Agenda for differential equations

1
Agenda for differential equations

• 1. Complex numbers
• 2. Differential calculus
• 3. Integral calculus
• 4. Modeling
• 5. Element equations
• 6. System equations
• 7. Differential equations
• 8. Solving differential equations

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1. Complex numbers

• Definition
• Arithmetic
• In-phase and quadrature

1. Complex numbers
3
Definition

• A complex number, z, consists of the sum of a
  real and imaginary number.
• The symbols i and j have the value of the square
  root of -1
• Example

imaginary axis
abi
b
a 3 b 4 z a bi z 3 4i
r
real axis
?
a
1. Complex numbers
4
Arithmetic (1 of 2)

• Addition (abj) (cdj) (ac)(bd)j
• Subtraction (abj) - (cdj) (a-c)(b-d)j
• Multiplication (abj)(cdj) (ac-bd)(cbda)j
• Conjugate conj(abj) a-bj
• Absolute abs(abj) sqrt(a2b2)
• Argument arg(abj) atan2(b,a)
• Division (abj)/(cdj) (abj) conj(cdj)/
• abs(cdj)2
• abj r x ej?
• where r abs(abj ) and ? arg(abj )

1. Complex numbers
5
Arithmetic (2 of 2)
Complex arithmetic using Excel
1. Complex numbers
6
In-phase and quadrature (IQ)

• In-phase component of signal that is in-phase
  with reference
• Quadrature component of signal that is 90
  degrees out of phase with reference

1. Complex numbers
7
2. Differential calculus

• Derivative of a function
• Elementary derivative operations
• Examples
• Critical points
• Partial differentiation

2. Differential calculus
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Derivative of a function
?f(x)
Lim f (x)
?x
0
?x
2. Differential calculus
9
Elementary derivative operations

• D k 0
• D xn nxn-1
• D ln x 1/x
• D eax a eax

2. Differential calculus
10
Examples (1 of 2)

• D k f(x) k D f(x)
• D (f(x) ? g(x)) D f(x) ? D g(x)
• D (f(x) g(x)) f(x) D g(x) g(x) D f(x)
• D (f(x)/g(x)) g(x) D f(x) - f(x) D g(x)/g(x)2
• D f(x)n nf(x)n-1 D f(x)
• D f (g(x)) Dg (f(g)) Dx g(x)

2. Differential calculus
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Examples (2 of 2)

• D sin x cos x
• D cos x -sinx
• D tan x sec2x
• D arcsin x 1/sqrt(1 - x2)
• D arctan x 1/(1 x2)

2. Differential calculus
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Critical points
f (x) 0 at critical point f (x) lt 0 at
maximum point f (x) gt 0 at minimum point f
(x) 0 at inflection point
f (x)
local maximum
inflection point
local minimum
global minimum singular point
x
2. Differential calculus
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Partial differentiation

• A partial derivative is a derivative that is
  taken with respect to only one variable
• z 4x3 - 5y2 2xy y -12
• ?z/ ?x 12x2 2y
• Partial derivatives are important in finite
  element computations

2. Differential calculus
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3. Integral calculus

• Integration
• Elementary integration operations
• Examples
• Integration by parts
• Initial values
• Definite integral

3. Integral calculus
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Integration

• Integration is the inverse operation of
  differentiation

f (x) dx f (x) C
3. Integral calculus
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Elementary integration operations
k dx k x C xm dx xm1/(m1) C e kx dx
ekx/k C
3. Integral calculus
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Examples
sin x dx -cos x C 1/x dx ln x C ln
x dx x ln x - x C dx/(k2 x2) I/k
arctan(x/k) C
3. Integral calculus
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Integration by parts (1 of 3)

