Classical control 2-converted

1. Classical Control
Sheet (2)
1.a) Write down the generalized form of KVL (constituitive equation) and KCL (continuity
equation) as verbal statements.
b) Consider the mechanical (mass-damper-spring) system you have studied at the
beginning of the course. Assume that F(t) is the driving force and x(t) is the system's
displacement from the equilibrium position. Further assume that for large signal
operation the damping force is proportional to the square of the velocity and that
the spring restoration force is also proportional to the square of the displacement.
Rewrite the governing differential equation.
c) Derive a suitable state model to represent that kind of nonlinear system. Can you
use frequency tools instead to analyze this system? Why?
2.a) The law of thermal conduction for a certain element is given by:
𝑄 = 𝑘𝐴
𝑑𝑇
𝑑𝑥
Where 𝑄 = ℎ𝑒𝑎𝑡 𝑓𝑙𝑜𝑤 𝑟𝑎𝑡𝑒 (𝑊),
𝑘 = 𝑡ℎ𝑒𝑟𝑚𝑎𝑙 𝑐𝑜𝑛𝑑𝑢𝑐𝑡𝑖𝑣𝑖𝑡𝑦 ( 𝑊 ∙ 𝑚−1 ∙ 𝐾−1
),
𝐴 = 𝑎𝑟𝑒𝑎 𝑛𝑜𝑟𝑚𝑎𝑙 𝑡𝑜 ℎ𝑒𝑎𝑡 𝑓𝑙𝑢𝑥 (𝑚2)
𝑑𝑇
𝑑𝑥
= 𝑡𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒 𝑔𝑟𝑎𝑑𝑖𝑒𝑛𝑡 ( 𝐾 ∙ 𝑚−1)
Assuming that
𝑑𝑇
𝑑𝑥
=
𝑇
𝑥
, write down an expression for the thermal resistance. Define
its physical units.
b) A basic element for a chemical system is identified by the concentration C and the
flux 𝜙. Which of them is the through variable and which is the across variable? Write
down the equation relating these two variables. Also write down the equation to
determine the dissipated power/ stored energy for that element.