Damping

Damping is particularly relevant in studying mechanical systems, such as springs and pendulums. The damping equation provides a mathematical representation of the damping force acting on a system. This force opposes the motion and helps dissipate energy, reducing the amplitude as time progresses.

The damping equation in one dimension is commonly described by the following second-order ordinary differential equation (ODE):

Where:

• m is the mass of the system
• c is the damping coefficient
• k is the spring constant
• x is the displacement of the system from its equilibrium position
• t is time

The term \( c \frac{dx}{dt} \) is the damping force.

The solution to this damping equation depends on the nature of the damping, which can be categorized into three cases:

1. Underdamped (c² < 4mk)

\[ x(t) = e^{-\frac{c}{2m}t} \left(A \cos\left(\omega_dt\right) + B \sin\left(\omega_dt\right)\right) \]

Where:

• \( \omega_d = \sqrt{\frac{4mk – c^2}{4m^2}} \) is the damped angular frequency
• A and B are constants determined by initial conditions

2. Critically damped (c² = 4mk)

\[ x(t) = (A + Bt) e^{-\frac{c}{2m}t} \]

Where:

• A and B are constants determined by initial conditions.

3. Overdamped (c² > 4mk)

\[ x(t) = C_1 e^{r_1t} + C_2 e^{r_2t} \]

Where:

• r₁ and r₂ are the roots of the characteristic equation \( mr^2 + cr + k = 0 \)
• C₁ and C₂ are constants determined by initial conditions

The values of A, B, C₁, C₂, ω_d, r₁, and r₂ depend on the specific initial conditions of the system.

Damping resonance is when a vibrating system is subject to damping forces, and the external force frequency matches the system’s natural frequency. Resonance amplifies the system’s response, and damping is crucial in determining the system’s behavior under these conditions. In the context of a damped harmonic oscillator, the resonance occurs when the frequency of the external force is close to the system’s natural frequency.

Consider a damped harmonic oscillator described by the equation

\[ m \frac{d^2x}{dt^2} + c \frac{dx}{dt} + kx = F_0 \cos(\omega t) \]

Where:

• m is the mass
• c is the damping coefficient
• k is the spring constant
• F₀ is the amplitude of the external force
• ω is the frequency of the external force
• x is the displacement from the equilibrium

The natural frequency ω₀ of the system is given by \( \omega_0 = \sqrt{\frac{k}{m}} \).

The response of the system to the external force is influenced by the damping ratio ζ, defined as

\[ \zeta = \frac{c}{2\sqrt{mk}} \]

The system exhibits different behaviors based on the value of ζ.

In the underdamped case (0 < ζ< 1), resonance occurs when the driving frequency ω is close to the natural frequency ω₀. The displacement response is maximized, and the system oscillates at the driving frequency. The resonant frequency ω_r in the underdamped case is given by \( \omega_r = \sqrt{\omega_0^2 – \zeta^2} \).

The system approaches resonance for the critically damped case (ζ = 1) without oscillations. The displacement response is rapid and reaches its maximum value faster than in the underdamped case.

In the overdamped case (ζ > 1), resonance is less pronounced, and a slow approach to the equilibrium position without oscillations characterizes the system response.

A mass-spring system has a mass (m) of 2 kg, a damping coefficient (c) of 8 Ns/m, and a spring constant (k) of 50 N/m. The system is subjected to an external force F(t) = 10 cos(2t) N. Determine the response of the system for t ≥ 0.

Solution:

The equation of motion for the damped harmonic oscillator is given by:

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

Substituting the given values, we get:

\[ 2 \frac{d^2x}{dt^2} + 8 \frac{dx}{dt} + 50x = 10 \cos(2t) \]

It is a non-homogeneous second-order linear differential equation. To solve it, we first find the characteristic equation by setting the homogeneous part equal to zero:

\[ 2r^2 + 8r + 50 = 0 \]

Solving this quadratic equation, we find two complex roots:

\[ r = -2 + 4i \quad \text{and} \quad r = -2 – 4i \]

The general solution for the homogeneous part is then:

\[ x_h(t) = e^{-2t}(A \cos(4t) + B \sin(4t)) \]

Now, we need to find a particular solution for the non-homogeneous part. Since the external force is a cosine function, we assume a particular solution of the form:

\[ x_p(t) = C \cos(2t) + D \sin(2t) \]

After finding x_p(t), the general solution for the entire system is given by the sum of the homogeneous and particular solutions:

\[ x(t) = x_h(t) + x_p(t) \]

Finally, applying initial conditions or using other information in the problem will allow us to determine the constants (A, B, C, D) and fully describe the system’s response over time.