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Chapter 19: Problem 128

Show that in the case of heavy damping \(\left(c>c_{c}\right),\) a body released from an arbitrary position with an arbitrary initial velocity cannot pass more than once through its equilibrium position.

Short Answer

Expert verified
For heavy damping, the system is overdamped and the body cannot oscillate, thus passing through equilibrium at most once.

Step by step solution

01

Understanding Heavy Damping

In a damped harmonic oscillator, the damping ratio is denoted by \( c \). For heavy damping, the condition \( c > c_c \) indicates an overdamped system where the damping is stronger than the critical damping.
02

Defining the Equation of Motion

The differential equation for a damped harmonic oscillator is given by \( m\ddot{x} + c\dot{x} + kx = 0 \). For heavy damping, where \( c > c_c \), we solve this equation to understand the system's behavior.
03

Finding the Characteristic Equation

We assume a solution of the form \( x(t) = e^{\lambda t} \), leading to the characteristic equation \( m\lambda^2 + c\lambda + k = 0 \).
04

Solving the Characteristic Equation for Heavy Damping

The roots of the characteristic equation are \( \lambda_{1,2} = \frac{-c \pm \sqrt{c^2 - 4mk}}{2m} \). In the case of heavy damping \( (c > c_c) \), \( \sqrt{c^2 - 4mk} \) is real and positive, providing two distinct real roots.
05

Expressing the General Solution

With distinct real roots \( \lambda_1 \) and \( \lambda_2 \), the general solution is \( x(t) = A e^{\lambda_1 t} + B e^{\lambda_2 t} \), where \( A \) and \( B \) are constants determined by initial conditions.
06

Analyzing the Motion

For the solution \( x(t) = A e^{\lambda_1 t} + B e^{\lambda_2 t} \), where both \( \lambda_1 \) and \( \lambda_2 \) are negative, the motion exponentially decays to zero without oscillating through the equilibrium.
07

Verifying Passage Through Equilibrium

The only possible time \( t \) when \( x(t) = 0 \) is when the initial conditions are such that the terms \( A e^{\lambda_1 t} \) and \( B e^{\lambda_2 t} \) cancel each other out immediately after release. After this, the function continues to decay towards zero.
08

Conclusion

In an overdamped system, the body cannot pass through the equilibrium position more than once, as it does not undergo oscillations.

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Key Concepts

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

Overdamped System
An overdamped system is a particular type of damped harmonic oscillator where the damping force is so strong that it causes the system to return to equilibrium without oscillating. When we talk about an overdamped system, we mean that the damping constant, represented by \(c\), is greater than the critical damping constant \(c_c\). In this case, the damping is more effective than necessary to bring the system to rest quickly.
In an overdamped scenario, the roots of the system's characteristic equation are real and distinct. This results in a motion that approaches zero exponentially without any back-and-forth movement. Think of it like a door that you open and release 鈥� instead of swinging back and forth before coming to rest (as would be the case in underdamping), it slowly returns directly to a closed position.
  • No oscillations occur.
  • The system gradually approaches rest.
  • It is more damped than strictly necessary (beyond critically damped).
Damped Harmonic Oscillator
A damped harmonic oscillator is a system that experiences a restraining force proportional to its velocity. In simpler terms, it's like having friction or air resistance that slows down its motion. This damping affects the system's ability to oscillate, influencing how fast or whether it returns to its point of equilibrium.
In the context of the damped harmonic oscillator, we often describe its behavior using the second-order differential equation:\[ m\ddot{x} + c\dot{x} + kx = 0 \]where:
  • \(m\) is the mass of the oscillator.
  • \(c\) is the damping coefficient.
  • \(k\) is the spring constant.
The damped harmonic oscillator's equation helps us describe various behaviors:
  • Under damped - where the system oscillates while eventually coming to rest.
  • Critically damped - the fastest return to equilibrium without overshooting.
  • Overdamped - described earlier, where the system returns to equilibrium slowly without any oscillations.
Equilibrium Position
The equilibrium position is the point where a system, such as a damped harmonic oscillator, experiences no net force. In this neutral position, the forces acting on the system are in perfect balance.
For all movements, including those of a damped harmonic oscillator, the equilibrium position is a goal or endpoint in its motion. The system may move away from it due to external forces or initial displacements but will naturally try to return to this point over time. In heavily damped situations, a body released from an arbitrary position with arbitrary initial velocity will slowly approach its equilibrium position without overshooting or oscillating.
  • An overdamped system ensures a single passage through this point.
  • The return to equilibrium is characterized by a swift yet non-oscillating behavior.
Understanding how systems behave in relation to their equilibrium position is crucial for designing mechanical or electrical systems, ensuring they perform desired functions efficiently and reliably.

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Most popular questions from this chapter

The force-deflection equation for a nonlinear spring fixed at one end is \(F=1.5 x^{112}\) where \(F\) is the force, expressed in newtons, applied at the other end and \(x\) is the deflection expressed in meters. (a) Determine the deflection \(x_{0}\) if a 4 -oz block is suspended from the spring and is at rest. (b) Assuming that the slope of the force-deflection curve at the point corresponding to this loading can be used as an equivalent spring constant, determine the frequency of vibration of the block if it is given a very small downward displacement from its equilibrium position and released.

An 8 -kg uniform disk of radius \(200 \mathrm{mm}\) is welded to a vertical shaft with a fixed end at \(B\). The disk rotates through an angle of \(3^{\circ}\) when a static couple of magnitude \(50 \mathrm{N} \cdot \mathrm{m}\) is applied to it. If the disk is acted upon by a periodic torsional couple of magnitude \(T=T_{m} \sin \omega_{f} t,\) where \(T_{m}=60 \mathrm{N} \cdot \mathrm{m},\) determine the range of values of \(\omega_{f}\) for which the amplitude of the vibration is less than the angle of rotation caused by a static couple of magnitude \(T_{m} .\)

A small \(2-\) kg sphere \(B\) is attached to the bar \(A B\) of negligible mass that is supported at \(A\) by a pin and bracket and connected at \(C\) to a moving support \(D\) by mans of a spring of constant \(k=3.6 \mathrm{kN} / \mathrm{m}\). Knowing that support \(D\) undergoes a vertical displacement \(\delta=\delta_{m} \sin \omega_{f} t,\) where \(\delta_{m}=3 \mathrm{mm}\) and \(\omega_{f}=15 \mathrm{rad} / \mathrm{s}\), determine \((a)\) the magnitude of the maximum angular velocity of bar \(A B,\) (b) the magnitude of the maximum acceleration of sphere \(B .\)

A \(6-\) lb slender rod is suspended from a steel wire that is known to have a torsional spring constant \(K=1.5 \mathrm{ft}\) lb/rad. If the rod is rotated through \(180^{\circ}\) about the vertical and released, determine (a) the period of oscillation, (b) the maximum velocity of end \(A\) of the rod.

A beam \(A B C\) is supported by a pin connection at \(A\) and by rollers at \(B\). A 120 -kg block placed on the end of the beam causes a static deflection of \(15 \mathrm{mm}\) at \(C\). Assuming that the support at \(A\) undergoes a vertical periodic displacement \(\delta=\delta_{m} \sin \omega_{f} t,\) where \(\delta_{m}=10 \mathrm{mm}\) and \(\omega_{f}=18 \mathrm{rad} / \mathrm{s}\), and the support at \(B\) does not move, determine the maximum acceleration of the block at \(C\). Neglect the weight of the beam and assume that the block does not leave the beam.
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