Question

A single degree of freedom system is represented as a \(2 \mathrm{~kg}\) mass attached to a spring possessing a stiffness of \(4 \mathrm{~N} / \mathrm{m}\). If the coefficients of static and kinetic friction between the block and the surface it moves on are respectively \(\mu_{s}=0.12\) and \(\mu_{k}=0.10\), determine the drop in amplitude between successive periods during free vibration. What is the frequency of the oscillations?

Short answer

Answer: The expression for the drop in amplitude between successive periods during free vibration is: $$ \Delta A = \sqrt{\frac{3.92 \mathrm{~N} \cdot A \cdot N}{2 \mathrm{~N} / \mathrm{m}}} $$ The frequency of the oscillations is \(0.159 \, \mathrm{Hz}\).

Step-by-step solution

Calculate natural frequency / angular frequency

The natural frequency (\(\omega_n\)) can be calculated as the square root of the stiffness divided by the mass. In this case, \(\omega_n = \sqrt{4/2}\). $$ \omega_n = \sqrt{\frac{4 \, \mathrm{N/m}}{2 \, \mathrm{kg}}} = 1 \, \mathrm{rad/s} $$

Calculate total friction force

To determine the total friction force, we first find the maximum static friction (\(F_s\)), which can be calculated as \(F_s = \mu_s \cdot m \cdot g\), where \(\mu_s\) is the static friction coefficient, \(m\) is the mass, and \(g\) is the acceleration due to gravity (approximately \(9.81 \, \mathrm{m/s^2}\)). Next, we find the maximum kinetic friction (\(F_k\)), which can be calculated as \(F_k = \mu_k \cdot m \cdot g\), where \(\mu_k\) is the kinetic friction coefficient. $$ F_s = 0.12 \cdot 2 \, \mathrm{kg} \cdot 9.81 \, \mathrm{m/s^2} = 2.36 \, \mathrm{N} $$ $$ F_k = 0.10 \cdot 2 \, \mathrm{kg} \cdot 9.81 \, \mathrm{m/s^2} = 1.96 \, \mathrm{N} $$ Since the system is in free vibration, the maximum friction will always be the kinetic friction force.

Calculate energy dissipated per cycle

The energy dissipated per cycle due to friction can be calculated as the work done by friction throughout one full oscillation. Assuming a simple harmonic motion for the system, this can be calculated as the product of the total friction force (\(F_k\)), the amplitude of oscillation (\(A\)), and the number of oscillation cycles (\(N\)). The energy, after \(N\) cycles, can be expressed as: $$ E_d = F_k \cdot A \cdot N $$

Calculate drop in amplitude between successive periods

Given the energy dissipation per cycle calculated in step 3, we can find the drop in amplitude between successive periods, denoted as \(\Delta A\). To do this, we set the energy dissipation equal to the change in potential energy of the system, which is: $$ \frac{1}{2} k (\Delta A)^2 = E_d $$ Solving for \(\Delta A\): $$ \Delta A = \sqrt{\frac{2 E_d}{k}} = \sqrt{\frac{2 F_k \cdot A \cdot N}{4 \mathrm{~N} / \mathrm{m}}} = \sqrt{\frac{2 \cdot 1.96 \mathrm{~N} \cdot A \cdot N}{4 \mathrm{~N} / \mathrm{m}}} $$

Find the frequency of oscillations

The frequency of the oscillations can be calculated as the natural frequency divided by \(2\pi\): $$ f = \frac{\omega_n}{2 \pi} = \frac{1 \, \mathrm{rad/s}}{2 \pi} = 0.159 \, \mathrm{Hz} $$ In conclusion, the drop in amplitude between successive periods during free vibration is given by the expression: $$ \Delta A = \sqrt{\frac{3.92 \mathrm{~N} \cdot A \cdot N}{2 \mathrm{~N} / \mathrm{m}}} $$ The frequency of the oscillations is \(0.159 \, \mathrm{Hz}\).

Key concepts

Single Degree of Freedom System

A single degree of freedom system is one where the motion is dictated by a single variable. Imagine a simple system where a mass is attached to a spring. Here, the movement of the mass, either stretching or compressing the spring, represents the single degree of motion. This simplification helps in analyzing more complex systems by focusing on one main motion.

Such systems are easier to model in terms of equations and are foundational in understanding more intricate mechanical and structural behaviors. The ability to predict how these systems move is crucial in fields like engineering and physics, where precise motion control is desired. Whenever you see such a system, know that its behavior can be predicted with great accuracy using mathematical models.

Natural Frequency

Natural frequency is a key characteristic of any vibrating system. It refers to the speed at which a system oscillates naturally when disturbed, without any external force applied.

For a mass-spring system, it is calculated using the formula:

\[ \omega_n = \sqrt{\frac{k}{m}} \] where \(k\) is the stiffness of the spring and \(m\) is the mass.

In the exercise, substituting the given values, we find the natural frequency \(\omega_n = 1 \, \text{rad/s}\). Understanding natural frequency helps engineers design systems that avoid resonant frequencies, which can lead to mechanical failures or inefficiencies.

Friction in Oscillatory Motion

Friction plays a significant role in the oscillations of a system. It tends to resist movement, turning some of the system's mechanical energy into heat.

For oscillatory motion, two types of friction are usually considered: static and kinetic.

• Static friction prevents the system from starting motion.
• Kinetic friction comes into play when the system is moving, affecting its speed and amplitude.

The problem provided uses kinetic friction to gauge the energy loss, as it is active during motion. This friction leads to a decline in motion over time, which crucially impacts the amplitude of successive vibrations.

Energy Dissipation in Vibrations

Energy dissipation in vibrations refers to the loss of energy from the vibrating system, mainly due to friction. As the system oscillates, energy is converted into different forms and eventually lost, usually as heat.

This can be calculated by analyzing the work done by the frictional forces over a complete cycle. Specifically, for the problem at hand:

\[ E_d = F_k \cdot A \cdot N \] Here, \(E_d\) is the energy dissipated per cycle, \(F_k\) is the kinetic friction force, \(A\) is the amplitude, and \(N\) stands for the number of cycles. Understanding this concept is essential when designing systems to minimize unwanted damping and maintain consistent operation.