Lecture 15 Discrete Forcing

Rose-Hulman Institute of TechnologyMechanical Engineering

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Today’s Objectives:

Students will be able to:

a) Integrate any single degree of freedom system with transient or discrete forcing

b) Have a basic understanding of response spectra

Transient dynamics and discrete forcing


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-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0 5 10 15 20 25 30 35

1979 El Centro Earthquake


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Discrete case (assumed periodic)

In the “real world” things aren’t regular and you may only have experimental data

-1.5

0

1.5

0 0.5 1 1.5 2

Time (s)

Forc

e

-1.5

0

1.5

0 0.5 1 1.5 2

Time (s)

Forc

e

T = N∆tN = number of data points

Note: We only have f(t) at a discrete number of data points. If we are given an acceleration time history we usually integrate it twice to get displacement. This is often done with earthquakes.


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We will use Simulink

Input time values

Input force values

You can also use data in your workspace

“repeating sequence” block


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Motion under a non-periodic (transient) force

Methods– Fourier Integral/Transform– Convolution Integral– Laplace Transform– Numerical methods (Simulink)

( ) ( ) ( ) ( )∫ −= −−t

t

d

dteFm

tx n

0dsin1 ττωτ

ωτζω

Convolution (or Duhamel) Integral (discussed more tomorrow)

• Represents the response of an underdamped SDOF system to an arbitrary excitation, f(t)• Can be solve analytically or numerically• Several examples are in the text


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Response (or shock) Spectrum (4.6 in text)

A plot of the maximum peak response (acceleration, velocity, displacement, etc.) of a SDOF system as a function of the natural frequency (or natural period) of the system.

– Widely used in Earthquake engineering– Used in environmental testing

m

ck

m

ckor

The book has examples of finding the response spectrum analytically

F(t)

y(t)

ωn

Peak

resp

onse


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Examples


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Generated a response spectrum with Matlab

0 0.5 1 1.5 2 2.5 3 3.5 41

1.2

1.4

1.6

1.8

2


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NS component of the 1940 El Centro Imperial Valley Earthquake

y- x z

222

=

−=++

where

yzzz

nnn ωωζ

ω&&

&&&


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Examples (from text)

Lecture 15 Discrete Forcing

Documents

Transcript of Lecture 15 Discrete Forcing