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Damped Harmonic Oscillation

In the previous chapter, we encountered a number of energy conserving physical systems which exhibit simple harmonic oscillation about a stable equilibrium state. One of the main features of such oscillation is that, once excited, it never dies away. However, the majority of the oscillatory systems that we encounter in everyday life suffer some sort of irreversible energy loss whilst they are in motion, which is due, for instance, to frictional or viscous heat generation. We would therefore expect oscillations excited in such systems to eventually be damped away. Let us examine an example of a damped oscillatory system.

Consider the mass-spring system investigated in Section 2.1. Suppose that, as it slides over the horizontal surface, the mass is subject to a frictional damping force which opposes its motion, and is directly proportional to its instantaneous velocity. It follows that the net force acting on the mass when its instantaneous displacement is $x(t)$ takes the form

f = - k\,x -m\,\nu\,\dot{x},
\end{displaymath} (55)

where $m>0$ is the mass, $k>0$ the spring force constant, and $\nu>0$ a constant (with the dimensions of angular frequency) which parameterizes the strength of the damping. The time evolution equation of the system thus becomes [cf., Equation (2)]
\ddot{x} + \nu\,\dot{x} + \omega_0^{\,2}\,x = 0,
\end{displaymath} (56)

where $\omega_0=\sqrt{k/m}$ is the undamped oscillation frequency [cf., Equation (6)]. We shall refer to the above as the damped harmonic oscillator equation.

Let us search for a solution to Equation (56) of the form

x(t) = a\,{\rm e}^{-\gamma\,t}\,\cos(\omega_1\,t-\phi),
\end{displaymath} (57)

where $a>0$, $\gamma>0$, $\omega_1>0$, and $\phi$ are all constants. By analogy with the discussion in Section 2.1, we can interpret the above solution as a periodic oscillation, of fixed angular frequency $\omega_1$ and phase angle $\phi$, whose amplitude decays exponentially in time as $a(t)=a\,\exp(-\gamma\,t)$. So, (57) certainly seems like a plausible solution for a damped oscillatory system. It is easily demonstrated that
$\displaystyle \dot{x}$ $\textstyle =$ $\displaystyle - \gamma\,a\,{\rm e}^{-\gamma\,t}\,\cos(\omega_1\,t-\phi) -\omega_1\, a\,{\rm e}^{-\gamma\,t}\,\sin(\omega_1\,t-\phi),$ (58)
$\displaystyle \ddot{x}$ $\textstyle =$ $\displaystyle (\gamma^2-\omega_1^{\,2})\,a\,{\rm e}^{-\gamma\,t}\,\cos(\omega_1\,t-\phi) +2\,\gamma\,\omega_1\, a\,{\rm e}^{-\gamma\,t}\,\sin(\omega_1\,t-\phi),$ (59)

so Equation (56) becomes
$\displaystyle 0$ $\textstyle =$ $\displaystyle \left[ (\gamma^2-\omega_1^{\,2}) -\nu\,\gamma + \omega_0^{\,2}\right]a\,{\rm e}^{-\gamma\,t}\,\cos(\omega_1\,t-\phi)$  
    $\displaystyle +\left[2\,\gamma\,\omega_1-\nu\,\omega_1\right] a\,{\rm e}^{-\gamma\,t}\,\sin(\omega_1\,t-\phi).$ (60)

Now, the only way in which the above equation can be satisfied at all times is if the coefficients of $\exp(-\gamma\,t)\,\cos(\omega_1\,t-\phi)$ and $\exp(-\gamma\,t)\,\sin(\omega_1\,t-\phi)$ separately equate to zero, so that
$\displaystyle (\gamma^2-\omega_1^{\,2}) -\nu\,\gamma+\omega_0^{\,2}$ $\textstyle =$ $\displaystyle 0,$ (61)
$\displaystyle 2\,\gamma\,\omega_1-\nu\,\omega_1$ $\textstyle =$ $\displaystyle 0.$ (62)

These equations can be solved to give
$\displaystyle \gamma$ $\textstyle =$ $\displaystyle \nu/2,$ (63)
$\displaystyle \omega_1$ $\textstyle =$ $\displaystyle (\omega_0^{\,2}-\nu^2/4)^{1/2}.$ (64)

