2P1 Mechanics

Question 1

How can you simplify a 2DoF mass-spring-mass system, which is not connected to a reference position?

Answer 1

By defining a new coordinate system in terms of the centre of mass, xG, and then rearranging. You can then see the acceleration of xG is 0. d/dt(p1)=0, because the lagranian does not depend on the q1, so there is no generalised force. This is the derivation of the conservation of total momentum.

Question 2

Under what situation will total energy (E=T+V), be conserved in a Lagrangian system?

Answer 2

When the lagranian does not depend explicitly on time.

Question 3

How do you find the motion of a 1DoF system?

Answer 3

Use the first integral of the equation of motion.

Multiply the eq of motion by qdot, then integrate with respect to time. This should look like the eq for total energy, which is a constant.

Question 4

When are you likely to be able to find a conservation law within Lagrangian mechanics?

Answer 4

When there is invarianve under a coordinate. i.e. you can define a new reference coordinate and everything is unchanged.

Question 5

How do you find equilibrium states in a Lagranian system for a 1DoF system?

Answer 5

Set either the acceleration to 0 or the V’(q) to 0. (same thing in essence)

Question 6

What is the concept of a phase portrait?

Answer 6

For a system with constant total energy, then you can plot a graph of velocity against position for difference energies. Each line represents 1 value of total energy.

Question 7

What are characteristic phase portraits from maxima and minima of effective potentials?

Answer 7

Minima, forms close circles (eyes). This is indicative of oscillation)

Maxima, they appear to diverge from, hyperbolic (or parabolic) looking curves).

Higher energy lines go over the top.

Question 8

What is the generalised form for oscillation in a 1DoF system, with an equilibria q0?

Answer 8

mdqddot = - v’‘(q0) dq = -V’(q0 +dq)

Question 9

What is the second integral of the equation of motion?

Answer 9

Integrate qdot with respact to time.

qdot = dq/dt = sqrt(2/m(E-V(q))

then seperate variables and integrate

Question 10

How can the concept of effective potential be applied in a central potential?

Answer 10

Central potential V(r), independent of θ

therefore define h as the pθ, since this is constant.

eq of motion =
mrddot = mr θdot^2 - V’(r) = pθ / (mr^3) - V’(r)

Veff’(r) = V’(r) - pθ/(mr^3)

integrate to find effective potential.

the pθ/2mr^2 is known as the centrifugal potential.

Question 11

What is the first integral for a single DoF system in terms of effective potential?

Answer 11

0.5 m rdot^2 + Veff(r) = E

Question 12

How do you find the frequency of oscillation for an effective potential system?

Answer 12

Veff(rmin+dr) = Veff(rmin) + 0.5 V’‘eff(rmin) dr^2

=> V’eff(rmin + dr) = V’‘eff(rmin) dr^2

Question 13

What is the consequence of oscillation for satellite mechanics?

Answer 13

Circular orbits => elliptical, frequency of vibration is the same as the orbital freuqncy.

Question 14

What is the conseuqnce of oscillation in the motion around a code?

Answer 14

the angular frequency of radial vibration is dependent on the angle of the cone. If this matches the spinning round the cone, then the motion closes, but if it doesn’t then the motion processes around the cone.

Question 15

What is the eq for total energy?

Answer 15

E = KE + Veff(q)

Question 16

What is noted about a bead on a freely rotating hoop?

Answer 16

There is infinite effective potential at θ =0, so no matter how much energy the ball has it cannot cross to the other side of the hoop.

Question 17

How can you simplify a 2-body problem in 3D space?

Answer 17

Write the lagranian in terms of vectors r1, and r2.

Establish new vectors, rG (centre of mass), and rS (seperation vector).

Seperate these Lagranians into two sub Lagrangians, and then solve these seperately.

Question 18

What is the general form of the kinetic energy in terms of the mass matrix.

Answer 18

T = 0.5 * qdot ^T *M *qdot

Question 19

What is the general form of the potential energy in terms of the stiffness matricies?

Answer 19

V(q0) - 0.5 * dq^T K dq

Question 20

What is the general form of each term of the mass and stiffness matrix?

