Phys 325 HW 6A due Thursday March 12, 2015 by 1pm Name:________________
1.] 20 points This upside-down pendulum has a
torsional spring κ at its base. It has governing
nonlinear ODE ML2θ&& = −κθ + MgL sin θ . As
discussed in class and in discussion motion near the
top, at θ small, has linearized ODE
ML2θ&& + (κ − MgL)θ = 0 and is unstable if κ < MgL.
Let us ask how this system behaves if κ is too small.
Take κ = MgL/2 for illustration.
The system has three equilibrium points ( one is θ = 0 which we know is unstable) The
other two are at θ = ±θeq. Find θeq. (give it in both radians and degrees.) You will need
to numerically solve the transcendental equation θeq for the point at which total torque is
zero ( or equivalently where total PE is minimum.
Let θ(t) = θeq + ε(t) where ε(t) is a small angle measuring deviation away from this
equilibrium but θeq is not necessarily small. Find the linear differential equation
governing small ε in the form meff d2ε/dt2 + keffε=0. You will need the Taylor Series
expansion for sin(θeq + ε(t)) Show that the motion near this equilibrium is stable. What
is its natural frequency?
You may find it useful to set g = L = m = 1 and then solve the resulting entirely
numerical problem. There is no loss of generality is doing so, as we can always
choose units to make it so. If, however, we wish to recover the parametric
dependence on these quantities, we can insert factors with the desired dimensions.
For example, if you want a frequency, you take your numerical answer and multiply
it by (g/L)1/2 – which has the right dimensions.
2] 20 points (you will need a calculator) A
lightly damped system is driven near
resonance. Consider parameters ζ = 0.02, k
= 2, and m = 2/49 acted on from quiescent
initial conditions: xo = vo = 0 by the force F
= sin 6.9t; Find G(6.9) and φ(6.9). Write
the deviation from static equilibrium to be of the form x = xparticular + xcomplementary and find
the two coefficients in xcomplementary needed to match initial conditions for total x. Make a
numerical plot of x(t) from t = 0 to t = 20. Use a time resolution dt that is finer than 1/ωn.
Check your plot: are the initial conditions met? (you may need to zoom in near the plot
origin) From the plot, judge whether steady-state has been achieved by time 20. What
was the duration of the transient?
Phys 325 HW 6B due Thursday March 12, 2015 by 1pm Name:________________
1. 15 points The pictured pendulum (with a massless rod of
length L and point mass M on its tip) oscillates in an oil bath;
the mass on the end suffers a drag force F =-cv proportional to
its speed v and directed opposed to its velocity vector. Ignore
buoyancy. a) Derive the differential equation for arbitrary
amplitude theta (but so that the mass remains submerged).
b) Linearize it & Check it for obvious errors: is the damping
positive? is the effective stiffness positive? What is its
damped frequency of vibration? (you may assume the system
is underdamped) c) For initial conditions θ = Θ, dθ/dt = Ω,
find the subsequent θ(t).
2. 25 points Consider the pictured mass-spring system
excited by a moving base y(t) and with a drag
force = -c times the velocity of the mass relative to
the lab. We wish to find the motion x(t)
representing the displacement from equilibrium
(Xeq=natural spring length) relative to the moving
base. We are interested in the steady state motion,
i.e, the motion x(t) after the effects of initial
conditions have died away. There are two stages
to this problem:
a) derivation of the differential equation. It should come out the form indicated here:
M eff &&
x + C eff x& + K eff x = D&&
y + E&y
You are advised to use F= ma to derive it. The sum of the forces has two terms, with
magnitudes: k times the stretch of the spring, and c times the velocity of the mass relative
to the lab. But be careful in your derivation: x and X represent motion relative to the
moving base; the mass's acceleration is therefore NOT d2x/dt2. Nor is its velocity relative
to the lab equal to dx/dt. Check the differential equation you derived. What are the five
coefficients M,C,K,D and E?: Does the effective force (the right hand side) scale sensibly
with dy/dt and d2y/dt2? If the base y is accelerating to the right, for example, is there an
apparent force acting on m to its left? Is your equation dimensionally consistent? Are the
effective stiffness and damping positive? Does the equation reduce to what you should
have if y(t) = 0?
b) obtaining the particular solution. Substitute y(t) = Yo sin { ωt } and use the
standard formulas for harmonic responses in terms of the quantities G(ω) and φ(ω). You
needn’t re-derive them, but you will have to substitute for all the forms of the
coefficients. ( You need not consider initial conditions; they only affect the constants in
the homogeneous part of the general solution xh, which dies away after enough time; the
question only asks for the steady state part of the solution. ) You may save a lot of
algebra if you remember that the particular solution associated with the sum of two forces
is the sum of the particular solutions.