• Integration by parts is an integration technique
  that is used when the function can be partitioned
  into two parts with favorable properties

f(x) dg(x) f(x)g(x) - g(x) df(x) C
3. Integral calculus
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Integration by parts (2 of 3)
dg(x) ex dx ex g(x)
f(x) x2 2x df(x)
x2 ex dx x2 ex - ex (2x) dx C
3. Integral calculus
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Integration by parts (3 of 3)
dg(x) ex dx ex g(x)
f(x) 2x 2 df(x)
ex (2x) dx 2x ex - ex (2) dx C
2x ex - 2 ex
x2 ex dx x2 ex - 2x ex 2 ex C
3. Integral calculus
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Initial values

• The constant of integration C can be found only
  if the value of the function is known at a point
• If there are multiple integrations involved, then
  multiple initial values are needed
• Example, if f(x) 4 when x 1 then

(3x2 - 2x)dx x3- x2 C 13 - 12 C 4 C 4
3. Integral calculus
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Definite integrals

• A definite integral is restricted to the region
  bounded by lower and upper limits

x2
f (x) dx f(x2 ) - f(x1)
x1
2
2x dx x2(2) - x2(1) 22 - 12 3
1
3. Integral calculus
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4. Modeling

• Approaches to finding a model
• Linear systems
• Nonlinear systems
• Guidelines for equations

4. Modeling
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Approaches to finding a model

• 1. Lumped parameters
• Break system into smaller elements
• For each element, use the physical laws that
  govern the element to write equations
• Build a model of the system from these lumped
  parameters
• 2. System identification
• Stimulate the system and observe its response
• Works only with existing systems

4. Modeling
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Linear systems (1 of 3)

• A system is linear if and only if it obeys the
  principle of superposition
• H(?x1 ? x2) ?H(x1) ?H(x2), where H is the
  system response

4. Modeling
26
Linear systems (2 of 3)
system response
H
x1
y H(x1 x2)
x2
x
4. Modeling
27
Linear systems (3 of 3)
y1 y2
slope K
y2
y1
x2
x1
x1 x2
4. Modeling
28
Nonlinear systems (1 of 3)

• Occasionally, application of physical laws to a
  system result in nonlinear equations.
• The nonlinearity may be overcome by finding a
  limited region of operation where linear
  operation takes place

4. Modeling
29
Nonlinear systems (2 of 3)
slope K
?(y1 y2)
y2
y1
c
x2
x1
x1 x2
4. Modeling
30
Nonlinear systems (3 of 3)
y2
?(y1 y2)
y1
c
x2
x1
x1 x2
4. Modeling
31
Guidelines for equations (1 of 4)

• 1. Understand the system -- sketch or describe in
  qualitative terms
• 2. Identify inputs and outputs, including
  disturbances
• 3. Express system in terms of elements that can
  be expressed mathematically
• 4. Develop equations for each element

4. Modeling
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Guidelines for equations (2 of 4)

• 5. Determine unknown parameter values by analysis
  or experiment
• 6. Adjust the model until it produces behavior
  like the actual system
• 7. Simplify the system if nonlinearities are
  involved

4. Modeling
33
Guidelines for equations (3 of 4)

• Ideally, the relationship should be linear
• A lumped-parameter model has time as its only
  independent variable. This fact allows ordinary
  differential equations to be used. If there are
  more independent variables, partial differential
  equations would need to be used, and they are
  more difficult
• Use idealized equivalent of the system e.g.
• Mass concentrated at a point rather than
  distributed
• Inductors have no resistance or capacitance

4. Modeling
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Guidelines for equations (4 of 4)

• The number of variables and the number of
  equations needs to be the same.
• Units need to be consistent
• Need to validate the model with prototypes or
  data from similar systems
• In practice, systems are not truly linear.
  Variations in the plant or transducers can make
  design much harder

4. Modeling
35
5. Element equations

• Proportional (P) relationship
• Integral (I) relationship
• Derivative (D) relationship
• PID
• Electrical components
• Rectilinear mechanical components
• Rotational mechanical components
• Fluid component
• Thermal components

