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.

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