STRUCTURAL DYNAMICS MULTI DEGREE OF FREEDOM SYSTEMS EQUATION OF MOTION

STRUCTURAL DYNAMICS
MULTI DEGREE OF FREEDOM SYSTEMS EQUATION OF MOTION

1. Class #8

Structural Dynamics

Multi-Degree of Freedom Systems
Equations of Motion

Dr. Tesfaye Alemu

2. Degrees of Freedom

e SDOF: Single Degree of Freedom System:
e One mass moving in one direction.

e MDOF: Multi-Degree of Freedom System:
e More than one mass moving in
one (same, lateral) direction.
e One mass moving in
more than one direction.

3. 1 234 5

WERD

„12345

NA
WT UM AN

4. Equations of Motion

e SDOF:
e Free Vibrations: Forcing function = F(t) = 0 = p(t)
e Undamped: energy dissipation = 0
e Damped: energy dissipation # 0
e Forced Vibrations: Forcing function = F(t) #0 # p(t)
e Undamped: energy dissipation = 0
e Damped: energy dissipation # 0

e MDOF:
e Free Vibrations: Forcing function = F(t) = 0 = p(t)
—— > Undamped: energy dissipation = 0
e Damped: energy dissipation # 0
e Forced Vibrations: Forcing function = F(t) #0 # p(t)
e Undamped: energy dissipation = 0
e Damped: energy dissipation # 0

5. Stiffness Coefficients for MDOF

e Multi-story structures can be divided
into 2 types:
e According to deformation characteristics

e Shear Structure be
e Floors move only horizontally,
e Rigid beam-column joints, NO Rotation

e Flexural Structure =, |
e Floors move horizontally and Rotate |
e Flexible (pinned) beam-column joints

6. Stiffness Coefficients for Shear Building

he

hy

DEM +Lr)

in

= GAG ba)
hy

k=

12E (1, +12)
hy

7. | rl, AG)
E a hi
Lar ha :
| IT) net)
E k, = 7 2 |
12E, UG + 1)
a
Palt) Fig ult
la ust) Fag
ie Feat pat)
Feat =
Fsor S3R

8. Forces Acting ona
Shear Building

[no Pr) ml] pp?” mb O1 ps”

u,(t Uat)
OI Sel, -à (0) 0-0]
ku] k,[u,(t)—u,(1)] k,|u;(t)—u,(r)]

9. Forces Acting ona
Shear Building

e For Mass #1:

im) Bro >
~mlii,(¢)]-¢,[4, (Qk. ba + Li Où 0+ koa (0-4. D+ p,(0)=0
= multi, (e)]-(c, + Ji (O]+ cl (1)]- (4, +8 Ju (]+k le = =p, (0)
Define: c,,=C,+C, k,, =k, +k,

— C2 = Cy Kyo =k,
= ma (O1 (ei lé OI cale OT Jin Ol- 4, oh (01==p, (0)
tml + (i Ol+ cola O]+& u O]+ kale = +20)

t

Ir a] pep Putt malito] po ml) 2°
7, u,(t up(t) ux(t)
C la, ala: li, ()- uy ote li, ()- uy ort)

k[u,@] k,[u,(t)—u,(1)] k[u,()-u,(0)]

10. Forces Acting ona
Shear Building

e For Mass # 2:

2F,=0 +=>

mal (cali (lu (+ ei (1) + lus (o) u,(1)]+ 20 =

Define: =C,, =+C, Ca =C,+C; —C ou
-k, =k Ta +k,

0

“ale (+ coli l(c, + (clé (Ok lu le im +k ie ]+klae)]=-D.)

maki, (e)]-(c,, Jl, ()]- E, ali, ( e)|- © ali, (-( x re kl 1-60 (01=-p,(0)
+ mi, (+ (cr (+ enfin 2 le li, QE 21 Ju, (+ Kal ll (0l=+p, t)

Pa(t)

alt)