Answer 20

Evaluated at q=q0

Mij = ∂2T/∂qidot∂qjdot
Kij = ∂2V/∂qi∂qj

Question 21

In the expanded Lagrangian, why are there no linear terms?

Answer 21

Because we are expanding around a stationary point.

Question 22

What is the generalised form for normal mode vibration with n degree of freedoms?

Answer 22

dq = sum to n [ Ci Ai cos (ωt + φ) ]

where Ai is the mode shapes

Question 23

What does ω=0 indicate?

Answer 23

Rigid body mode,

dq = (C1 + C2t)A

where A is the mode shape

Question 24

Conditions for orthogonal mode shapes?

Answer 24

K and M are both symmetric and one is the multiple of the identity matrix.

Question 25

What can we do with orthogonal mode shapes?

Answer 25

C1 = A1.(inital condiitons) where A1 is the orthonormal mode shape.

Question 26

What happens if we get that ω^2 <0?

Answer 26

We get imaginary natural frequency, this leads to a mode which grows exponetially over time scale τ.

Question 27

What is generally true when you have one imaginary root?

Answer 27

You are at a saddle point in potential energy, but this is still unstable.

Question 28

List 5 methods for computing simple integrals (which would be used for 1DOF systems)

Answer 28

Left Riemann sum sum v(tn)Δτ Right Riemann sum sum v(t_n+1)Δt Midpoint method sum v(tn + Δt/2)Δt O(N^2) O(N^3) for each rectangle but N rectangles) Trapezoidal method (also O(N^2) but usually worse) Simpson's method, uses lots more data points, has O(Δt^4)

Question 29

What is forward Euler?

Answer 29

yn+1 = yn + fn Δt each has O(Δt^2) xn+1 = xn + vn Δt vn+1 = vn + an Δt Can be a highly unstable method and also only O(Δt)

Question 30

What is backward Euler, or semi-implicit euler?

Answer 30

xn+1 = xn + vn+1 Δt vn+1 = vn + an+1 Δt numerically stable, so good but eq need to be rearranged to be able to solve)

Question 31

What is semi-implicit?

Answer 31

Update velocity based on previous acceleration Update position with updated velocity, conserves energy almost perfectly. (not-terribly accurate but reliable)

Question 32

What is leapfrog method?

Answer 32

It uses velocities at the midpoint between acceleration and position steps (only works for a(x,t) not a(x,v,t) has error of O(Δt^3) xn+1 = xn + v (n+.5) Δt v (n+1.5) = v (n+.5) + an+1 Δt

Question 33

What is a Poincare section?

Answer 33

It is the section of the much larger (θ1, θ2, θ1dot, θ2dot) 4D phase portrait. Since we can't draw this, we extract simply (θ2,θ2dot) for everytime that θ1 =0. A single point indicates periodic behaviour. Loops or lines indiciates quasi periodic behaviour, but predictable behaviour. Eventually it gets to chaotic non-perioidic behvaiour.

Question 34

How does a time to flip plot appear when plotted on a θ1/θ2 graph?

Answer 34

As a fractal with infinitely small patterns, large eye in the middle which never flips.

Question 35

What's the Noether theorem?

Answer 35

Time invariance leads to energy conservation.

Question 36

How do you add a driving force to a Lagranian?

Answer 36

xf(t) if it is with x. Consider generalised force is dL/dx so d/dx(xf(t)) = f(t), consider the direction of thuis force, Consider potential energy change if a force does work xF.

Question 37

What happens with resonance when the driving force exactly matches the resonant freq?

Answer 37

The solution grows linearly in time with the form Ate^(iωt)`

Question 38

How does parametric resonance work?

Answer 38

Property of the system (such as spring constant) varies perioidically with time. This leads to a time varying resonant frequnecy. When ω = 2ω0, you get exponentially increasing amplitude. qddot + ω^2 q = h ω0^2cos(ωt)q if q = A cos(ω0t) we can apply multiplication formula to show that the term on the RHS appears as a harmonic driving force with frequency ω , which therefore creates resonance.