FREE VIBRATIONRESPONSE E OF A DAMPED SYSTEM UNDER HARMONIC FORCE

Thammasat UniversityCIVIL Engineering Department

MR. BHAKAPONG BHADRAKOM

1

Free Vibration of a Cantilever tower

Mathematical Analysis

The fundamental frequency for the beam with no axial load is

L h

b

fn = the natural frequency (Hz)

E = the modulus of elasticity (N/m2)

I = the area moment of inertia (m4)L = the length (m)m = the mass per unit length (Kg/m)

Fixed - support

= ( )

2

Natural vibration modes and frequencies of cantilever beams

Free Vibration of a Cantilever tower3

Problem 1Cantilever tower, A reinforced-concrete 130m high, has a uniform hollow circular cross section with outside diameter 8 m and wall thickness 0.7 m and the damping is estimated as 2%. The unit weight of concrete is 2549.49 kg/m^3 and its elastic modulus Ec = 30 GPa

Free Vibration of a Cantilever tower4

140m

8 m

0.7 m

1. Determine the natural frequencySolution

8 m

0.7 m

Length : =Cross-section area : = = 4 3.3= 16.Mass/mater : = = . × 16. =.Moment of inertia : = 4 4 3.3 = .Flexural rigidity : = 3.The natural period : = 3.2 = .= = 1 /= 1 = 3.

Free Vibration of a Cantilever tower

5

Frequency (rad/sec)

Frequency (Hz)

The natural period (sec)

Free Vibration of a Cantilever tower

= ( )== 1

Mode Frequency (Hz) Frequency (rad/sec) The natural period (sec)

1 3.516 0.2539 1.5955 3.93792 22.03 1.5910 9.9971 0.62853 61.70 4.4562 27.9992 0.22444 120.90 8.7318 54.8639 0.1145

6

Modal AnalysisSAP2000

1. File menu > New Model

2. Select the Grid Only button, and the form

Modal Analysis SAP2000Begin a New Model

3. The program will appear as shown in Figure. Note that the grids appear in two view windows tiled vertically, an X-Z “Plan” View on the left and a 3-D View on the right.

Modal Analysis SAP2000Begin a New w w Model

Define menu > Materials command to display the Define Materials

Modal Analysis SAP2000Define a a a Material

Define > Section properties > frame section

Modal Analysis SAP2000Define e e Sections

Assign menu > Joint > Restraints command to access the Joint Restraints

Modal Analysis SAP2000Restraints

Modal Analysis SAP2000Analysis s Options

13

Modal Analysis SAP2000Run the e Analysis

Analyze menu > Run Analyze or F5

mode 1 2 3 4

Theory (sec) 3.9379 0.6285 0.2244 0.1145

Sap 2000 (sec) 3.938 0.6284 0.2244 0.11454

Modal Analysis SAP200015

Time-history analysis

16

Time-history analysis

Harmonic Loads

A harmonic force is = , where is the amplitude or maximum value of the force and its frequency is called the exciting frequency or forcing frequency; = is the exciting period or forcing period

17

Time-history analysis

= sin sinh + sin + sinhcos + cosh (cosh + cos )Shape function cantilever beam

, = + + + …..= ++ +

Time-history analysis

=M

M =

M =

= + ( )

= d = .

Time-history analysis

Mode 1

= ( ) d = .M = = , .23 /M = = 14, ,612.M = = 8, , .

Length =Cross-section

area = 16.

Mass/mater = = .Moment of

inertia= .

Flexural rigidity = 3.

= 2

Time-history analysis

Mode 1 = sin sinh + sin 1. + sinh 1.cos 1. + cosh 1. (cosh + cos )Time (t) U(140,t)

= * (t)0 0.000126535 0.000253

0.1 0.014758313 0.0295140.2 0.017222903 0.0344430.3 0.005488423 0.0109760.4 -0.010770875 -0.021540.5 -0.018150345 -0.03630.6 -0.010566136 -0.021130.7 0.005729107 0.011457. . .. . .. . .

300 0.000126535 0.000253

= 1.

-0.04-0.03-0.02-0.01

00.010.020.030.04

200 202 204D

ispla

cem

ent (

m)

Time (sec)

Theroy mode 1

Time-history with SAP2000

Time-history analysis with SAP2000

DEFINE

Functions

Sine

Time History

Step 1Select the ADD FUNCTION

Time-history analysis with SAP2000

DEFINE

LOAD CASE

Model

Add new Load case

Step 2

Time-history analysis with SAP2000

DEFINE

LOAD CASE

Time history

Add new Load case

Step 3

Time-history analysis with SAP2000

Running Analysis

• Select Run from the Analyze menu to analyze the structure or (F5)

Time-history analysis with SAP2000

Display

Show plot function

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

200 202 204 206 208 210

Disp

lace

men

t (m

)

Time (sec)

Sap2000 Theroy mode 1

Time-history analysis with SAP2000

Time-history analysis with SAP2000

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

200 202 204 206 208 210

DISP

LACE

MEN

T (M

)

TIME (SEC)

Sap2000 mode 1+2+3 Theroy mode 1+2+3

Time-history analysis with SAP2000

Work shop

31

Workshop

Length : =area : = =Density : = .Modulus of elasticity : =

hb

A simply supported bridge with a single span of length L has a deck of uniform cross section with mass m per meter length and flexural rigidity EI. Neglecting damping, Determine the natural vibration frequencies for the first two three modes.

32

= = 1. = .643= 4 = 6.222 = .= = 13. = .

Mode 1

Mode 2

Mode 3

Workshop

Mode n Natural vibration frequencies The natural period

Numerical results

Thank you

Thammasat UniversityCIVIL Engineering Department