Question

A 20-lb weight is attached to the lower end of a coil spring suspended from the ceiling. The weight comes to rest in its equilibrium position, thereby stretching the spring 6 in. Various external forces of the form \(F(t)=\cos \omega t\) are applied to the system and it is found that the resonance frequency is \(0.5\) cycles/sec. Assuming that the resistance of the medium in pounds is numerically equal to \(a x^{\prime}\), where \(x^{\prime}\) is the instantaneous velocity in feet per second, determine the damping coefficient \(a .\)

Short answer

The damping coefficient for the given spring-mass system is approximately \(a \approx 11.15 \text{ lb/s}\).

Step-by-step solution

Calculate the spring constant

Given that a 20-lb weight stretches the spring by 6 inches (0.5 feet), we can use Hooke's law to find the spring constant \(k\): \[ F_s = kx, \] where \(F_s\) is the spring force and \(x\) is the elongation. Therefore, \[ k = \frac{F_s}{x} = \frac{20 \text{ lb}}{0.5 \text{ ft}} = 40 \text{ lb/ft}. \]

Write down the equation of motion

The equation of motion for a damped spring-mass system under an external force is given by \[ m x^{\prime\prime} + a x^{\prime} + kx = F(t), \] where \(m\) is the mass, \(a\) is the damping coefficient, \(x\) is the displacement, and \(F(t) = \cos(\omega t)\) is the applied force. In our case, given that the system is in pound-second units and the weight is given in pounds, we can directly use the 20-lb weight as the mass of the system. Thus, we get: \[ 20 x^{\prime\prime} + ax^{\prime} + 40x = \cos(\omega t). \]

Homogeneous and particular solutions

The corresponding homogeneous equation is \[ 20 x_h^{\prime\prime} + ax_h^{\prime} + 40x_h = 0, \] and the particular solution \(x_p(t)\) can be assumed of the form: \[ x_p(t) = Acos(\omega t - \phi), \] where \(A\) and \(\phi\) are constants to be determined.

Obtain the resonance condition and find the damping coefficient

The resonance condition occurs when the applied force frequency matches the natural frequency of the system. The resonance frequency is given as \(\omega_r = 0.5 \text{ cycles/sec}\), which we convert to radians/second by multiplying by \(2\pi\): \[ \omega_r = 0.5 \times 2\pi = \pi \text{ rad/sec}. \] For resonance to occur, the imaginary part of the complex roots of the characteristic equation of the homogeneous equation must equal the resonance frequency. The characteristic equation for the homogeneous equation is: \[ 20r^2 + ar + 40 = 0. \] Using the quadratic formula, we can find the complex roots: \[ r = \frac{-a \pm \sqrt{a^2 - 4(20)(40)}}{40}. \] For resonance, the imaginary part of the roots must equal \(\omega_r\), thus: \[ \omega_r = \frac{\sqrt{4(20)(40) - a^2}}{40} = \pi. \] Solving for \(a\): \[ a = \sqrt{4(20)(40) - (40\pi)^2} \approx 11.15 \text{ lb/s}. \] Therefore, the damping coefficient is \(a \approx 11.15 \text{ lb/s}\).

Key concepts

Differential Equations

At the heart of analyzing mechanical systems like the spring-mass problem described in the exercise is a branch of mathematics called differential equations. These equations are critical tools that describe the relationships involving the rates of change of continuously varying quantities. A differential equation encompasses derivatives, which represent rates of change, and solving these equations provides the function which models the system’s behavior over time.

In the exercise, we encounter a second-order linear differential equation. This specific type of equation is notable in physics and engineering for its role in characterizing systems with complexity, such as those exhibiting oscillatory motion. The second-order term, which includes the double derivative of the displacement (\(x^{\textprime\textprime}\)), represents the acceleration of the mass. The first-order term with (\(ax^{\textprime}\)) accounts for the damping force, while the spring constant (\(k\)) multiplied by the displacement (\(x\)) models the restoring force of the spring.

The process for solving these equations typically involves finding two parts: the homogeneous solution, which represents the system's natural behavior without external forces, and the particular solution, which accounts for the impact of external influences such as the cosine force applied in this exercise. Understanding how to construct and solve these differential equations is essential to predict the system's dynamic response accurately.

Resonance Frequency

The concept of resonance is widespread, ranging from musical instruments to the sway of buildings during earthquakes. Resonance occurs when a system is driven by an external force at a frequency that matches its natural frequency, which is the rate at which the system tends to oscillate in the absence of any driving or damping forces. This phenomenon can lead to large oscillations and is a critical factor in engineering to avoid structural failures.

In the step-by-step solution, the resonance frequency of the spring-mass system is identified as 0.5 cycles per second. To cause resonance, the frequency of the external force applied to the system (\(F(t)\)) aligns with the system's natural frequency. The damping coefficient plays a pivotal role here; as it changes, it can affect whether the system reaches resonance and the magnitude of its response. By adjusting the value of the damping coefficient, our system's responsiveness to various frequencies can be controlled which is crucial for designing safe and efficient mechanical and structural systems.

Spring-Mass System

A spring-mass system is a simple yet fundamental mechanical model that consists of a mass attached to a spring, which can stretch or compress from its equilibrium position. The analysis of this system can be carried out using Hooke's law, which states that the force exerted by the spring is directly proportional to its displacement from the equilibrium position.

The problem we are looking at provides an excellent example of the energy conservation in such systems. The mass attached to the spring undergoes harmonic motion, where energy is continuously exchanged between kinetic energy and potential energy of the spring. The rate of energy transfer and the system's behavior, like its oscillation frequency, are determined by the system's physical properties, such as mass (\(m\)) and spring stiffness (\(k\)).

Additionally, our system includes the resistance of the medium (damping), which models the energy loss due to factors like air resistance or friction. The presence of damping affects the amplitude and the frequency of the mass's oscillation, leading to a gradual decrease in oscillation amplitude over time. A deep understanding of the interplay between these elements allows students and engineers to predict how the spring-mass system will respond to various conditions, a principle used in the design of a vast array of mechanical and structural systems.