As an example, I'm considering a beam hinged at one end which is subjected to a prescribed angular velocity Omega. (EJ = bending stiffness , m = mass per unit length , l = length).
Let's assume i want to use Lagrange equations with a Ritz approx (Assumed modes method).
Kinetic energy:
T = 1/2 int_0^l ( s/t m s/t) dx
Potential energy:
K = 1/2 int_0^l (w/xx EJ w/xx) dx
where s/t indicates the derivative of s over time, s being the displacement of a general point of the beam. w is the bending displacement.
Now i would compute s in terms of the rigid angle of the beam theta and the bending displacement w, so s = s(x , theta , w).
Considering the beam in its initial position on the x-axis from x = 0 to x = l :
initial set of point P_0 = (x , w(x,t))
using rotation matrix R(theta) P_1 = R*P_0
s = P_1 - P_0
Now i would apply the Ritz approx to the displacement w with a trial function like N(x) = x/l ( hoping it is suitable).
and i have my equations of motion in the unknowns theta and w, where theta is the integral between 0 and t of the angular speed Omega.
Would this procedure be right? If not, how should I treat a problem like this one?
Hope I was clear enough, not easy without LaTeX.
Thank you
Eduardo Polli
If you have any prescribed motion then, you need to include it in the Kinetic Energy expression. So for your case, you should add an extra term that would correspond to the rotational angular velocity. If you are using Lagrangian equations to find the equation of motions, then expand the K.E (using Taylor series) to higher-order terms (cubic?). The rest of the solving process would remain the same.
Regards,
Pranoy Nair
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