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 drop in amplitude between successive periods is \(A_n (1 - e^{-0.512})\), and the frequency of oscillations is 0.225 Hz.

Step-by-step solution

Calculate the damping constant

To find the damping constant, we need to use the kinetic friction coefficient, mass of the block, and gravitational acceleration (9.81 m/s²). The damping constant (c) can be calculated using the formula: \(c = \mu_k * m * g\) Where: \(m = 2 \mathrm{~kg}\) (mass of the block) \(\mu_k = 0.10\) (kinetic friction coefficient) \(g = 9.81 \mathrm{~m/s^2}\) (gravitational acceleration) \(c = 0.10 * 2 * 9.81 = 1.962 \mathrm{~N \cdot s / m}\)

Calculate the natural frequency

To find the natural frequency (wn) of the system, we will use the formula: \(w_n = \sqrt{\frac{k}{m}}\) Where: \(k = 4 \mathrm{~N / m}\) (spring stiffness) \(m = 2 \mathrm{~kg}\) (mass of the block) \(w_n = \sqrt{\frac{4}{2}} = \sqrt{2} \mathrm{~rad/s}\)

Calculate the damping ratio

To find the damping ratio (ζ), we will use the formula: \(\zeta = \frac{c}{2 \sqrt{mk}}\) Where: \(c = 1.962 \mathrm{~N \cdot s / m}\) (damping constant) \(m = 2 \mathrm{~kg}\) (mass of the block) \(k = 4 \mathrm{~N / m}\) (spring stiffness) \(\zeta = \frac{1.962}{2 \sqrt{2 \cdot 4}} = 0.245\)

Calculate the logarithmic decrement

To find the logarithmic decrement (δ) in amplitude, we will use the formula: \(\delta = 2 \pi \frac{\zeta}{\sqrt{1-\zeta^2}}\) Where: \(\zeta = 0.245\) (damping ratio) \(\delta = 2 \pi \frac{0.245}{\sqrt{1 - 0.245^2}} = 0.512\)

Calculate the drop in amplitude between successive periods

To find the drop in amplitude between successive periods, we will use the formula: \(A_{n+1} = A_n e^{-\delta}\) Where: \(A_n\) is the amplitude at period n \(A_{n+1}\) is the amplitude at period n+1 \(\delta = 0.512\) (logarithmic decrement) \(A_{n+1} = A_n e^{-0.512}\) The drop in amplitude between successive periods is equal to \(A_n - A_{n+1} = A_n (1 - e^{-0.512})\).

Calculate the frequency of oscillations

To find the frequency of oscillations, we will use the formula: \(f = \frac{w_n}{2 \pi}\) Where: \(w_n = \sqrt{2} \mathrm{~rad/s}\) (natural frequency) \(f = \frac{\sqrt{2}}{2 \pi} = 0.225 \mathrm{~Hz}\) The frequency of oscillations is 0.225 Hz. In conclusion, the drop in amplitude between successive periods is \(A_n (1 - e^{-0.512})\), and the frequency of oscillations is 0.225 Hz.

Key concepts

Damping Constant

The damping constant is a measure of how much energy a system loses per cycle of its motion. In the context of a single degree of freedom system, it represents the frictional forces or resistive forces acting on the system. These forces dissipate the system's energy, reducing its amplitude over time.

To calculate the damping constant (c), you need to know the kinetic friction coefficient (\(\mu_k\)), the mass of the block (m), and gravitational acceleration (g). The formula is:

• c = \(\mu_k \times m \times g\)

This gives us an idea of how much the friction opposes motion. It is crucial for understanding how quickly oscillations will die down in a damped system.

Natural Frequency

The natural frequency of a system is the frequency at which it vibrates when disturbed from an equilibrium position without any damping. For a simple mass-spring system, it's calculated using the spring's stiffness (k) and the mass of the block (m).

Mathematically, natural frequency (w_n) is expressed as:

• \(w_n = \sqrt{\frac{k}{m}}\)

This formula provides a key insight into the system's dynamics, showing how stiffness and mass affect the speed of oscillations. It highlights that lighter and stiffer systems have higher natural frequencies.

Damping Ratio

The damping ratio (\(\zeta\)) is a dimensionless measure that describes how oscillations in a system decay after a disturbance. It's derived from the damping constant (c), the mass (m), and the spring stiffness (k).

The formula is:

• \(\zeta = \frac{c}{2 \sqrt{mk}}\)

It describes how underdamped, critically damped, or overdamped a system is.

• For \(\zeta < 1\), the system is underdamped and oscillates.
• For \(\zeta = 1\), it is critically damped and returns to equilibrium without oscillating.
• For \(\zeta > 1\), the system is overdamped and returns to equilibrium slowly.

The damping ratio helps in designing systems to ensure their performance based on the required damping behavior.

Logarithmic Decrement

The logarithmic decrement (\(\delta\)) quantifies the rate at which the amplitude of an underdamped system's oscillations decreases. It uses the damping ratio (\(\zeta\)) to find how much the oscillations decay over successive cycles.

The formula is:

• \(\delta = 2\pi \frac{\zeta}{\sqrt{1 - \zeta^2}}\)

This tells us the relative drop in amplitude from one cycle to the next, assuming the oscillations aren't damped to the point of being critically or overdamped.

By understanding the logarithmic decrement, engineers can predict how long it will take for an oscillating system to settle, which is crucial in applications like automotive suspension and building design.