Nonlinear Pendulum

Trigonometry

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Nonlinear Pendulum

Differential Equation of Oscillations

Pendulum is an ideal model in which the material point of mass m is suspended on a weightless and inextensible string of length L. In this system, there are periodic oscillations, which can be regarded as a rotation of the pendulum about the axis O (Figure 1).

Figure 1.

Dynamics of rotational motion is described by the differential equation

ε=d2αdt2=MI,

where ε is the angular acceleration, M is the moment of the force that causes the rotation, I is the moment of inertia about the axis of rotation.

In our case, the torque is determined by the projection of the force of gravity on the tangential direction, that is

M=mgLsinα.

The minus sign indicates that at a positive angle of rotation α (counterclockwise), the torque of the forces causes rotation in the opposite direction.

The moment of inertia of the pendulum is given by

I=mL2.

Then the dynamics equation takes the form:

d2αdt2=mgLsinαmL2=gsinαL,d2αdt2+gLsinα=0.

In the case of small oscillations, one can set sinαα. As a result, we have a linear differential equation

d2αdt2+gLα=0ord2αdt2+ω2α=0,

where ω=gL is the angular frequency of oscillation.

The period of small oscillations is described by the well-known formula

T=2πω=2πLg.

However, with increasing amplitude, the linear equation ceases to be valid. In this case, the correct description of the oscillating system implies solving the original nonlinear differential equation.

Period of Oscillation of a Nonlinear Pendulum

Suppose that the pendulum is described by the nonlinear second order differential equation

d2αdt2+gLsinα=0.

We consider the oscillations under the following initial conditions

α(t=0)=α0,dαdt(t=0)=0.

The angle α0 is the amplitude of oscillation.

The order of the equation can be reduced, if we find a suitable integrating factor. Multiply this equation by the integrating factor dαdt. This leads to the equation

d2αdt2dαdt+gLsinαdαdt=0,ddt[12(dαdt)2gLcosα]=0.

After integration we obtain the first order differential equation:

(dαdt)22gLcosα=C.

Given the initial conditions, we find the constant C:

C=2gLcosα0.

Then the equation becomes:

(dαdt)2=2gL(cosαcosα0).

Next, we apply the double angle identity

cosα=12sin2α2,

which leads to the following differential equation:

(dαdt)2=4gL(sin2α02sin2α2),dαdt=2gLsin2α02sin2α2.

Integrating this equation, we obtain

d(α2)sin2α02sin2α2=gLdt.

We denote sinα02=k and introduce the new variable θ instead of the angle α:

sinα2=sinα02sinθ=ksinθ.

Then

d(sinα2)=cosα2d(α2)=1sin2α2d(α2)=1k2sin2θd(α2)=kcosθdθ.

It follows that

d(α2)=kcosθdθ1k2sin2θ.

In the new notation, our equation can be written as

kcosθdθ1k2sin2θkcosθ=gLdt,dθ1k2sin2θ=gLdt.

Next, we discuss the limits of integration. The passage of the arc from the lowest point α=0 to the maximum deviation α=α0 corresponds to a quarter of the oscillation period T4. It follows from the relationship between the angles α and θ that sinθ=1 or θ=π2 at α=α0. Therefore, we obtain the following expression for the period of oscillation of the pendulum:

gLT4=0π2dθ1k2sin2θorT=4Lg0π2dθ1k2sin2θ.

The integral on the right cannot be expressed in terms of elementary functions. It is the so-called complete elliptic integral of the 1st kind:

K(k)=0π2dθ1k2sin2θ.

The function K(k) is computed in most mathematical packages. Its graph is shown below in Figure 2.

Figure 2.

The function K(k) can also be represented as a power series:

K(k)=π2{1+(12)2k2+(1324)2k4+(135246)2k6++[(2n1)!!(2n)!!]2k2n+},

where the double factorials (2n1)!! and (2n)!! denote the product, respectively, of odd and even natural numbers.

Note that if we restrict ourselves to the zero term of the expansion, assuming that K(k)π2, we obtain the known formula for the period of small oscillations:

T0=4LgK(k)4Lgπ2=2πLg.

Further terms of the series for n1 are just allow to consider the anharmonicity of the oscillations of the pendulum and the nonlinear dependence of the period T on the oscillation amplitude α0.