1. Introduction
The measurement of the period of a simple pendulum is a popular experiment for undergraduates. Its simplicity is subjected to a singular and important condition that the initial angular displacement
θm
rendered to the bob is small. Mathematically, this reduces the non-linear equation of motion [^{1}]
d2θdt2=-ω02sinθ
(1)
to a linear equation of motion
d2θdt2=-ω02θ,
(2)
where
ω0≡g∕L
,
L
being the pendulum length and
g
being the local acceleration of gravity. An invariable problem arising by the imposition of this condition is the characterization of the small-angle regime. Would it be
θm=3∘
,
5∘
,
10∘
, or more? A more inquisitive mind would ask how different would the motion of the pendulum be if this condition is not adhered. In fact, the time period is found to increase with the initial angular displacement
θm
according to the Bernoulli's power series [^{2}]
T=T01+116θm2+113072θm4+…,
(3)
where
T0
is the period of the pendulum in the limit of small-angle oscillations, in which it exhibits a simple harmonic motion (SHM). It is only for small initial displacements, namely
θm≪π∕2
rad, that the pendulum actually oscillates harmonically, and Eq. (3) reduces to
T=T0
. Beyond this small-angle regime, the number of terms to be included in the above series increases with
θm
. The literature is rich on this topic, presenting distinct approximations for the large-angle period [^{3}–^{19}].
Here in this paper, we address the distinction between small and large-angle oscillations based upon Fourier series analysis. This method has not been explored in literature, except by the works by Gil and co-workers in Ref. [^{20}] and Simon and Riesz in Ref. [^{21}], standing out for its simplicity, in contrast to Borghi's more complex treatment [^{22}]. The use of Fourier Series is of academic interest, especially at graduate level where the concepts of Mechanics, Electronics, Data Acquisition, and Mathematical Methods are taught. In fact, a consolidated approach can be developed following our method for studying the nonlinear oscillations of a simple pendulum.
2. Fourier series analysis of the nonlinear pendulum motion
Before we address the problem in hand, for completeness, we introduce the basic ideas involved. Mathematically, the solution of Eq. (2) that describes small-angle oscillations is
θ(t)=Asin(ω0t)+Bcos(ω0t).
(4)
Importantly, it can be noted that the solution is written in terms of trigonometric functions of a single frequency. Let us assume that the pendulum is released from a state of rest at an initial displacement
θm
, with
0<θm≤π∕2
rad, and
θ°(t=0)=0
. The solution then reduces to
θ(t)=θmcos(ω0t).
(5)
The mathematical description of the pendulum oscillations in this regime of small amplitudes then reduces to a single cosine term, thus only one frequency suffices and that is why this motion is called a ‘Simple Harmonic Motion’. The question of importance that now remains is, “What would be the nature of the motion beyond the small-angle regime, when Eq. (1) becomes a poor approximation?” Clearly, even when the pendulum motion has to be described by Eq. (1), a restoring force exists that tries to bring the pendulum back to its mean position, resulting in periodic oscillations. Its motion might not be ‘simple’ since the restoring force is not directly proportional to the displacement but it would still present a to-and-fro motion, hence a periodic motion. Mathematically, any periodic function which is both bounded and continuous by parts can be expanded in a trigonometric series (i.e., the Fourier Series), as given by [^{23}]
s(ωt)=a0+∑n=1∞ancos(nωt)+∑n=1∞bnsin(nωt),
(6)
where
ω
is the fundamental frequency and
nω
are its integral multiples, referred to as the higher harmonics. The fundamental frequency here is related to the SHM oscillation's frequency through Eq. (3), and is given as
ω=ω01+116θm2+113072θm4+…,
(7)
for
0<θm≤π∕2
rad. The term
a0
in Eq. (6) of course acts as a constant term, which does not vary with time. Then, by identifying the
θm
value for which the higher harmonics become significant, one would be in a position to identify the transition from SHM to anharmonic motion. In the case of SHM, for the initial condition
θ(t=0)=θm
and
θ°(t=0)=0
, one can expect
a1=θm
and
b1=0
, thus enabling us to identify the small-angle condition. This method is unique when compared to those found in literature [^{24}]. In order to study the mentioned transition, we have solved Eq. (1) numerically using a short Scilab code,^{1} as given in Appendix A. In Fig. 1, it is shown the numerical solution of the differential equation in Eq. (1) for both small and large-angle oscillations (panel a), as well as the increase of the time period with the angular amplitude (panel b). To avoid ambiguity, the large-angle oscillation data was generated for
θm=45∘
. The fundamental frequency
ω
of the pendulum oscillations was fixed as
1
rad/s, which corresponds to a time period of
T0=2π≈6.28
s. We have truncated the series to just three terms, which is accurate for oscillations with
θm≤0.7
rad.
We now proceed to obtain the Fourier coefficients
a0
,
an
and
bn
of the series given by Eq. (6) using the Discrete Fourier Series (DFS) program given in Appendix B. The interesting result we find on the onset is that the coefficients associated with the sine terms
bn
are all zero. It may be pointed out here that only periodic graphs of an even function, which by definition are symmetrical about the
y
-axis (i.e. it is a mirror image about the
y
-axis), would give
bn=0
[^{23}]. Thus, the solution of Eq. (1) for the initial condition
θ(t=0)=θm
is an even function.
