Pendulum (mathematics)
A simple pendulum is an idealization of the real pendulum. It is a basic model for understanding of pendulum oscillations.
Mathematical pendulum are characterized by two major features:
- There is no friction in any form, that is not resistance to flow in the thread or internal friction or suspension.
- The total mass of the pendulum is concentrated on a single point. The thread is considered to be no mass, the mass distribution of the pendulum body is represented by its center of mass.
In practice, you can get a mathematical pendulum thus approach that you used a very long and thin thread and a possible small and heavy pendulum bob. Since the maximum deflection decreases in this configuration only after a long time of observation, there is a largely friction- free state.
With a small deflection, the frequency of the pendulum oscillation exclusively depends on the length of the pendulum and the case of acceleration. But larger deflection angle influence increasingly the pendulum frequency.
Contrary to what one might initially assume, the mass of the pendulum does not matter.
Mathematical Description
Equation of motion
Based on the forces the equation of motion of the pendulum is situated below.
Due to gravity (g = acceleration of gravity ) results in deflection of a string pendulum of mass m a force FR (t ) which acts tangentially to the circular aerial tramway. The radial component of the movement is irrelevant, since it acts in the direction of the thread. The restoring force increases with the deflection angle φ with respect to the rest position. Since the mathematical pendulum has only one degree of freedom, a scalar equation is sufficient.
When viewing a swinging pendulum yarn shows that speed decreases with increasing deflection and, after reaching the apex of the direction change. Speed change means that the pendulum mass is accelerated, and more specifically a tangential acceleration takes place as a circular path of movement is present. The equation of motion is according to Newton's 2nd law.
The tangential acceleration can be expressed by the angular acceleration.
In the unperturbed oscillation represents the restoring force of the pendulum is the only external force dar. After changing and shortening of the mass creates a nonlinear second order differential equation.
Small Amplitude: Harmonic oscillation
For small angles, the small angle approximation is valid:
By substituting this results in a linear second order of the general form, the general solution of the wave equation leads to the differential equation.
Here, denote the angular amplitude and φ0 the zero phase angle at the time. In addition, the natural angular frequency and the associated period are shown.
Alternatively, the period of time without deriving the proportionality factor, even with the Buckinghamschen Π theorem.
Exact solution
Since pendulum in reality be deflected more than infinitesimal, they do not behave linearly, ie Oscillations with finite amplitude are anharmonic. The general differential equation is not solvable elementary and requires knowledge of elliptic integrals. This allows the general solution for the period in a number of developing:
Alternatively, the elliptic integral occurring via the arithmetic- geometric mean M evaluate:
In addition, the damping due to frictional losses in a real pendulum is greater than zero, so that the deflections decrease approximately exponentially with time.
Conservation laws
When mathematical pendulum of energy conservation theorem of mechanics applies. On the way from the maximum deflection to the rest position, the potential energy decreases, and more specifically by the associated weight of the tangential force component performs acceleration work, so the kinetic energy increases. After passing through the minimum of one component of the weight acts against the movement direction. It is done lifting work.
From this, can derive the differential equation:
The sum is constant in time, ie
This equation has two solutions: