Undamped oscillator with driving force

In summary, to find the position of an undamped harmonic oscillator subject to a driving force, we can use the general solution x(t) = A\cos(\omega t - \phi) + Be^{-\alpha t} and solve for the constants using the differential equation and initial conditions. The final solution is given by x(t) = \frac{F_0}{\sqrt{k^2 + (m\omega)^2}}\cos(\sqrt{\frac{k}{m}}t - \arctan\left(\frac{m\omega}{k}\right))e^{-bt}.
  • #1
thenewbosco
187
0
just wondering how to go about this one:
An undamped harmonic oscillator is subject to a driving force [tex]F_0e^{-bt}[/tex]. It starts from rest at the origin (x=0) at time t=0.

assuming a general solution [tex]x(t)=A cos(\omega t - \phi) + Be^{-\alpha t}[/tex] where A, [tex]\omega[/tex], [tex]\phi[/tex], B, and [tex]\alpha[/tex] are real constants, find the position x(t) as a function of time.

I was thinking to take this general solution, differentiate, and apply initial conditions to the two equations, however all i could solve for is alpha in terms of omega.
Then i thought to differentiate again and plug into the equation
[tex]ma=-kx + F_{driving}[/tex] however i cannot see how i can solve for all the constants with these two equations..any help?
 
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  • #2
To solve this problem, you need to first solve the differential equation that describes the motion of the oscillator. This is done by taking the second derivative of the position with respect to time, which yields the equation:m\ddot{x} + kx = F_0e^{-bt}where m is the mass of the oscillator, k is the spring constant, and b is the damping coefficient.Assuming a solution of the form x(t) = A\cos(\omega t - \phi) + Be^{-\alpha t}, where A, \omega, \phi, B, and \alpha are real constants, we can substitute it into the differential equation and solve for the constants. After some algebraic manipulation, we can find that:A = \frac{F_0}{\sqrt{k^2 + (m\omega)^2}}\omega = \sqrt{\frac{k}{m}}\phi = \arctan\left(\frac{m\omega}{k}\right)B = 0\alpha = bTherefore, the solution of the position of the undamped harmonic oscillator under this driving force is:x(t) = \frac{F_0}{\sqrt{k^2 + (m\omega)^2}}\cos(\sqrt{\frac{k}{m}}t - \arctan\left(\frac{m\omega}{k}\right))
 
  • #3


I would suggest approaching this problem by using the equations of motion for an undamped harmonic oscillator with a driving force. This would involve using the second-order differential equation:

m\ddot{x} + kx = F_0e^{-bt}

Where m is the mass of the oscillator, k is the spring constant, and F_0e^{-bt} is the driving force. From this equation, we can find the general solution for x(t) as:

x(t) = A\cos(\omega t - \phi) + \frac{F_0}{k-b^2m}e^{-bt}

Where A and \phi are determined by the initial conditions of the oscillator. From this solution, we can see that the position of the oscillator is a combination of the natural oscillation (A\cos(\omega t - \phi)) and the forced oscillation (\frac{F_0}{k-b^2m}e^{-bt}). The amplitude and phase of the natural oscillation are determined by the initial conditions, while the amplitude of the forced oscillation is determined by the driving force and the parameters of the oscillator (mass, spring constant, and damping coefficient).

To solve for the constants A and \phi, we can use the initial conditions given in the problem (starting from rest at the origin at t=0). This would give us the following equations:

x(0) = A + \frac{F_0}{k-b^2m} = 0

\dot{x}(0) = -\omega A + b\frac{F_0}{k-b^2m} = 0

Solving these equations for A and \phi, we get:

A = \frac{-F_0}{k-b^2m}

\phi = \tan^{-1}\left(\frac{b\omega}{k-b^2m}\right)

Substituting these values back into the general solution for x(t), we get:

x(t) = \frac{-F_0}{k-b^2m}\cos(\omega t - \tan^{-1}\left(\frac{b\omega}{k-b^2m}\right)) + \frac{F_0}{k-b^2m}e^{-bt}

Therefore, the position x(t) as a function of time can be determined using this equation. It is important to note that the constants A, \
 

Related to Undamped oscillator with driving force

1. What is an undamped oscillator with driving force?

An undamped oscillator with driving force is a physical system that exhibits periodic motion without any energy loss (undamped) and is subjected to an external force (driving force) that causes the system to oscillate with a specific frequency.

2. What are the key components of an undamped oscillator with driving force?

The key components of an undamped oscillator with driving force are the oscillator itself, which can be a simple pendulum or a mass-spring system, and the driving force, which can be a periodic force or a force that varies with time.

3. How does the driving force affect the behavior of an undamped oscillator?

The driving force affects the behavior of an undamped oscillator by changing its amplitude, frequency, and phase. When the driving force has the same frequency as the natural frequency of the oscillator, resonance can occur, resulting in significantly amplified oscillations.

4. What is the mathematical equation that describes an undamped oscillator with driving force?

The mathematical equation that describes an undamped oscillator with driving force is the differential equation known as the harmonic oscillator equation. It states that the second derivative of the position of the oscillator with respect to time is equal to the negative of its natural frequency squared multiplied by its displacement from equilibrium, plus the driving force.

5. How is the behavior of an undamped oscillator with driving force affected by changes in the driving force or oscillator parameters?

The behavior of an undamped oscillator with driving force is affected by changes in the driving force or oscillator parameters in different ways. Changes in the driving force can alter the amplitude, frequency, and phase of the oscillations, while changes in the oscillator parameters, such as mass or spring constant, can change the natural frequency and alter the behavior of the oscillator. These changes can result in different types of motion, such as resonance, beats, or chaotic motion.

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