Question

What is the damped vibrations? Establish the differential equation of motion for damped harmonic oscillator​

Answer 1

Answer:

Damped harmonic oscillators are vibrating systems for which the amplitude of vibration decreases over time. ... These are second-order ordinary differential equations which include a term proportional to the first derivative of the amplitude.

Answer 2

Damped vibrations refer to the oscillations of a system where the amplitude of the vibration gradually decreases over time due to the presence of damping forces.

In a damped vibration system, the energy of the system is dissipated by the damping force, resulting in a decrease in the amplitude of the oscillations. The damping force can be caused by various factors, such as friction, air resistance, or viscosity.

The motion of a damped harmonic oscillator can be described by the following differential equation:

$m \times \frac {d^2x}{dt^2} + c \times \frac{dx}{dt} + kx = F(t)$

where:

"m" is the mass of the oscillator

"k" is the spring constant

"c" is the damping coefficient

$F(t)$  is an external force applied to the oscillator, if any

The first term on the left-hand side represents the acceleration of the mass, the second term represents the damping force, and the third term represents the spring force. The damping force is proportional to the velocity of the oscillator and is in the opposite direction of the velocity.

This equation can be derived using Newton's second law, which states that the force acting on an object is equal to its mass times its acceleration. In the case of a damped harmonic oscillator, the force is the sum of the spring force and the damping force, and the acceleration is the second derivative of the displacement with respect to time.

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