Feature Showcase - External Force

In version 0.8 I have added external and actuated forces and moments. What took me so long, you ask? I don't know, I guess it seemed like low hanging fruit, so I didn't really prioritize it, but here they are.

I added a video to the MOMDYN Youtube channel demonstrating an external force to create a form of the classic Duffing oscillator. The video doesn't cover much of the theory, as its more focused on using the app, but to suffice to say you can get some wild and chaotic dynamics out of a seemingly simple system. Read on after the jump for more discussion of the dynamics you see in the video.

The system created in the video includes a large mass (the "Base") that is attached to a spring-damper that allows translational motion in the vertical Y-direction, with an attached small pendulum mass (the "Bob") that has rotational motion about the out-of-plane Z-axis, centered at the mass center of the Base. The mass matrix and forcing vector derived in MOMDYN is below.

The mass matrix will multiply by a vector containing the derivatives w.r.t. time of the generalized speeds, (s_A, s_C), so the pair of equations of motion are:

(m+m_{Base}) s_A' - m y_D sin(q_C) s_C' = -c_A s_A - k_A (-e_A + q_A) + m y_D s_C^2 cos (q_C) + 500 y_B sin(10 t) / sqrt(y_B^2)

-m y_D sin(q_C) s_A' + m y_D^2 s_C' = -c_C s_C - k_C q_C

In the first equation, we'll note that m_{Base} >> m, so we'll assume it is effectively zero, and we'll replace the sum of m + m_{Base} with M. Also, the vector component y_B is exactly 1 in our scenario, and we can replace the generalized speed s_A with q_A', and replace the force magnitude and frequency with F and ω. Thus, we can reduce and rearrange the first equation to a simple mass-spring-damper equation

M q_A'' + c_A q_A' + k_A (q_A - e_A) = F sin(ω t) ,

where the steady-state response is 

...  = A sin (ω t)

We can also rearrange the second equation to give

m y_D^2 q_C'' + c_C q_C' + k_c q_C = m y_D sin(q_C) q_A'' .

A Taylor series expansion of the right hand side with respect to the generalized coordinate q_C, retaining up to the third order terms, and substituting in the steady-state response of the base acceleration, gives

m y_D^2 q_C'' + c_C q_C' + k_c q_C = m y_D (q_C - q_C^3 / 6) A sin (ω t)