Fundamentals of Fundamentals of Structural Vibration Speaker: - - PDF document

SLIDE 1

Fundamentals of Fundamentals of Structural Vibration

Speaker: Speaker:

• Prof. FUNG Tat Ching

Date & Time: Wed 20 August 2014, 1:30 - 5:30 pm

School of Civil and Environmental Engineering N T h l i l U i it

Venue: CEE Seminar Room D (N1-B4C-09B)

1

Nanyang Technological University

Topics in Fundamentals of Structural Vibration (1.5 hrs)

SDoF Systems

Dynamic Equilibrium

MDoF Systems

Mode Shapes

y q

Natural Freq/Period Damping ratio

p

Modal decomposition Modal responses

Damping ratio

Phase lag Disp Resp Factor

Modal responses

Disp Resp Factor Response Spectrum

2

SLIDE 2

Course Outline for CV6103

Lecture Course Content Chapters in Textbook 1 Si l D f F d S t

(21 hours)

1 Single-Degree-of-Freedom Systems Equations of motion Free vibration 1.2 – 1.6 2.1 – 2.2 2 Single-Degree-of-Freedom Systems Response to harmonic and periodic excitations 3.1 – 3.4 3.12 – 3.13 3 Single-Degree-of-Freedom Systems g g y Response to arbitrary, step and pulse excitations 4.1 – 4.11 4 Multi-Degree-of-Freedom Systems Equations of motion 9.1 – 9.2 Equations of motion Natural vibration frequencies and modes 9.1 9.2 10.1 – 10.7 5 Multi-Degree-of-Freedom Systems Free vibration response 10 8 – 10 15 Free vibration response 10.8 – 10.15 6 Multi-Degree-of-Freedom Systems Forced vibration response 12.1 – 12.7 7 S t ith G li d D f F d

3

7 Systems with Generalized Degrees of Freedom Generalized coordinates and their applications 14.3, 17.1

Textbook and References

• Main Text

Ch A K D i f St t Th

• Chopra, A. K., Dynamics of Structures: Theory

and Applications to Earthquake Engineering, Prentice Hall 4rd Edition 2011 Prentice Hall, 4rd Edition, 2011.

• References
• References
• Clough, R. W., and Penzien, J.,

Dynamics of Structures, Dynamics of Structures, McGraw-Hill, 1993.

• Meirovitch, L. Fundamentals of

, Vibrations, McGraw-Hill, 2001.

4

SLIDE 3

Why Is There A Need To Do Dynamic Analysis?

• Static analysis
• External Load = Internal Force
• External Load Internal Force

Magnitude of loading & stiffness

Dynamic analysis

• Dynamic analysis
• External Load ≠ Internal Force

f & ff

Magnitude of loading & stiffness Frequency characteristics of loading, and the dynamic

properties of structures (mass stiffness damping) properties of structures (mass, stiffness, damping)

5

Examples of SDOF Systems

• Water tank
• Mass concentrated at one location
• Mass concentrated at one location
• Supports assumed to be massless
• C

b d l d SDOF t

• Can be modeled as a SDOF system
• Pendulum
• Rod is assumed to be massless
• Only allowed to rotate about hinge
• Can be modelled as a SDOF system

6

• Can be modelled as a SDOF system

SLIDE 4

Equation of Motion

Damping force Restoring force

p f f u m

S D

= + + & &

External force

Newton’s Second Law of Motion:

External force

u m f f p

D S

& & = − −

D’Alembert’s Principle (Dynamic equilibrium):

Inertia force

(Dynamic equilibrium):

u m f f f f p

I I D S

& & = = − − − with

7

Single-degree-of-freedom

c u(t)

(SDoF) Systems

m

(mass)

p(t)

Typical representation:

(mass)

p( ) k

Mass-spring-damper system External force p(t) Assumptions: External force p(t) Restoring force ku

d

Linear elastic restoring force Li i d i Damping force

2

du

t u c d d

Linear viscous damping I ti f

) (t k & & &

2

dt m

8

Inertia force

) (t p ku u c u m = + +

SLIDE 5

Undamped Free Vibration

• Equation of Motion:

