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Last update: 2010 Ahmed Elgamal

Multi-Degree-Of-Freedom (MDOF) Systems and Modal Analysis

Ahmed Elgamal

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Ahmed Elgamal

SDOF Shear Building (rigid roof)

m = lumped mass = mroof + 2 (1/2 mcol)

3

c

3

ccol

h

24EI

h

12EI22kk

gumuckuum

2

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Ahmed Elgamal

2-Story Shear Building (2-DOF system)

m1

m2

k1

k2

u1

u2

col22 m

2

12roofmm

col11 m

2

14floormm

col22 2kk col11 2kk

g212222 um)u(ukum

g

2

1

2

1

22

221

2

1

2

1u

m

m

u

u

kk

kkk

u

u

m0

0m

g11121211 umuk)u(ukum

gu1mukum or,

where 1 or 1 is the Identity Matrix

gu m1kuum 3

Ahmed Elgamal

m1

m3

k1

k2

u1

u2

u3

k3

m2

g323333 um)u(ukum

g212232322 um)u(uk)u(ukum

g11121211 umuk)u(ukum

g

3

2

1

3

2

1

33

3322

221

3

2

1

3

2

1

u

m

m

m

u

u

u

kk0

kkkk

0kkk

u

u

u

m00

0m0

00m

3-Story Shear Building (3-DOF system)

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Ahmed ElgamalFor a N-DOF system,

n

m

2

1

m

m

m

m

nn

n

1m1mmm

3322

221

kk000

k|

0|

0k-kkk-00

|

|

0

0

00kkkk

00k-kk

m

k

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Ahmed ElgamalEquation of motion for a N-DOF system,

n

m

2

1

n

k

2

1

u

|

|

u

|

|

|

|

u

u

m

m

m

m

n

m

2

1

nn

n

1m1mmm

3322

221

u

|

|

u

|

|

|

|

u

u

kk000

k|

0|

0k-kkk-00

|

|

0

0

00kkkk

00k-kk

g

n

m

2

1

u

m

|

|

m

|

|

|

|

m

m

with initial conditions:

gu m1kuucum

uu

uu

(t=0)

(t=0)

or (with damping included):

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Natural Frequencies of a N-DOF system

Similar to the SDOF system, MDOF systems have natural frequencies. A 2-DOF

has 2 natural frequencies w1 and w2, and a n-DOF system has natural frequencies

w1 , w2 , …, wn

Similar to the SDOF, free vibration involves the system response in its

natural frequencies. The corresponding Free Vibration Equation is (with

no damping):

0kuum

In free vibration, the system will oscillate in a steady-state harmonic

fashion, such that:

uu2ω

tcosbtsinau ωω e.g. gives uu 2-ω7

Ahmed Elgamal

substituting for u , we get:

0 u km-2ω

or

0u m-k2ω Equation 1

The above equation represents a classic problem in

Math/Physics, known as the Eigen-value problem.

The trivial solution of this problem is u = 0 (i.e., nothing is

happening, and the system is at rest). 8

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Ahmed Elgamal

For a non-trivial solution (which will allow for computing the

natural frequencies during free vibration), the determinant of

0 m-k 2ω 0 m-k λor 2λ ωwhere

For a 2-DOF system for instance (see next page), the

above determinant calculation will result in a quadratic

equation in the unknown term l . If this quadratic equation

is solved (by hand), two roots are found (l1 and l2), which

define w1 and w2 (the natural resonant frequencies of this

2-DOF system).

m-k2ω must be equal to zero such that:

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Ahmed Elgamal

2-Story Shear Building (2-DOF system)

m1

m2

k1

k2

u1

u2 g

2

1

2

1

22

221

2

1

2

1u

m

m

u

u

kk

kkk

u

u

m0

0m

0

0

u

u

mkk

kmkk

2

1

222

2121

l

l

0u m-k l

0

mkk

kmkk

222

2121

l

l

0)k)(k()mk)(mkk 2222121( ll

0)kk())kk(mkm(()mm( 2121221

2

21 ll

02 cba lla

acbb

2

4

2

21l

Solve for the l1 and l2 using the standard approach

ZY

XW

Determinant = WZ-XY

Note: For

Set Determinant = 0:

or,

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Ahmed Elgamal

For a general N-DOF system:

Matlab or similar computer program can be used to solve the

determinant equation (of order equal to the NDOF system), defining

NDOF roots or NDOF natural frequencies

w1 , w2 ,…, wNDOF

Note:

These resonant (natural) frequencies w1 , w2 , … are conventionally

ordered lowest to highest (e.g., w1 = 8 radians, w2 = 14 radians, and

so forth).

