General Solution to the Driven Harmonic Oscillator Problem

Abstract

In this paper, we present a comprehensive solution to the problem of the driven or forced harmonic oscillator under the influence of an external periodic force. We provide a method of solution based on the elimination of the first derivative in a second-order linear differential equation. This approach is particularly useful for students in elementary physics or mechanics courses.

Keywords

Forced Oscillator, Driven Oscillator, Ordinary Differential Equation, Harmonic Oscillator

Introduction

The harmonic oscillator is one of the fundamental problems in classical and quantum physics and mechanics.

Its solution is well-documented in the classical context, with various formal methods available in the literature.

In this paper, we aim to provide a general solution to the problem of the forced harmonic oscillator. We consider the external force to be a time-dependent function, which is not limited to sinusoidal functions.

The differential equation governing the forced or driven harmonic oscillator is given as:

$$mfrac{d^2x}{dt^2} + rfrac{dx}{dt} + kx = F(t)$$ .

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..........(1)

Where:

• $$m$$ is the mass of the oscillator
• $$r$$ is the damping coefficient
• $$k$$ is the proportionality constant
• $$F(t)$$ is a time-dependent function (which can be sinusoidal, cosine, versine, vercosine, hacoversine, or hacovercosine).

Equation (1) can be written as:

$$mfrac{d^2x}{dt^2} + 2bfrac{dx}{dt} + n^2x = F_1(t)$$

Where:

• $$2b = frac{r}{m}$$
• $$n^2 = frac{k}{m}$$
• $$F_1(t) = F(t)$$

Comparing this equation with the standard form $$x'' + Px' + Qx = R$$, we have:

$$P = 2b$$

$$Q = n^2$$

$$R = F_1(t)$$

Now, let's remove the first derivative from the given equation:

$$mfrac{d^2x}{dt^2} + 2bfrac{dx}{dt} + n^2x = F_1(t)$$

We choose:

$$v(t) = frac{dx}{dt}$$

Let the complete solution be:

$$x(t) = uv$$

Then it is given by the normal equation:

$$mfrac{d^2x}{dt^2} + 2bfrac{dx}{dt} + n^2x = F_1(t)$$

Where:

$$frac{d^2u}{dt^2} = frac{F_1(t)}{m} - 2bfrac{du}{dt} - n^2u$$ .

..........(2)

Let's denote:

$$D = m$$

Now, we can find the auxiliary equation and its roots:

$$D = m$$

$$Rightarrow D = 0$$

So, the complementary solution (C.F.) is:

$$C.F. = u_c = Ae^pt + Be^-pt$$

Where:

$$p = -b pm sqrt{b^2 - n^2}$$

Particular Integral

For the particular integral (P.I.), we have:

$$P.I. = u_p$$

Assuming the external force is a versine function, we can write:

$$P.I. = u_p = f(t)$$

Where:

$$f(t) = F_1(t)$$

Now, let's solve for the P.I. using the appropriate method. Once we have the P.I., we can write the complete solution as:

$$x(t) = uv = (Ae^pt + Be^-pt) + u_p$$

Now, by applying the given initial conditions, we can determine the values of $$A$$ and $$B$$, and ultimately, we can find the displacement of the oscillator at any time $$t$$.

Conclusion

In this paper, we have presented a general solution to the problem of the driven or forced harmonic oscillator. We considered an external force that can be any time-dependent function, not limited to sinusoidal functions. By transforming the original differential equation and using the method of elimination of the first derivative, we obtained a complete solution for the system.

This solution provides valuable insights for students studying elementary physics or mechanics, as it demonstrates how to approach and solve complex differential equations that arise in real-world scenarios. By understanding the dynamics of a driven harmonic oscillator, students can apply these principles to a wide range of physical systems, further enhancing their problem-solving skills in the field of physics.

References

1. Dr. P.G. Kshirsagar, A Text Book for Engineering Physics
2. H. K. Das, Advance Engineering Mathematics, Publication S.K.Chand
3. Sharma, D., Poudel, T. N., & Adhakari, H. P. (2067). Engineering Mathematics II. Kathmandu: Sukanda Pustak Bhawan.
4. Harper, C. (1999). Introduction to Mathematical Physics. New Delhi: Prentice Hall of India Private Limited.