Prof. A. Meher PrasadProf. A. Meher Prasad

Department of Civil EngineeringDepartment of Civil EngineeringIndian Institute of Technology MadrasIndian Institute of Technology Madras

email: prasadam@iitm.ac.in

Dynamic Equations of Motion

Force excited system

[ ] { } [ ] { } [ ] { } ( ){ }m x c x K x P t+ + =&& &

Ground excited system

[ ] { } [ ] { } [ ] { } [ ] { }gm u c u K u m x+ + = -&& && &&

where is the relative displacement of the structure w.r.t ground.

{ }u

gxx&&gyx&&

gzx&&Non-moving reference

Ground Acceleration vector

{ } { } { } { }1 1 1g x gx y gy z gzx x x x= + +&& && && &&

where,gxx&& gyx&& gzx&& are the ground accelerations in x,y,z

directions respectively.

{ } { } { }1 1 1x y z, , are null vectors except that those elements are equal to 1, which corresponds to x,y,z translational DOF.

Let

System equations reduce to following uncoupled equations

where participation factors,

Note: aj = bj = 0 since initial conditions are zero i.e

{ } { } ( )1

n

j jj

u q t=

= f￥

( ) ( )2 22j j j j j j j jx gx jy gy jz gzq p q t p q t p C x C x C x￩+ z + = - + +￫&& & && && &&

{ } [ ]

{ } [ ] { }2

1

T

j xyz

jx Tjy j j jjz

m I

Cp m

￬f ￭�

￮=f f

{ } { } { }0 0 0u u= =&

Modal Superposition applied to GES

Solution to uncoupled equation of motion can be expressed as,

( )

20

max

( ) ( ) ( ) ( )

( ) sin ( )1

( , )

( , ) ( , ) ( , )

where

Maximum values of

The maximum relative displac

x tt t tz

x

z z z

- -

= + +

-￩= -￫-

=

= + +

&& j j

y yz z

y yz z

j jx jx jy jy jz jz

tp tj

j x g x dj

j

i x a x j j

j jx ax j j jy ay j j jz az j j

q t C A t C A t C A t

pA x e p t d

A S T

q C S T C S T C S T

maxmax(

in mo

)

dement of the

f=

th th

ij ij j

i DOF the s

q

j e

U

i

In general , for design the response quantities of interest are:

R = maximum values of (u , fs, Δ, V, M)

Equivalent lateral loadsStorey shears

Storey Moments

Storey driftsRelative displacements

• uses mode shapes to reduce size, uncouple the equations of motion.

• Summation of individual modal response in frequency domain.

( )

22

2 2

222

2 22

max

g

g

d xd u dum c Ku m

dt dt dt

d xd upu p u

dt dt

U u t

x

+ + = -

+ + = -

= = max deformationof spring

Modal Frequency Response Analysis

Damping

m

k

)(txg&&

Ground Excited MDOF System

[ ] { } [ ] { } [ ] { } [ ] { }.. . ..

gM u C u K u M x+ + = -

{ }u { } { }gx x-

{ } { } { } { }.. .. .. ..

g xg yg zgx x x x= + +

.. .. ..

&, ,xg yg zgx x x

= relative displacement of the structure w r t ground

=

Ground acceleration vector :

where, are ground accelerations in x, y & z directions respectively

Reference

base

..

zgx

..

xgx

..

ygx

x

yz

1) SRSS

2) CQC

3) Double Sum

4) Grouping

2

1( )

N

jjR R

=S;

Serious errors for closely spaced frequencies and for 3-D structures ,which include torsional contribution.

SRSS :

Square Root of Sum of Squares .It gives most probable maximum response.

Modal combination rules

**Since the maximum response in each mode would not necessarily occur at the same instant of time, over conservative to add separate modal maximum responses.

CQC :

Complete Quadratic Combination Rule (Wilson, Der Kiureghion & Baya 1981). It is based on random vibration theory.

312 2

1 1

2 2 2 2 2 2

( )

8( ) ( );

(1 ) 4 (1 ) 4( )

where

a

z z z rz ra r

r z z r r z z r

= =S S

+= =

- + + + +

are the maximum responses inthe modes

;

j

N N

i ij ji j

th thi

i j i j jij

i j i j i

R R R

R i and j

p

p

Note: All cross modal terms included very good agreement with full modal superposition extra computation minimal.

• Structures subjected to time varying forces or enforced motions

• Discretization in Space- Time

Finite Element Method In Structural Dynamics

Methods of solution

• Time domain

• Frequency domain

3) Explicit methods

Equation of motion at tn conditionally stable

6) Implicit methods

Equation of motion at tn+1 can be unconditionally stable.

1) Direct integration method Response at discrete interval of time (usually equally spaced).

Process of marching along time evaluated from values at previous time stations.

