1.4

a) InfinitelyMrigidMhorizontalMmembe

rMStiffness:

12𝐸𝐼 12M∙M(30M∙M10Z)M∙M171
𝑘M =M 2M#M (M ,M= =M 21,100M𝑙𝑏/𝑖𝑛.
𝐿 (12M∙M15)(


NaturalMfrequency:

3

, 0M𝑘 21,100M×M386
𝜔M = =M =M 18.05M𝑟𝑎𝑑/𝑠𝑒𝑐
𝑚M 25,000
0

𝜔
𝑓M =M =M 2.87M𝑐𝑝𝑠
2𝜋

b) FlexibleMhorizontalMmemberMconsistingMofMW18X30

ComputeMtheMstiffnessMbyMmomentMdistributionMmethod.MDisplaceMtheMframeMhorizontallyM
byMoneMinchMandMdetermineMtheMstiffnessMofMtheMframeMasMtheMsumMofMtheMshearMforcesMin
MbothMcolumns.MTakeMadvantageMofMtheMsymmetryMbyMmodifyingMtheMstiffnessMofMhorizonta

lMmemberMbyMfactorM3/2.


DistributionMfactors
4𝐸𝐼M 4𝐸M
𝑘 =M =M ∙M171M →M 171M →M 0.1244
^_
𝐿 𝐿
4𝐸𝐼M3M 4𝐸M M 3
𝑘 =M =M𝐿M ∙M802M →M 1203M →M 0.8756
^a
𝐿M 2 2M
∙


FixedMendMmoments:
0.1244 0.8756
C
950 B
𝑀 =M 𝑀 6M∙M(30M∙M10Z)M∙M170. -118 -832
^_ _^ M 9 M M -832M
6𝐸𝐼 X
=M = (12M∙M15)X
𝐿
832
=M 950M(𝑘M−M𝑖𝑛.M)

ShearMforce:
832M+M891M
𝑘M =M ∙M2M =M 19.14M𝑘𝑖𝑝/
950 A D
𝑖𝑛. -59
180
891
NaturalMfrequency:

1M 𝑘 1 19,140M×M386
0 M =M
𝑓M =M =M 2.74M𝑐𝑝𝑠
2𝜋 𝑚M 2𝜋M 25,000
0


4