Appendix A: Simple Harmonic Motion978-3-319-45726-0...Appendix B: Pendulum Problem B.1 Definition A pendulum is a mass (or bob) on the end of a string of negligible mass that, when
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Appendix A: Simple Harmonic Motion
A.1 We Start with Hooke’s Law
Harmonic oscillator is depicted here and Hooke’s law defines the following
equation:
F ¼ �kx:
But using Newton’s second law of motion, we can write
B. Zohuri, Dimensional Analysis Beyond the Pi Theorem,DOI 10.1007/978-3-319-45726-0
249
F ¼ ma;
where F is the sum of forces on the object, m is the mass, and a is the instantaneousacceleration. Because we are only concerned with changes in speed, and because
the bob is forced to stay in a circular path, we apply Newton’s equation to the
length
q
point of suspension
Massless Rod
equilibriumposition
Bob’s Trajectory
Massivebob
amplitude
Fig. B.1 Simple gravity
pendulum assumes no air
resistance and no friction
Fig. B.2 Force diagram of
a simple gravity pendulum
250 Appendix B: Pendulum Problem
tangential axis only. The short violet arrow represents the component of the
gravitational force in the tangential axis, and trigonometry can be used to determine
its magnitude. Thus,
F ¼ �mg sin θ ¼ maa ¼ �g sin θ;
where g is the acceleration due to gravity near the surface of the earth. The negativesign on the right-hand side implies that θ and a always point in opposite directions.This makes sense because when a pendulum swings further to the left, we would
expect it to accelerate back toward the right.
This linear acceleration a along the red axis can be related to the change in angleθ by the arc length formulas; s is arc length:
s ¼ lθ;
υ ¼ ds
dt¼ l
dθ
dt;
a ¼ d2s
dt2¼ l
d2θ
dt2:
Thus,
ld2θ
dt2¼ �g sin θ
or
d2θ
dt2þ g
lsin ¼ 0; ðB:2Þ
This is the differential equation which, when solved for θ(t), will yield the motion of
the pendulum. It can also be obtained via the conservation of mechanical energy
principle: any given object, which fell a vertical distance h, would have acquired
kinetic energy equal to that which it lost to the fall. In other words, gravitational
potential energy is converted into kinetic energy. Change in potential energy is
given by
ΔU ¼ mgh
change in kinetic energy (body started from rest) is given by
ΔK ¼ 1
2mυ2:
Since no energy is lost, those two must be equal:
Appendix B: Pendulum Problem 251
1
2mυ2 ¼ mgh;
υ ¼ffiffiffiffiffiffiffiffi2gh
p:
Using the arc length formula above, this equation can be rewritten in favor of dθdt
dθ
dt¼ 1
l
ffiffiffiffiffiffiffiffi2gh
p;
where h is the vertical distance the pendulum fell. Consider Fig. B.3. If the
pendulum starts its swing from some initial angle θ0, then y0, the vertical distancefrom the screw, is given by
y0 ¼ l cos θ0
similarly, for y1, we have
y1 ¼ l cos θ
then h is the difference of the two
h ¼ l cos θ � cos θ0ð Þ
substituting this into the equation for dθdt gives
� 2g=lð Þ sin θffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2g=lð Þ cos θ � cos θ0ð Þp dθ
dt
¼ 1
2
� 2g=lð Þ sin θffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2g=lð Þ cos θ � cos θ0ð Þp ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2g
lcos θ � cos θ0ð Þ
r¼ �g
lsin θ
d2θ
dt2¼�g
lsin θ;
which is the same result as obtained through force and dimensional analysis.
Here we discuss briefly how to handle and solve a partial differential equation of
high order by reducing to and ordinary differential equation using self-similar
methods given by George W. Bluman and J. D. Cole.
C.1 Self-Similar Solutions by Dimensional Analysis
Consider the diffusion problem from the last section, with point-wise release
(reference: Similarity Methods for Differential Equations (Applied Mathematical
Sciences, Vol. 13)—Paperback (Dec. 2, 1974) by George W. Bluman and J.D. Cole
(Sect. 2.3):
∂c∂t
¼ D∂2
c
∂x2þ Q0δ xð Þδ tð Þ
c x; 0ð Þ ¼ 0, c �1, tð Þ ¼ 0:
8<:Initial release within infinitely narrow neighborhood of x ¼ 0, such that Π xð Þ=d¼ δ xð Þ and L=d ! 1. Note Q0 has different dimension as the previous Q because
of the cross-sectional area S and time contained in δ(t).
