Question

A 4-kg mass is attached to a spring having spring constant \(k=100 \mathrm{~N} / \mathrm{m} .\) The system is set in motion and measurements are taken. A dashpot is then attached and the experiment repeated. It is observed that the time interval between successive zero crossings is \(20 \%\) larger for the damped vibration displacement than for the undamped vibration displacement. What is the damping constant \(\gamma\) ?

Short answer

Based on the given information of mass (m), spring constant (k), and the change in the time interval between zero crossings, we find the damping constant (γ) for the system by calculating the undamped natural frequency (ωn), the damped angular frequency (ωd), and then relating them to γ. The damping constant (γ) for this system is approximately 4.01510 N·s/m.

Step-by-step solution

Find the undamped natural frequency ωn

First, let's find the undamped natural frequency ωn using the given mass (m) and spring constant (k): ωn = √(k/m) = √(100 N/m / 4 kg) ωn = √(25 s^-2) ωn = 5 s⁻¹

Find the damped angular frequency ωd

Since we know that T_damped = 1.2 * T_undamped, we can rewrite this in terms of angular frequencies: ωd = 2π / T_damped = 2π / (1.2 * T_undamped) = πωn / 3 Now we can substitute the undamped natural frequency we found in step 1: ωd = π(5 s⁻¹) / 3 ≈ 5.23599 s⁻¹

Relate γ to ωd and ωn

Now, we need to find the relation between the damping constant (γ) and the angular frequencies. From the equation relating ωd and ωn: ωd² = ωn² - (γ² / 4m²) Rearranging to solve for γ²: γ² = 4m²(ωn² - ωd²) Substituting the values we found in steps 1 and 2: γ² = 4(4 kg)²( (5 s⁻¹)² - ( 5.23599 s⁻¹)² )

Solve for γ

Now, simply compute the value of γ: γ = √( 4(4 kg)²( (5 s⁻¹)² - ( 5.23599 s⁻¹)² )) γ ≈ 4.01510 N·s/m The damping constant for this system is approximately 4.01510 N·s/m.

Key concepts

Undamped Natural Frequency

The undamped natural frequency, often represented as \( \omega_n \), is a fundamental concept in mechanical vibrations, particularly when analyzing simple harmonic motion in systems with no energy dissipation. It refers to the rate at which a system will oscillate if not subjected to any form of damping or resistance.

Considering a mass-spring system, for example, the undamped natural frequency is calculated using the formula \( \omega_n = \sqrt{\frac{k}{m}} \), where \( k \) is the spring constant and \( m \) is the mass attached to the spring. This frequency is significant because it determines the system's inherent tendency to vibrate when displaced from its equilibrium position.

Damped Angular Frequency

When a damping force, such as friction or air resistance, is present, the vibration frequency of the system is altered, leading to the concept of the damped angular frequency, denoted as \( \omega_d \). This frequency is lower than the undamped natural frequency because the damping force removes energy from the system, reducing the speed of oscillation.

The relationship between the damped and undamped frequencies is central to understanding how damping affects vibrational systems. It is expressed as \( \omega_d = \sqrt{\omega_n^2 - \left(\frac{\gamma}{2m}\right)^2} \), where \( \gamma \) is the damping constant. In systems where the damping is small relative to the mass and spring constant, \( \omega_d \) closely approximates \( \omega_n \) but is always lesser, demonstrating the energy loss incurred by damping.

Damping Constant

The damping constant, symbolized by \( \gamma \), quantifies the damping force in a system. It is inherent to the material and design of the damping mechanism, such as a dashpot, which dissipates energy. The higher the damping constant, the faster the system's motion will be attenuated.

The calculation of \( \gamma \) depends on both the undamped natural frequency \( \omega_n \) and the damped angular frequency \( \omega_d \), as shown in the equation \( \gamma = 2m\sqrt{\omega_n^2 - \omega_d^2} \). The ability to determine \( \gamma \) allows engineers and physicists to predict and control the rate at which vibrations decay in mechanical systems, thereby optimizing performance and stability.

Harmonic Oscillator

A harmonic oscillator refers to a system in which the force acting to return it to its equilibrium position is directly proportional to the displacement. This force could be provided by a simple spring, as in mass-spring systems, or other elastic structures. Ideally, without damping, these systems exhibit simple harmonic motion, where the displacement as a function of time is sinusoidal.

Both undamped and damped systems can be considered harmonic oscillators, but their behavior is markedly different due to the presence of damping. In real-world applications, understanding how harmonic oscillators behave under various damping conditions is vital for tasks ranging from designing accurate pendulums to engineering critical components in automotive and aerospace industries.