) (t p ku u c u m = + + & & &

• with c = 0, p(t) = 0

k ω

+ ku u m & &

2

+ u u ω & &

• Initial conditions:

) ( ), ( u u u u & & = =

m

n =

ω

= + ku u m

= + u u

n

ω

• r
• Exact Solution:

u ) ( & t u t u t u

n n n

ω ω ω sin ) ( cos ) ( ) ( + =

2

⎞ ⎛

See Page 46 in Chopra

( )

θ ω − = t u t u

n

cos ) (

max

• r

9

( )

2 2 max

) ( ) ( ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + =

n

u u u ω & ) ( ) ( tan 1 u u

n

ω θ &

−

=

Free Vibration of a System without y Damping

umax

umax umax 10

SLIDE 6

Periods of Vibration of Common Structures

Common Structures Period 20-story moment resisting frame 1.9 sec 10-story moment resisting frame 1.1 sec 1-story moment resisting frame 0.15 sec 20-story braced frame 1.3 sec 10-story braced frame 0.8 sec 1-story braced frame 0.1 sec Gravity dam 0.2 sec

11

Suspension bridge 20 sec

Viscously Damped Free Vibration

= + + ku u c u m & & & k c & & &

Equation of Motion:

= + + u m u m u

Divided by m:

c c

Let

cr n

c c m c = = ω ζ 2

(reasons will be clear later)

zeta

2

2

= + + u u u

n n

ω ζω & & &

Hence

m k

n =

ω

Note: as before

km m c

n cr

2 2 = = ω

c

12

critical damping ratio

SLIDE 7

Type of Motion

ζ :damping ratio

Three scenarios ζ

ccr : critical damping coefficient

• ζ < 1, i.e. c < ccr ⇒ under-damped (oscillating)
• ζ = 1, i.e. c = ccr ⇒ critically damped

13

• ζ > 1, i.e. c > ccr ⇒ over-damped

Typical Damping Ratio

Structure ζ Welded steel frame 0.010 e ded stee a e 0 0 0 Bolted steel frame 0.020 Uncracked prestressed concrete 0.015 Uncracked reinforced concrete 0.020 Cracked reinforced concrete 0 035 Cracked reinforced concrete 0.035 Glued plywood shear wall 0.100 p y Nailed plywood shear wall 0.150 Damaged steel structure 0.050 Damaged concrete structure 0.075

14

Structure with added damping 0.250

SLIDE 8

Effects of Damping in Free Vibration

⎟ ⎞ ⎜ ⎛ + u u

t

ζω

ζω

i ) ( ) ( ) ( ) ( &

2

1 ζ ω ω =

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + + =

−

t u u t u e t u

D D n D t

n

ω ω ζω ω

ζω

sin ) ( ) ( cos ) ( ) (

1 ζ ω ω − =

n D

T

15

2

1 ζ − =

n D

T T

max

u = ρ

Decay of Motion

One way to measure damping is from rate of decay from free vibration TD decay from free vibration TD

( )

θ ω

ζω

− =

−

t u e t u

D t

n

cos ) (

max

(exactly)

( )

⎟ ⎞ ⎜ ⎛

−

2 exp exp πζ ζω

ζω t i

T e u

n

Since peaks are separated by TD ,

( y) ( )

( )

⎟ ⎟ ⎠ ⎜ ⎜ ⎝ − = = =

+ − + 2 1

1 exp exp ζ ζ ζω

ζω D n T t i i

T e u

D n

16

SLIDE 9

Type of Excitations

• Harmonic / Periodic Excitations
• Commonly encountered in engineering

Steady-state Responses

• Commonly encountered in engineering

Unbalanced rotating machinery Wave loading Wave loading

• Basic components in more general periodic

excitations excitations

Fourier series representation

More General Excitations

T i t

• More General Excitations
• Step/Ramp Forces

Transient Responses

• Pulses Excitations

17

Equation of Motion

) (t p ku u c u m = + + & & &

Resonance

• Linear
• ⇒u(t) can be replaced
• ⇒u(t) can be replaced

by u(t) + any free vibration responses p

• For example,

c = 0 u(t) = u sinωt

2 / >

n

ω ω

Slowly Rapidly loaded

• c = 0, u(t) = u0sinωt
• (-mω2 + k) u0 sinωt = p(t)

loaded loaded

• ⇒ p(t) = p0 sinωt

1 p

k

18

( )