1. Dimensional analysis
cf g ¼ ML�3, Df g ¼ L2T�1, Q0f g ¼ ML�2 (mass release per unit cross-sectional
area) xf g ¼ L, tf g ¼ T. Thus, we expect 2Pi groups:
Π1 ¼ffiffiffiffiffiDt
p
Q0
c, Π2 ¼ xffiffiffiffiffiDt
p
and the solution to the PDE problem must be of the form Π1 ¼ f Π2ð Þ or
ODE Eq. C.1, along with condition Eqs. C.2 and C.3, will uniquely determine f(ξ), from which we get c(x, t). We are not concerned with the actual solution of
the new ODE problem. Rather, the interesting question is how did we manage toturn a PDE to an ODE.
3. Discussion
(a) The problem admits a self-similar solution: if x is scaled by the diffusion
length (Dt)1/2, then the c(x, t) profiles at different times can be collapsed onto
each other if c is scaled by Q0/(Dt)1/2
(b) This means that x and t are not really two independent variables; as far as c isconcerned, they can be rolled into one independent variable ξ.
(c) Similarity solutions are “happy coincidences” in physical process. Can we
always find them for any PDEs? No. This problem is special in that there is
no inherent length scale. Thus, we are not able to form dimensionless groups
for each of the variables x, t and c; instead, we have to combine them and end
up with only 2Pi groups. That is how we ended up with ODE. If we had the
release length dS or the domain length L, the self-similar will be ruined.
(d) Can we always find similarity solutions by dimensional analysis?
No. However, we will study another example next and then introduce the
general “stretching transformation” idea for detecting similarity solutions.
–5 –4 –3 –2 –1 0 1 2
m–0, s 2–0.2,s 2=1.0s 2=5.0s 2=0.5
m–0,m–0,m=–2,
3 4 5x
C.2 Similarity Solutions by Stretching Transformation
It is rare that similarity solutions can be obtained from dimensional analysis. In this
section, we introduce the idea of stretching transformation which is a more general
procedure for seeking out similarity in PDE problems. The materials are based on
Barenblatt (Sect. 5.2) and Bluman and Cole (Sect. 2.5).
As a concrete example, we will take Prandtl’s boundary-layer equation for flow
over a flat semi-plane. After the boundary-layer approximation (that viscosity acts
only within a thin layer, that the gradient in the flow direction (x) is much smaller
than in the transverse direction (y), and that the pressure is constant in the
y direction), the governing equations are
u∂u∂x
þ υ∂u∂y
¼ v∂2
u
∂y2
∂u∂x
þ ∂υ∂y
¼ 0
u x; 0ð Þ ¼ 0, υ x; 0ð Þ ¼ 0
u x;1ð Þ ¼ U1, u 0; yð Þ ¼ U1
8>>>>>><>>>>>>:where U1 is the free-stream velocity and v is the kinematic viscosity. If you recall
your fluid mechanics, this problem does have a similarity solution (Blasius’ssolution), and the PDE can be reduced to ODE. (Try to distinguish the velocity υfrom the viscosity v. We could use different symbols but these are the conventional
ones).
1. Would dimensional analysis work?Let us write out the dimensions of all the variables and parameters:
uf g ¼ υf g ¼ U1f g ¼ L=T, vf g ¼ L2=T, xf g ¼ yf g ¼ L
There are two independent dimensions involved (L and T ), and we can constructfour
u ¼ U1f ζð Þ, υ ¼ffiffiffiffiffiffiffiffiffiffivU1x
rg ζð Þ:
Plugging this into the original PDE will show that, indeed, we have reduced the
PDE problem to a couple of ODEs, whose solution is detailed in Fluid Mechan-ics textbooks. For another example of such “ingenious¨ dimensional analysis,
see the Rayleigh problem analyzed in the next section (see also Bluman and
Cole, p. 195). We typically seek to increase the number of independent dimen-
sions (as done above) or decrease the number of dimensional parameters
(as done in Bluman and Cole’s example).
2. Stretching transformationThe “ingenious” dimensional analysis method is specific to the problems. There
is, however, a general scheme for seeking out possible similarity solutions. The
scheme sometimes goes by the name of “renormalization groups” or “invariant
transformation groups” and is based on rather formalistic mathematical manip-
ulations. We will skip the proofs and focus on the technique itself.
Since the essence of similarity is that the solution is invariant after certain
scaling of the independent and dependent variables, we consider the following