Structural Dynamics-Material and Structures-Lecture Handout, Exercises of Structures and Materials

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VI.

(Introduction to)

Structural Dynamics

Fall, 2002

Can actually identify three “categories” of response:things also move, this includes structures. Thus far have considered only static response. However,

C. Wave PropagationB. Dynamic A. (Quasi) - Static [“quasi” because the load must first be applied]

What is the key consideration in determining which regime one is in?

--> the frequency of the forcing function

Example

: Mass on a Spring

Figure 19.

Representation of mass on a spring

Fall, 2002

B) Push with an oscillating magnitude

Figure 19.

Representation of force with oscillating magnitude

The response also oscillates

Figure 19.

Representation of oscillating response

C) Whack mass with a hammer

Force is basically a unit impulse

Figure 19.

Representation of unit impulse force

Force has

very high

frequencies

Response is (structural) waves in spring with no global

deflection

--> These are not well-defined borderlines

depends on specifics of configuration

actually transition regions, not borders

interactions between behaviors

Figure 19.8 So illustration is:

Representation of regions of structural response versus

frequency of forcing function

(Structural)

Wave

(Quasi) - Static

Dynamics

Propagation

Static

f(natural

f(speed of

frequency of

waves in

structure)

material)

= region of transition

Fall, 2002

(Structural) Dynamics -- 16.221 (graduate course).Waves -- Unified Statics -- Unified and 16.20 to date

Consider the simplest ones… Look at what we must include/add to our static considerations

The Spring-Mass System

Figure 19.9 Are probably used to seeing it as:

General representation of spring-mass system

q

where:

d

(

)

d t

(derivative with respect to time)

Drawing the free body diagram for this configuration:

Figure 19.

Free body diagram for spring-mass system

∑ F = 0 ⇒ F −

k q

m

m q

k q

F t

system (no damping) Basic spring-mass

This is a 2nd order Ordinary Differential Equation in

time

When the Ordinary/Partial Differential Equation is in

space

, need

Boundary

Conditions. Now that the Differential Equation is in time,

need

Initial

Conditions.

Figure 19.12 constant c which produces a force in proportion to the velocity:For the spring-mass system, this is represented by a dashpot with a

Representation of spring-mass system with damping

[Force/length]

[Force/length/time]

q ˙ q

Multi-Mass System

For example, consider two masses linked by springs:

Each mass has stiffness, (k

i) mass (m

i) and force (F

i) with

associated deflection, q

i

Figure 19.

Representation of multi-mass (and spring) system

Consider the free body diagram for each mass:

q

Mass

Figure 19.

Face body diagram of Mass 1 in multi-mass system

∑

F

yields:

F

1

k 2 (^) ( q 2

− (^) q

1 (^) )

−

k q

− m 1 1 = 0

1

1

q q

Write in matrix form:

m 1

1 (^) 

( k 1

(^) k 2 (^) )

k 2 (^)  ^ q 1 (^) 

F

1 (^) 

m 2 (^)  (^)  ˙˙ 2 (^) 

− k 2 k 2 

q 2 (^) 

F

2 (^) 

or:

mq

˙˙

kq

F

mass

stiffness matrix

matrix

Note that the stiffness matrix is symmetric (as it has been in all

other considerations)

k

ij

k

ji

This formulation can then be extended to 3, 4….n masses with

m

i = mass of unit i

k i = stiffness of spring of unit i

q i = displacement of unit i

F

i = force acting on unit i

etc.

Fall, 2002

First issue -- what causes such response are: about why these considerations are important in structures. Will next consider solutions to this equation. But first talk

Dynamic Structural Loads

Generic sources of dynamic loads:

Wind (especially gusts)

Impact

Unsteady motion (inertial effects)

Servo systems

How are these manifested in particular types of structures?

A response which is comprised of two parts: What does this all result in? Earthquakes and Buildings Civil StructuresAutomobiles, Trains, etc.

rigid-body motion

elastic deformation and vibration of structure

Note that:

of the static valuesPeak dynamic deflections and stresses can be several times that

Dynamic response can (quickly) lead to fatigue failure

(Helicopter = a fatigue machine!)

Discomfort for passengers

(think of a car without springs)

simple spring-mass case and learn: Before dealing with the continuous structural system, first go back to the So there is a clear need to study structural dynamics

Solutions for spring-mass systems

systemHow to model a continuous system as a discrete spring-mass

then…

Extend the concept to a continuous system