This document discusses mathematical modeling of physical systems. It explains that to understand and control complex systems, quantitative mathematical models using differential equations are needed. These equations describe the dynamic behavior of physical processes and are obtained using physical laws. Examples of modeling mechanical systems using Newton's laws and electrical systems using Kirchhoff's laws are provided. Specifically, the document presents the differential equations for a spring-mass damper system and a parallel RLC electrical circuit. It also shows the equations for modeling glider performance.

1. AE 313
AE Systems & Control
02 MATHEMATICAL MODELLING
Dr. Syed Saad Azhar Ali
75-110
syed.ali@kfupm.edu.sa

2. Control Systems
• To understand and control complex systems, one must obtain
quantitative mathematical models of these systems.
• The term model, as it is used and understood by control engineers,
means a set of differential equations that describe the dynamic
behavior of the process.
• The differential equations describing the dynamic performance of a
physical system are obtained by utilizing the physical laws of the
process.
• For mechanical systems, one utilizes Newton’s laws, and for
electrical systems Kirchhoff’s voltage and current laws.
• We will consider a few examples.

3. DIFFERENTIAL EQUATIONS OF PHYSICAL SYSTEMS
– To model any physical system,
• Physical laws
• Mathematical equivalent of the laws
• Differential equations

4. Mechanical System
– Newton’s Laws
• F = ma
– Equations of Motion
– Forces
• Tension
• Friction
– And many more

6. Mechanical System
Spring-Mass Damper
• Displacement of the mass M (y)
• The mass is suspended with
spring (spring constant k)
• moves inside the walls with
friction (coefficient b)
• r(t) is the applied force that
– Moves -> Ma
– Takes care of spring -> ky
– Overcomes the friction -> bv
Ma bv ky
𝒓 𝒕 = 𝑴𝒂 + 𝒃𝒗 + 𝒌𝒚
𝒓 𝒕 = 𝑴
𝒅𝟐𝒚
𝒅𝒕𝟐
+ 𝒃
𝒅𝒚
𝒅𝒕
+ 𝒌𝒚 This is the differential equation
for the spring-mass damper
𝒓 𝒕 = 𝑴𝒂 + 𝒃𝒗 + 𝒌𝒚

7. Electrical System
Parallel RLC Circuit
• The total current is divided in all
the parallel branches
• r(t) is divided in
– resistance -> iR(t) = v(t)/R
– inductance -> iL(t) =
1
𝐿
v(t)𝑑𝑡
– Capacitance-> iC(t) = C
𝑑v(t)
𝑑𝑡
This is the called the integro-
differential equation
𝒓 𝒕 = 𝒊𝑹 + 𝒊𝑳 + 𝒊𝑪
𝒅𝒓(𝒕)
𝒅𝒕
= C
𝒅𝟐
v(t)
𝑑𝑡2 +
1
𝑅
𝒅v(t)
𝒅𝒕
+
1
𝐿
v(t)
𝒓 𝒕 =
v(t)
𝑹
+
1
𝐿
v(t)𝑑𝑡 + C
𝑑v(t)
𝑑𝑡
Differentiating the equation

8. Glider
Glider
𝑇𝑎𝑛𝜃 =
1
𝐿 𝐷
𝑣 =
2𝑐𝑜𝑠𝜃
𝜌 ∗ 𝑪𝑳
∗
𝑊
𝑆
Higher L/D yields smaller
glide angle
W/S = Wing loading
𝜌 = Density
𝑪𝑳 = lift coefficient
Aircraft performance and design
Book by John D. Anderson

9. 9
Project 1 (Competition.. ?)
• Group Project
• Competition (I will try to get some prize as well)
• Design the glider
• Get theoretical results
• Competition day - we will get real results
• 5%
8/23/2023