2

/ 1

n

k p u ω ω − =

m k

n =

ω

SLIDE 10

Excitation

t p ω sin t p ω sin

u

n

) ( 2 . ω ω = = k p u u

n

) ( ) ( ω = = &

Response

k ( )

t k p t up ω ω ω sin / 1 1 ) (

2

= 19

( )

k

n

ω ω / 1−

Undamped Resonant Systems

• For resonance, ω = ωn

( )

t t t p t u ω ω ω sin cos 1 ) ( =

Derivation: See Page 70 & 72 in Chopra

• Response grows indefinitely

( )

t t t k t u

n n n

ω ω ω sin cos 2 ) ( − − =

) ( , ) ( = = u u &

p g y

• Becomes infinite after infinite duration

20

SLIDE 11

Harmonic Vibration of Viscous Damping

Equation of Motion

Sinusoidal force

t p ku u c u m ω sin = + + & & &

Equation of Motion

m k

n =

ω

Excitation frequency Amplitude of force

S

t p ku u c u m ω sin + + m

Amplitude of force

Particular Solution

t D t C t up ω ω cos sin ) ( + =

Derivation: See Page 73 in Chopra

2 2 2 2

] / 2 [ ] ) / ( 1 [ ) / ( 1

n

k p C ζ ω ω + − =

p

Page 73 in Chopra

2 2 2

] / 2 [ ] ) / ( 1 [

n n

k ω ζω ω ω + − / 2

n

p D ω ζω − =

2 2 2

] / 2 [ ] ) / ( 1 [

n n

k D ω ζω ω ω + − =

21

Steady-State Solution

The particular solution can also be written as:

( )

φ ω = t u t u sin ) (

2 2

D C u + =

( )

φ ω − = t u t up sin ) (

where

D tan 1 − =

−

φ D C u +

Using previously derived results for C and D, where

C tan φ

[ ]

2 max

1 k p u u = =

( )

[ ]

( ) [ ]

2 2 2 max

/ 2 / 1

n n

k ω ω ζ ω ω + −

( ) ( )

2 1

/ 1 / 2 tan

n

ω ω ω ω ζ φ =

−

22

( )

/ 1

n

ω ω −

SLIDE 12

2 . = ζ

• Static response exactly in phase with force
• Static response exactly in-phase with force
• Dynamic response has a time lag, φ/ω

23

General Solution

Complementary solution is the free damped vibration response: vibration response:

( )

t B t A e t u

D D t

n

ω ω

ζω

sin cos ) ( + =

−

2

1 ζ ω ω − =

n D

( )

t B t A e t u

D D c

ω ω sin cos ) ( +

Complete solution:

Recall: A, B derived by satisfying initial conditions

) ( ) ( ) ( t u t u t u

p c

+ =

satisfying initial conditions

( )

t D t C t B t A e

D D t

n

ω ω ω ω

ζω

cos sin sin cos + + + =

−

Transient Steady state

24 Derivation: See Page 73 in Chopra

SLIDE 13

Example 1

ω /ωn= 0.2, ζ = 0.05, u(0) = 0, k p u

n

/ ) ( ω = &

Observe how the transient response decays due to damping, leaving only the steady state part

25

Example 2

ω = ωn (resonant response), ζ = 0.05 With d i h

st

u

With damping, response approach

ζ 2

max st

u =

26

But can still be larger for another value of ω!

SLIDE 14

Example 3

27

Significance of Steady-state Solutions

• In certain problems, e.g. wave loads on an
• ffshore structure, the load is assumed to be
• ffshore structure, the load is assumed to be

in place for a sufficiently long time, so that the transient response has completely decayed transient response has completely decayed.

• The interest is in the steady-state solution.

