6.2: Spring Problems II
( \newcommand{\kernel}{\mathrm{null}\,}\)
Free Vibrations with Damping
In this section we consider the motion of an object in a spring–mass system with damping. We start with unforced motion, so the equation of motion is
my″+cy′+ky=0.
Now suppose the object is displaced from equilibrium and given an initial velocity. Intuition suggests that if the damping force is sufficiently weak the resulting motion will be oscillatory, as in the undamped case considered in the previous section, while if it is sufficiently strong the object may just move slowly toward the equilibrium position without ever reaching it. We’ll now confirm these intuitive ideas mathematically. The characteristic equation of Equation ??? is
mr2+cr+k=0.
The roots of this equation are
r1=−c−√c2−4mk2mandr2=−c+√c2−4mk2m.
We saw in Section 5.2 that the form of the solution of Equation ??? depends upon whether c2−4mk is positive, negative, or zero. We’ll now consider these three cases.
Underdamped Motion
We say the motion is underdamped if c<√4mk. In this case r1 and r2 in Equation ??? are complex conjugates, which we write as
r1=−c2m−ω1iandr2=−c2m+ω1i,
where
ω1=√4mk−c22m.
The general solution of Equation ??? in this case is
y=e−ct/2m(c1cosω1t+c2sinω1t).
By the method used in Section 6.1 to derive the amplitude–phase form of the displacement of an object in simple harmonic motion, we can rewrite this equation as
y=Re−ct/2mcos(ω1t−ϕ),
where
R=√c21+c22,Rcosϕ=c1,andRsinϕ=c2.
The factor Re−ct/2m in Equation ??? is called the time–varying amplitude of the motion, the quantity ω1 is called the frequency, and T=2π/ω1 (which is the period of the cosine function in Equation ??? is called the quasi–period. A typical graph of Equation ??? is shown in Figure 6.2.1 . As illustrated in that figure, the graph of y oscillates between the dashed exponential curves y=±Re−ct/2m.
Overdamped Motion
We say the motion is overdamped if c>√4mk. In this case the zeros r1 and r2 of the characteristic polynomial are real, with r1<r2<0 (see ???), and the general solution of ??? is
y=c1er1t+c2er2t.
Again lim as in the underdamped case, but the motion isn’t oscillatory, since y can’t equal zero for more than one value of t unless c_1=c_2=0. (Exercise 6.2.23.)
Critically Damped Motion
We say the motion is critically damped if c=\sqrt{4mk}. In this case r_1=r_2=-c/2m and the general solution of Equation \ref{eq:6.2.1} is
y=e^{-ct/2m}(c_1+c_2t). \nonumber
Again \lim_{t\to\infty}y(t)=0 and the motion is nonoscillatory, since y can’t equal zero for more than one value of t unless c_1=c_2=0. (Exercise 6.2.22).
Suppose a 64 lb weight stretches a spring 6 inches in equilibrium and a dashpot provides a damping force of c lb for each ft/sec of velocity.
- Write the equation of motion of the object and determine the value of c for which the motion is critically damped.
- Find the displacement y for t>0 if the motion is critically damped and the initial conditions are y(0)=1 and y'(0)=20.
- Find the displacement y for t>0 if the motion is critically damped and the initial conditions are y(0)=1 and y'(0)=-20.
Solution a
Here m=2 slugs and k=64/.5=128 lb/ft. Therefore the equation of motion Equation \ref{eq:6.2.1} is
\label{eq:6.2.4} 2y''+cy'+128y=0.
The characteristic equation is
2r^2+cr+128=0, \nonumber
which has roots
r={-c\pm\sqrt{c^2-8\cdot128}\over4}. \nonumber
Therefore the damping is critical if
c=\sqrt{8\cdot128}=32\mbox{ lb-sec/ft}. \nonumber
Solution b
Setting c=32 in Equation \ref{eq:6.2.4} and cancelling the common factor 2 yields
y''+16y+64y=0. \nonumber
The characteristic equation is
r^2+16r+64y=(r+8)^2=0. \nonumber
Hence, the general solution is
\label{eq:6.2.5} y=e^{-8t}(c_1+c_2t).
