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Chapter 19: Problem 142

Show that for a small value of the damping factor \(c / c_{c}\), the maximum amplitude of a forced vibration occurs when \(\omega_{f} \approx \omega_{n}\) and that the corresponding value of the magnification factor is \(\frac{1}{2}\left(c / c_{c}\right) .\)

Short Answer

Expert verified
The maximum amplitude occurs near resonance (\(\omega_f \approx \omega_n\)), and the magnification factor is \(\frac{1}{2}(c/c_c)\) for small damping.

Step by step solution

01

Understanding the Forced Vibration and Natural Frequency

The system undergoes forced vibration due to an external force applied with a frequency \( \omega_f \). The natural frequency of the system is denoted \( \omega_n \). We want to find the condition when the maximum amplitude occurs, particularly for small damping factor \( c/c_c \).
02

Considering Resonance in the System

For small values of the damping ratio, the system exhibits resonance when the applied frequency \( \omega_f \) is close to the natural frequency \( \omega_n \). In this context, resonance maximizes the system's amplitude response.
03

Deriving the Magnification Factor Formula

The magnification factor \( M(\omega_f) \) relates the forced vibration amplitude to the static deflection and is given by:\[M(\omega_f) = \frac{1}{\sqrt{\left(1 - \left(\frac{\omega_f}{\omega_n}\right)^2\right)^2 + \left(2\zeta \frac{\omega_f}{\omega_n}\right)^2}}\]where \( \zeta = \frac{c}{c_c} \) is the damping ratio.
04

Simplifying for Small Damping Ratio

For \( c/c_c \) being small, the damping term \( 2\zeta \frac{\omega_f}{\omega_n} \approx 2\zeta \) becomes significantly smaller than 1, simplifying the equation to:\[M(\omega_f) \approx \frac{1}{\sqrt{1 - 2\left(\frac{\omega_f}{\omega_n}\right) + 4\zeta^2}}\]
05

Approximating Condition for Maximum Magnification

By examining the simplified magnification factor expression we can see that the effect of resonance (maximum amplitude) occurs when \( \omega_f \approx \omega_n \), due to the small damping influence being minimal, thus simplifying to:\[M(\omega_f) \approx \frac{1}{2\zeta} = \frac{1}{2}\left( \frac{c}{c_c} \right)\]
06

Conclusion

We have demonstrated that the maximum amplitude in forced vibrations, for small damping ratios, occurs near the system's natural frequency where \( \omega_f \approx \omega_n \). At this point, the magnification factor is approximately \( \frac{1}{2}\left( \frac{c}{c_c} \right) \).