5. Element equations
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Proportional (P) relationship
v(t)
i(t)
i(t)
a
b
R
i(t) current (A) through variable v(t)
voltage (V) across variable R resistance
(?) i(t) 1/R v(t) through variable constant
across variable
5. Element equations
37
Integral (I) relationship
v(t)
i(t)
i(t)
a
b
L
i(t) current (A) through variable v(t)
voltage (V) across variable L inductance
(H) i(t) 1/L v(t) dt through variable
constant ( across variable) dt
5. Element equations
38
Derivative (D) relationship
v(t)
i(t)
i(t)
a
b
C
i(t) current (A) through variable v(t)
voltage (V) across variable C capacitance
(F) i(t) C d/dt v(t) through variable
constant d/dt( across variable)
5. Element equations
39
PID

• Proportional (P) -- through variable is
  proportional to across variable
• Integral (I) -- through variable is proportional
  to integral of across variable
• Derivative (D) -- through variable is
  proportional to derivative of across variable

5. Element equations
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Electrical components

• Across variable potential difference v (V)
• Through variable current I (A)

P -- Resistor I -- Inductor D -- Capacitor
R(?) L(H) C(F)
5. Element equations
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Rectilinear mechanical components

• Across variable linear velocity v(m/s)
• Through variable force f(N)

P -- Linear damper I -- Linear spring D --
Mass
B(N/ms-1) K(N/m) M(kg)
5. Element equations
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Rotational mechanical components

• Across variable angular velocity ?(rad/s)
• Through variable torque T(Nm)

P -- Angular damper I -- Angular spring D --
Inertia
B(Nm/rads-1) K(Nm/rad) J(Nm/rads-2)
5. Element equations
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Fluid components

• Across variable pressure head h(m)
• Through variable volume flow rate q(m 3s-1)

P -- fluid resistance D -- fluid capacity
1/R(m2/s) A(m2)
5. Element equations
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Thermal components

• Across variable temperature difference ?(K)
• Through variable heat flow rate q(W)

P -- thermal resistance D -- thermal capacity
1/R(W/K) C(J/K)
5. Element equations
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6. System equations

• Example -- suspension

6. System equations
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Example -- suspension
body displacement x(t)
body mass
spring, k
shock absorber, b
m d2x/dt2 -b dx/dt - k x
wheel
6. System equations
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7. Differential equations (de)

• Definition of de
• Order of a de
• Linear de
• Linear de with constant coefficients
• Nonlinear de
• Homogeneous de
• Nonhomongeneous de
• Auxiliary equation

7. Differential equations
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Definition of de

• A differential equation is a mathematical
  expression combining a function (e.g., yf(x))
  and one or more of its derivatives
• Examples
• dy/dx - 5 y 0
• d2y/dx2 - 3 dy/dx 2y 0
• d2y/dx2 - (x25) dy/dx2 y sin 2x

7. Differential equations
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Order of a de

• The order of a differential equation is the order
  of the highest derivative in the equation
• Examples
• dy/dx - 5 y 0 -- 1st
• d2y/dx2 - 3 dy/dx 2y 0 -- 2nd
• d2y/dx2 - (x25) dy/dx2 y sin 2x -- 2nd

7. Differential equations
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Linear de

• A linear differential equation is an equation
  consisting of a sum of terms each made of a
  multiplier and either the function or its
  derivatives
• Examples
• dy/dx - 5 y 0 -- linear
• d2y/dx2 - 3 dy/dx 2y 0 -- linear
• d2y/dx2 - (x25) dy/dx2 y sin 2x --
  nonlinear

7. Differential equations
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Linear de with constant coefficients

• If the multipliers are constant, then the
  differential equation is said to have constant
  coefficients
• Examples
• dy/dx - 5 y 0 -- constant coefficients
• dy/dx - 5 xy 0 -- non- constant

7. Differential equations
52
Nonlinear de

• If the function or one of its derivatives is
  raised to a power or embedded in another
  function, the differential equation is nonlinear
• Example
• d2y/dx2 - (x25) dy/dx2 y sin 2x --
  nonlinear