Differentiating this yields
\label{eq:6.2.6} y'=-8y+c_2e^{-8t}.
6.2.2 Imposing the initial conditions y(0)=1 and y'(0)=20 in the last two equations shows that 1=c_1 and 20=-8+c_2. Hence, the solution of the initial value problem is
y=e^{-8t}(1+28t). \nonumber
Therefore the object approaches equilibrium from above as t\to\infty. There’s no oscillation.
Solution c
Imposing the initial conditions y(0)=1 and y'(0)=-20 in Equation \ref{eq:6.2.5} and Equation \ref{eq:6.2.6} yields 1=c_1 and -20=-8+c_2. Hence, the solution of this initial value problem is
y=e^{-8t}(1-12t). \nonumber
Therefore the object moves downward through equilibrium just once, and then approaches equilibrium from below as t\to\infty. Again, there’s no oscillation. The solutions of these two initial value problems are graphed in Figure 6.2.2 .
Find the displacement of the object in Example 6.2.1 if the damping constant is c=4 lb–sec/ft and the initial conditions are y(0)=1.5 ft and y'(0)=-3 ft/sec.
Solution
With c=4, the equation of motion Equation \ref{eq:6.2.4} becomes
\label{eq:6.2.7} y''+2y'+64y=0
after cancelling the common factor 2. The characteristic equation
r^2+2r+64=0 \nonumber
has complex conjugate rootsr={-2\pm\sqrt{4-4\cdot64}\over2}=-1\pm3\sqrt7i. \nonumber
Therefore the motion is underdamped and the general solution of Equation \ref{eq:6.2.7} isy=e^{-t}(c_1\cos3\sqrt7t+c_2\sin3\sqrt7t). \nonumber
Differentiating this yieldsy'=-y+3\sqrt7e^{-t}(-c_1\sin3\sqrt7t+c_2\cos3\sqrt7t). \nonumber
Imposing the initial conditions y(0)=1.5 and y'(0)=-3 in the last two equations yields 1.5=c_1 and -3=-1.5+3\sqrt7c_2. Hence, the solution of the initial value problem is
\label{eq:6.2.8} y=e^{-t}\left({3\over2}\cos3\sqrt7t-{1\over2\sqrt7} \sin3\sqrt7t\right).
The amplitude of the function in parentheses is
R=\sqrt{\left(\frac{3}{2} \right)^{2}+\left(\frac{1}{2\sqrt{7}} \right)^{2}}=\sqrt{\frac{9}{4}+\frac{1}{4\cdot 7}}=\sqrt{\frac{64}{4\cdot 7}}=\frac{4}{\sqrt{7}} \nonumber
Therefore we can rewrite Equation \ref{eq:6.2.8} asy={4\over\sqrt7}e^{-t}\cos(3\sqrt7t-\phi), \nonumber
where\cos\phi={3\over2R}={3\sqrt7\over8}\quad \text{and} \quad\sin\phi=-{1\over2\sqrt7R}= -{1\over8}. \nonumber
Therefore \phi \cong -0.125 radians.
Let the damping constant in Example 6.2.1 be c=40 lb–sec/ft. Find the displacement y for t>0 if y(0)=1 and y'(0)=1.
Solution
With c=40, the equation of motion Equation \ref{eq:6.2.4} reduces to
\label{eq:6.2.9} y''+20y'+64y=0
after cancelling the common factor 2. The characteristic equation
r^2+20r+64=(r+16)(r+4)=0 \nonumber
has the roots r_1=-4 and r_2=-16. Therefore the general solution of Equation \ref{eq:6.2.9} is
\label{eq:6.2.10} y=c_1e^{-4t}+c_2e^{-16t}.
Differentiating this yields
y'=-4e^{-4t}-16c_2e^{-16t}. \nonumber
The last two equations and the initial conditions y(0)=1 and y'(0)=1 imply that
\begin{array}{rlrl} c_1&+&c_2&=1\\[4pt] -4c_1&-&16c_2&=1. \end{array} \nonumber
The solution of this system is c_1=17/12, c_2=-5/12. Substituting these into Equation \ref{eq:6.2.10} yields
y={17\over12}e^{-4t}-{5\over12}e^{-16t} \nonumber
as the solution of the given initial value problem (Figure 6.2.3 ).
