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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



Observed Response of Linear SDOF

(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



Properties of Structural DAMPING (2)

AREA =

ENERGY

DISSIPATED

DAMPING FORCE

DAMPING

DISPLACEMENT

Damping vs Displacement response is

Elliptical for Linear Viscous Damper


CONCEPT of ENERGY ABSORBED and DISSIPATED

F

ENERGY

DISSIPATED

ENERGY

ABSORBED

F

u

u

LOADING

YIELDING

+

ENERGY

RECOVERED

ENERGY

DISSIPATED

F

F

u

u

UNLOADING

UNLOADED


Development of Effective Earthquake Force

Ground Motion Time History


RELATIVE

TOTAL

M

M

Somewhat Meaningless

Total Base Shear


Equation of Motion:

Undamped Free Vibration

Initial Conditions:

Assume:

Solution:


Undamped Free Vibration (2)

T = 0.5 seconds

1.0

Circular Frequency

(radians/sec)

Period of Vibration

(seconds/cycle)

Cyclic Frequency

(cycles/sec, Hertz)


Periods of Vibration of Common Structures

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


Damped Free Vibration

Equation of Motion:

Initial Conditions:

Assume:

Solution:



Undamped Harmonic Loading

Equation of Motion:

= Frequency of the forcing function

= 0.25 Seconds

po=100 kips


Undamped Harmonic Loading

Equation of Motion:

Assume system is initially at rest

Particular Solution:

Complimentary Solution:

Solution:


Undamped Harmonic Loading

LOADING FREQUENCY

Define

Structure’s NATURAL FREQUENCY

Transient Response

(at STRUCTURE Frequency)

Dynamic Magnifier

Steady State

Response

(At LOADING Frequency)

Static Displacement



Response Ratio: Steady State to Static

(Signs Retained)

In Phase

Resonance

180 Degrees Out of Phase


Response Ratio: Steady State to Static

(Absolute Values)

Resonance

Slowly

Loaded

Rapidly

Loaded

1.00


Damped Harmonic Loading

Equation of Motion:

po=100 kips


Damped Harmonic Loading

Equation of Motion:

Assume system is initially at rest

Particular Solution:

Complimentary Solution:

Solution:


Damped Harmonic Loading

Transient Response, Eventually Damps Out

Solution:

Steady State Response



Resonance

Slowly

Loaded

Rapidly

Loaded


Alternative form of the equation of motion
Alternative Form of theEquation of Motion

Equation of Motion:

Divide by m:

but

and

or

Therefore:


General dynamic loading
General Dynamic Loading

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

  • Duhamel’s Integral

  • Time-stepping methods


Development of an Elastic Displacement

Response Spectrum, 5% Damping

El Centro Earthquake Record

Maximum Displacement Response Spectrum

T=0.6 Seconds

T=2.0 Seconds



2

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



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