7. Differential equations
53
Homogeneous de

• A homogeneous differential equation is one in
  which each term contains either the function or
  its derivatives. In other words, the sum of the
  derivative terms is zero
• Examples
• dy/dx - 5 y 0 -- homogeneous
• d2y/dx2 - 3 dy/dx 2y 0 -- homogeneous

7. Differential equations
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Nonhomogeneous de

• A nonhomogeneous differential equation is a sum
  of derivative terms that doesnt equal zero
• Example
• d2y/dx2 - (x25) dy/dx2 y sin 2x --
  non-homogeneous

7. Differential equations
55
Auxiliary equation

• The auxiliary equation is the polynomial formed
  by replacing all derivatives in a linear,
  constant coefficient, homogeneous differential
  equation with variables raised to the the power
  of the respective derivatives
• Example
• d2y/dx2 - 3 dy/dx 2y 0 has an auxiliary
  equation of s2 - 3s 2 0

7. Differential equations
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8. Solving differential equations

• Introduction
• Examples
• Alternate expression

8. Solving differential equations
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Introduction

• There are a large number of types of differential
  equations
• Many types have closed form solutions others do
  not
• A type of differential equations of importance to
  engineering is the linear, non-homogeneous
  differential equation with constant coefficients

8. Solving differential equations
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Example 1

• de Dy - 2y 0
• auxiliary equation S - 2 0
• root 2
• solution y C e2x
• if y(0) 10, then C 10

8. Solving differential equations
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Example 2

• de D2y 3 Dy 2y 0
• auxiliary equation S2 3S 2 0
• roots -1, -2
• solution y C1 e-2x C2 e-x
• if y(0) 0, Dy(0) -1, then C1 1 and C2 -1

8. Solving differential equations
60
Example 3

• de D2y y 0
• auxiliary equation S2 1 0
• roots i, -i
• solution y C1 cos x C2 sin x

8. Solving differential equations
61
Example 4

• de D2y 2Dy 2y 0
• auxiliary equation s2 2s 2 0
• roots -1 i, -1 - i
• solution y C1 e-x cos x C2 e-x sin x

8. Solving differential equations
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Example 5

• de D2y 2Dy y 0
• auxiliary equation S2 2S 1 0
• roots -1 , -1
• solution y (C1 C2 x ) e-x

8. Solving differential equations
63
Example 6

• de D5y 0
• auxiliary equation S5 0
• roots 0, 0, 0, 0, 0
• solution y C1 C2 x C3x2 C4 x3
  C5x4

8. Solving differential equations
64
Example 7

• de D4y 4 D3y 8 D2y 8 Dy 4 y 0
• auxiliary equation s4 4 s3 8 s2 8 s 4
  (s2 2s 2)( s2 2s 2) 0
• roots -1 i, -1 - i, -1 i, -1 - i
• solution y (C1 C2 x) e-x cos x (C3 C4
  x) e-x sin x

8. Solving differential equations
65
Example 8 (1 of 2)

• de D2y Dy - 2y 2x -40 cos 2x
• homogeneous auxiliary equation s2 s - 2 0
• homogeneous roots 1, -2
• homogeneous solution yc C1 ex C2 e-2x
• particular roots 0, 0, 2i, -2i
• particular solution yp A Bx C cos 2x E
  sin 2x
• total solution y yc yp

8. Solving differential equations
66
Example 8 (2 of 2)

• -2 yp -2A -2Bx -2C cos 2x -2E sin 2x
• D yp B 2Ecos2x - 2C sin 2x
• D2 yp -4C cos 2x -4E sin 2x
• constant terms -2A B 0
• X terms -2B 2
• cos x terms -2C 2E -4C -40
• sin x terms -2E -2C -4E 0
• constants A -0.5. B -1, C 6, E -2

8. Solving differential equations
67
Example 9 (1 of 2)

• de D2y y sin x
• homogeneous auxiliary equation s2 1 0
• homogeneous roots i, -i
• homogeneous solution yc C1 cos x C2 sin x
• particular roots i, -i
• particular solution yp Ax cos x Bx sin x
• total solution y yc yp

