9.6 KiB
Chapter 5 - Oscillations
Section 5.1 - Hooke's Law
Theorem. Hooke's Law. The force exerted by a spring has the form F_x = -kx, where x is the displacement from its equilibrium length and k is the spring constant.
From this, knowing F = -\nabla U, we see that the potential energy of a spring is U(x) = \frac{1}{2} kx^2.
Section 5.2 - Simple Harmonic Motion
From F = m\ddot{x} = -kx, we see that \ddot{x} = -\frac{k}{m} x. We now establish \omega = \sqrt{\frac{k}{m}}, or the angular frequency. This allows us to state
\ddot{x} = -\omega^2 x
This may also be written as \ddot{x} + \omega^2 x = 0, leading to the characteristic equation r = \pm \omega i. Now, we can write the solutions as x(t) = e^{\pm i \omega t}, so that x(t) = C_1 e^{i \omega t} + C_2 e^{-i \omega t} by superposition.
We know from Differential Equations that for r = a \pm bi, we can apply Euler's formula to see the solution is e^a(A \cos(b t) + B \sin(bt)), where A = C_1 + C_2 and B = i(C_1 - C_2). Applying this to the spring, we see that
x(t) = B_1 \cos (\omega t) + B_2 \sin(\omega t)
The displacement function being real thus mandates that B_2 is real.
Notably, at t = 0, we see that x(0) = B_1, and \dot{x}(0) = B_2.
Definition. The period \tau is the time in which a function repeats itself. The period for \sin and \cos are \tau = 2\pi, so the period for this oscillator is $\tau = \frac{2\pi}{\omega} = 2\pi \sqrt{\frac{m}{k}}.
Now, let us simplify. We can write x as
x(t) = \sqrt{B_1^2 + B_2^2}[\frac{B_1}{\sqrt{B_1^2 + B_2^2}} \cos(\omega t) + \frac{B_2}{\sqrt{B_1^2 + B_2^2}} \sin(\omega t)]
Now, if we have a right triangle with legs B_1 and B_2, we can define some \delta in said triangle such that \cos(\delta) = \frac{B_1}{\sqrt{B_1^2 + B_2^2}} and \sin(\delta) = \frac{B_2}{\sqrt{B_1^2 + B_2^2}}. Note that by this definition, \tan(\delta) = \frac{B_2}{B_1}. We can thus simplify the equation for x to
x(t) = \sqrt{B_1^2 + B_2^2}[\cos(\delta) \cos(\omega t) + \sin(\delta) \sin(\omega t)] = \sqrt{B_1^2 + B_2^2} \cos(\omega t - \delta)
A different way to see this is to write C_1 = \frac{1}{2}(B_1 - iB_2) and C_2 = \frac{1}{2}(B_1 + iB_2). We can then say that C_2 = C_1^*. Then, x(t) = C_1 e^{i\omega t} + C_1^* e^{-i\omega t}
We can then solve to see x(t) = 2\Re(C_1 e^{i \omega t}), as x(t) \in \mathbb{R}. Then, we can see that 2C_1 = B_1 - i B_2 = \frac{1}{\sqrt{B_1^2 + B_2^2}} e^{-i \delta}. This implies that x(t) = \Re 2C_1 e^{i \omega t} = \Re \frac{1}{\sqrt{B_1^2 + B_2^2}} A e^{i(\omega t - \delta)}
If we consider energy, we see that U = \frac{1}{2}kx^2 = \frac{1}{2}kA^2 \cos^2(\omega t - \delta), and that T = \frac{1}{2}m \dot{x}^2 = \frac{1}{2}m \omega^2 A^2 \sin^2(\omega t - \delta) = \frac{1}{2} k A^2 \sin^2 (\omega t - \delta), as \omega^2 = k/m from the original differential equation.
Note that this implies the total energy is a constant, where E = T + U = \frac{1}{2}kA^2.
