02 Mathematical Modelling.pptx

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