Pendulum (mechanics)

Pendulum (mechanics)
A pendulum is a body suspended from a fixed support so that it swings freely back and forth under the influence of gravity. When a pendulum is displaced sideways from its resting, equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back toward the equilibrium position. When released, the restoring force acting on the pendulum's mass causes it to oscillate about the equilibrium position, swinging it back and forth. The mathematics of pendulums are in general quite complicated. Simplifying assumptions can be made, which in the case of a simple pendulum allow the equations of motion to be solved analytically for small-angle oscillations.
Simple gravity pendulum
A simple gravity pendulum is an idealized mathematical model of a real pendulum. This is a weight (or bob) on the end of a massless cord suspended from a pivot, without friction. Since in this model there is no frictional energy loss, when given an initial displacement it will swing back and forth at a constant amplitude. The model is based on these assumptions:
The rod or cord on which the bob swings is massless, inextensible and always remains taut.
The bob is a point mass.
Motion occurs only in two dimensions, i.e. the bob does not trace an ellipse but an arc.
The motion does not lose energy to friction or air resistance.
The gravitational field is uniform.
The support does not move.
The differential equation which represents the motion of a simple pendulum is
where is the magnitude of the gravitational field, is the length of the rod or cord, and is the angle from the vertical to the pendulum.
Small-angle approximation
The differential equation given above is not easily solved, and there is no solution that can be written in terms of elementary functions. However, adding a restriction to the size of the oscillation's amplitude gives a form whose solution can be easily obtained. If it is assumed that the angle is much less than 1 radian (often cited as less than 0.1 radians, about 6°), or
then substituting for into using the small-angle approximation,
yields the equation for a harmonic oscillator,
The error due to the approximation is of order (from the Taylor expansion for ).
Let the starting angle be . If it is assumed that the pendulum is released with zero angular velocity, the solution becomes
The motion is simple harmonic motion where is the amplitude of the oscillation (that is, the maximum angle between the rod of the pendulum and the vertical). The corresponding approximate period of the motion is then
which is known as Christiaan Huygens's law for the period. Note that under the small-angle approximation, the period is independent of the amplitude ; this is the property of isochronism that Galileo discovered.
Rule of thumb for pendulum length
gives
If SI units are used (i.e. measure in metres and seconds), and assuming the measurement is taking place on the Earth's surface, then , and (0.994 is the approximation to 3 decimal places).
Therefore, relatively reasonable approximations for the length and period are:
where is the number of seconds between two beats (one beat for each side of the swing), and is measured in metres.
Arbitrary-amplitude period
For amplitudes beyond the small angle approximation, one can compute the exact period by first inverting the equation for the angular velocity obtained from the energy method,
and then integrating over one complete cycle,
or twice the half-cycle
or four times the quarter-cycle
which leads to
Note that this integral diverges as approaches the vertical
so that a pendulum with just the right energy to go vertical will never actually get there. (Conversely, a pendulum close to its maximum can take an arbitrarily long time to fall down.)
This integral can be rewritten in terms of elliptic integrals as
where is the incomplete elliptic integral of the first kind defined by
Or more concisely by the substitution
expressing in terms of ,
Here is the complete elliptic integral of the first kind defined by
For comparison of the approximation to the full solution, consider the period of a pendulum of length 1 m on Earth (= ) at initial angle 10 degrees is
The linear approximation gives
The difference between the two values, less than 0.2%, is much less than that caused by the variation of with geographical location.
From here there are many ways to proceed to calculate the elliptic integral.
Legendre polynomial solution for the elliptic integral
Given and the Legendre polynomial solution for the elliptic integral:
where denotes the double factorial, an exact solution to the period of a simple pendulum is:
Figure 4 shows the relative errors using the power series. is the linear approximation, and to include respectively the terms up to the 2nd to the 10th powers.
Power series solution for the elliptic integral
Another formulation of the above solution can be found if the following Maclaurin series:
is used in the Legendre polynomial solution above.
The resulting power series is:
more fractions available in the On-Line Encyclopedia of Integer Sequences with having the numerators and having the denominators.
Arithmetic-geometric mean solution for elliptic integral
Given and the arithmetic–geometric mean solution of the elliptic integral:
where is the arithmetic-geometric mean of and .
This yields an alternative and faster-converging formula for the period:
The first iteration of this algorithm gives
This approximation has the relative error of less than 1% for angles up to 96.11 degrees. Since the expression can be written more concisely as
The second order expansion of reduces to
A second iteration of this algorithm gives
This second approximation has a relative error of less than 1% for angles up to 163.10 degrees.
Approximate formulae for the nonlinear pendulum period
Though the exact period can be determined, for any finite amplitude rad, by evaluating the corresponding complete elliptic integral , where , this is often avoided in applications because it is not possible to express this integral in a closed form in terms of elementary functions. This has made way for research on simple approximate formulae for the increase of the pendulum period with amplitude (useful in introductory physics labs, classical mechanics, electromagnetism, acoustics, electronics, superconductivity, etc. The approximate formulae found by different authors can be classified as follows:
‘Not so large-angle’ formulae, i.e. those yielding good estimates for amplitudes below rad (a natural limit for a bob on the end of a flexible string), though the deviation with respect to the exact period increases monotonically with amplitude, being unsuitable for amplitudes near to rad. One of the simplest formulae found in literature is the following one by Lima (2006): , where .