Forced Vibrations with Damping
Now we consider the motion of an object in a spring-mass system with damping, under the influence of a periodic forcing function F(t)=F_0\cos\omega t, so that the equation of motion is
\label{eq:6.2.11} my''+cy'+ky=F_0\cos\omega t.
In Section 6.1 we considered this equation with c=0 and found that the resulting displacement y assumed arbitrarily large values in the case of resonance (that is, when \omega=\omega_0=\sqrt{k/m}). Here we’ll see that in the presence of damping the displacement remains bounded for all t, and the initial conditions have little effect on the motion as t\to\infty. In fact, we’ll see that for large t the displacement is closely approximated by a function of the form
\label{eq:6.2.12} y=R\cos(\omega t-\phi),
where the amplitude R depends upon m, c, k, F_0, and \omega. We’re interested in the following question:
Assuming that m, c, k, and F_0 are held constant, what value of \omega produces the largest amplitude R in Equation \ref{eq:6.2.12}, and what is this largest amplitude?
To answer this question, we must solve Equation \ref{eq:6.2.11} and determine R in terms of F_0,\omega_0,\omega, and c. We can obtain a particular solution of Equation \ref{eq:6.2.11} by the method of undetermined coefficients. Since \cos\omega t does not satisfy the complementary equation
my''+cy'+ky=0, \nonumber
we can obtain a particular solution of Equation \ref{eq:6.2.11} in the form
\label{eq:6.2.13} y_p=A\cos\omega t+B\sin\omega t.
Differentiating this yields
y_p'=\quad\omega (-A\sin\omega t+B\cos\omega t) \nonumber
and
y_p''=-\omega^2(A\cos\omega t+B\sin\omega t). \nonumber
From the last three equations,
my''_p+cy'_p+ky_p=(-m\omega^2A+c\omega B+kA)\cos\omega t+ (-m\omega^2 B-c\omega A+kB)\sin\omega t, \nonumber
so y_p satisfies Equation \ref{eq:6.2.11} if
\begin{array}{lll} (k-m\omega^2) A+\quad c\omega B &=F_0\\[4pt] -c\omega A\quad+(k-m\omega^2)B&= 0. \end{array} \nonumber
Solving for A and B and substituting the results into Equation \ref{eq:6.2.13} yields
y_p={F_0\over(k-m\omega^2)^2+c^2\omega^2} \left[(k-m\omega^2)\cos\omega t+c\omega\sin\omega t\right], \nonumber
which can be written in amplitude–phase form as
\label{eq:6.2.14} %\springsteady y_p={F_0\over\sqrt{(k-m\omega^2)^2+c^2\omega^2}} \cos(\omega t-\phi),
where
\label{eq:6.2.15} \cos\phi={k-m\omega^2\over\sqrt {(k-m\omega^2)^2+c^2\omega^2}}\quad \text{and} \quad \sin\phi={c\omega\over\sqrt{(k-m\omega^2)^2+c^2\omega^2}}.
To compare this with the undamped forced vibration that we considered in Section 6.1 it is useful to write
\label{eq:6.2.16} k-m\omega^2=m\left({k\over m}-\omega^2\right)= m(\omega_0^2-\omega^2),
where \omega_0=\sqrt{k/m} is the natural angular frequency of the undamped simple harmonic motion of an object with mass m on a spring with constant k. Substituting Equation \ref{eq:6.2.16} into Equation \ref{eq:6.2.14} yields
\label{eq:6.2.17} y_p={F_0\over\sqrt{m^2(\omega^2_0-\omega^2)^2+ c^2\omega^2}}\cos(\omega t-\phi).