28

SLIDE 15

Maximum Response and Phase Angle

( )

t t i ) (

St d t t l ti i

( )

φ ω − = t u t up sin ) (

Steady state solution is

1

( )

[ ]

( ) [ ]

2 2 2

/ 2 / 1 1 k p u ω ω ζ ω ω + =

( )

[ ]

( ) [ ]

/ 2 / 1

n n

ω ω ζ ω ω + −

( )

/ 2 ω ω ζ

Called Deformation Response Factor (DRF) Rd in Chopra

( ) ( )

2 1

/ 1 / 2 tan

n n

ω ω ω ω ζ φ − =

−

( )

d

p textbook (Page 76)

( )

n

φ is the phase lag

29

Time lag = φ/ω

DRF and Phase

R

For ω /ωn << 1

• Slowly varying

Resonance

y y g

• DRF ≅ 1
• u0 ≅ p0/k
• Displacement in-phase with force
• Response dominated by stiffness

DRF

For ω /ωn >> 1

• R

idl i

D

Slowly Rapidly loaded

• Rapidly varying
• DRF → 0

φ

2

ω p p

n ⎟

⎞ ⎜ ⎛

loaded loaded

• Displacement anti-phase with force

Phase φ

2

ω ω ω m p k p u

n

= ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ≅

• Response dominated by mass

P

30

SLIDE 16

When ω ≈ ωn

• Forcing freq ≈ natural freq
• Resonance

DAF is very large close to max at ω

• DAF is very large, close to max at ωn
• p

k p u

0 /

= ≅ ( )

/ = k p u

• Response dominated by damping

n

c u ω ζ 2 = ≅ ( )

2 max

1 2 ζ ζ − u

• Response dominated by damping
• Displacement is 90° out of phase with force

when ω = ωn

• This is the scenario we want to avoid! (but

This is the scenario we want to avoid! (but not always possible)

31

Periodic Excitation

p t

) ( ) ( t jT t

T0 T0 T0

/ 2 T π ω =

Separate into harmonic components using Fourier series

) ( ) ( t p jT t p = +

j : integer in (-∞, ∞ ) Separate into harmonic components using Fourier series

) sin( ) cos( ) ( t j b t j a a t p

j j

ω ω

∑ ∑

∞ ∞

+ + = ) ( ) ( ) (

1 1

j j p

j j j j

∑ ∑

= =

Note: Arbitrary excitations can also be transformed into Fourier

32

Note: Arbitrary excitations can also be transformed into Fourier

series with appropriate technique, such as FFT.

SLIDE 17

Response To Arbitrary Time- Varying Forces

E ti f M ti

) (t p ku u c u m = + + & & &

Equation of Motion Initial conditions:

) ( p

) ( , ) ( = = u u &

p(t): varying arbitrarily with time p(t): varying arbitrarily with time e.g. step forces (with finite rise time), pulses etc pulses etc. Interested in the max response.

33

p Max response Response/Shock spectrum

Simple Examples

p p Step Force p0 ) ( ≥ = t p t p t Ramp or li l p p linearly increasing force p0 ) ( ≥ = t t t p t p

r

Step force t tr p Step force with finite rise time p0

( )

⎩ ⎨ ⎧ ≥ ≤ = / ) (

r r

t t p t t t t p t p t tr ⎩ ≥

r

t t p

34

SLIDE 18

Dynamic Response to Step Forces

⎤ ⎡ ⎞ ⎛ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − + − =

−

t t e k p t u

D D t

n

ω ζ ζ ω

ζω

sin 1 cos 1 ) (

2

⎦ ⎣ ⎠ ⎝ ζ 1

35

Dynamic Response of Ramp

t p0

• r Linearly Increasing Force

⎞ ⎛ t t p sin

r

t p0

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − =

r n n r

t t t t k p t u ω ω sin ) (

p ) ( = =

st

k p u ζ 5 . 2 =

n r

T t ζ

n

36

SLIDE 19

Dynamic Response to Step

p

Force With Finite Rise Time

p0

( )

⎩ ⎨ ⎧ ≥ ≤ = / ) (

r r r

t t p t t t t p t p

t tr

⎩

r

t t p

Phase 1 Phase 2

Consider undamped response:

Phase 1 Phase 2

1. Ramp phase:

r n

t t t t t t k p t u ≤ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − = sin ) ( ω ω

Consider undamped response:

r n r

t t k ⎟ ⎠ ⎜ ⎝ ω p ⎫ ⎧ 1 2. Constant phase:

( ) [ ]

r r n n r n

t t t t t t k p t u > ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ − − − = ω ω ω sin sin 1 1 ) (

37

Step Force With Finite Rise Time

38

SLIDE 20

Maximum Deformation

T t ) / sin(π

n r n r d

T t T t R / ) / sin( 1 π π + =

39

Response Spectrum

Single Pulse Excitations

F

• Example of pulse excitations:
• underground explosions

Force Blast

• verpressure
• underground explosions
• Idealized by simple shapes

E

Time

• E.g.

40

SLIDE 21

Response Spectrum

⎨ ⎧ < = =

2 1

/ / sin 2

n d n d

T t T t u R π ⎩ ⎨ ≥ = =

2 1 ,

/ 2

n d st d

T t u R

Also called Shock Spectrum for single pulse

41

Also called Shock Spectrum for single pulse

Multi-Degree-of-Freedom Systems

p (t) k c (t) u (t) p (t) p (t)

4 4 4 4 3

k c u (t) u (t) p ( ) p (t)

3 3 2 3 3 2

k c u (t) u (t) p (t)

2 2 1 2 1

k c

1 1 1

42

SLIDE 22

General Approach for Complex Structures

• Elastic resisting forces
• Same as static analysis (i e ku=p)
• Same as static analysis (i.e. ku p)
• fS = ku

D i f ll th i l

• Damping forces: usually rather simple
• u

c fD & =

• Inertia forces: usually simple
• u

m fI & & =

• Equations of motion

u m fI ) ( ) ( ) ( ) ( t t t t k & & &

43

) ( ) ( ) ( ) ( t t t t p ku u c u m = + + & & &

Arbitrary u(0)

• NOT Simple Harmonic Motion (SHM)
• Frequency of motion cannot be defined

Modal coordinators

• u1 and u2 are not proportional

(⇒deflected shapes varies with time)

Modal coordinators

44

SLIDE 23

When u(0) = φ1

• can be SHM with appropriate initial conditions
• u1 proportional to u2

1 2

• φ1 is a natural mode

45

When u(0) = φ2

• can be SHM with appropriate initial conditions
• u1 proportional to u2

1 2

• φ2 is a natural mode

46

SLIDE 24

Natural Frequency

The natural period of vibration Tn = the time required for one cycle of the harmonic motion in q y

• ne of these natural modes.

π 2 1

n n

T ω π 2 =

n n

T f 1 =

f = natural cyclic frequency of vibration fn = natural cyclic frequency of vibration ωn = natural circular frequency of vibration An N-DOF system has N number of natural periods and N number of natural modes and N number of natural modes.

47

How to Find the natural

periods and natural modes?

• EoM:
• Natural frequency ω and mode shape φ can

) ( ) ( ) ( ) ( t t t t p ku u c u m = + + & & &

• Natural frequency ωn and mode shape φn can

be obtained by solving the following eigenvalue problem eigenvalue problem

m k = −

n n

φ ] [

2

ω

• Different modes can be shown to be
• rthogonal wrt the m and k matrices, i.e.
• rthogonal wrt the m and k matrices, i.e.

=

r T n

φ φ m =

r T n φ

φ k (n ≠ r)

48 n T n n n T n n

K Μ φ φ φ φ k m = = ,

SLIDE 25

Vibration Analysis of MDOF Systems

• EoM: with
• Time Stepping Methods

) ( ) ( ) ( ) ( t t t t p ku u c u m = + + & & & ) ( and ) ( u u &

• Time Stepping Methods
• E.g. Central Difference, Newmark’s method etc.
• Modal Decomposition
• MDoF problem ⇒ a number of SDoF problems

