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The inverted pendulum was a central component in the design of several early seismometers due to its inherent instability resulting in a measurable response to any disturbance. [8] The inverted pendulum model has been used in some recent personal transporters, such as the two-wheeled self-balancing scooters and single-wheeled electric unicycles.
A classic example along these lines is the inverted pendulum. [42] Other examples are vibrations of a string with periodically varying tension [41] stability of railroad rails as trains drive over them; seasonally forced population dynamics; the phenomenon of parametric resonance in forced oscillators; motion of ions in a quadrupole ion trap [43]
While the syntax is similar to other modeling languages such as gProms, APMonitor solves DAEs of any index without rearrangement or differentiation. [6] As an example, an index-3 DAE is shown below for the pendulum motion equations and lower index rearrangements can return this system of equations to ODE form (see Index 0 to 3 Pendulum example).
Distance in the Poincaré half-plane model of the hyperbolic plane from the apex of a semicircle to another point on it is the inverse Gudermannian function of the central angle. The angle of parallelism function in hyperbolic geometry is the complement of the gudermannian, Π ( ψ ) = 1 2 π − gd ψ . {\displaystyle {\mathit {\Pi }}(\psi ...
A simple gravity pendulum [1] is an idealized mathematical model of a real pendulum. [2] [3] [4] It is a weight (or bob) on the end of a massless cord suspended from a pivot, without friction. Since in the model there is no frictional energy loss, when given an initial displacement it swings back and forth with a constant amplitude. The model ...
The Van der Pol oscillator was originally proposed by the Dutch electrical engineer and physicist Balthasar van der Pol while he was working at Philips. [2] Van der Pol found stable oscillations, [3] which he subsequently called relaxation-oscillations [4] and are now known as a type of limit cycle, in electrical circuits employing vacuum tubes.
The mathematical MIMO-model which is needed in order to simulate a ballbot and to design a sufficient controller which stabilizes the system, is very similar to an inverted pendulum on a cart. The LEGO NXT Ballbot, [ 38 ] Adelaide Ballbot, [ 40 ] Rezero [ 32 ] and Kugle [ 51 ] include actuator models in their robot models, whereas CMU Ballbot ...
Verlet integration (French pronunciation:) is a numerical method used to integrate Newton's equations of motion. [1] It is frequently used to calculate trajectories of particles in molecular dynamics simulations and computer graphics.