Question

A mass of \(20 \mathrm{gm}\) stretches a spring \(5 \mathrm{~cm}\). The spring is attached to a dashpot with damping constant 400 dyne sec/cm. Determine the displacement for \(t>0\) if the mass is initially displaced \(9 \mathrm{~cm}\) above equilibrium and released from rest.

Short answer

Answer: The displacement function for the mass is given by \(x(t) = e^{-0.628 \times 14 t}(9 \cos(10.131 t) + 4.148 \sin(10.131 t))~\mathrm{cm}\), where t represents the time elapsed since the system was released from its initial conditions.

Step-by-step solution

(Step 1: Find the spring constant)

The spring constant can be found using Hooke's Law: \(F = k \Delta x\) Given, mass (m) = 20g and displacement (\(\Delta x\)) = 5cm, the force (F) can be calculated as: \(F = m \times g = 20 \times 980 \mathrm{~dyne}\) We can then find the spring constant (k) as: \(k = \frac{F}{\Delta x} = \frac{20 \times 980}{5} = 3920 \mathrm{~dyne/cm}\)

(Step 2: Calculate the damping ratio)

Now, we have mass (m), damping constant (c) and spring constant (k). We can find the damping ratio (zeta) using the following formula: \(zeta = \frac{c}{2 \sqrt{m k}}\) Using the given values, \(m = 20\mathrm{~gm}\), \(c = 400\mathrm{~dyne~sec/cm}\) and \(k = 3920\mathrm{~dyne/cm}\), we can calculate zeta as: \(zeta = \frac{400}{2\sqrt{20 \times 3920}} \approx 0.628\)

(Step 3: Identify the type of damping and find the solution)

Since the damping ratio is less than 1 (\(zeta < 1\)), the system is underdamped. The general solution for an underdamped system is: \(x(t) = e^{-zeta \omega_n t}(A \cos(\omega_d t) + B \sin(\omega_d t))\) where \(\omega_n = \sqrt{\frac{k}{m}}\), \(\omega_d = \omega_n \sqrt{1 - zeta^2}\), and A and B are constants that will be determined based on the initial conditions. Calculating \(\omega_n\) and \(\omega_d\): \(\omega_n = \sqrt{\frac{3920}{20}} = 14 \mathrm{~rad/sec}\) \(\omega_d = 14 \sqrt{1 - 0.628^2} \approx 10.131 \mathrm{~rad/sec}\)

(Step 4: Apply initial conditions to find A and B)

Given initial conditions: mass is initially displaced 9cm above equilibrium and released from rest. At \(t=0\), \(x(0) = 9 \mathrm{~cm}\). So, \(9 = e^{-0}(A \cos(0) + B \sin(0))\) \(A = 9\) Also, initial velocity is 0, i.e. \(\frac{dx}{dt}(0) = 0\). Taking derivative of \(x(t)\) and applying this condition: \(\frac{dx}{dt} = -zeta \omega_n e^{-zeta \omega_n t}(A \cos(\omega_d t) + B \sin(\omega_d t)) + e^{-zeta \omega_n t}(-A \omega_d \sin(\omega_d t) + B \omega_d \cos(\omega_d t))\) at \(t=0\), \(\frac{dx}{dt}(0) = 0\): \(0 = -0.628 \times 14 \times 9 + B \times 10.131\) \(B \approx 4.148\)

(Step 5: Write the specific solution)

Now, we have A, B, zeta, \(\omega_n\) and \(\omega_d\). We can write the specific solution for the displacement as: \(x(t) = e^{-0.628 \times 14 t}(9 \cos(10.131 t) + 4.148 \sin(10.131 t))~\mathrm{cm}\) This is the displacement function for displacement of the mass for \(t>0\), with initial conditions of 9 cm above equilibrium and released from rest.

Key concepts

Hooke's Law

Hooke's Law is a fundamental principle that describes how the force exerted by a spring is directly proportional to the displacement that stretches or compresses it. The general form of Hooke's Law is given by the formula:
\[ F = k \Delta x \]
where:

• \(F\) is the force applied to the spring,
• \(k\) is the spring constant that measures the stiffness of the spring,
• \(\Delta x\) is the displacement of the spring from its equilibrium position.

In the given exercise, we see this law applied to determine the spring constant \( k \) from the known force exerted by a mass and the resulting displacement. By understanding Hooke's Law, students can analyze spring-mass systems and predict how they will respond to different forces.

Damping Ratio

The damping ratio is a dimensionless measure describing how oscillations in a system decay over time. It's defined by the symbol \( zeta \) and calculated using the formula:
\[ zeta = \frac{c}{2 \sqrt{mk}} \]
where:

• \(c\) is the damping constant,
• \(m\) is the mass of the object,
• \(k\) is the spring constant.

The damping ratio provides insight into the behavior of the system. For instance, if \( zeta < 1 \), the system is underdamped and will oscillate with decreasing amplitude. If \( zeta = 1 \), it's critically damped, returning to equilibrium without oscillating, while \( zeta > 1 \) means the system is overdamped, also not oscillating but returning to equilibrium slower than in the critically damped case. The exercise demonstrates the calculation of the damping ratio, which foreshadows the system's behavior by considering the mass, damping constant, and spring constant.

Underdamped System

An underdamped system occurs when the damping ratio \( zeta \) is less than 1, revealing that the system will exhibit oscillatory behavior with an amplitude gradually decreasing over time. It’s a common characteristic of many physical systems, like the suspension in automobiles or seismic dampers in buildings.

For an underdamped system, the displacement function typically has the form:
\[ x(t) = e^{-zeta \omega_n t}(A \cos(\omega_d t) + B \sin(\omega_d t)) \]
where \( \omega_n \) is the natural frequency of the system, \( \omega_d \) is the damped frequency, and \( A \) and \( B \) are constants determined by the initial conditions. The exponential term reflects the damping effect reducing the amplitude, while the sinusoidal components represent the oscillatory nature of the motion. In the exercise, the underdamped nature of the system is shown when the calculated damping ratio is found to be less than 1.

Initial Conditions

Initial conditions in the context of mechanical vibrations refer to the starting position and velocity of a vibrating system when time is zero. These conditions are crucial because they influence how the system will behave over time and determine the specific solution to the system's displacement function.

In our example, there are two initial conditions:

• The initial displacement from equilibrium: \( x(0) = 9 \, \mathrm{cm} \),
• The initial velocity, which is zero as the mass is released from rest: \( \frac{dx}{dt}(0) = 0 \).

Applying these conditions enables us to solve for the constants \( A \) and \( B \) in the displacement equation, which are necessary to articulate the precise motion of the mass attached to the spring-damper system.

Displacement Function

The displacement function describes the position of a mass in a vibratory system as a function of time. It provides a mathematical representation of the motion and is critical for predicting how the system will behave in the future.

In underdamped systems, the displacement function typically combines exponential decay with oscillatory sine and cosine functions to reflect both the damping and the restoring forces at play. Formulating the displacement function involves deriving expressions for \( \omega_n \) (the natural frequency) and \( \omega_d \) (the damped frequency), along with calculating constants from the initial conditions — such as in the step-by-step solution provided. By applying the values of \( A \) and \( B \) attained from initial conditions to the displacement function \( x(t) \), students can determine the system's motion for any given time after the initial release.