Thus, the solution to the damped harmonic oscillator equation is written
x(t) = a\,{\rm e}^{-\nu\,t/2}\,\cos\left(\omega_1 \,t-\phi\right),
\end{displaymath} (65)

assuming that $\nu< 2\,\omega_0$ (since $\omega_1^{\,2}=\omega_0^{\,2}-\nu^2/4$ clearly cannot be negative). We conclude that the effect of a relatively small amount of damping, parameterized by the damping constant $\nu$, on a system which exhibits simple harmonic oscillation about a stable equilibrium state is to reduce the angular frequency of the oscillation from its undamped value $\omega_0$ to $(\omega_0^{\,2}-\nu^2/4)^{1/2}$, and to cause the amplitude of the oscillation to decay exponentially in time at the rate $\nu/2$. This modified type of oscillation, which we shall refer to as damped harmonic oscillation, is illustrated in Figure 6. [Here, $T_0=2\pi/\omega_0$, $\nu\,T_0=0.5$, and $\phi = 0$. The solid line shows $x(t)/a$, whereas the dashed lines show $\pm a(t)/a$.] Incidentally, if the damping is sufficiently large that $\nu\geq 2\,\omega_0$, which we shall assume is not the case, then the system does not oscillate at all, and any motion simply decays away exponentially in time (see Exercise 3).

Figure 6: Damped harmonic oscillation.
\epsfysize =3in

Note that, although the angular frequency, $\omega_1$, and decay rate, $\nu/2$, of the damped harmonic oscillation specified in Equation (65) are determined by the constants appearing in the damped harmonic oscillator equation, (56), the initial amplitude, $a$, and the phase angle, $\phi$, of the oscillation are determined by the initial conditions. In fact, if $x(0)=x_0$ and $\dot{x}(0)=v_0$ then it follows from Equation (65) that

$\displaystyle x_0$ $\textstyle =$ $\displaystyle a\,\cos\phi,$ (66)
$\displaystyle v_0$ $\textstyle =$ $\displaystyle - \frac{\nu}{2}\,a\,\cos\phi + \omega_1\,a\,\sin\phi,$ (67)

$\displaystyle a$ $\textstyle =$ $\displaystyle \left[x_0^{\,2} + \frac{(v_0+\nu\,x_0/2)^2}{\omega_1^{\,2}}\right]^{1/2},$ (68)
$\displaystyle \phi$ $\textstyle =$ $\displaystyle \tan^{-1}\left(\frac{v_0+\nu\,x_0/2}{\omega_1\,x_0}\right).$ (69)

Note, further, that the damped harmonic oscillator equation is a linear differential equation: i.e., if $x(t)$ is a solution then so is $a\,x(t)$, where $a$ is an arbitrary constant. It follows that the solutions of this equation are superposable, so that if $x_1(t)$ and $x_2(t)$ are two solutions corresponding to different initial conditions then $a\,x_1(t)+b\,x_2(t)$ is a third solution, where $a$ and $b$ are arbitrary constants.

Multiplying the damped harmonic oscillator equation (56) by $\dot{x}$, we obtain

\dot{x}\,\ddot{x} + \nu\,\dot{x}^{\,2}+ \omega_0^{\,2}\,\dot{x}\,x=0,
\end{displaymath} (70)

which can be rearranged to give
\frac{dE}{dt} = - m\,\nu\,\dot{x}^{\,2},
\end{displaymath} (71)

E = \frac{1}{2}\,m\,\dot{x}^{\,2}+\frac{1}{2}\,k\,x^2
\end{displaymath} (72)

is the total energy of the system: i.e., the sum of the kinetic and potential energies. Clearly, since the right-hand side of (71) cannot be positive, and is only zero when the system is stationary, the total energy is not a conserved quantity, but instead decays monotonically in time due to the presence of damping. Now, the net rate at which the force (55) does work on the mass is
P = f\,\dot{x} = -k\,\dot{x}\,x-m\,\nu\,\dot{x}^2.
\end{displaymath} (73)

Note that the spring force (i.e., the first term on the right-hand side) does negative work on the mass (i.e., it reduces the system kinetic energy) when $\dot{x}$ and $x$ are of the same sign, and does positive work when they are of the opposite sign. On average, the spring force does no net work on the mass during an oscillation cycle. The damping force, on the other hand, (i.e., the second term on the right-hand side) always does negative work on the mass, and, therefore, always acts to reduce the system kinetic energy.

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Next: Quality Factor Up: Damped and Driven Harmonic Previous: Damped and Driven Harmonic
Richard Fitzpatrick 2010-10-11