Lagrange's Equation

Question 1

What are generalised coordinates

Answer 1

these are any set of coordinates (q1,q2, .., qn) that are independent and equal in number to the number of degrees of freedom of a system n

Question 2

What is kinetic energy dependent on

Answer 2

kinetic energy is dependent on displacement and velocity

Question 3

What is potential energy a function of

Answer 3

potential energy is a function of displacement

Question 4

What is Lagrange’s Equation

Answer 4

d/dt( dT/dq(dot)i) - dT/dqi + dU / dqi = Qi

This part is equivalent to conservation of energy principle when summed for all qi:
d/dt( dT/dq(dot)i) - dT/dqi + dU / dqi

This is work done by nonconservative generalised forces. When conservation it equals 0:
Qi

Question 5

What is the transverse vibration of strings/cables

Answer 5

Uses wave equation to describe motion of a string in free vibration

Question 6

What is tranverse vibration equation

Answer 6

d^2y/dx^2 = 1/c^2 x d^2y/dt^2

Question 7

How to solve tranverse vibration equation

Answer 7

• assume solution in the form y(x,t) = Y(x)G(t)
• substitute into EoM
• have ‘x’ varibales and ‘t’ variables on opposite sides
• assume both equal to a constant and solve then combine

y = (Asin(wx/c) +Bcos(wx/c))(Csin(wt) + Dcos(wt))

Question 8

What does the solution for fixed string imply

Answer 8

• fundamental frequency is solely dependent on the length of the string and the wave propagation speed
• each Wn is a multiple of the fundamental frequency
• the mode shapes are always sine wave