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Key Concepts

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

Natural Frequency
In the realm of mechanical systems, the natural frequency is a crucial concept. It refers to the frequency at which a system tends to oscillate in the absence of any driving or damping forces. Think of it as the system's "comfort zone" where it vibrates naturally.
When a system is subjected to forced vibration, such as an external periodic force, the natural frequency plays a vital role in determining how the system will respond.
The symbol for natural frequency is typically designated as \(\omega_n\), and it is central in analyzing dynamic systems, especially in evaluating situations like resonance.
Damping Factor
The damping factor is a term used to describe how oscillations in a system diminish over time. It is crucial for understanding how energy is lost in the system, often due to friction or resistance.
The damping factor is usually expressed as a ratio of the actual damping to the critical damping (denoted as \(c/c_c\)), which defines the threshold at which oscillations cease before transitioning to aperiodic behavior.
  • If the damping factor is zero, the system will continue to oscillate indefinitely.
  • A small damping factor leads to slowly diminishing vibrations.
  • When the damping factor is large, the system quickly returns to rest without oscillating much.
Understanding the damping factor helps in ensuring a system remains stable and doesn't exhibit excessive vibrations.
Magnification Factor
The magnification factor is a measure that indicates how much the amplitude of a system's vibration increases due to forced vibration, as compared to its static position. It is a dimensionless parameter and is key when analyzing how different frequencies affect vibrational amplitude.
Represented by \(M(\omega_f)\), this factor helps in understanding how a system might react under varying conditions of external force. Specifically, it is defined as the ratio of the amplitude due to vibration to the amplitude due to a static load.
  • At or near resonance, the magnification factor can reach very high values.
  • A small magnification factor implies that the system is not significantly deviated from its rest position by the external force.
By working with the magnification factor, engineers can design systems to avoid excessive responses that might lead to structural failure.
Resonance
Resonance occurs in a system when the frequency of an external force matches the natural frequency of the system itself, often resulting in maximum amplitude of vibration. It's a phenomenon that can drastically increase the energy in a system, leading to potentially destructive vibrations.
In practical terms, think of a child on a swing being pushed at just the right moments to go higher and higher 鈥� that's resonance in action.
For forced vibrations, the resonance condition is typically when the forcing frequency \(\omega_f\) is approximately equal to the natural frequency \(\omega_n\).
  • In engineering designs, care must be taken to ensure that operating frequencies stay away from natural frequencies to avoid destructive resonance.
  • Resonant peaks in amplitudes can cause damage, hence the reason why damping is crucial in controlling those extremes.
Recognizing resonance and managing it through design or materials is critical in maintaining the integrity of mechanical systems.
Damping Ratio
The damping ratio is a dimensionless measure describing how oscillations in a system decay over time, similar to the damping factor but normalized in terms of critical damping. It's denoted by \(\zeta\), and it essentially dictates the behavior of the system's oscillations.
  • A damping ratio of zero signifies no damping, implying perpetual oscillation.
  • A value less than one indicates underdamping, where the system oscillates with decreasing magnitude.
  • A damping ratio equal to one signifies critical damping鈥攄amping just enough to prevent oscillation.
  • Values greater than one indicate overdamping, where the system returns to equilibrium without oscillating.
Understanding the damping ratio is essential in applications such as earthquake-resistant building design, where preventing resonance and ensuring stability are paramount.

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Most popular questions from this chapter

A 150 -kg electromagnet is at rest and is holding \(100 \mathrm{kg}\) of scrap steel when the current is turned off and the steel is dropped. Knowing that the cable and the supporting crane have a total stiffiness equivalent to a spring of constant \(200 \mathrm{kN} / \mathrm{m}\), determine \((a)\) the frequency, the amplitude, and the maximum velocity of the resulting motion, \((b)\) the minimum tension that will occur in the cable during the motion, \((c)\) the velocity of the magnet 0.03 s after the current is turned off.

A 4 -kg collar can slide on a frictionless horizontal rod and is attached to a spring with constant \(k\). It is acted upon by a periodic force of magnitude \(P=P_{m} \sin \omega_{f} t,\) where \(P_{m}=9 \mathrm{N}\) and \(\omega_{f}=5 \mathrm{rad} / \mathrm{s}\). Determine the value of the spring constant \(k\) knowing that the motion of the collar has an amplitude of \(150 \mathrm{mm}\) and is \((a)\) in phase with the applied force, \((b)\) out of phase with the applied force.

From mechanics of materials, it is known that for a simply supported beam of uniform cross section, a static load \(P\) applied at the center will cause a deflection of \(\delta_{A}=P L^{3} / 48 E I,\) where \(L\) is the length of the beam, \(E\) is the modulus of elasticity, and \(I\) is the moment of inertia of the cross-sectional area of the beam. Knowing that \(L=15 \mathrm{ft}, E=30 \times 10^{6}\) psi, and \(I=2 \times 10^{-3} \mathrm{ft}^{4}\), determine \((a)\) the equivalent spring constant of the beam, \((b)\) the frequency of vibration of a \(1500-\mathrm{lb}\) block and to the center of the beam. Neglect the mass of the beam and assume that the load remains in contact with the beam.

Two small weights \(w\) are attached at \(A\) and \(B\) to the rim of a uniform disk of radius \(r\) and weight \(W\). Denoting by \(\tau_{0}\) the period of small oscillations when \(\beta=0\), determine the angle \(\beta\) for which the period of small oscillations is \(2 \tau_{0}\).

The \(100-\) -lb platform \(A\) is attached to springs \(B\) and \(D,\) each of which has a constant \(k=120 \mathrm{lb} / \mathrm{ft}\). Knowing that the frequency of vibration of the platform is to remain unchanged when an \(80-\) -lb block is placed on it and a third spring \(C\) is added between springs \(B\) and \(D,\) determine the required constant of spring \(C .\)
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