Abstract
A geometric model of the rotational-translational motion of a dumbbell in a central force field is proposed, provided that its center of mass moves in a circle of a given radius. The term dumbbell denotes a geometric object with two masses spaced apart at a certain distance, connected by a weightless rod (like a sports dumbbell). The idea of modeling is based on the compilation and solution of a system of Lagrange differential equations of the second kind. To compose the equations, the Lagrangian was used to describe the rotational-translational motion of a dumbbell in a central force field. The system of equations is formed with respect to five functions denoting five generalized coordinates. These coordinates connect the fixed and mobile coordinate systems that provide the rotational-translational movement of the dumbbell. The system of Lagrange differential equations of the second kind was solved numerically in the environment of the maple mathematical processor. The results obtained correspond to each of five different coordinate functions that can be plotted graphically. Also obtained are approximate descriptions of the derivatives of these functions depending on the parameters of the dumbbell and the initial conditions of its motion. These results made it possible to construct graphs of phase trajectories of functions of five coordinate coordinates, with the help of which it is possible to determine the nature of the dumbbell movement. The obtained time dependences for the functions of generalized coordinates make it possible to compose an algorithm for computer animation of the rotational-translational motion of the dumbbell. In this case, the parameters of the dumbbell and the initial conditions of its movement will be taken into account. Examples of modeling the trajectories of the centers of mass of the dumbbell weights are given. It is advisable to use the results obtained to illustrate the position of the dumbbell in space when studying its orbital dynamics. Computer animations of a dumbbell movement in zero gravity will make it possible to analyze the influence of the angular velocities of rotation on its movement. At this stage of research, it is advisable to use the results obtained as the basis for laboratory or coursework of the departments of geometric modeling and computer graphics.
Highlights
-: sol := dsolve({ODE1, ODE2, ODE3, ODE4, ODE5} union initial, numeric, method=rkf45); solu := solv := solU := solV := solr :=
Stability of Relative Equilibria of an Orbit Tether with Impact Interactions Taken into Account
Spinning top - represented by dumbbell http://aias.us/blog/wp-content/uploads/2017/02/3712.pdf
Summary
-: sol := dsolve({ODE1, ODE2, ODE3, ODE4, ODE5} union initial, numeric, method=rkf45); solu := solv := solU := solV := solr := Subs(sol, u(t)): subs(sol, v(t)): subs(sol, U(t)): subs(sol, U(t)): subs(sol, r(t)): dsolu := subs(sol, diff(u(t),t)): dsolv := subs(sol, diff(v(t),t)): dsolU := subs(sol, diff(U(t),t)): dsolV := subs(sol, diff(V(t),t)): dsolr := subs(sol, diff(r(t),t)): plot([solu(t), dsolu(t),t=0..T], color=red, thickness=3, labels = [u, Du], axes=BOXED, font = [TIMES, ROMAN, 16], labelfont = [TIMES, ROMAN, 16]). For i from 0 to N do u1[i] := solu(T*i/N); v1[i] := solv(T*i/N); U1[i] := solU(T*i/N); V1[i] := solV(T*i/N); r1[i] := solr(T*i/N); end do: for i from 0 to N do Rx[i] := r1[i]*cos(V1[i])*sin(U1[i]); Ry[i] := r1[i]*sin(V1[i])*sin(U1[i]); Rz[i] := r1[i]*cos(U1[i]); display3d(curve([seq([Rx[i],Ry[i],Rz[i]], i=0..N)], color=black, thickness = 2),scaling=CONSTRAINED); Rx_1,Ry_1,Rz_1
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