{ } { } { }, ,&x x x&& &

Time Domain Methods

1. Unconditional stability when applied to linear problems

2. No more than one set of implicit equations

3. Second-order accuracy

4. Self-starting

5. Controllable algorithmic dissipation in the higher modes.

Desirable attributes:

Modal Superposition Method

• Transformation of co-ordinates results in a set of uncoupled

SDOF equations in terms of modal co-ordinates.

• Solve SDOF equations

• Useful for many problems where the response can be

approximated very well by using few eigen modes.

Time Domain Methods

• Suitable for linear problem subjected to sinusoidal or

oscillatory forces -

2

2

gd x

dt

{ } ( ){ } { }0ji ti t

jj

P t P e or P t P e = =

Frequency Domain Methods

• Response {x}ei t is a complex number having magnitude

and phase w.r.t the applied force.

• Structural excitation computed at discrete excitation

frequencies.

• Solve coupled matrix equation using Complex Algebra.

Direct Frequency Response Analysis

Multiple support ExcitationMultiple support Excitation

Super structure free Dof

Support Dof

[ ]

{ }{ } [ ]

{ }{ } [ ]

{ }{ }

{ }{ }0

f ffff fr ff fr ff fr

rf rr rf rr rf rrr r r

u u uM M C C K K P

M M C C K Ku u u

+ + =

&& &

&& &

{ }{ }{ }

=

=

=

Support motion(restrained dof)

Super structure free dof

Nodal force excitation vector

r

f

u

u

P

Decompose {uf} into pseudo static and dynamic parts

{uf}= {us} + {ud}

Considering only static response ( i.e. stiffness matrix alone)

{ } { } { }

{ } { } [ ] { }1

0ff s fr r

s ff fr r r

K u K u

u K K u i u-

+ =

= - =

Influence matrix

Describes influence of support displacement on structural displacement

jth column of [i]=structural displacements due to unit support displacement url only (l th base displacement)

{ } { } { } { } { } { } { }

{ } { } { } { } { } { }{ } { } { } { }

{ } [ ] { }{ } [ ] { }

.

ff f ff f ff f fr r fr r fr r

ff d ff d ff d ff s ff s ff s

fr r fr r fr r

s r

s r

M u C u K u M u C u K u P

i e

M u C u K u M u C u K u

M u C u K u P

But

u i u

so u i u

+ + + + + =

+ + + + + + + + =

=

=

&& & && &

&& & && &

&& &

& &

&{ } [ ] { }s ru i u=& &&

{ } { } { }0ff s fr rK u K u + = (By definition )and

{ } { } { }{ } { } { } { } { }

ff d ff d ff d

ff fr r ff fr r

M u C u K u

P M i M u C i C u

+ + = - + - +

&& &

&& &&

i.e.

If assume light damping

{ } { } { } { } { } { }

[ ]

{ } [ ] ( ){ } [ ] [ ] [ ]

0

￩￩ ￩ ￩ ￩ ￩+ + = - +￫ ￫ ￫ ￫ ￫￫

￩ ﾻ￫

￩= F F F￫

For lumped mass system

Carryout Mode superposition

and

&& & &&ff d ff d ff d ff fr r

fr

d ff

M u C u K u P M i M u

M

u q t MT

= I

{ } { } { } [ ] { }22z f f ￩￩ ￩+ + = - +￫ ￫￫&& & &&

T T

o j j j j j j j ff fr rq p q p q P M i M u

Uncoupled equations of motion are,

( )gxx t&&

610bigM m

( ) { }( )big gxP t M x t= &&

1su&& 2

su&& 3su&&

A big mass (much bigger than the total mass of the structure (~106total mass) is added to each degree of freedom at moving bases.

As more big masses are applied, more low frequency modes have to be extracted.

The desired base motion is obtained by applying a point force to each degree of freedom at moving bases by

N Ns big sP M u= &&

Where Mbig=big mass and is the applied acceleration prescribed for degree of freedom N associated with moving supports

Nsu&&

The combined equation of motion is

[ ] { } [ ] { } [ ] { } { } { }{ } { } big

s

i is s

M u C u K u P P

P M u

+ + = +

= && &

&&with

Where is the diagonal matrix containing the big masses for moving base ‘i’ and is the base motion applied to this base

bigiM

{ }isu&&

The mass matrix [M] now contains the mass of the structure as well as the big masses associated with the secondary base.

The modal equations { } [ ] { } 1T

j jMf f =

{ } { } { } { }22T T

o j j j j j j j sq p q p q P Pz f f+ + = +&& &with

1.0001.0006.766252.28360.0

4.7876

5.2909

108

0.99991.0006.766152.28360.0

4.7876

5.2909

106

0.99951.00036.764152.282310-10

4.7871

5.2910

104

0.95241.03356.553152.055210-9

4.8011

5.3025

102

Response peaks (m/s2)

X1 max X2 max X3 max X4 max

Natural frequency

Ratio of large mass to structure