The parameter values were taken as and, i.e., the masses and lengths of the pendula were identical: and. In this case, the pendulum was dragged into the initial configuration and and then released from rest. Figure 2 contains the animated behavior of the double pendulum for one such simulation. Provided a set of initial conditions and, we may now numerically compute the evolution of each pendulum’s angular displacement and then construct the motion of the overall double pendulum.
![motion animation matlab 2012 motion animation matlab 2012](https://matplotlib.org/3.3.2/_images/sphx_glr_double_pendulum_thumb.gif)
![motion animation matlab 2012 motion animation matlab 2012](https://blogs.mathworks.com/pick/files/animation_example_2.png)
Which is a form of the equations of motion that is suitable for numerical integration in MATLAB. To do so, we introduce the state vector such that If we arrange the angles and in a column vector, then the system of differential equations in ( 5) can be written in the convenient second-order matrix-vector form, whereīefore we can numerically integrate the double pendulum’s equations of motion in MATLAB, we must express the equations in first-order form. Where the prime symbol denotes differentiation with respect to the dimensionless time. Respectively, yields the following pair of nondimensional, second-order, ordinary differential equations governing the double pendulum’s behavior: Where the Lagrangian depends on the double pendulum’s kinetic energyĮvaluating ( 1) and then introducing the dimensionless mass, length, and time parameters We can obtain the equations of motion for the double pendulum by applying balances of linear and angular momenta to each pendulum’s concentrated mass or, equivalently, by employing Lagrange’s equations of motion in the form Consequently, the rotations of the pendula are characterized by the rotation tensors and. The dynamic behavior of the double pendulum is captured by the angles and that the first and second pendula, respectively, make with the vertical, where both pendula are hanging vertically downward when and. The first pendulum, whose other end pivots without friction about the fixed origin, has length and mass, while the second pendulum’s length and mass are and, respectively.
![motion animation matlab 2012 motion animation matlab 2012](https://upload.wikimedia.org/wikipedia/commons/4/43/Coupled_oscillators.gif)
Referring to Figure 1, the planar double pendulum we consider consists of two pendula that swing freely in the vertical plane and are connected to each other by a smooth pin joint, where each pendulum comprises a light rigid rod with a concentrated mass on one end.