Section 5.3 - Two-Dimensional Oscillators
Definition. An isotropic harmonic oscillator is an oscillator in which \mathbf{F} = -k \mathbf{r}. That is, F_x = -kx, F_y = -ky$, and F_z = -kz. Then, with \omega = \sqrt{k/m}, we see that x(t) = A_x \cos(\omega t - \delta_x), and y(t) = A_y \cos(\omega t - \delta_y). We can redefine the origin of time such that \delta_x = 0 and \delta_y = delta.
Definition. An anisotropic oscillator is one in which k_x \neq k_y. Then, we see that x(t) = A_x \cos(\omega_x t) and y(t) = A_y \cos(\omega_y t - \delta).
Section 5.4 - Damped Oscillations
We now consider a damping term, in which \mathbf{F}_{damping} = -b \hat{\mathbf{x}}. Then, we write
m\ddot{x} + b\dot{x} + kx = 0
An important note is that this similar to an RLC circuit, in which
L\ddot{q} + R\dot{q} + \frac{1}{C}q=0
Definition. We will define a damping constant 2\beta = \frac{b}{m}. Additionally, we will rename the constant \omega_0^2 = k/m, which denotes the natural frequency, or the frequency the system operates at for b = 0, that is, there are no resistive forces present. This allows us to rewrite the differential equation as
\ddot{x} + 2 \beta \dot{x} + \omega_0^2 x = 0
Solving the characteristic (or, as this textbook calls it, the auxiliary equation), we see that r = -\beta \pm \sqrt{\beta^2 - \omega_0^2}. Then, we can write x(t) = e^{-\beta t}(C_1 e^{\sqrt{\beta^2 - \omega_0^2}t} + C_2 e^{-\sqrt{\beta^2 - \omega_0^2}t}).
Applying \beta = 0, we recover the solution for an undamped oscillator.
Now, consider \beta \leq \omega_0, which we call underdamping. In this case, \sqrt{\beta^2 - \omega_0^2} = i\sqrt{\omega_0^2 - \beta^2} = i \omega_1, where \omega_1 = \sqrt{\omega_0^2 - \beta^2}. We can then write x(t) = e^{-\beta t}(C_1 e^{i\omega_1 t} + C_2 e^{-i\omega_1 t}) = Ae^{-\beta t}\cos(\omega_1 t - \delta).
Conversely, when \beta > \omega_0, we see overcamping. Here, x(t) = C_1 e^{-\beta + \sqrt{\beta^2 - \omega_0^2}} + C_2 e^{-\beta - \sqrt{\beta^2 - \omega_0^2}}.
When \beta = \omega_0, we are critically damped, in which case x(t) = C_1 e^{-\beta t} + C_2 t e^(-\beta t).
Section 5.5 - Driven Damped Oscillations
Consider an external force F(x), such that m\ddot{x} + b\dot{x} + kx = F(x). Then, for f(t) = F(t) / m,
\ddot{x} = 2\beta\dot{x} + \omega_0^2 x = f(t)
We can write the differential operator D = \frac{d^2}{dt^2} + 2\beta \frac{d}{dt} + \omega_0^2, so that the differential equation becomes Dx = f. Notably, this operator is linear.
Definition. The equation Dx = 0 is the homogenous equation.
Definition. Some x_p(t) that satisfies Dx = f is a particular solution.
Definition. Any x_h(t) that satisfies Dx = 0 is a homogenous solution.
Note that if x_p is a solution to a differential equation, by the linearity of D, so is x_p + x_h.
Now consider f(t) = f_0 \cos(\omega t), or a driving force with frequency \omega. Then, by the method of undetermined coefficients, our particular solution is x_p = A \cos (\omega t) + B \sin (\omega t). However, we're not allowed to use that. We are only allowed to assume there is a second solution y(t) that satisfies f(t) = f_0 \sin(\omega t) instead that also satisfies the differential equation for the original function.
