Question

A spring is stretched 6 in. by a mass that weighs 8 lb. The mass is attached to a dashpot mechanism that has a damping constant of \(0.25 \mathrm{lb}-\) sec/ft and is acted on by an external force of \(4 \cos 2 t\) lb. (a) Determine the steady-state response of this system. (b) If the given mass is replaced by a mass \(m,\) determine the value of \(m\) for which the amplitude of the steady-state response is maximum.

Short answer

** The steady-state response of the mass-spring-dashpot system is given by the displacement function \(x(t) = -\frac{1}{2} \cos (2t - \phi)\).

Step-by-step solution

Calculate the Spring Constant k

Given that the spring is stretched 6 inches (0.5 ft) by a mass of 8 pounds, we can use Hooke's Law to determine the spring constant, k. Hooke's Law states that F = -kx, where F is the force applied, k is the spring constant, and x is the displacement: \(8 = k \times 0.5\) Solving for k, we get: \(k = 16 \thinspace \mathrm{lb/ft}\)

Define the Equation of Motion

Now we need to find the equation of motion for the mass-spring-dashpot system. Let x(t) be the displacement of the mass as a function of time. The equation of motion for a system like this can be written as: \(mx''(t) + cx'(t) + kx(t) = F(t)\) where m is the mass, c is the damping constant, x'(t) is the velocity, x''(t) is the acceleration and F(t) is the external force acting on the system. For this problem, since we are seeking the steady-state response, let \(x(t) = X \cos (2t - \phi)\). Plugging this in, we have: \(-4m(X \cos (2t - \phi)) - 0.5(2)(-2X \sin (2t - \phi)) + 16(X \cos (2t - \phi)) = 4 \cos 2t\)

Solve for Steady-State Response

To determine the steady-state response, compare the coefficients of the trigonometric functions: \(-4mX + 16X = 4\) \(-4X = 2\) Solving for X, we get: \(X = -\frac{1}{2}\) Thus, the steady-state displacement of the system is: \(x(t) = -\frac{1}{2} \cos (2t - \phi)\)

Determine the Mass for Maximum Amplitude

To determine the mass for which the amplitude of the steady-state response is maximum, we need to find the value when the derivative of the amplitude is equal to 0. Differentiating X with respect to m, we have: \(\frac{dX}{dm} = \frac{d}{dm}\left(-\frac{1}{2}\right) = 0\) Since the amplitude X doesn't depend on the mass m, X is constant and thus, the amplitude does not reach a maximum value for any mass value.

Key concepts

Mass-Spring-Dashpot System

In a mass-spring-dashpot system, we have three main components: the mass, the spring, and the dashpot (or damper). This setup is common in mechanical systems and models the behavior of materials under mechanical stress. The mass is the object under consideration that is subject to forces. The spring provides a restoring force based on its extension or compression, aligning with Hooke's Law. The dashpot or damper introduces a force that counteracts motion, proportional to the velocity of the mass. This combination is crucial for understanding dynamic systems that involve vibrations and oscillations. Such systems are used in engineering to model car suspensions, building dynamics in response to earthquakes, and many other scenarios. They help predict how an object will respond to various forces over time, especially in scenarios with both constant and external time-varying forces.

Hooke's Law

Hooke's Law is a fundamental principle that describes the behavior of springs in a mechanical system. It states that the force exerted by a spring is directly proportional to the amount it is stretched or compressed from its natural length. Mathematically, it is expressed as:

• \( F = -kx \)

where \( F \) is the force applied to the spring, \( k \) is the spring constant, and \( x \) is the displacement from the spring's equilibrium position. The negative sign indicates that the force is restorative, opposing the direction of displacement. The spring constant \( k \) quantifies the stiffness of the spring - a higher \( k \) means a stiffer spring. In our original exercise, we calculated that \( k = 16 \text{ lb/ft} \), indicating this is how resistant the spring is to being stretched by the given weight of 8 pounds.

Equation of Motion

The equation of motion for a mass-spring-dashpot system allows us to analyze the movement of the mass over time. Representing physical laws, this equation describes how forces result in motion for a given system. For a basic mass-spring-dashpot system, the equation of motion is:

• \( mx''(t) + cx'(t) + kx(t) = F(t) \)

Here, \( m \) is the mass, \( x''(t) \) represents acceleration (the second derivative of displacement \( x(t) \)), \( c \) is the damping constant, \( x'(t) \) is velocity (the first derivative of displacement), and \( F(t) \) is any external force applied. This equation incorporates the effects of inertia, damping, and spring force into a single formula. Solving it gives us insight into the system’s behavior and with specific components, how displacement varies over time under constant, damping, and forcing conditions. For the steady-state response, specific techniques allow simplifications, showing us how the system will behave when under a periodic external force.

Damping Constant

The damping constant \( c \) in a mass-spring-dashpot system reflects the damper's ability to dissipate energy, essential for controlling oscillations and stabilizing the system. It represents how much frictional resistance is exerted against motion, slowing it down. The unit of the damping constant is typically force per unit velocity, such as \( \text{lb-sec/ft} \). A large damping constant means that the system reaches its equilibrium position more quickly, as it strongly resists movement. Conversely, a small damping constant means the system might oscillate more and take longer to stabilize. In the exercise example, the damping constant is \( 0.25 \text{ lb-sec/ft} \), which provides enough resistance but does not fully prevent oscillations, allowing us to examine the steady-state behavior under an oscillating force. This balance between damping and restoring forces determines how quickly the system settles back into equilibrium and how it behaves under periodic or sudden forces, crucial for real-world applications like vehicle suspension systems and building stability.