L

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Prob. 2 The full answer does depend on the
initial spring length Lo. Solve the problem for
LLo.
L
L
L
L
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Damped Vibrations Equation of Motion
Only two parameters
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Three Classes of Solutions
Overdamped
Damping is strong
Critical Damping
Damping is weak
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DAMPED VIBRATIONS - Strategy for Problem
Solving
Draw F.B.D.
Write down general solution, for appropriate
value of ?
Apply initial conditions to determine constants
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Forced Vibrations
Free Vibrations Provide initial conditions, then
let go
Forced Vibrations External force continuously
driving system
The driving force can be periodic in time
(force oscillates)
or non-periodic
Here study harmonic case sinusoidal force or
displacement in time General cases study
numerically or analytically building on harmonic
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Types of Forcing
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Resonance
When forcing frequency ? is near a natural
frequency ?n of the system, amplitude of the
motion becomes very large At ? ?n , amplitude
only limited by damping
This is Resonance
The effect of forcing on the system is a matter
of timing
Pushing bowling ball pendulum will increase the
oscillation amplitude IF we push in sequence
with the natural frequency of the swing (Pushing
in this way always adds energy to the system)
Pushing out of sequence, the force sometimes
opposes motion and slows it down (extracts energy)
With damping, damping force increases as speed
increases, so eventually a steady-state
amplitude is achieved where energy put in by the
forcing is exactly balanced by the energy
extracted by the damping
If damping is small, steady-state amplitude can
be very large
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Resonance Sneak Preview
Resonance can have catastrophic effects
but is critical to many technologies
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Analyze basic system
mass, spring, dashpot, and harmonic
force
MATLAB Lets just look at the numerical
solutions..
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Finding a Particular Solution
Guess
i.e. displacement will be at same frequency as
force, but perhaps out of phase
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Finding a Particular Solution
(1)
(2)
M (Dynamic Amplitude of Vibration)/(Static
Amplitude)
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The Complete Solution
For z lt 1 (underdamped case), the complete
solution can be written as
Always dies out over time
Persists forever
Transient solution
Steady-state solution
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Steady State Solution
Define Magnification Factor
(Dynamic Amplitude of Vibration)/(Static
Amplitude)
Function only of two quantities ?/?n and ?
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Magnification Factor
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The Steady State Solution Phase
What is
when ?
90o, regardless of z !
? The displacement is 90o out of phase with the
force when
Well, look at the velocity
Significant?
So the velocity is ALWAYS 90o out of phase with
the displacement (at any forcing frequency) since
sine is 90o out of phase with cosine.
So when
The velocity is IN-PHASE with the force!
? When the mass is moving to the right, the force
is to the right when the mass is moving to the
left, the force is to the left (or up/down etc)
? When we force the system at its natural
frequency, energy is being added to the system in
the most efficient way possible we are not
fighting the natural dynamics of the system
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Types of Forcing
All of these situations are of practical
interest. Some subtle but important distinctions
to consider, so we will look at each. BUT
Strategy is simple derive Equation of Motion
and put into the Standard Form
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Base Excitation the Earthquake Problem
Here, base supporting object is subjected to
motion. How does the object respond?
Draw F.B.D. and get equation of motion.
Forces in the spring, dashpot are proportional to
the motion RELATIVE to the base
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Now in the standard form but with a new
driving force
(Displacement Amplitude of body)/(Displacement
Amplitude of Base)
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Harmonic Base Excitation
Displacement transmission ratio