Question

A 3.00 -kg mass is vibrating on a spring. It has a resonant angular speed of \(2.40 \mathrm{rad} / \mathrm{s}\) and a damping angular speed of \(0.140 \mathrm{rad} / \mathrm{s} .\) If the sinusoidal driving force has an amplitude of \(2.00 \mathrm{~N}\), find the maximum amplitude of the vibration if the driving angular speed is (a) \(1.20 \mathrm{rad} / \mathrm{s}\), (b) \(2.40 \mathrm{rad} / \mathrm{s}\), and (c) \(4.80 \mathrm{rad} / \mathrm{s}\).

Short answer

Answer: The maximum amplitudes of the vibration for the given driving angular speeds are approximately 0.289 m, 0.989 m, and 0.0602 m, respectively.

Step-by-step solution

Understand the mechanical system

A mass attached to a spring and being driven by a sinusoidal force can be considered a damped, driven harmonic oscillator. Its equation of motion is given by: \(m \ddot{x} + b \dot{x} + kx = F_0 \cos(\omega_d t)\) where \(m\) is the mass, \(x\) is the displacement, \(b\) is the damping coefficient, \(k\) is the spring constant, \(F_0\) is the amplitude of the driving force, and \(\omega_d\) is the driving angular speed. We are given the resonant angular speed, damping angular speed, and amplitude of the driving force. Our goal is to find the maximum amplitude of the vibration (\(A_{max}\)) for different driving angular speeds.

Find the damping ratio and the quality factor

The damping ratio (\(\zeta\)) and quality factor (Q) can be found using the resonant angular speed (\(\omega_0\)) and the damping angular speed (\(\omega_d\)): \(\zeta = \frac{\omega_d}{2 \omega_0}\) \(Q = \frac{1}{2 \zeta}\) Use the given values and find \(\zeta\) and \(Q\): \(\zeta = \frac{0.140}{2 \cdot 2.40} \approx 0.0292\) \(Q = \frac{1}{2 \cdot 0.0292} \approx 17.1\)

Find A_{max} using the maximum amplitude formula

The maximum amplitude of the vibration (\(A_{max}\)) can be found using the following formula: \(A_{max} = \frac{F_0}{m\sqrt{(\omega_0^2 - \omega_d^2)^2 + (2 \zeta \omega_0 \omega_d)^2}}\) Calculate \(A_{max}\) for the three cases using the given driving angular speeds and the previously calculated values for \(\zeta\) and \(Q\): (a) \(\omega_d = 1.20 \mathrm{rad/s}\) \(A_{max(a)} = \frac{2.00}{3.00\cdot\sqrt{(2.40^2 - 1.20^2)^2 + (2 \cdot 0.0292 \cdot 2.40 \cdot 1.20)^2}} \approx 0.289 \mathrm{m}\) (b) \(\omega_d = 2.40 \mathrm{rad/s}\) \(A_{max(b)} = \frac{2.00}{3.00\cdot\sqrt{(2.40^2 - 2.40^2)^2 + (2 \cdot 0.0292 \cdot 2.40 \cdot 2.40)^2}} \approx 0.989 \mathrm{m}\) (c) \(\omega_d = 4.80 \mathrm{rad/s}\) \(A_{max(c)} = \frac{2.00}{3.00\cdot\sqrt{(2.40^2 - 4.80^2)^2 + (2 \cdot 0.0292 \cdot 2.40 \cdot 4.80)^2}} \approx 0.0602 \mathrm{m}\) So the maximum amplitudes of the vibration for the given driving angular speeds are: (a) \(A_{max} \approx 0.289 \mathrm{m}\), (b) \(A_{max} \approx 0.989 \mathrm{m}\), and (c) \(A_{max} \approx 0.0602 \mathrm{m}\).

Key concepts

Resonant Angular Speed

In the context of a damped driven harmonic oscillator, the resonant angular speed, often denoted as \( \omega_0 \), is a critical parameter. It is the angular frequency at which the system would oscillate if there was no driving force and no damping. Essentially, it represents the natural frequency of the mass-spring system. When the driving angular speed approaches the resonant angular speed, the oscillator experiences a condition known as resonance, which leads to a significant increase in the amplitude of oscillation. It's crucial to note that the resonant angular speed is dependent solely on the properties of the system—the mass and the stiffness of the spring.

Damping Angular Speed

While dealing with damped harmonic oscillators, we encounter the damping angular speed, represented as \( \omega_d \). This term is not the frequency of damping, but rather it is related to the damping coefficient \( b \) in the differential equation that models the motion. Specifically, \( \omega_d \) can be used alongside \( \omega_0 \) to calculate the damping ratio. Damping is what causes the oscillation amplitude to decrease over time. A higher damping angular speed means the oscillations will die out more quickly, while lower values indicate less energy loss per cycle. Understanding how damping angular speed affects the motion is essential for characterizing the transient response of the system.

Maximum Amplitude Formula

The maximum amplitude formula plays a pivotal role in determining the maximum displacement from the equilibrium point that the oscillator will reach under a specific driving force. It's given by:
\[ A_{max} = \frac{F_0}{m\sqrt{(\omega_0^2 - \omega_d^2)^2 + (2 \zeta \omega_0 \omega_d)^2}} \]
where \( F_0 \) is the amplitude of the driving force, \( m \) is the mass of the object attached to the spring, \( \omega_0 \) and \( \omega_d \) are the natural and driving angular speeds respectively, and \( \zeta \) is the damping ratio. This formula takes into account the essential factors affecting an oscillator's amplitude in a driven, damped scenario. Notably, when the driving frequency is equal to the natural frequency, the system reaches its resonant condition, usually resulting in the highest amplitude.

Damping Ratio

The damping ratio, symbolized as \( \zeta \), is a dimensionless measure that describes how oscillations in a system decay after a disturbance. Specifically, it's the ratio of the damping angular speed \( \omega_d \) to twice the resonant angular speed \( \omega_0 \):
\[ \zeta = \frac{\omega_d}{2 \omega_0} \]
A damping ratio less than 1 indicates an underdamped system, where oscillations gradually decrease over time. A damping ratio equal to 1 describes a critically damped system, which returns to equilibrium in the shortest possible time without oscillating. Lastly, a damping ratio greater than 1 indicates an overdamped system, where the return to equilibrium is slower than in the critically damped case. The damping ratio offers insights into the behavior of the system's response under various forcings.

Quality Factor

The quality factor, or Q-factor, is a parameter that measures the damping of an oscillator. It is inversely proportional to the damping ratio and is defined as:
\[ Q = \frac{1}{2 \zeta} \]
The Q-factor provides a quantitative way to express the 'sharpness' of the resonance peak of an oscillating system. In other words, it tells us how narrow or broad the resonance curve is. A high Q-factor means the system has low energy loss and sustains oscillations longer, typically resulting in a sharp resonance peak. Conversely, a low Q-factor indicates higher energy dissipation and a broader resonance curve. Engineers and scientists use the Q-factor to describe how underdamped a system is and to optimize the system's behavior for various applications, such as in filters and resonators.