Legrange

Question 1

What are generalised coordinates

Answer 1

Any set of coordinates that are independent and equal in number the number of degrees of freedom

Question 2

What would be the generalised coordinate in a pendulum

Answer 2

theta

Question 3

How is kinectic energy notated

Answer 3

T

Question 4

How is potential energy notated

Answer 4

U

Question 5

How is kinectic energy stored

Answer 5

in the system due to velocity

Question 6

How is potential energy stored

Answer 6

in the form of strain energy in deformation

Question 7

For a conservative system how are KE and PE related

Answer 7

T + U = constant

d(T + U) = 0

Question 8

How can we find natural frequency of a SDOF systemusing energy methods

Answer 8

T + U = constant
T1 + U1 = T2 + U2
t=1 static equilibrium U1 = 0
t=2 max displacement v=0 T2 = 0
thus T1 = U2
if harmonic motion
Tmax = Umax

Question 9

KE of rotating object and linear velocity object

Answer 9

rotating 0.5Jtheta.^2

linear 0.5mv^2

Question 10

PE of spring

Answer 10

0.5kx^2 (force * distance)

Question 11

What is the lagarangian

Answer 11

L = T - U

Question 12

What is Lagrange’s equation for an n degree of freedom system

Answer 12

d/dt * (dL/dqi.) - dL/dqi = Qi
or
d/dt * (dT/dqi.) - dT/dqi + dU/dqi = Qi

Question 13

What can you instantly cancel from lagrange equation of motion

Answer 13

dT/dqi as KE is not a function of displacement

d/dt(dU/dqi.) as no physical system where potential energy is function of acceleration

Question 14

What is Qi in lagranges equation for motion

Answer 14

nonconservative generalised forces

Question 15

How to find equations of motion using lagrange

Answer 15

Find total KE and PE of system
KE look at each mass and sum
PE look at each spring and sum
Apply lagrange equation for each generalised coordinate