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Simple harmonic motion can be considered the one-dimensional projection of uniform circular motion. If an object moves with angular speed ω around a circle of radius r centered at the origin of the xy-plane, then its motion along each coordinate is simple harmonic motion with amplitude r and angular frequency ω.
A simple pendulum exhibits approximately simple harmonic motion under the conditions of no damping and small amplitude. Assuming no damping, the differential equation governing a simple pendulum of length l {\displaystyle l} , where g {\displaystyle g} is the local acceleration of gravity , is d 2 θ d t 2 + g l sin θ = 0. {\displaystyle ...
Diagram showing the periodic orbit of a mass-spring system in simple harmonic motion. (Here the velocity and position axes have been reversed from the standard convention in order to align the two diagrams) Given a dynamical system (T, M, Φ) with T a group, M a set and Φ the evolution function
There are two main descriptions of motion: dynamics and kinematics.Dynamics is general, since the momenta, forces and energy of the particles are taken into account. In this instance, sometimes the term dynamics refers to the differential equations that the system satisfies (e.g., Newton's second law or Euler–Lagrange equations), and sometimes to the solutions to those equations.
The motion is simple harmonic motion where θ 0 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
The solution to this equation of motion for the balance is simple harmonic motion; i.e., a sinusoidal motion of constant period: = + () Thus, the following equation for the periodicity of oscillation can be extracted from the above results: T = 2 π I κ {\displaystyle T=2\pi {\sqrt {\frac {I}{\kappa }}}\,} This period controls the rate ...
Illustration of how a phase portrait would be constructed for the motion of a simple pendulum Time-series flow in phase space specified by the differential equation of a pendulum. The X axis corresponds to the pendulum's position, and the Y axis its speed.
Figure 2: A simple harmonic oscillator with small periodic damping term given by ¨ + ˙ + =, =, ˙ =; =.The numerical simulation of the original equation (blue solid line) is compared with averaging system (orange dashed line) and the crude averaged system (green dash-dotted line). The left plot displays the solution evolved in time and ...
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