# Initial Conditions of an Undamped Forced Harmonic Oscillator

• ColdFusion85
In summary, the equation of motion for an undamped harmonic oscillator is x(t) = Acos(\omega_0*t) + Bsin(\omega_0*t) + \frac{F_0}{m}\frac{cos(\omega*t)}{\omega_0^2-\omega^2}. Setting A and B equal to zero results in x(0) = \frac{F_0}{m}\frac{1}{\omega_0^2 - \omega^2} and v(0)=0. The first two terms represent the motion of the system at its natural frequency, while the third term decays exponentially. If there
ColdFusion85
The equation of motion of an undamped harmonic oscillator with driving force $$F=F_ocos(\omega*t)$$ is

$$x(t) = Acos(\omega_0*t) + Bsin(\omega_0*t) + \frac{F_0}{m}\frac{cos(\omega*t)}{\omega_0^2-\omega^2}$$

I am to determine the initial conditions such that the undamped oscillator begins steady state motion immediately. Is steady state motion simply when $$\frac{F_0}{m}\frac{cos(\omega*t)}{\omega_0^2-\omega^2} = Acos(\omega*t-\theta)?$$

I really have no idea how to approach this problem. Any help would be appreciated. No answers, just hints. Thanks.

ColdFusion85 said:
The equation of motion of an undamped harmonic oscillator with driving force $$F=F_ocos(\omega*t)$$ is

$$x(t) = Acos(\omega_0*t) + Bsin(\omega_0*t) + \frac{F_0}{m}\frac{cos(\omega*t)}{\omega_0^2-\omega^2}$$

I am to determine the initial conditions such that the undamped oscillator begins steady state motion immediately. Is steady state motion simply when $$\frac{F_0}{m}\frac{cos(\omega*t)}{\omega_0^2-\omega^2} = Acos(\omega*t-\theta)?$$

I really have no idea how to approach this problem. Any help would be appreciated. No answers, just hints. Thanks.

My first impression is that a couple of derivatives of x(t) with respect to t would be helpful.

it appears that one gets the correct answer if you set A and B equal to zero. if i do so, i get $$x(0) = \frac{F_0}{m}\frac{1}{\omega_0^2 - \omega^2}$$ and the derivative produces a sin term which would make v(0)=0, which is correct as well. Are the first two terms (Acoswt and B sinwt) the damping or some sort of interfering expression that goes away when the motion becomes steady state?

ColdFusion85 said:
it appears that one gets the correct answer if you set A and B equal to zero. if i do so, i get $$x(0) = \frac{F_0}{m}\frac{1}{\omega_0^2 - \omega^2}$$ and the derivative produces a sin term which would make v(0)=0, which is correct as well. Are the first two terms (Acoswt and B sinwt) the damping or some sort of interfering expression that goes away when the motion becomes steady state?

Some justification for setting those terms equal to zero would be good. Those terms represent the motion of the system at its natural frequency. They are solutions to the homgeneous differential equation of a harmonic oscillator. If there were no driving force and you set the system in motion, those terms would be the ones to keep, with A and B established by the initial displacement and velocity of the system.

I think the satement of the problem is weak. If the oscillator is truly undamped, then "steady state" is a misnomer. There really would not be a decay to steady state motion in that case. The natural frequency oscillations would last forever if they were ever excited. If there were any damping it would show up as a decaying exponential in front of those first two terms. So when they say steady state, what they mean is the steady state of a lightly damped oscillator where you gradually remove the damping as steady state is approached. In any case, what they are calling "steady state" is achieved when those first two terms are gone. Damping would kill them eventually if they were present. You can also kill them with the intial conditions making A = B = 0 as you have done.

OlderDan said:
Some justification for setting those terms equal to zero would be good. Those terms represent the motion of the system at its natural frequency. They are solutions to the homgeneous differential equation of a harmonic oscillator. If there were no driving force and you set the system in motion, those terms would be the ones to keep, with A and B established by the initial displacement and velocity of the system.

Yeah, I recall dealing with such cases of harmonic oscillators in Differential Equations. I think I got this now. Thanks Dan.

## 1. What is an undamped forced harmonic oscillator?

An undamped forced harmonic oscillator is a physical system where a mass is attached to a spring and is subject to an external force. The mass is able to oscillate back and forth with a certain frequency and amplitude.

## 2. What are the initial conditions of an undamped forced harmonic oscillator?

The initial conditions of an undamped forced harmonic oscillator include the initial displacement, velocity, and acceleration of the mass. These values determine the starting point of the oscillation and can affect the amplitude and frequency of the motion.

## 3. How do initial conditions affect the behavior of an undamped forced harmonic oscillator?

The initial conditions of an undamped forced harmonic oscillator can greatly influence its behavior. For example, a larger initial displacement will result in a higher amplitude of oscillation, while a larger initial velocity will result in a higher frequency of oscillation.

## 4. Can the initial conditions be changed during the motion of an undamped forced harmonic oscillator?

Yes, the initial conditions can be changed during the motion of an undamped forced harmonic oscillator. This can occur if an external force is applied or if the system experiences any other type of disturbance. The new initial conditions will then affect the future behavior of the oscillator.

## 5. How can the initial conditions of an undamped forced harmonic oscillator be calculated or measured?

The initial conditions of an undamped forced harmonic oscillator can be calculated or measured through various methods. For example, the initial displacement can be measured using a ruler or other measuring device, while the initial velocity can be calculated using the formula v = dx/dt (where x is the displacement and t is time). The initial acceleration can also be calculated using the formula a = d^2x/dt^2 (where x is the displacement and t is time).

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