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Ahmed Elgamal

Mode ShapesSteady State vibration at any of the resonant frequencies wn

takes place in the form of a special oscillatory shape, know

as the corresponding mode shape fn

To define these mode shapes (one for each identified wn), go

ahead and substitute the value of wn for w in Eq. 1

and solve for the vector u which will define the

corresponding mode shape fn :

0 m-k nn f2ω

0u m-k2ω

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Ahmed Elgamal2-DOF system ( 2 mode shapes f1 and f2)

u1

u2m2

m1

11φ

21φ

f12

f22

u1

u2m2

m1

11φ

21φ

f1 f2

Note: Any mode shape fn only defines relative amplitudes of motion of the

different degrees of freedom in the MDOF system. For instance, if you are solving

a 2-DOF system, you might end up with something like (when solving for the first

mode):

f11 - 2f21 = 0, only defining a ratio between amplitudes of f11 and f21

(for instance, if f11 = 1, then f21 = 0.5, or if you choose f11 = 2, then f21 = 1, and so

forth).

Generally, go ahead and make fmn= 1 (where m is top floor Dof and n is mode

shape number) and solve for the other degrees of freedom in the vector fn

21

11

1f

ff

22

12

2f

ff

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Ahmed Elgamal

Note: When you substitute any of the wn values into Eq. 1, the determinant of the matrix (k-

wn2m) automatically becomes = 0, since this wn is a root of the determinant equation (i.e.,

the matrix becomes singular).

The determinant being zero is a necessary condition for obtaining a vector u (the mode shape

fn) that is not equal to zero (i.e., a solution other than the trivial solution of u = 0.

2-Story shear building

(one node in mode 2)4-story shear building (4-DOF system)

Note one node in mode 2, two in mode 3, and 3 in mode 4

3-Story Shear Building

node

nodenode

node

Sample Mode shape Configurations

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Ahmed Elgamal

nn

T

n Mφφ m

n

2

nnn

T

n MK ωφφ k

1.0n

T

n φφ m2

nn

T

n ωφφ k

Properties of fn

b) For any mode fn , modal mass Mn is defined by:

d) If then

c) For any mode fn , modal stiffness Kn is defined by:

To do that, multiply each component of mode fn bynM

1

(not )0r

T

nr

T

n φφφφ mk 0r

T

n φφ

a) Mode shapes are orthogonal such that (for any nr)

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Ahmed Elgamal

Solution of by Mode superposition

Example of a 2-DOF system ( 2 mode shapes and )1φ 2φ

u1

u2m2

m1

11φ

21φ f22

f12

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To benefit from the mode orthogonality property, multiply by FT to get:

Substituting

Results in

Ahmed Elgamal

Φqu

2

1

2221

1211

2

22

12

1

21

11

2

1

q

qqq

u

u

φφ

φφ

φ

φ

φ

φ Φqq 21 φφ

in the equation of motiongu m1kuum

gu m1kΦqmΦ q

gu m1ΦkΦΦqmΦΦTTT q

gT

2

T

1

2

1

2

T

21

T

2

2

T

11

T

1

2

1

2

T

21

T

2

2

T

11

T

1 uq

q

q

q

m1

m1

kk

kk

mm

mm

φ

φ

φφφφ

φφφφ

φφφφ

φφφφor

Modal Analysis (Solution of MDOF equation of motion by Mode Superposition)

The solution u will be represented by a summation of the mode shapes fn, each

multiplied by a scaling factor qn (known as the generalized coordinate) . For instance,

for the 2-DOF system:

In the above, F is known as the modal matrix. As such, changes in the displaced shape

of the structure u with time will be captured by the time histories of the vector q

u =

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Ahmed Elgamal

Due to the orthogonality property of mode shapes (see previous slide), the

matrix equation becomes un-coupled and we get:

g22

2

2222

g11

2

1111

uLqMqM

uLqMqM

ω

ω

or,

g

2

22

2

22

g

1

11

2

11

uM

Lqq

uM

Lqq

ω

ω

The terms L1/M1 and L2/M2 are known as modal participation factors. These terms

control the influence of on the modal response. You may notice that (if both modes

are normalized to 1.0 at roof level for example) L1/M1 > L2/M2 since f11 and f21 are

of the same sign while f12 and f22 are of opposite signs. Therefore, the first mode is

likely to play a more prominent role in the overall response (frequency content of the

input ground motion also affects this issue).

