Dinamika ljudskog tijela

master's thesis

• Author: Marin Smoljan
• Mentor: Marko Vrdoljak (mentor)
• Committee member: Marko Vrdoljak (predsjednik povjerenstva)
• Committee member: Pavle Goldstein (član povjerenstva)
• Committee member: Eduard Marušić-Paloka (član povjerenstva)
• Committee member: Hrvoje Šikić (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2025-02-26, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom radu je demonstrirano kako možemo modelirati pokrete ljudskog tijela koristeći Lagrangeov pristup gibanju sustava krutih tijela, te način na koji možemo od skice pokreta doći do Euler-Lagrangeovih jednadžbi, u koje su dodane i generalizirane sile i momenti. Krenuli smo od definiranja krutih tijela s pripadnim relativnim koordinatnim sustavima i opisom njihovog međusobnog odnosa. Računanjem kinetičke i potencijalne energije svakog tijela smo odredili Lagrangeovu funkciju, odnosno Euler-Lagrangeove jednadžbe za dvije generalizirane koordinate. Time smo dobili model u kojem se dva međusobno spojena kruta tijela gibaju pod utjecajem sile teže, to jest dvostruko njihalo. Da bi u model dodali i sile mišića, definirali smo generalizirane sile \(Q_i\) za svaku generaliziranu koordinatu i ubacili ih na desnu stranu Euler-Lagrangeovih jednadžbi. Sustav diferencijalnih jednadžbi drugog reda smo svođenjem na sustav diferencijalnih jednadžbi prvog reda numerički rješili u programu Octave, koristeći funkciju ode45. Program je napravio i animaciju pokreta, te ispisao graf s generaliziranim koordinatama i brzinama kroz vrijeme. Pokazali smo jedan zanimljiv način korištenja event funkcije za zaustavljanje programa, te na kraju iz već izračunatih sila, i numerički izračunate akceleracije dobili i reakcijske sile u šaci i laktu.

Abstract (english)

In this work, it was demonstrated how we can model the movements of the human body using Lagrange's approach to the motion of a system of rigid bodies, and the way we can get from the diagram of motion to the Euler-Lagrange equations, to which generalized forces and moments are added. We started by defining rigid bodies with the corresponding relative coordinate systems and a description of their mutual relationship. By calculating the kinetic and potential energy of each body we determined the Lagrange function, that is, the Euler-Lagrange equations for two generalized coordinates. This resulted in a model in which two connected rigid bodies move under the influence of gravity, that is, a double pendulum. To add the muscle forces to the model, we have defined the generalized forces \(Q_i\) for each generalized coordinate and added them to the right side of the Euler-Lagrange equations. By reducing the system of second-order differential equations to a system of the first-order differential equations we have numerically solved it in the Octave program, using the function ode45. The program also created an animation of the movement, and printed a graph with generalized coordinates and speeds through time. We have shown an interesting way of using the event function for stopping the program, and finally, from the already known forces, and numerically calculated accelerations we have calculated reaction forces in the hand and elbow.

• Keywords: dinamika; kruta tijela; Lagrangeova mehanika; tenzor tromosti; generalizirane sile
• Keywords (english): dynamics; rigid bodies; Lagrange mechanics; the moment of inertia; generalized forces
• Language: croatian
• URN:NBN: urn:nbn:hr:217:031620
• Study programme: Title: Financial and Business Mathematics; Study programme type: university; Study level: graduate; Academic / professional title: sveučilišni magistar matematike (sveučilišni magistar matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2025-02-12 07:56:44