8. Solving differential equations
68
Example 9 (2 of 2)

• yp Ax cos x Bx sin x
• D yp A cos x - Ax sin x B sin x Bx cos x
• D2 yp -2A sin x - Ax cos x 2B cos x - Bx sin
  x
• cos x terms 2B 0
• sin x terms -2A 1
• constants A -0.5, B 0

8. Solving differential equations
69
Example 10 (1 of 1)

• de D3y - Dy 4 e-x 3 e2x
• homogeneous auxiliary equation s3 - s 0
• homogeneous roots 0, 1, -1
• homogeneous solution yc C1 C2 ex C3 e-x
• particular roots -1, 2
• particular solution yp Ax e-x B e2x
• total solution y yc yp

8. Solving differential equations
70
Example 10 (2 of 2)

• yp Ax e-x B e2x
• D yp A e-x - Ax e-x 2 B e2x
• D2 yp -2A e-x Ax e-x 4 B e2x
• D3 yp 3A e-x - Ax e-x 8 B e2x
• e-x terms -A 3A 4
• e2x terms -2B 8B 3
• constants A 2. B 0.5

8. Solving differential equations
71
Example 11

• In the previous problem, y(0) 0, Dy(0) -1, D2
  y(0) 2
• Determine C1, C2, C3
• Use the general solution y C1 C2 ex C3
  e-x 2x e-x 0.5 e2x
• Dy C2 ex - C3 e-x - 2x e-x 2e-x e2x
• D2 y C2 ex C3 e-x 2x e-x - 4e-x 2e2x
• y(0) 0 C1 C2 C3 0.5
• Dy(0) -1 C2 - C3 3
• D2 y(0) 2 C2 C3 -2
• C1 -4.5, C2 0, C3 4

8. Solving differential equations
72
Example 12 (1 of 3)

• de D2 y 2D y 2y cos x
• homogeneous auxiliary equation s2 2s 2 0
• homogeneous roots -1i, -1-i
• homogeneous solution yc C1 e-x cos x C2 e-x
  sin x
• particular roots i, -i
• particular solution yp A cos x B sin x
• total solution y yc yp

8. Solving differential equations
73
Example 12 (2 of 3)

• yp A cos x B sin x
• D yp - A sin x B cos x
• D2 yp - A cos x - B sin x
• cos x terms -A 2B 2A 1
• sin x terms -B -2A 2B 0
• constants A 0.2, B 0.4

8. Solving differential equations
74
Example 12 (3 of 3)

• Use the general solution y C1 e-x cos x C2
  e-x sin x 0.2 cos x 0.4 sin x
• initial conditions y(0) 1, D y(0) 0
• Dy - C1 e-x cos x - C2 e-x sin x - C1 e-x sin
  x C2 e-x cos x - 0.2 sin x 0.4 cos x
• y(0) 1 C1 0.2
• Dy(0) 0 - C1 C2 0.4
• C1 0.8, C2 0.4
• y(x) 0.8 e-x cos x 0.4 e-x sin x 0.2 cos x
  0.4 sin x

8. Solving differential equations
75
Alternate expression (1 of 3)

• It is sometimes desirable to express a
  higher-order differential equation as a set of
  first-order equations
• Matrix representation
• Computer solutions

8. Solving differential equations
76
Alternate expression (2 of 3)

• Example
• D3y 2 D2Y 5Dy 10y r
• Choose
• y1 y
• y2 Dy Dy1
• y3 D2y Dy2
• Single equation replaced by three equations
• Dy1 y2
• Dy2 y3
• Dy3 r - 10 y1 - 5y2 - 2y3

8. Solving differential equations
77
Alternate expression (3 of 3)

• Matrix format

Dy1 Dy2 Dy3
y1 y2 y3
0 1 0 0 0 1 -10 -5 -2
0 0 r


8. Solving differential equations