The solution of an initial value problem
my''+cy'+ky=F_0\cos\omega t, \quad y(0)=y_0,\quad y'(0)=v_0, \nonumber
is of the form y=y_c+y_p, where y_c has one of the three forms
\begin{aligned} y_c&= e^{-ct/2m}(c_1\cos\omega_1t+c_2\sin\omega_1t),\\[4pt] y_c&= e^{-ct/2m}(c_1+c_2t),\\[4pt] y_c&= c_1e^{r_1t}+c_2e^{r_2t}\,(r_1,r_2<0).\end{aligned} \nonumber
In all three cases \displaystyle \lim_{t\to\infty} y_c(t)=0 for any choice of c_1 and c_2. For this reason we say that y_c is the transient component of the solution y. The behavior of y for large t is determined by y_p, which we call the steady state component of y. Thus, for large t the motion is like simple harmonic motion at the frequency of the external force.
The amplitude of y_p in Equation \ref{eq:6.2.17} is
\label{eq:6.2.18} R={F_0\over\sqrt{m^2(\omega^2_0-\omega^2)^2+c^2\omega^2}},
which is finite for all \omega; that is, the presence of damping precludes the phenomenon of resonance that we encountered in studying undamped vibrations under a periodic forcing function. We’ll now find the value \omega_{\max} of \omega for which R is maximized. This is the value of \omega for which the function
\rho (\omega)=m^2(\omega^2_0-\omega^2)^2+c^2\omega^2 \nonumber
in the denominator of Equation \ref{eq:6.2.18} attains its minimum value. By rewriting this as
\label{eq:6.2.19} \rho (\omega)=m^2(\omega^4_0+\omega^4)+ (c^2-2m^2\omega^2_0)\omega^2,
you can see that \rho is a strictly increasing function of \omega^2 if
c\ge\sqrt{2m^2\omega^2_0}=\sqrt{2mk}. \nonumber
(Recall that \omega^2_0=k/m). Therefore \omega_{\max}=0 if this inequality holds. From Equation \ref{eq:6.2.15}, you can see that \phi=0 if \omega=0. In this case, Equation \ref{eq:6.2.14} reduces to
y_p={F_0\over\sqrt{m^2\omega^4_0}}={F_0\over k}, \nonumber
which is consistent with Hooke’s law: if the mass is subjected to a constant force F_0, its displacement should approach a constant y_p such that ky_p=F_0. Now suppose c<\sqrt{2mk}. Then, from Equation \ref{eq:6.2.19},
\rho'(\omega)=2\omega(2m^2\omega^2+c^2-2m^2\omega^2_0), \nonumber
and \omega_{\max} is the value of \omega for which the expression in parentheses equals zero; that is,
\omega_{\max}=\sqrt{\omega^2_0-{c^2\over2m^2}} =\sqrt{{k\over m}\left(1-{c^2\over2km}\right)}. \nonumber
(To see that \rho(\omega_{\max}) is the minimum value of \rho(\omega), note that \rho'(\omega)<0 if \omega <\omega_{\max} and \rho'(\omega)>0 if \omega>\omega_{\max}.) Substituting \omega=\omega_{\max} in Equation \ref{eq:6.2.18} and simplifying shows that the maximum amplitude R_{\max} is
R_{\max}={2mF_0\over c\sqrt{4mk-c^2}} \quad \text{if} \quad c< \sqrt{2mk}. \nonumber
We summarize our results as follows.
Suppose we consider the amplitude R of the steady state component of the solution of
my''+cy'+ky=F_0\cos\omega t \nonumber
as a function of \omega.
- If c\ge\sqrt{2mk}, the maximum amplitude is R_{\max}=F_0/k and it is attained when \omega= \omega_{\max}=0.
- If c<\sqrt{2mk}, the maximum amplitude is
\label{eq:6.2.20} R_{\max}={2m F_0\over c\sqrt{4mk-c^2}},
and it is attained when\label{eq:6.2.21} \omega=\omega_{\max}=\sqrt{{k\over m}\left(1-{c^2\over 2km}\right)}.
Note that R_{\max} and \omega_{\max} are continuous functions of c, for c\ge0, since Equation \ref{eq:6.2.20} and Equation \ref{eq:6.2.21} reduce to R_{\max}=F_0/k and \omega_{\max}=0 if c=\sqrt{2km}.