Energy of Weaker damped oscillator system
The reduction of the amplitude of the damped oscillator system indicated a continuous decrease in the system’s energy.
The total energy of the system any time ‘t’ is given as
E(t)= K.E + P.E
E(t)=½mv²+ ½kx²
E(t)=½m(dx/dt)²+ ½kx²
Let us consider an underdamped system. Therefore, the displacement of an oscillatory system w.r.t time is given as
X(t)=e-μtsin(qt+θ)
Therefore, dx/dt =d/dt {e-μtsin(qt+θ)}
dx/dt=a0(e-μtqcos(qt+θ)– μe-μtsin(qt+θ))
dx/dt=a0e-μtq(cos(qt+θ) – μ/qsin(qt+θ))….(3)
For underdamping oscillations
(µ<<ω0) and q=√ω0²- μ² = μ<<q = μ/q<<1 = μ/qsin(qt+θ)<<<1
Therefore, equation (3) become
dx/dt=a0e-μtq(cos(qt+θ)….(4)
Using (1).(2)and(4) we have
K.E=½m(dx/dt)²= ½m[a0e-μt.q(cos(qt+θ)]²…..(5)
P.E = ½kx²= ½k[a0e-μt.sin(qt+θ)]²……(6)
Substituting equations (5)and (6) in (1), we get,
E(t) = ½m[a0e-μtq(cos(qt+θ)]² + ½k[a0e-μtsin(qt+θ)]²
E(t) = ½a0²e-2μt(mq²cos²(qt+θ)+k.sin²(qt+θ))
E(t) = ½a0²(e-2μt)m(q²cos²(qt+θ)+k/m.sin²(qt+θ))….(7)
We know that q=√ω0²-μ²=q²=ω0²-μ²
If ω0²>>μ²=q²≈ω0²=k/m
Using this value in equation (7), we get
E(t) = ½a0²(e-2μt).m(k/m.cos²(qt+θ)+k/m.sin²(qt+θ)).
E(t) = ½a0²(e-2μt).k(cos²(qt+θ)+sin²(qt+θ)).
E(t) = ½a0²(e-2μt).k. (at t=0,E(0)=½a0²k=E0)
E(t) = E0(e^-2μt)………(9)
Equation 9 represents the energy of the damping oscillator system at any time t.
As the system continuously works against the damping force, the energy of the damped oscillator system also decreases with time.
Damped Oscillation – Definition, Equation, Types, Examples
Damped Oscillation means the oscillating system experiences a damping force, causing its energy to decrease gradually. The level of damping affects the frequency and period of the oscillations, with very large damping causing the system to slowly move toward equilibrium without oscillating.
In this article, we will look into damped oscillation, damped oscillator, damping force, general equation derivation, application and type of damped oscillation, etc.
Table of Content
- What is Damped Oscillation?
- Damped Oscillation Differential Equation
- Damped Harmonic Oscillator
- Types of Damped Oscillator
- Effects of Damping
- Damped Oscillation Example