‘Very large-angle’ formulae, i.e. those which approximate the exact period asymptotically for amplitudes near to rad, with an error that increases monotonically for smaller amplitudes (i.e., unsuitable for small amplitudes). One of the better such formulae is that by Cromer, namely: .
Of course, the increase of with amplitude is more apparent when , as has been observed in many experiments using either a rigid rod or a disc. As accurate timers and sensors are currently available even in introductory physics labs, the experimental errors found in ‘very large-angle’ experiments are already small enough for a comparison with the exact period and a very good agreement between theory and experiments in which friction is negligible has been found. Since this activity has been encouraged by many instructors, a simple approximate formula for the pendulum period valid for all possible amplitudes, to which experimental data could be compared, was sought. In 2008, Lima derived a weighted-average formula with this characteristic:
where , which presents a maximum error of only 0.6% (at ).
Arbitrary-amplitude angular displacement
The Fourier series expansion of is given by
where is the elliptic nome and the angular frequency.
If one defines
can be approximated using the expansion
(see ). Note that for we have , thus the approximation is applicable even for large amplitudes.
Equivalently, the angle can be given in terms of the Jacobi elliptic function with modulus
For small and , so the solution is well-approximated by the solution given in Pendulum (mechanics)#Small-angle approximation.
Examples
The animations below depict the motion of a simple (frictionless) pendulum with increasing amounts of initial displacement of the bob. The small graph above each pendulum is the corresponding phase plane diagram; the horizontal axis is displacement and the vertical axis is velocity.
Damped, driven pendulum
The above discussion focuses on a pendulum bob only acted upon by the force of gravity. Here we instead suppose a damping force, e.g. air resistance, as well as a sinusoidal driving force acts on the body. This system is a damped, driven oscillator, and is chaotic.
Equation (1) can be written as
(see the Torque derivation of Equation (1) above).
We can now add a damping term and a forcing term to the right hand side to get
where we have assumed the damping couple to be proportional to the angular velocity (this is true for low-speed air resistance, see also Drag (physics)). and are constants defining the amplitude of forcing and the degree of damping respectively. is the angular frequency of the driving oscillations.
We can now simplify by dividing through by and redefining the constants appropriately to obtain
This equation exhibits chaotic behaviour. The exact motion of this pendulum can only be found numerically and is highly dependent on initial conditions, e.g. the initial velocity and the starting amplitude. However, the small angle approximation outlined above can still be used under the required conditions to give an approximate analytical solution.
Compound pendulum
A compound pendulum (or physical pendulum) is one where the rod is not massless, and may have extended size; that is, an arbitrarily shaped rigid body swinging by a pivot. In this case the pendulum's period depends on its moment of inertia around the pivot point.
The equation of torque gives:
where:
is the angular acceleration.
is the torque
The torque is generated by gravity so:
where:
is the mass of the body
is the distance from the pivot to the object's center of mass
is the angle from the vertical
Hence, under the small-angle approximation ,
where is the moment of inertia of the body about the pivot point.
The expression for is of the same form as the conventional simple pendulum and gives a period of
And a frequency of
If the initial angle is taken into consideration (for large amplitudes), then the expression for becomes:
and gives a period of:
where is the maximum angle of oscillation (with respect to the vertical) and is the complete elliptic integral of the first kind.
Physical interpretation of the imaginary period
The Jacobian elliptic function that expresses the position of a pendulum as a function of time is a doubly periodic function with a real period and an imaginary period. The real period is, of course, the time it takes the pendulum to go through one full cycle. Paul Appell pointed out a physical interpretation of the imaginary period: if is the maximum angle of one pendulum and is the maximum angle of another, then the real period of each is the magnitude of the imaginary period of the other.
Coupled pendula
Coupled pendulums can affect each other's motion, either through a direction connection (such as a spring connecting the bobs) or through motions in a supporting structure (such as a tabletop). The equations of motion for two identical simple pendulums coupled by a spring connecting the bobs can be obtained using Lagrangian mechanics.
The kinetic energy of the system is:
where is the mass of the bobs, is the length of the strings, and , are the angular displacements of the two bobs from equilibrium.
The potential energy of the system is:
where is the gravitational acceleration, and is the spring constant. The displacement of the spring from its equilibrium position assumes the small angle approximation.
The Lagrangian is then
which leads to the following set of coupled differential equations:
Adding and subtracting these two equations in turn, and applying the small angle approximation, gives two harmonic oscillator equations in the variables and :
with the corresponding solutions
where
and , are constants of integration.
Expressing the solutions in terms of and alone:
If the bobs are not given an initial push, then the condition requires , which gives (after some rearranging):
See also
Harmonograph
Conical pendulum
Cycloidal pendulum
Double pendulum
Inverted pendulum
Kapitza's pendulum
Rayleigh–Lorentz pendulum
Elastic pendulum
Mathieu function
Pendulum equations (software)
References
Further reading
External links
Mathworld article on Mathieu Function
Differential equations
Dynamical systems
Horology
Mathematical physics
Mathematics