dynamics lecture_part 2-mshafik

8/3/2019 Dynamics Lecture_Part 2-Mshafik

1/44

DYNAMICS 2

Multi Degrees of Freedom (MDOF)Systems Using Modal Superposition

By

Moustafa M Shafik Mohamed


2/44

Objectives

Outline of the modal superposition method

To introduce the Modal superposition as a powerful technique in solvingthe MDOF systems thru a solved example

Applying the modal superposition for solving MDOF systems usingSAP2000

Introduction to the Response spectrum method

Application of the Modal Superposition in Floor Vibration Problems


3/44

*Outline of Modal Superposition Method 1

*This section is extracted from Structural

Analysis By Coates, Coutie and Kong


4/44

Outline of Modal Superposition Method 2


5/44



6/44


Similarly,

Transposing the I mode equation,

Because of the symmetry ofM ,

Equation 2

Equation 1


7/44


Pre-multiplying 1 by

, Post-multiplying 2 by

, then subtracting yields ,


8/44


The above equation shows the different modes areorthogonal to each other with respect to the Mass Matrix.Similarly, we can prove that they are orthogonal to eachother with respect to the Stiffness Matrix.

By using this property, the coupled D Eqs of equilibrium

can be decoupled and solved as a group of SDOFsystems. Then,

The final answer is obtained by combining again alldifferent modes.

The steps are,


9/44

Outline of Modal SuperpositionMethod 7


10/44

Outline of Modal SuperpositionMethod 8

Then solve the individual (uncoupled) SDOF systems

Then obtain the final answer by combining all the modes using,


11/44

Solved Example of 2DOF system 1

l d E l f 2DOF


12/44

olved Example of 2DOFsystem 2

l d E l f 2DOF


13/44


l d E l f 2DOF


14/44


l d E l f 2DOF


15/44


l d E l f 2DOF


16/44


l d E l f 2DOF


17/44


l d E l f 2DOF


18/44


l d E l f 2DOF


19/44


l d E m l f 2DOF


20/44



21/44



22/44



23/44



24/44

Solved Example of 2DOF system 13a

2

1tan

l

l

Mk

c

222

02

)/2(])/(1[

/)sin(cossin

slsl

ldd

tM

c

ffff

ktPtBtAey

where the Phase Angle or the Angle of Lag of the response

The complete solution becomes

tPkydt

dyc

dt

ydM lsin02

2

The Equation of Motion is:


25/44

Solved Example of 2DOF system 13b

222)/2(])/(1[

1

slsl ffffDLF


26/44



27/44



28/44



29/44



30/44



31/44



32/44



33/44



34/44



35/44

Response Spectrum Analysis 1


36/44



37/44



38/44



39/44



40/44


Floor Vibration using Modal


41/44

Floor Vibration using ModalSuperposition

nnnnnn

nn

F

F

xKxKxm

xKxKxm 1

11

..

1

1111

..

11

.

....

....

.Equilibrium Equation,

Decouple using,

FXKXM..

j

m

j

jiix

1

Consider thefirst mode only,

FKMTTT

1

111

1

..

11)()(

1

1

1

2

11

..

M

FT

Fl Vib i i M d l


42/44


Normalize the maximum

Eigenvector value to 1 andconsider only one load at midspan,

For a simple beam, themodal mass of mode 1 is

given by,

max1

FFT

1

max

1

2

11

..

M

F

21

mLM

The solution of thisequation is given by,

DlfM

FDLf

K

Fx

1

2

1

ma x

1

max

max

2/12222})(4])(1{[

1

s

L

s

L

f

f

f

fDLF

2

1DLf

For sinusoidal loadingfunction, the DLF isgiven by,

For resonance case,


43/44


Considering the loading function givenin section 2, zero initial conditions, andthe matching between the forcefrequency and the floor frequency, thedynamic response in terms of

acceleration is given by

In terms of acceleration, theresponse is given by,

2

1

2

2

1

max2max

..

ml

Fx

1001

%maxmax

..

xmgl

F

g

x

10029.0

%

)35.0(max

..

xW

e

g

xf


44/44

Floor Vibration Using Modal Superposition)(sin tax

)(cos.

tax

)(sin2

..

tax

)(cos3

...

tax

..

...

x

x

)()()(......

fLogxLogxLog

)()()(.....

fLogxLogxLog

mXCY

dynamics lecture_part 2-mshafik

Documents