gu

NDOF

1j

φ2

jiji mM

NDOF

1j

φ jiji mLFor a diagonal mass matrix:

u1

u2m2

m1

11φ

21φ

f12

f22

u1

u2m2

m1

11φ

21φ

f1 f2

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Ahmed Elgamal

Note that the original coupled matrix Eq. of motion, has now become

a set of un-coupled equations. You can solve each one separately (as

a SDOF system), and compute histories of q1 and q2 and their time

derivatives. To compute the system response, plug the q vector back

into Equation 2 and get the u vector

(the same for the time derivatives to get relative velocity and

acceleration).

The beauty here is that there is no matrix operations involved, since

the matrix equation of motion has become a set of un-coupled

equation, each including only one generalized coordinate qn.

Φqu

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Ahmed Elgamal

Now, you can add any modal damping you wish (which is

another big plus, since you control the damping in each mode

individually). If you choose = 0.02 or 0.05, the equations

become:

Damping in a Modal Solution

i

g

i

ii

2

iiiii uM

Lqq2q ωω , i = 1, 2, … NDOF

OK, go ahead now and solve for qi(t) in the above uncoupled equations

(using a SDOF-type program), and the final solution is obtained from:

Φqu

qΦu

qΦu

g

t u 1uu 20

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Ahmed Elgamal

Φqqu

21

2

1

32

22

31

21

1211

2

32

22

12

1

31

21

11

3

2

1

q

qqq

u

u

u

φφ

φ

φ

φ

φφφ

φ

φ

φ

φ

φ

φ

Modal Analysis (3-DOF system)

The solution u will be represented by a summation of the mode shapes fn, each

multiplied by a scaling factor qn (known as the generalized coordinate) . For instance,

for the 3-DOF system:

In the above, F is known as the modal matrix. As such, changes in the displaced shape

of the structure u with time will be captured by the time histories of the vector q

Note: If a two mode solution is sought, the system above becomes:

Φqqu

321

3

2

1

333231

232221

131211

3

33

23

13

2

32

22

12

1

31

21

11

3

2

1

q

q

q

qqq

u

u

u

φφφ

φφφ

φφφ

φφφ

φ

φ

φ

φ

φ

φ

φ

φ

φ

111

31

21

11

3

2

1

qq

u

u

u

φ

φ

φ

φ

u

Note: If a single (1st or fundamental) mode solution is sought, the system above

becomes:

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Ahmed Elgamal

Multi-Degree-Of-Freedom (MDOF) Response Spectrum Procedure

1. Once you have generalized coordinates and uncoupled equations, use response

spectrum to get maximum values of response (ri)max for each mode separately.

Calculate expected max response ( ) usingr 2

maximax rr root sum

square formula

where i = 1, 2, … N degrees of freedom of interest (maybe first 4 modes at most) and

r is any quantity of interest such as |umax| or SD

(note that summing the maxima from each mode directly is typically too

conservative and is therefore not popular; because the maxima occur at different

time instants during the earthquake excitation phase)

See A. Chopra “Dynamics of Structures” for improved formulae to estimate .maxr22

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Ahmed Elgamal

Response Spectrum Modal Responses

Max relative displacement |un| or |ujn| (jth floor, nth mode)

jnnd

n

njn S

M

Lu φ (Sdn is Sd evaluated at frequency or period Tn)nω

Estimate of maximum floor displacement

M

1n

2

jnj uu (M = number of modes of interest)

Maximum Equivalent static force fn or fjn (jth floor, nth mode)

jnjna

n

njn mS

M

Lf φ

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Ahmed Elgamal

Therefore, modal base shear V0n and moment M0n

N

1j

jn0n fV

# of floors

base

N

1j

jjn0n dfM

where dj = Distance from floor j to base

Estimate of maximum base shear and moment:

M

1n

2

0n0 VV

M

1n

2

0n0 MM

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Ahmed Elgamal

Damping Matrix for MDOF Systems

Mass-proportional damping

c = aom

Stiffness-proportional damping

c = a1k

Classical damping (Rayleigh damping)

Stiffness proportional damping appeals to intuition

because it generates damping based on story

deformations. However, mass proportional damping may

be needed as will be shown below.

gu m1kuucum

kmc 10 aa

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In any modal equation, we have

where,

Therefore, ao can be specified to obtain any desired zn for

a given mode n such that Cn = a0 Mn

or

(e.g. at w1 = 2 radians/s, z1 = .05) find a0

Ahmed Elgamal

0qKqCqM nnnnnn

nnnn M2ζC ω

n0nnn MaM2ζ ω nn0 2ζa ω

Mass-proportional damping: c = ao m

Defining a0 to obtain a desired modal damping zn in mode n

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Ahmed Elgamal

With a0 defined by a0 = 2 zn wn, this form of mass proportional

damping will change with frequency according to z = a0 / 2w as

shown in the figure below.

o

2

az

ω

z

nω

mc oa

1ω 2ω 4ω3ω

o

2

a

nω

mc oa

1ω 2ω 4ω3ω

27

In any modal equation, we have

where, and

Therefore, ao can be specified to obtain any desired n for

a given mode n such that Cn = a1Kn , or:

or

(e.g. at w1 = 2 radians/s, z1 = .05) find a1

Ahmed Elgamal

0qKqCqM nnnnnn

n

2

nn MK ωnnnn M2ζC ω

n1n MaMζ 2

nn ωω2 n nn1 2ζa /ω

Stiffness-proportional damping: c = a1 k

Defining a1 to obtain a desired modal damping zn

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Ahmed Elgamal

With a1 defined by a1 = 2 zn / wn , this form of stiffness proportional

damping will change with frequency according to z = a1w / 2 as shown

in the figure below (damping increases linearly with frequency.

kc 1a

2

az 1ω

z

ω1ω 2ω

3ω29

Physically, we often observe (in first approximation) a nearly equal

value of damping for the first few modes of structural response

(e.g., first 1- 4 modes or so), and we want to model that. Therefore,

we use (Classical or Rayleigh damping):

Now we choose damping ratios zi and zj for two modes (natural

frequencies wi and wj) and solve for the coefficients a0 and a1 (two

equations in two unknowns).

Ahmed Elgamal

kmc 10 aa

n1n0n MaMaMζ 2

nn ωω2 n

n

2

n ωω 2/)aa(ζ 10 n

)2/a()2/a(ζ 10 nn ωω n

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Ahmed Elgamal

z

ω

Frequency range

of interest

iω jω

w1 for example 2nd or , 3rd resonance for example

Combined

Stiffness-proportional damping

Mass-proportional damping

nearly uniform damping

kc 1a

mc 0a

kmc 10 aa

Variation of Classical (Rayleigh) Damping with Frequency

Damping defined by z = (a0/2w)+(a1w/2) results in the variation shown by the

combined curve below, which has the desirable feature of being somewhat uniform

within a frequency range of interest (say 1 Hz to 7 Hz or 2 to 14 in radians/s).

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Notes

1) For a choice of zi = zj = zsame damping ratio in the

two modes, we get

,

2) Classical damping and is attractive because of

combination of mass and stiffness, allowing the no-

damping free-vibration mode shapes to un-couple the

matrix equation of motion.

Ahmed Elgamal

ji

ji

0

2ζa

ωω

ωω

ji

1

2ζa

ωω

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Ahmed Elgamal

Caughey damping

The above procedure was generalized by Caughey to allow for more

control over damping in the specified modes of interest (i.e. to be

able to specify z for more than 2 modes i and j)

In this generalization, you can stay within the scope of classical

damping by using

1N

0i

i 1

ia kmmc

to find coefficients to match zi modal damping ratios, see for

instance “Dynamics of Structures” by A. Chopra.ia

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Ahmed Elgamal

Disadvantages:

1. c can become a full matrix instead of being a banded

matrix (if m and k are banded) as with c = a0m + a1k

2. You must check to ensure that you don’t end up with a

negative zi in some mode where you have not specifically

specified damping (because damping variation with

frequency might display sharp oscillations).

In summary, c = a0m + a1k is the usual choice at present

despite the limitations discussed above.

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