Then, we set z(t) = x(t) + i y(t) and rewrite the differential equation in terms of z, where f(t) = f_0 e^{i \omega t}. We can then try the solution C e^{i \omega t} to see that C = \frac{f_0}{\omega_0^2 - \omega^2 + 2 i \beta \omega}. Then, we can define some A, \delta such that C = Ae^{-i \delta}, and applying A^2 = C C^*, we see that A^2 = \frac{f_0^2}{(\omega_0^2 - \omega^2)^2 + 4 \beta^2 \omega^2}, and \delta = \arctan(\frac{2\beta \omega}{\omega_0^2 - \omega^2}).
We can then take the real part of z to see x(t) = A \cos(\omega t - \delta).
Definition. Any term that dies out exponentially as time passes is called transient. The motion that remains is called the attractor, as it can arise from many different initial conditions.
Section 5.6 - Resonance
When \beta is small, we see the second term in the denominator becomes small. Then, if the driving and natural frequency are very different, the amplitude of the resulting wave becomes small. Conversely, if the driving and natural frequency are close together, the amplitude of the wave becomes large.
Notably, if we try and force the oscillator to move at a frequency \omega, when close to \omega_0, the oscillator responds well, but fails to respond significantly otherwise. This is known as resonance.
Then, we can see that \omega_2 = \sqrt{\omega_0^2 - 2 \beta^2} is the frequency at which the response is maximum. This then lets us see that A_{max} \approx \frac{f_0}{2\beta \omega_0}.
We can then calculate the FWHM, or full-width at half maximum, and the HWHM, or the half width at half maximum. These are the distance between the two points in which A^2 is at half its maximum value. Note that \omega \approx \omega_0 \pm \beta, so \text{FWHM} \approx 2\beta and \text{HWHM} \approx \beta.
We can then calculate the sharpness of the peak as the natural frequency over the FWHM, or Q = \frac{\omega_0}{2\beta} = \pi \frac{1 / \beta}{2\pi \omega_0} = \pi \frac{\text{decay time}}{\text{period}}.
Additionally, we see that \delta = \arctan(\frac{2\beta \omega}{\omega_0^2 - \omega^2}). At resonance, this implies that \delta = \pi / 2.
Section 5.7 - Fourier Series
Skipped.
Section 5.8 - Fourier Series Solution for the Driven Oscillator
If we set f(t) = f_1(t) + f_2(t), we can write Dx_1 = f_1 and Dx_2 = f_2 such that D_x = D(x_1 + x_2) = f_1 + x_2 = f so then D_x = f.
Then, we let f(t) = \sum_{n=0}^\infty f_n \cos(n \omega t), where f_n is a constant.
We then find that \omega = 2\pi / \tau.
We can then set x_n(t) = A_n \cos(n \omega t - \delta_n), where
A_n = \frac{f_n}{\sqrt{(\omega_0^2 - n^2 \omega^2)^2 + 4\beta^2n^2\omega^2}}
We also see that
\delta_n = \arctan(\frac{2\beta n \omega}{\omega_0^2 - n^2 \omega^2})
Section 5.9 - The RMS Displacement; Parseval's Theorem
We know that x_{rms} = \sqrt{\langle x^2 \rangle}. Then, we see that \langle x^2 \rangle = \frac{1}{\tau} \int_{-\tau/2}^{\tau/2}x^2 dt. In this case,
\langle x^2 \rangle = \frac{1}{\tau} \int_{-\tau/2}^{\tau/2} \sum_n \sum_m A_n \cos(n\omega t - \delta_n) A_m \cos(n \omega t - \delta_m) dt
We can then apply identities to see that
\langle x^2 \rangle = A_0^2 + \frac{1}{2} \sum_{n=1}^\infty A_n^2
This assumes that A_0 is defined differently than A_{n \geq 1}.
This is known as Parseval's theorem, and allows us to find the response of an oscillator.