kind of damping which stops the oscillation in the shortest possible time
Answers
Answer:
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Explanation:
The simple harmonic oscillator describes many physical systems throughout the world, but early studies of physics usually only consider ideal situations that do not involve friction. In the real world, however, frictional forces – such as air resistance – will slow, or dampen, the motion of an object. Sometimes, these dampening forces are strong enough to return an object to equilibrium over time.
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Damped Harmonic Motion: Illustrating the position against time of our object moving in simple harmonic motion. We see that for small damping, the amplitude of our motion slowly decreases over time.
The simplest and most commonly seen case occurs when the frictional force is proportional to an object’s velocity. Note that other cases exist which may lead to nonlinear equations which go beyond the scope of this example.
Consider an object of mass m attached to a spring of constant k. Let the damping force be proportional to the mass’ velocity by a proportionality constant, b, called the vicious damping coefficient. We can describe this situation using Newton’s second law, which leads to a second order, linear, homogeneous, ordinary differential equation. We simply add a term describing the damping force to our already familiar equation describing a simple harmonic oscillator to describe the general case of damped harmonic motion.
F
net
=
m
d
2
x
dt
2
+
b
dx
dt
+
kx
=
0
=
d
2
x
dt
2
+
b
m
dx
dt
+
k
m
x
=
0
=
d
2
x
dt
2
+
γ
dx
dt
+
ω
2
0
x
=
0
ω
2
0
=
k
m
,
γ
=
b
m
Fnet=md2xdt2+bdxdt+kx=0=d2xdt2+bmdxdt+kmx=0=d2xdt2+γdxdt+ω02x=0ω02=km,γ=bm
This notation uses
d
2
x
dt
2
d2xdt2, the acceleration of our object,
dx
dt
dxdt, the velocity of our object,
ω
0
ω0, undamped angular frequency of oscillation, and ɣ, which we can call the damping ratio.
The type of damping that stops oscillation in the shortest possible time is called "critical damping."
Critical damping occurs when a damped harmonic oscillator returns to its equilibrium position without oscillating back and forth. In other words, it is the minimum amount of damping required to eliminate all oscillations as quickly as possible.
Mathematically, critical damping occurs when the damping factor (b) is equal to the square root of 4 times the mass (m) times the spring constant (k), or b = 2sqrt(mk). Any greater amount of damping will cause the oscillator to return to equilibrium more slowly, while any less damping will cause the oscillator to overshoot and oscillate before returning to equilibrium.
Critical damping is important in a variety of engineering applications, such as controlling the motion of vehicles, machinery, and buildings subjected to vibrations. By adjusting the amount of damping, engineers can ensure that the system returns to equilibrium quickly and smoothly, without causing excess wear and tear on the system.
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