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# Idealized Single Degree of Freedom Structure - PowerPoint PPT Presentation

Idealized Single Degree of Freedom Structure. F(t). Mass. t. Damping. Stiffness. u(t). t. Equation of Dynamic Equilibrium. Observed Response of Linear SDOF ( Development of Equilibrium Equation ). Damping Force, Kips. Inertial Force, kips. Spring Force, kips. SLOPE = k = 50 kip/in.

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Idealized Single Degree of Freedom Structure

F(t)

Mass

t

Damping

Stiffness

u(t)

t

(Development of Equilibrium Equation)

• Damping Force, Kips

Inertial Force, kips

Spring Force, kips

SLOPE = k

= 50 kip/in

SLOPE = c

= 0.254 kip-sec/in

SLOPE = m

= 0.130 kip-sec2/in

AREA =

ENERGY

DISSIPATED

DAMPING FORCE

DAMPING

DISPLACEMENT

Damping vs Displacement response is

Elliptical for Linear Viscous Damper

F

ENERGY

DISSIPATED

ENERGY

ABSORBED

F

u

u

YIELDING

+

ENERGY

RECOVERED

ENERGY

DISSIPATED

F

F

u

u

Ground Motion Time History

TOTAL

M

M

Somewhat Meaningless

Total Base Shear

Undamped Free Vibration

Initial Conditions:

Assume:

Solution:

T = 0.5 seconds

1.0

Circular Frequency

Period of Vibration

(seconds/cycle)

Cyclic Frequency

(cycles/sec, Hertz)

20 story moment resisting frame T=2.2 sec.

10 story moment resisting frame T=1.4 sec.

1 story moment resisting frame T=0.2 sec

20 story braced frame T=1.6 sec

10 story braced frame T=0.9 sec

1 story braced frame T=0.1 sec

Equation of Motion:

Initial Conditions:

Assume:

Solution:

Equation of Motion:

= Frequency of the forcing function

= 0.25 Seconds

po=100 kips

Equation of Motion:

Assume system is initially at rest

Particular Solution:

Complimentary Solution:

Solution:

Define

Structure’s NATURAL FREQUENCY

Transient Response

(at STRUCTURE Frequency)

Dynamic Magnifier

Response

Static Displacement

Linear Envelope

(Signs Retained)

In Phase

Resonance

180 Degrees Out of Phase

(Absolute Values)

Resonance

Slowly

Rapidly

1.00

Equation of Motion:

po=100 kips

Equation of Motion:

Assume system is initially at rest

Particular Solution:

Complimentary Solution:

Solution:

Transient Response, Eventually Damps Out

Solution:

Slowly

Rapidly

Alternative Form of theEquation of Motion

Equation of Motion:

Divide by m:

but

and

or

Therefore:

For SDOF systems subject to general dynamic loads, response may be obtained by:

• Duhamel’s Integral

• Time-stepping methods

Response Spectrum, 5% Damping

El Centro Earthquake Record

Maximum Displacement Response Spectrum

T=0.6 Seconds

T=2.0 Seconds

2

3

1

3

1

NEHRP Recommended Provisions

Use a Smoothed Design Acceleration Spectrum

“Short Period” Acceleration

SDS

“Long Period” Acceleration

Spectral Response Acceleration, Sa

SD1

T0

TS

T = 1.0

Period, T