Exponentially driven harmonic oscillator

In summary: Great. So in summary, we have an un-damped harmonic oscillator with a natural frequency ##\omega_0## subjected to a driving force of ##F(t)=ame^{-bt}##. The equation of motion for this system is given by ##m\ddot{x}+kx=F(t)##. By dividing through by ##m##, we get the differential equation ##\ddot{x}+\omega_0^2x=ae^{-bt}##. To find the general solution, we use the method of summing the homogenous solution with the particular solution. The homogenous solution is of the form ##A\cos{\omega_0t}+B\sin{\omega_0t}##.
  • #1
vbrasic
73
3

Homework Statement


An un-damped harmonic oscillator natural frequency ##\omega_0## is subjected to a driving force, $$F(t)=ame^{-bt}.$$ At time, ##t=0##, ##x=\dot{x}=0##. Find the equation of motion.

Homework Equations


##F=m\ddot{x}##

The Attempt at a Solution


We have $$m\ddot{x}+kx=F(t)=ame^{-bt}.$$ Dividing through by ##m## we have $$\ddot{x}+\omega_0^2x=ae^{-bt}.$$ From a course in differential equations, I know that the solution to this is the sum of the homogenous solution with the particular solution. The homogenous solution is of the form $$A\cos{\omega_0t}+B\sin{\omega_0t}.$$ I am having a bit of trouble finding the particular solution. Because the solution is an exponential, I assume a solution of the form ##x(t)=ce^{-bt}.## Differentiating twice, we have $$\ddot{x}=b^2ce^{-bt}.$$ Plugging into the expression gives, $$b^2ce^{-bt}+\omega_0^2ce^{-bt}=ae^{-bt}.$$ From this we have that $$a=b^2c+\omega_0^2c.$$ I'm not sure if I'm on the right track here. Any help would be appreciated.
 
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  • #2
First of all, you have a typo in your differential equation. I suggest you check the dimensions of each term to find it.

vbrasic said:
Plugging into the expression gives, $$b^2ce^{-bt}+\omega_0ce^{-bt}=ae^{-bt}.$$ From this we have that $$a=b^2c+\omega_0c.$$ I'm not sure if I'm on the right track here. Any help would be appreciated.

What is your unknown in finding the particular solution? Can you find it from your equation? Does what you get solve the differential equation?
 
  • #3
Orodruin said:
First of all, you have a typo in your differential equation. I suggest you check the dimensions of each term to find it.

Should have been ##\omega_0^2.## Also, I'm not solving for ##a##. Rather I should be solving for the ##c## of the particular solution. So from the $$(b^2+\omega_0^2)c=a,$$ we get $$c=\frac{a}{b^2+\omega_0^2}.$$ We verify this is indeed the particular solution. So the most general solution is, $$A\cos{\omega_0t}+B\sin{\omega_0t}+\frac{a}{b^2+\omega_0^2}e^{-bt}.$$ From here we would apply the initial conditions?
 
  • #4
Indeed, applying the initial conditions will fix your unknown constants ##A## and ##B##.
 

1. What is an exponentially driven harmonic oscillator?

An exponentially driven harmonic oscillator is a physical system that exhibits oscillatory motion, where the acceleration of the system is proportional to its displacement from its equilibrium position, and is also influenced by an external force that varies exponentially with time.

2. What are the key characteristics of an exponentially driven harmonic oscillator?

The key characteristics of an exponentially driven harmonic oscillator are its natural frequency, damping coefficient, and driving amplitude. The natural frequency determines the rate at which the oscillator oscillates without any external force, the damping coefficient describes how quickly the oscillations decrease in amplitude due to friction or other dissipative forces, and the driving amplitude controls the strength and frequency of the external force.

3. What is the equation of motion for an exponentially driven harmonic oscillator?

The equation of motion for an exponentially driven harmonic oscillator is given by:
m * d^2x/dt^2 + b * dx/dt + kx = F0 * e^(γt)
where m is the mass of the oscillator, b is the damping coefficient, k is the spring constant, and F0 and γ are the amplitude and decay rate of the driving force, respectively.

4. How does the damping coefficient affect the motion of an exponentially driven harmonic oscillator?

The damping coefficient affects the motion of an exponentially driven harmonic oscillator by determining the rate of decrease in amplitude of the oscillations. A larger damping coefficient results in faster decay of the oscillations, leading to a shorter period and lower amplitude of the motion.

5. What are some real-world examples of an exponentially driven harmonic oscillator?

Some real-world examples of an exponentially driven harmonic oscillator include a pendulum, a mass-spring system, and a guitar string. These systems exhibit oscillatory motion when acted upon by an external force, such as gravity, a hand plucking a string, or a car driving over a speed bump.

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