Problem 45 - Precalculus Mathematics for Calculus

Question

Motion of a Building A strong gust of wind strikes a tall building, causing it to sway back and forth in damped harmonic motion. The frequency of the oscillation is 0.5 cycle per second, and the damping constant is \(c=0.9 .\) Find an equation that describes the motion of the building. (Assume that \(k=1\) and take \(t=0\) to be the instant when the gust of wind strikes the building.)

Step-by-Step Solution

Answer

The motion is described by \(x(t) = e^{-0.45t} (A \cos(\pi t) + B \sin(\pi t))\).

Step 1: Identify the components of damped harmonic motion

In damped harmonic motion, the motion can be described by the differential equation: \(m\frac{d^2x}{dt^2} + c\frac{dx}{dt} + kx = 0\). Here, \(m\) is the mass, \(c\) is the damping constant, and \(k\) is the stiffness constant. We are given \(c = 0.9\) and \(k = 1\). The frequency \(f\) is given as 0.5 cycles per second.

Step 2: Convert frequency to angular frequency

The angular frequency \(\omega_d\) in radians per second is given by \(\omega_d = 2\pi f\). Substitute \(f = 0.5\) to get \(\omega_d = 2\pi \times 0.5 = \pi\) radians per second.

Step 3: Understand the form of the solution

The general solution for damped harmonic motion is \(x(t) = e^{-\frac{c}{2m}t} \left(A \cos(\omega_d t) + B \sin(\omega_d t)\right)\), where \(A\) and \(B\) are constants determined by initial conditions.

Step 4: Determine the effective angular frequency for underdamped motion

The effective angular frequency for underdamped motion is \(\omega_d = \sqrt{\frac{k}{m} - \left(\frac{c}{2m}\right)^2}\). Since \(\omega_d = \pi\), rearrange to find \(\frac{k}{m} - \left(\frac{c}{2m}\right)^2 = \pi^2\). Assuming mass \(m=1\), we verify \(\omega_d = \pi\).

Step 5: Reconstruct the equation of motion

Substitute \(m=1\), \(c=0.9\), \(\omega_d = \pi\), and \(k=1\) into the motion equation: \(x(t) = e^{-0.45t} \left(A \cos(\pi t) + B \sin(\pi t)\right)\). The coefficients \(A\) and \(B\) depend on initial conditions.

Key Concepts

Oscillation Frequency

The oscillation frequency is a fundamental characteristic of any repeating motion in a system, such as a building swaying during gusty winds.
This frequency tells us how many cycles the system completes in one second. In our example, the frequency is given as 0.5 cycles per second.

• The oscillation frequency can often be converted to an angular frequency, which is more commonly used in calculations involving radians.


• Angular frequency, denoted by \( \omega_d \), is calculated by multiplying the oscillation frequency by \(2\pi\).

For this problem, the angular frequency is calculated as \( \omega_d = 2\pi \times 0.5 = \pi\) radians per second.
Understanding the frequency is crucial as it connects to the speed and acceleration of the damped harmonic motion.

Damping Constant

The damping constant, represented by \( c \), plays a significant role in damped harmonic motion. It quantifies the amount of resistance that slows down the motion over time.
In a physical system, this might be due to friction or air resistance, and for our building example, it's given as 0.9.

• The damping constant influences how quickly the oscillating motion diminishes or reduces in amplitude.


• A larger damping constant means faster reduction in the amplitude of motion, leading to a halt.

Therefore, by observing the sway of a building, one can understand how severely the wind's effect fades with its influence over time.
In our equation, the term related to the damping constant appears exponentially as \( e^{-\frac{c}{2m}t} \), which describes the decay of the oscillations.

Motion Equation

In the context of damped harmonic motion, the motion equation combines all the important factors affecting the oscillating system: mass, damping, stiffness, frequency, and time.
It describes how the system behaves over time following an initial disturbance.

• The general form of such an equation is: \[ x(t) = e^{-\frac{c}{2m}t} \left(A \cos(\omega_d t) + B \sin(\omega_d t)\right) \]


• In our example, by substituting given values \( m=1, c=0.9, k=1, \) and \( \omega_d = \pi \), the equation of motion becomes: \[ x(t) = e^{-0.45t} \left(A \cos(\pi t) + B \sin(\pi t)\right) \]

The constants \( A \) and \( B \) depend on the initial conditions, such as the initial position and speed of the building's sway.
This formula captures how the motion of the building evolves over time, taking into account all damping and restoring forces.