MechE 132, Problem Set #8

MechE 132, Problem Set #8 Version 1

Spring, 2014, UC Berkeley

Below are 4 problems. I will post more problems over the course of the weekend, but this should be good to get started. These are due on Friday, April 11, at 5PM.

1. This purpose of this problem is to remind you how first-order Taylor series (ie., linear approximation) are used to approximate a function of many variables. The temperature in a particular 3-dimensional solid is a function of position, and is known to be

T(x,y,z) = 42 + (x− 2)2 + 3 (y − 4)2 − 5 (z − 6)2 + 2yz

(a) Find the first order approximation (linearization) of the temper- ature near the location (x̄ = 4, ȳ = 6, z̄ = 0). Use δx,δy and δz as your deviation variables.

(b) What is the maximum error between the actual temperature and the first order approximation formula for |δx| ≤ 0.3, |δy| ≤ 0.2, |δz| ≤ 0.1? Solve this numerically, by simply sampling a dense grid over the 3-dimesional cube, and determining the maximum error.

(c) More generally, suppose that x̄ ∈ R, ȳ ∈ R, z̄ ∈ R. Find the first order approximation of the temperature near the location (x̄, ȳ, z̄).

2. The pitching-axis of a tail-fin controlled missile is governed by the non- linear state equations

α̇(t) = K1Mfn (α(t),M) cos α(t) + q(t) q̇(t) = K2M

2 [fm (α(t),M) + Eu(t)]

Here, the states are x1 := α, the angle-of-attack, and x2 := q, the angular velocity of the pitching axis. The input variable, u, is the deflection of the fin which is mounted at the tail of the missile. K1, K2, and E are physical constants, with E > 0. M is the speed (Mach) of the missile, and fn and fm are known, differentiable functions (from wind-tunnel data) of α and M. Assume that M is a constant, and M > 0.

1