Damped Simple Harmonic Motion: Damped simple harmonic motion is a type of motion where a system oscillates with a decreasing amplitude due to the presence of a damping force or friction. The motion is described by a differential equation that includes a damping term, and the solutions depend on the damping coefficient and the initial conditions. Over time, the system’s energy is dissipated by the damping force, leading to a decrease in amplitude and a shift in the frequency of the oscillation.

What is Damping?

Damping can be defined as energy dissipation by restraining the vibratory motion like mechanical oscillations, alternating electric current, and noise. Damping can be understood as the resistance offered to the oscillation of a body. In automobiles, shock absorbers and carpet pads act like damping devices. Damping can be of two types:

• Natural Damping
• Artificial Damping

Expression for Damped Simple Harmonic Motion

The expression for damped simple harmonic motion is:

x(t) = A*e^(-γt)cos(ω_dt + φ)

where:

• x(t) is the displacement of the oscillator as a function of time
• A is the amplitude of the oscillation at t = 0
• γ is the damping coefficient, which determines how quickly the amplitude decreases
• ω_d is the damped angular frequency, which depends on the natural angular frequency of the oscillator (ω₀) and the damping coefficient (ω_d = √(ω₀² – γ²))
• φ is the phase angle, which depends on the initial conditions.

The expression shows that the amplitude of the oscillation decays exponentially with time, and the frequency of the oscillation decreases slightly due to the damping.

Examples of Damped Harmonic Motion

Damped harmonic motion is a type of motion in which a system oscillates around a stable equilibrium point with a gradually decreasing amplitude. This type of motion can be observed in many physical systems, including:

• Simple Pendulum: A pendulum is a weight suspended from a fixed point that swings back and forth under the influence of gravity. When the pendulum is in motion, it experiences damping due to air resistance and friction in the support, which causes its amplitude to decrease over time.
• RLC Circuit: An RLC circuit is an electrical circuit containing a resistor, an inductor, and a capacitor. When an electrical current is passed through the circuit, the current oscillates back and forth between the capacitor and the inductor due to the stored energy. The resistance in the circuit causes damping, which gradually reduces the amplitude of the oscillations.
• Mass-Spring-Dashpot System: This system consists of a mass connected to a spring and a dashpot (a device that provides resistance to motion). When the mass is set in motion, it oscillates around the equilibrium point due to the force of the spring. The dashpot provides damping, which causes the amplitude of the oscillations to decrease over time.
• Water Waves: When a disturbance is created in a body of water, it can create waves that propagate through the water. These waves are damped due to the viscosity of the water, which causes energy to be dissipated and the amplitude of the waves to decrease over time.

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By Team Learning Mantras