In Fig. 2, it is shown how the coefficient
a1
varies with
θm
. For SHM, i.e. for small values of
θm
, and
θ°(t=0)=0
, we know that Eq. (6) should reduce to Eq. (5), with
a1=θm
. That is, for small-angle oscillations
a1
would be equal to
θm
, the initial angular displacement. As
θm
becomes larger, a deviation from the proportionality is expected. However, pin pointing would not be possible as this deviation from SHM to anharmonic behavior would be gradual. To reflect on this, the values of
a1
shown in Fig. 2 are listed in Table I along with the deviation from
θm
(let us call this deviation the ‘error’), which is given by
Err(%)=θm-a1θm×100,
(8)
as usual. As seen in Table I, below, the magnitude of deviation in the second place after decimal for small angles becomes 1% and more for
θm≥0.8
rad. On using the freeware CurveExpert 1.4 to fit data points of Fig. 2, a low standard deviation of
0.00375
and a very good correlation coefficient of
0.9999
is found. The trend is
a1=θm+0.0181θm2-0.037θm3.
(9)
Table 1 Results of the DFS simulations are listed for various
θm
. All coefficients and deviations in
a1
with respect to
θm
are rounded-off at the second decimal place.
Err(%)
is as given in Eq. (8).
θm
(rad) |
0.1 |
0.2 |
0.3 |
0.8 |
0.9 |
1.0 |
1.1 |
1.2 |
1.3 |
1.4 |
1.5 |
1.6 |
1.7 |
a0
(rad) |
0.00 |
0.00 |
0.00 |
0.00 |
0.00 |
0.00 |
0.00 |
0.00 |
0.00 |
0.00 |
0.00 |
0.00 |
0.00 |
a1
(rad) |
0.1 |
0.2 |
0.29 |
0.79 |
0.88 |
0.98 |
1.07 |
1.16 |
1.25 |
1.33 |
1.42 |
1.49 |
1.55 |
a3
(rad) |
0.00 |
0.00 |
0.00 |
0.00 |
0.01 |
0.01 |
0.02 |
0.03 |
0.04 |
0.05 |
0.07 |
0.10 |
0.13 |
a5
(rad) |
0.00 |
0.00 |
0.00 |
0.00 |
0.00 |
0.00 |
0.00 |
0.00 |
0.00 |
0.00 |
0.00 |
0.00 |
0.01 |
Err (%) |
-0.08 |
-0.02 |
0.05 |
1.03 |
1.38 |
1.77 |
2.23 |
2.80 |
3.41 |
4.31 |
5.27 |
6.65 |
8.44 |
The deviation then increases with
θm
according to
Err(%)=3.7θm2-1.81θm.
(10)
At panel (b) of Fig. 2 it is shown that the coefficient
a0
tends to zero, with
a3
and
a5
giving a significant contribution for
θm>0.8
rad and
θm>1.6
rad, respectively. In Ref. [^{21}], Simon also presented a Fourier series analysis for large-angle pendulum oscillations and reported that only odd harmonic terms contribute to the solution. They explained this feature as being due to the symmetry of the problem, however Kreyszig [^{23}] explains that odd or even coefficients disappear if mixed initial or boundary conditions are imposed. For example (as in this case), while one condition is position at a given time,
θ(t=0)=θm
, the second condition is velocity at a given position,
dθ∕dt=0
at
θ=θm
. Hence, the Fourier analysis shows that simulated pendulum undergoing periodic oscillations can be fully described by
θ(t)≈a1cos(ωt)+a3cos(3ωt)+a5cos(5ωt),
(11)
where, from the above discussions, it is clear that the coefficients are functions of
θm
and, for small-angle approximation,
a1(θm→0)=θm
,
a3(θm→0)=0
and
a5(θm→0)=0
. As can be seen, the result in Eq. (11) differs from that found by Simon in Ref. [^{21}], so its conclusion that the simple pendulum is highly harmonic for all amplitudes is in contrast to our findings. The pertinent question as to what is a ‘small angle’ then resurfaces.
The need for determining “What a small angle is?” in undergraduate classes usually arises in experiments conducted to determine the local acceleration of gravity
g
. It is clear from the above arguments that Eq. (2) is valid only for
θm→0
and beyond this limit some error is always induced. It is then important to investigate the error committed in
g
with the increase of
θm
. For this we use the data of Fig. 1.
At panel (a) of Fig. 3, it is shown the variation of
g
with
θm
, whereas at panel (b) the error in the evaluation of
g
is depicted. The acceptable small angle is, therefore, the error acceptable in our result. If we demand an acceptable error of
0.5%
, we find that the acceptable small-angle boundary would be
0.211
rad or
12∘
. This boundary for the small-angle regime is in good agreement with earlier reports [^{24}]. Interestingly, if we revisit Eq. (10), we find that the minimum of this expression is at
θm≈0.244
rad or
≈14∘
which again indicates that the simple harmonic approximation for Eq. (1) within experimental limits is valid for
θm<12-14∘
. This method, hence, gives an alternative way to study the transition from SHM to anharmonic motion in the simple pendulum.