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Chapter 3: Problem 2

A spring-mass system is subjected to a harmonic force whose frequency is close to the natural frequency of the system. If the forcing frequency is \(39.8 \mathrm{~Hz}\) and the natural frequency is \(40.0 \mathrm{~Hz},\) determine the period of beating.

Short Answer

Expert verified
The period of beating for the given spring-mass system is \(5 \mathrm{~seconds}\).

Step by step solution

01

Calculate the beat frequency

To find the beat frequency, subtract the natural frequency from the forcing frequency: \(f_B = |f_{forcing} - f_{natural}| = |39.8 - 40.0|\)
02

Compute the difference

Subtract the given values: \(f_B = |-0.2| = 0.2 \mathrm{~Hz}\)
03

Find the period of beating

The period of beating (T) can be found by taking the reciprocal of the beat frequency: \(T = \frac{1}{f_B} = \frac{1}{0.2}\)
04

Compute the period

Calculate the period of beating: \(T = 5 \mathrm{~seconds}\) Therefore, the period of beating for the given spring-mass system is 5 seconds.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Spring-Mass System
Imagine a playground swing. When you push it, it swings back and forth. A spring-mass system works similarly. It consists of a mass attached to a spring. When the mass is moved, the spring stretches or compresses, resulting in back-and-forth motion. This system can oscillate indefinitely if there is no external interference, like friction, to stop it.

In a spring-mass system:
  • Mass: The object that moves due to the spring's force.
  • Spring: Elastic component that stores energy when stretched or compressed.
  • Oscillation: The repetitive motion of the mass due to the spring's elasticity.
These systems are widely studied because they help us understand many physical phenomena, such as mechanical vibrations and sound waves in musical instruments.
Harmonic Force
A harmonic force is like someone pushing you on a swing at regular intervals. It's a periodic external force applied to a system to keep it oscillating. For the spring-mass system, this force can affect how the system moves.

Characteristics of a harmonic force include:
  • Periodic: It repeats itself in regular intervals.
  • Consistent: The force is applied with the same strength each time.
The frequency at which this force is applied is critical. When the frequency of the harmonic force matches the natural frequency of the system, very large oscillations can occur鈥攎uch like pushing a swing higher and higher each time.
Natural Frequency
Every spring-mass system has a natural frequency. This is the rate at which the system naturally oscillates when disturbed, without any external influence. You can think of it like the swing's preferred rhythm. Natural frequency is determined by:
  • Mass: Heavier objects generally oscillate slower.
  • Spring Constant: A stiffer spring increases the frequency.
It's essential to recognize that if an external force (like the harmonic force) hits the system at this natural frequency, resonance occurs鈥攎eaning the system will oscillate with maximum energy and amplitude. In our example, this means the system will swing as high as it can.
Period of Beating
When two frequencies (like the forcing and natural frequencies) are close but not identical, a phenomenon called 'beating' occurs. This results in a varying amplitude that grows and shrinks over time, similar to hearing the volume of a musical note fluctuate. The 'period of beating' is the time it takes for the beat pattern to develop fully and then start over. It is the inverse of the beat frequency, which is the absolute difference between the two frequencies.

To determine the period of beating:
  • Calculate the difference in frequencies: \(f_B = |f_{forcing} - f_{natural}|\)
  • Find the reciprocal: \(T = \frac{1}{f_B}\)
Thus, if one frequency is 39.8 Hz and another is 40.0 Hz, the period of beating will be 5 seconds, indicating the time it takes for the complete beat cycle to occur.

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Most popular questions from this chapter

A weight of \(50 \mathrm{~N}\) is suspended from a spring of stiffness \(4000 \mathrm{~N} / \mathrm{m}\) and is subjected to a harmonic force of amplitude \(60 \mathrm{~N}\) and frequency \(6 \mathrm{~Hz}\). Find (a) the extension of the spring due to the suspended weight, (b) the static displacement of the spring due to the maximum applied force, and (c) the amplitude of forced motion of the weight.

The base of a damped spring-mass system, with \(m=25 \mathrm{~kg}\) and \(k=2500 \mathrm{~N} / \mathrm{m}\), is subjected to a harmonic excitation \(y(t)=Y_{0} \cos \omega t\). The amplitude of the mass is found to be \(0.05 \mathrm{~m}\) when the base is excited at the natural frequency of the system with \(Y_{0}=0.01 \mathrm{~m}\). Determine the damping constant of the system.

When an exhaust fan of mass \(380 \mathrm{~kg}\) is supported on springs with negligible damping, the resulting static deflection is found to be \(45 \mathrm{~mm}\). If the fan has a rotating unbalance of \(0.15 \mathrm{~kg}-\mathrm{m},\) find (a) the amplitude of vibration at \(1750 \mathrm{rpm},\) and \((\mathrm{b})\) the force transmitted to the ground at this speed.

A spring-mass system consists of a mass weighing \(100 \mathrm{~N}\) and a spring with a stiffness of \(2000 \mathrm{~N} / \mathrm{m}\). The mass is subjected to resonance by a harmonic force \(F(t)=25 \cos \omega t \mathrm{~N}\). Find the amplitude of the forced motion at the end of (a) \(\frac{1}{4}\) cycle, (b) \(2 \frac{1}{2}\) cycles, and (c) \(5 \frac{3}{4}\) cycles.

Consider an automobile traveling over a rough road at a speed of \(v \mathrm{~km} / \mathrm{hr}\). The suspension system has a spring constant of \(40 \mathrm{kN} / \mathrm{m}\) and a damping ratio of \(\zeta=0.1\). The road surface varies sinusoidally with an amplitude of \(Y=0.05 \mathrm{~m}\) and a wavelength of \(6 \mathrm{~m}\). Write a MATLAB program to find the displacement amplitude of the automobile for the following conditions: (a) mass of the automobile \(=600 \mathrm{~kg}\) (empty), \(1000 \mathrm{~kg}\) (loaded), (b) velocity of the automobile \((v)=10 \mathrm{~km} / \mathrm{h}, 50 \mathrm{~km} / \mathrm{h}, 100 \mathrm{~km} / \mathrm{h}\).
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