Question

If you kick a harmonic oscillator sharply, you impart to it an initial velocity but no initial displacement. For a weakly damped oscillator with mass \(m\), spring constant \(k\), and damping force \(F_{\gamma}=-b v\), find \(x(t)\), if the total impulse delivered by the kick is \(J_{0}\).

Short answer

Question: Determine the displacement of a weakly damped oscillator with mass \(m\), spring constant \(k\), and damping force \(b\) after receiving an initial impulse \(J_0\). Answer: The displacement of the weakly damped oscillator after receiving the impulse \(J_0\) is given by: \(x(t) = \frac{J_{0}}{2mr_2} (e^{(r_1t)} - e^{(r_2t)})\) Where \(r_1=\frac{-b+\sqrt{4mk-b^2}}{2m}i\) and \(r_2=\frac{-b-\sqrt{4mk-b^2}}{2m}i\).

Step-by-step solution

Write down the equation for the damped harmonic oscillator

The equation for a damped harmonic oscillator is given by: \(m\ddot{x}(t) + b\dot{x}(t) + kx(t) = 0\) Where \(m\) is the mass, \(k\) is the spring constant, \(b\) is the damping constant, and \(x(t)\), \(\dot{x}(t)\), and \(\ddot{x}(t)\) are the displacement, velocity, and acceleration of the oscillator, respectively.

Determine the initial conditions

The problem states that there is no initial displacement, that is \(x(0) = 0\). The initial velocity is determined by the total impulse delivered by the kick, by using the impulse-momentum theorem, we have: \(\displaystyle\int_{-\epsilon}^\epsilon F_{\gamma} dt = m\Delta{v}\) Since the impulse is delivered very fast, \(\displaystyle\int_{-\epsilon}^\epsilon F_{\gamma} dt \approx 0\), therefore, \(m\Delta{v} = J_0\) And the initial velocity is, \(\dot{x}(0) = \frac{J_{0}}{m}\)

Solve the differential equation for x(t)

To solve the damped harmonic oscillator equation, we will use the ansatz \(x(t) = e^{rt}\), where \(r\) is a complex number. Substituting this into the original equation, we have: \(m\ddot{x}(t) + b\dot{x}(t) + kx(t) = m( r^2e^{rt}) + b(re^{rt}) + ke^{rt} = 0\) Factoring out \(e^{rt}\): \((mr^2+br+k)e^{rt}=0\) Since \(e^{rt}\) is never zero, the term in parenthesis must be equal to zero: \(mr^2+br+k=0\) This is the characteristic equation for the damped harmonic oscillator. The roots of the equation can be real or complex, depending on the discriminant (\(\Delta\)): \(\Delta=b^2-4mk\) In our case, it is specified that the damping is weak, which means that we have an underdamped system (\(b^2<4mk\)). Thus, the roots of the characteristic equation are complex: \(r=\frac{-b\pm\sqrt{-4mk+b^2}}{2m}=\frac{-b\pm\sqrt{4mk-b^2}}{2m}i\)

Express x(t) in terms of complex exponentials

Now that we have the complex roots, the solution to the differential equation is given by: \(x(t) = Ae^{(r_1t)} + Be^{(r_2t)}\) Where \(A\) and \(B\) are constants that we will determine from the initial conditions, and \(r_1=\frac{-b+\sqrt{4mk-b^2}}{2m}i\) and \(r_2=\frac{-b-\sqrt{4mk-b^2}}{2m}i\). Applying the initial conditions: \(x(0) = A + B = 0\) \(\dot{x}(0) = r_1A + r_2B = \frac{J_{0}}{m}\) With these initial conditions, we find that \(A=-B\) and \(B=\frac{J_{0}}{2mr_2}\).

Final solution for x(t)

Plugging the values of \(A\) and \(B\) into the equation for \(x(t)\), we get: \(x(t) = \frac{J_{0}}{2mr_2} (e^{(r_1t)} - e^{(r_2t)})\) This is the displacement of the weakly damped oscillator after receiving the impulse \(J_0\).

Key concepts

Differential Equations

Invariant in the mathematical modeling of natural phenomena, differential equations are equations that relate functions with their derivatives. In physics, they often describe the rate at which something changes over time, or over another variable. Differential equations are a cornerstone of dynamic systems and control theory, playing a critical role in disciplines ranging from engineering to economics.

For the damped harmonic oscillator, the differential equation \(m\ddot{x}(t) + b\dot{x}(t) + kx(t) = 0\) incorporates both the resistance offered by damping and the restorative force provided by the spring. The displacements over time (\

Impulse-Momentum Theorem

The impulse-momentum theorem is a fundamental principle in physics derived from Newton's second law of motion. It states that the impulse applied to an object is equal to the change in momentum of the object. In mathematical terms, the theorem is represented by \( \int_{-\epsilon}^{\epsilon} F\ dt = m\Delta{v} \), where \( F \) is the force applied, \( m \) is the mass, and \( \Delta{v} \) is the change in velocity.

In the exercise provided, an impulse \( J_{0} \) is delivered to the damped harmonic oscillator, which sets it in motion. With no initial displacement, the impulse solely contributes to the initial velocity. Here, the theorem helps determine the starting conditions needed to solve the differential equation for the post-impulse motion of the system. },{

Complex Exponentials

The concept of complex exponentials extends the idea of exponential functions into the complex plane. These functions take the form \( e^{rt} \), where \( r \) can be a complex number. They are indispensable in solving differential equations with complex roots, as they elegantly handle oscillatory behavior.

In our damped harmonic oscillator, complex exponentials arise from the characteristic roots which are complex because the system is underdamped. When writing the solution for \( x(t) \), complex exponentials help to elucidate the oscillatory nature of the motion combined with an exponential decay, due to the term \( e^{(-bt/2m)} \) originating from the real part of the complex roots. Moreover, Euler's formula, \( e^{ix} = \cos(x) + i\sin(x) \), can be applied to write these solutions in terms of sines and cosines for easier interpretation and analysis. },{

Characteristic Equation

The characteristic equation of a differential equation is a key tool in finding the solutions. It emerges from substituting a trial solution, often involving exponentials, into a linear homogeneous differential equation. For the damped harmonic oscillator, the characteristic equation is \( mr^2 + br + k = 0 \), and its roots specify the nature of the system's response over time.

Whether the roots are real or complex dictates whether the system experiences overdamping, critical damping, or underdamping. In the scenario of our exercise, where the damping is weak (\( b^2 < 4mk \)), the characteristic equation yields complex roots, explaining the resulting underdamped oscillations after the impulse. This equation is the fingerprint of the harmonic system's response and is crucial for understanding its temporal behavior.