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Mathematics LibreTexts

7.9: The Period of the Nonlinear Pendulum

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Recall that the period of the simple pendulum is given by

T=2πω=2πLg

for

ωgL

This was based upon the solving the linear pendulum Equation 7.5.2. This equation was derived assuming a small angle approximation. How good is this approximation? What is meant by a small angle?

We recall that the Taylor series approximation of sinθ about θ=0 :

sinθ=θθ33!+θ55!+

One can obtain a bound on the error when truncating this series to one term after taking a numerical analysis course. But we can just simply plot the relative error, which is defined as

 Relative Error =|sinθθsinθ|.

A plot of the relative error is given in Figure 7.9.1. Thus for θ0.4 radians (or (23)) we have that the relative error is about 2.6%.

(Relative error in sinθ approximation). We would like to do better than this. So, we now turn to the nonlinear pendulum equation 7.5.1 in the simpler form

¨θ+ω2sinθ=0

clipboard_e5dcfee3d26b06b320f2d0f5a46dc73f4.png
Figure 7.9.1: The relative error in percent when approximating sinθ by θ.

(Solution of nonlinear pendulum equation). We next employ a technique that is useful for equations of the form

¨θ+F(θ)=0

when it is easy to integrate the function F(θ). Namely, we note that

ddt[12˙θ2+θ(t)F(ϕ)dϕ]=(¨θ+F(θ))˙θ

For the nonlinear pendulum problem, we multiply Equation 7.9.4 by ˙θ,

¨θ˙θ+ω2sinθ˙θ=0

and note that the left side of this equation is a perfect derivative. Thus,

ddt[12˙θ2ω2cosθ]=0

Therefore, the quantity in the brackets is a constant. So, we can write

12˙θ2ω2cosθ=c

Solving for ˙θ, we obtain

dθdt=2(c+ω2cosθ)

This equation is a separable first order equation and we can rearrange and integrate the terms to find that

t=dt=dθ2(c+ω2cosθ)

Of course, we need to be able to do the integral. When one finds a solution in this implicit form, one says that the problem has been solved by quadratures. Namely, the solution is given in terms of some integral.

In fact, the above integral can be transformed into what is known as an elliptic integral of the first kind. We will rewrite this result and then use it to obtain an approximation to the period of oscillation of the nonlinear pendulum, leading to corrections to the linear result found earlier.

We will first rewrite the constant found in Equation 7.9.5. This requires a little physics. The swinging of a mass on a string, assuming no energy loss at the pivot point, is a conservative process. Namely, the total mechanical energy is conserved. Thus, the total of the kinetic and gravitational potential energies is a constant. The kinetic energy of the mass on the string is given as

T=12mv2=12mL2˙θ2

The potential energy is the gravitational potential energy. If we set the potential energy to zero at the bottom of the swing, then the potential energy is U=mgh, where h is the height that the mass is from the bottom of the swing. A little trigonometry gives that h=L(1cosθ). So,

U=mgL(1cosθ)

(Total mechanical energy for the nonlinear pendulum). So, the total mechanical energy is

E=12mL2˙θ2+mgL(1cosθ)

We note that a little rearranging shows that we can relate this result to Equation Equation 7.9.5. Dividing by m and L2 and using the definition of ω2=g/L, we have

12˙θ2ω2cosθ=1mL2Eω2

Therefore, we have determined the integration constant in terms of the total mechanical energy,

c=1mL2Eω2

We can use Equation 7.9.6 to get a value for the total energy. At the top of the swing the mass is not moving, if only for a moment. Thus, the kinetic energy is zero and the total mechanical energy is pure potential energy. Letting θ0 denote the angle at the highest angular position, we have that

E=mgL(1cosθ0)=mL2ω2(1cosθ0)

Therefore, we have found that

12˙θ2ω2cosθ=ω2cosθ0

We can solve for ˙θ and integrate the differential equation to obtain

t=dt=dθω2(cosθcosθ0)

Using the half angle formula,

sin2θ2=12(1cosθ)

we can rewrite the argument in the radical as

cosθcosθ0=2[sin2θ02sin2θ2]

Noting that a motion from θ=0 to θ=θ0 is a quarter of a cycle, we have that

T=2ωθ00dθsin2θ02sin2θ2

This result can now be transformed into an elliptic integral. 1 We define

z=sinθ2sinθ02

and

k=sinθ02

Then, Equation 7.9.8 becomes

T=4ω10dz(1z2)(1k2z2)

This is done by noting that dz=12kcosθ2dθ=12k(1k2z2)1/2dθ and that sin2θ02sin2θ2=k2(1z2). The integral in this result is called the complete elliptic integral of the first kind.

Note 1

Elliptic integrals were first studied by Leonhard Euler and Giulio Carlo de’ Toschi di Fagnano (16821766), who studied the lengths of curves such as the ellipse and the lemniscate,

(x2+y2)2=x2y2.

We note that the incomplete elliptic integral of the first kind is defined as

F(ϕ,k)ϕ0dθ1k2sin2θ=sinϕ0dz(1z2)(1k2z2)

(The complete and incomplete elliptic integrals of the first kind). Then, the complete elliptic integral of the first kind is given by K(k)=F(π2,k), or

K(k)=π/20dθ1k2sin2θ=10dz(1z2)(1k2z2)

Therefore, the period of the nonlinear pendulum is given by

T=4ωK(sinθ02)

There are table of values for elliptic integrals. However, one can use a computer algebra system to compute values of such integrals. We will look for small angle approximations.

For small angles (θ0π2), we have that k is small. So, we can develop a series expansion for the period, T, for small k. This is simply done by using the binomial expansion,

(1k2z2)1/2=1+12k2z2+38k2z4+O((kz)6)

Inserting this expansion into the integrand for the complete elliptic integral and integrating term by term, we find that

T=2πLg[1+14k2+964k4+]

The first term of the expansion gives the well known period of the simple pendulum for small angles. The next terms in the expression give further corrections to the linear result which are useful for larger amplitudes of oscillation. In Figure 7.9.2, we show the relative errors incurred when keeping the k2 (quadratic) and k4 (quartic) terms as compared to the exact value of the period.

clipboard_e91f2acd02f014440e7c1d35c43047fc3.png
Figure 7.9.2: The relative error in percent when approximating the exact period of a nonlinear pendulum with one (solid), two (dashed), or three (dotted) terms in Equation 7.9.11.

This page titled 7.9: The Period of the Nonlinear Pendulum is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by Russell Herman via source content that was edited to the style and standards of the LibreTexts platform.

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