Ir GO] Pt mi, (+)] K mil >

7. ut uo(t ux(t)
C la, ala: li, ()- uy ote li, ()- uy ort)
k[u,@] k,[u,(t)—u,(1)] k[u,()-u,(0)]

11. Forces Acting ona
Shear Building

e For Mass #3:

1 2F,=0 =>=+
—malii,(+)]-c.[ú,()-ú,()]-x,[u,(+)-u,(1)]+ p,(+)=0
—molii,(+)]+c,[ú, (]-c,[a,(0]+ k,[u, ()]-4,[u, Ok =P; (e)
Den SC CS —k,, =k,
SO = ae; —k,; =—kz

-m,[ü,(t )I- (calli, (s))- Cala (e )- (a lu, (e))- k,slus()|=-p;(1)
|. tmliis()]+ (cs, AO) AO) CS A0) alu, = +750)

ml

pa(t) Palt)

| ii, (a) P,(t) m, lii, ()] a Ws lii, ()]

7. ut uo(t us(t)
C la, ala: li, ()- uy ote li, ()- uy ort)
k[u,@] k,[u,(t)—u,(1)] k[u,()-u,(0)]

12. Forces Acting ona

Shear Building

Writing the force equations for all 3 masses in
matrix form.

0

Note:

0 m,

0 O m
Symmetry of Mass, Damping, and Stiffness matrices

o : Ci2 iR x ky
ii,(t) 20, O ü,(t) + Ka kn

Gin

0 © C3 últ) 0 ky

ü,(e)
C33 = C, ky =k =

kz =k,

C=C, FC

Con = C2 +C,

0
ky,
Rz

ka, = ky =k

u(t) p(t)
„&)|=| pole)
u(t) pst)

—k, ky, =k, +k,
ky =k, +k,

q la, 9)
k, la, ()]

RD

[no li, 9)
do

Pa(t)

OO Pr
k,[u,(t)- u, ()]

mlü, (0)]
C3 la, ()- uy ote |
k alu, ()- u, ol

Pa(t)

13. ci T 1)]
ku]

Forces Acting ona
Shear Building

0
0 m,

m,

rm

0 0 m
writing in

[mlfi(e)]+[c

0
0

G1 Ci

Ji} +{eTa] =

for seismic excitation :

ii(t) 1 12
ii,(t) Gn E
3 ü,() 0 cy,

Ca

matrix vector

[56]
[p(e)] =-mii, (+)

0
CG

C3

ui(t) ky ky Ou
3 últ) + ky ko Ko |
3 ui,(t) 0 k, k;]lu
form:

|

m 0
0 m 0
0 0 mI

oT

)| Fr)
t)|=| pale)
1)] Lt)

Miran wo EE
[u,(t) u, ()]

Po(t)

ml, ()]

G li, ()- uy el
k alu, ( )- u, ol

14. MDOF - Equations of Motion

Assumptions: (Same as for SDOF)

Massless
frame

Mass of structure is concentrated at roof and floors.
Frame has zero mass.
Beams & columns are inextensible axially.
Motion of structure depends on
e Lateral Stiffness (spring)
e Relative displacement between base and mass
e Energy Dissipation (damper, dashpot)
e Relative velocity between base and mass
Applied Force Ground Motion

Mass

Pit)

‘Viscous
amper

15. MDOF - Equations of Motion

Assumptions: (same as for SDOF)

e Three separate PURE components
e Mass component,
e Roof, floors, walls, columns, beams
e lumped together into ONE mass per floor level or roof level.
e Stiffness component.
e Lateral stiffness of walls, columns, beam-column joints
e lumped together into ONE stiffness per level.
e Stiffness depends on relative displacement between base and mass.
e Damping component
+ Energy (kinetic & strain) dissipation of all elements
e lumped together into ONE damper per level.
e Damping depends on relative velocity between base and mass.
Applied Force Ground Motion
Mass =
Pit)
Massless Viscous
frame damper
7 2
(a) @) pa] Ue