The collisions in Example 8 in Ch.11 are inelastic and mechanical energy will be lost in a collision. The mechanical energy loss in the first collision is
p2/2m− p2/2(M+ m) = (p2/2m)(M/(M+ m))
and in the second
p2/2(M+ m) − p2/2(M+ 2m).
(a) Show that the two energy losses are the same if m/M = 1 +√
2. Compare the two energy losses as a function of m/M.
(b) Suppose that n shots are fired into the block under conditions of maximum amplitude gain as explained in Example 8. What will be the amplitude of the oscillator after the n:th shot?
2.3 Free Damped Motion of a Linear Oscillator
2.3.1 Energy Considerations
The mechanical energy in the harmonic motion of a mass-spring oscillator is the sum of the kinetic energy Mu2/2 of the mass M and the potential energy V of the spring. If the displacement from the equilibrium position is ξ , the force required for this displacement is Kξ . The work done to reach this displacement is the potential energy
V (ξ )=
ξ 0
Kξ dξ = Kξ2/2. (2.18)
In the harmonic motion, there is a periodic exchange between kinetic and potential energy, each going from zero to a maximum value E, where E = Mu2/2+ Kξ2/2 is the total mechanical energy. In the absence of friction, this energy is a constant of motion.
To see how this follows from the equation of motion, we write the harmonic oscil-lator equation (2.13) in the form
M˙u + Kξ = 0, (2.19)
where u= ˙ξ is the velocity, and then multiply the equation by u. The first term in the equation becomes Mu˙u = d/dt[Mu2/2]. In the second term, which becomes Kuξ, we use u= ˙ξ so that it can be written Kξ ˙ξ = d/dt[Kξ2/2]. This means that Eq. 2.19 takes the form
d/dt[Mu2/2+ Kξ2/2] = 0. (2.20) The first term, Mu2/2, is the kinetic energy of the mass M, and the second term, Kξ2/2, is the potential energy stored in the spring. Each is time dependent but the sum, the total mechanical energy, remains constant throughout the motion. Although no new physics is involved in this result (since it follows from Newton’s law), the conservation of mechanical energy is a useful aid in problem solving.
In the harmonic motion, the velocity has a maximum when the potential energy is zero, and vice versa, and the total mechanical energy can be expressed either as the maximum kinetic energy or the maximum potential energy. The average kinetic and potential energies (over one period) are the same.
When a friction force is present, the total mechanical energy of the oscillator is no longer conserved. In fact, from the equation of motion M˙u + Kξ = −Ru it follows
by multiplication by u (see Eq. 2.20) that
d/dt[Mu2/2+ Kξ2/2] = −Ru2. (2.21) Thus, the friction drains the mechanical energy, at a rate−Ru2, and converts it into heat.4
As a result, the amplitude of oscillation will decay with time and we can obtain an approximate expression for the decay by assuming that the average potential and kinetic energy (over one period) are the same, as is the case for the loss-free oscillator.
Thus, with the left-hand side of Eq. 2.21 replaced by d(Mu2]/dt, and the right-hand side by Ru2, the time dependence of u2 will be
u(t)2 ≈ u(0)2e−(R/M)t. (2.22) The corresponding rms amplitude then will decay as exp[−(R/2M)t].
2.3.2 Oscillatory Decay
After having seen the effect of friction on the time dependence of the average energy, let us pursue the effect of damping on free motion in more detail and determine the actual decay of the amplitude and the possible effect of damping on the frequency of oscillation.
The idealized oscillator considered so far had no other forces acting on the mass than the spring force. In reality, there is also a friction force although in many cases it may be small. We shall assume the friction force to be proportional to the velocity of the oscillator. Such a friction force is often referred to as viscous or dynamic.
Normally, the contact friction with a table, for example, does not have such a simple velocity dependence. Often, as a simplification, one distinguishes merely between a
‘static’ and a ‘dynamic’ contact friction, the magnitude of the latter often assumed to be proportional to the magnitude of the velocity but with a direction opposite that of the velocity. The ‘static’ friction force is proportional to the normal component of the contact force and points in the direction opposite that of the horizontal component of the applied force.
A friction force proportional to the velocity can be obtained by means of a dashpot damper, as shown in Fig. 2.5. It is in parallel with the spring and is simply a ‘leaky’
piston which moves inside a cylinder. The piston is connected to the mass M of the oscillator and the force required to move the piston is proportional to its velocity relative to the cylinder (neglecting the mass of the piston). The cylinder is attached to the same fixed support as the spring, as indicated in Fig. 2.5. The fluid in the cylinder is then forced through a narrow channel (a ‘leak’) between the piston and the cylinder and it is the viscous stresses in this flow which are responsible for the friction force. Therefore, this type of damping is often referred to as viscous.
The friction on a body moving through air or some other fluid in free field will be proportional to the velocity only for very low speeds and approaches an approximate square law dependence at high speeds.
4When the concept of energy is extended to include other forms of energy other than mechanical, the law of conservation of energy does bring something new, the first law of thermodynamics which can be regarded as a postulate, the truth of which should be considered as an experimental fact.
Figure 2.5:Oscillator with dash-pot damper.
With a friction force proportional to the velocity, the equation of motion for the oscillator becomes linear so that a solution can be obtained in a simple manner. For dry contact friction or any other type of friction, the equation becomes non-linear and the solution generally has to be found by numerical means, as will be demonstrated in Section 2.7.3.