u ⎫ ⎧ + + + =

N Nq

q q L

2 2 1 1

φ φ φ u Φ q

1 −

= Φq = ⎪ ⎬ ⎫ ⎪ ⎨ ⎧ = = ∑

n N r r

q q M L

1 1

] [ φ φ φ

T T n n

q φ φ φ m mu =

49

q ⎪ ⎭ ⎬ ⎪ ⎩ ⎨

∑

= N n r r r

q q

1 1

] [ φ φ φ

n n

φ φ m

Uncoupled Equations

• EoM:

with

• Transformed into N SDoF Systems, each

) ( ) ( ) ( ) ( t t t t p ku u c u m = + + & & & ) ( and ) ( u u &

y ,

) ( ) ( ) ( ) ( t P t q K t q C t q M

n n n n n n n

= + + & & &

classical damping

t P ) ( ) ( , , t P K Μ

T n n n T n n n T n n

p k m φ φ φ φ φ = = =

n T r nr

C φ φ c = K

With initial conditions

n n n n n n n n

M t P t q t q t q ) ( ) ( ) ( 2 ) (

2

= + + ω ω ζ & & &

n n n

Μ K = ω

• With initial conditions

T n

φ mu ) (

T n

φ u m ) ( & &

50 n T n n n

q φ φ φ m ) (

, = n T n n n

q φ φ φ m u m ) (

,

& =

SLIDE 26

Displacement Responses

Once q1(t), …, qN(t) are determined, the response u1(t) uN(t) in u(t) can be obtained response u1(t), …, uN(t) in u(t) can be obtained from

∑

=

N

t q t ) ( ) ( φ u

∑

=

=

n n n

t q t

1

) ( ) ( φ u and subsequently the internal forces can also and subsequently the internal forces can also be calculated if required Caution: The expression could be very lengthy.

51

Modal Contribution

It is useful to define the contribution of the nth mode to u(t) as ) ( ) ( t q t

n n n

φ = u mode to u(t) as Then internal force due to un(t) can be evaluated first and then sum up for all the modes later first and then sum up for all the modes later. Further Improvement p Since qn(t) is a scalar function, the internal force due to φn can be evaluated first (static analysis) and φn ( y ) then times qn(t) before sum up for all the modes later.

52

SLIDE 27

How To Calculate the Internal Forces?

• Directly from u(t) or un(t) or φn

Alternatively the same internal forces can be

• Alternatively, the same internal forces can be
• btained by considering the same structure

s bjected to the eq i alent static forces k (t) subjected to the equivalent static forces ku(t)

• r kun(t) or kφn

53

Equivalent Static Force

) ( ) ( ) ( ) ( t t t t k & & & ) ( ) ( ) ( t t t u c u m p & & & − − ) ( ) ( t t

S

ku f = ) ( ) ( ) ( ) ( t t t t p ku u c u m = + +

p5(t) m5 m k5 u5(t) fS5(t) k5 ) (

5 5

t u m & & p4(t) m4 m k4 u4(t) fS4(t) k4 ) (

4 4

t u m & & p3(t) p (t) m3 m2 k3 u3(t) fS3(t) (t) f (t) k3 ) (

3 3

t u m & & ) (t & & p2(t) p (t) m1 k2 u2(t) fS2(t) u (t) f (t) k2 ) (

2 2

t u m ) (t u m & & p1(t) k1 u1(t) fS1(t) k1 ) (

1 1

t u m 54 V(t) V(t)

SLIDE 28

) ( ) ( ) ( ) (

2

t q t q t t

n n n n n n n

φ φ m k ku f ω = = =

1 2 1 1 or

: Forces φ φ m k ω

1 1 1

• r

: Forces φ φ m k ω q1(t) M1 P1(t) r (t) = r st q (t) K1, ζ1 r1

st

r1(t) = r1 q1(t)

2 N N N

φ φ m k

2

• r

: Forces ω

qN(t) MN PN(t) KN, ζN rN

st

rN(t) = rN

st qN(t)

55

Recap

SDOF Systems

Dynamic Equilibrium

MDOF Systems

Mode Shapes

y q

Natural Freq/Period Damping ratio

p

Modal decomposition Modal responses

Damping ratio

Phase lag DAF/DRF

Modal responses

DAF/DRF Response Spectrum

56