With dξ/dt ≡ ˙ξ, we shall express the friction force as −R ˙ξ and the equation of motion for the mass element in an oscillator becomes M ¨ξ = −Kξ − R ˙ξ or, with K/M= ω20,
F ree oscillat ions, damped oscillat or
¨ξ + (R/M)˙ξ + ω20ξ = 0 ξ(t )= Ae−γ tcos(ω0t− φ) γ = R/2M, ω0 =
ω20− γ2
. (2.23)
The general procedure to solve a linear differential equation is aided considerably with the use of complex variables (Section 2.3.3). For the time being, however, we use a ‘patchwork’ approach to construct a solution, making use of the result obtained in the decay of the energy in Eq. 2.22 from which it is reasonable to assume that the solution ξ(t) will be of the form given in Eq. 2.23, where γ , and ω0are to be determined. Thus, we insert this expression for ξ(t) into the first equation in 2.23 and write the left-hand side as a sum of sin(ω0t )and cos(ω0t )functions. Requiring that each of the coefficients of these functions be zero to satisfy the equation at all times, we get the required values of γ and ω0in Eq. 2.23. Actually, the value of γ is the same as obtained in Eq. 2.22. The damping makes the ω0 lower than ω0.
When there is no friction, i.e., γ = 0, the solution reduces to the harmonic motion discussed earlier, where A is the amplitude and φ the phase angle. The damping produces an exponential decay of the amplitude and also causes a reduction of the frequency of oscillation. If the friction constant is large enough to that ω0= 0, the motion is non-oscillatory and the oscillator is then said to be critically damped. If γ > ω0, the frequency ω0 formally becomes imaginary and the solution has to be reexamined, as will be done shortly. As it turns out, the general solution then consists of a linear combination of two decaying exponential functions.
2.3.3 Use of Complex Variables. Complex Frequency
With the use of complex variables in solving the damped oscillator equation, there is no need for the kind of patchwork that was used in Section 2.3.2. We merely let the mathematics do its job and present us with the solution.
It should be familiar by now, that the complex amplitude ξ(ω) of ξ(t) is defined by
ξ(t )= {ξ(ω)e−iωt}. (2.24)
The corresponding complex amplitudes of the velocity and the acceleration are then−iωξ(ω) and −ω2ξ(o)and if these expressions are used in Eq. 2.23 we obtain the following equation for ω
ω2+ i2γ − ω02= 0 (2.25)
in which γ = R/2M.
Formally, the solution to this equation yields complex frequencies ω= −iγ ±
ω2− γ2. (2.26)
The general solution is a linear combination of the solutions corresponding to the two solutions for ω, i.e.,
ξ(t )= e−γ t{A1eiω0t + A2e−iω0t}, (2.27) where ω0 =
ω2− γ2 and A1 and A2 are complex constants to be determined from initial conditions. We distinguish between the three types of solutions which correspond to γ < ω0, γ > ω0, and γ = ω0.
Oscillatory decay, γ < ω0. In this case, ω0is real, and the oscillator is sometimes referred to as underdamped; the general solution takes the form
ξ(t )= A e−γ tcos(ω0t− φ) (2.28) which is the same as in Eq. 2.23. The constants A and φ are determined by the initial conditions of the oscillator.
Overdamped oscillator, γ > ω0. The frequency ω0 now is purely imaginary, ω0= i
γ2− ω02, and the two solutions to the frequency equation (6.18) become
ω+= −i(γ −
γ2− ω20)≡ −iγ1
ω−= −i(γ +
γ2− ω200≡ −iγ2. (2.29) The motion decays monotonically (without oscillations) and the corresponding gen-eral solution for the displacement is the sum of two exponential functions with the decay constants γ1and γ2,
ξ(t )= C1e−γ1t+ C2e−γ2t, (2.30) where the two (real) constants are to be determined from the initial conditions.
Critically damped oscillator, γ = ω0. A special mention should be made of the ‘degenerate’ case in which the two solutions to the frequency equation are the same, i.e., when γ1 = γ2 = ω0. To obtain the general solution for ξ in this case
requires some thought since we are left with only one adjustable constant. The general solution must contain two constants so that the two conditions of initial displacement and velocity can be satisfied (formally, we know that the general solution to a second order differential equation has two constants of integration). To obtain the general solution we can proceed as follows.
We start from the overdamped motion ξ = C1exp(−γ1t )+C2exp(−γ2t ). Let γ2= γ1+ and denote temporarily exp(−γ2t )by f (γ2, t ). Expansion of this function to the first order in yields f (γ2, t )= f (γ1, t )+(∂f/∂)0= exp(−γ t)−t exp(−γ1t ). The expression for the displacement then becomes ξ = (C1+ C2)exp(−γ1t )− t (C2)exp(−γ1t ), or
ξ = (C + Dt)e−ω0t, (2.31)
where C= C1+C2and D= −C2, C2being adjusted in such a way that D remains finite as → 0. Direct insertion into the differential equation ¨ξ + 2γ ˙ξ + ω02ξ = 0 (Eq. 2.23) shows that this indeed is a solution when γ = ω0.
In summary, the use of complex amplitudes in solving the frequency equation (6.18) and accepting a complex frequency as a solution, we have seen that it indeed has a physical meaning; the real part being the quantity that determines the period of oscillation (for small damping) and the imaginary part, the damping. In this manner, the solution for the displacement emerged automatically from the equation of motion.