Modeling: Mass-Spring System (1)

(1)

Modeling: Mass-Spring System (1)

• Newton’s 2

^nd

Law

• Spring-Mass System

 

my F

(2)

– Static equilibrium

--- F₀ balances the weight W=mg

– Restoring force

• Undamped System --- assume no damping

Hooke’s law

1

  F ky

0

 

0

F ks

Modeling: Mass-Spring System (2)

0

  

0

  0

F W ks mg

   0 my ky

Datum

(3)

– General sol.

Or

where

– Period

Frequency (cycle/sec, Hz)

2 2

, tan

   B

C A B

 A

Modeling: Undamped System (1)

   cos 

0

 

y t C  t 

   cos

⁰

 sin

⁰

,  0  

y t A  t B  t  k m

Harmonic oscillation

2  

0 0

2

 

(4)

Modeling: Undamped System (2)

(5)

Modeling: Damped System (1)

– Damping force existing

– Governing eqn.

– Characteristic eqn.

   F cy

Damping constant, >0

     0 my cy ky

2

 c  k  0

m m

 

(6)

Modeling: Damped System (2)

– Roots of the characteristic eqn.

Or

– Three cases

I. Distinct real roots II. Real double root

III. Complex conjugate roots

2 1,2

1 4

2 2

  c  

c mk

m m



2

 4

c mk

2

 4

c mk

2

 4

c mk

Overdamping Critical damping Underdamping

  

  

(7)

Damped System: I. Overdamping

– General sol.

--- The mass will be at rest y = 0 since the damping takes energy from the system.

 

1 ^ ^ 2 ^ ^

   



^t



^t

y t c e

^{ }

c e

^{ }

(8)

Damped System: II. Critical Damping

– General sol.

--- May or may not pass over static equilibrium y = 0.

– Border between non-oscillatory and oscillatory motion

  

1 2



 

 ^t

y t c c t e

^

(9)

Damped System: III. Underdamping (1)

– Pure imaginary

– Complex conjugate roots

– General sol.

where

2 2

2

, 1 4

2 4

 

    k  c

i mk c

m m m

  

1,2

   i



  

  

^ ^t

 cos

^

 sin

^



y t e

^

A  t B  t

 



cos



 Ce

^^t

 t  

2 2

, tan

   B

C A B

 A

(10)

Damped System: III. Underdamping (2)

– Damped oscillations – Frequency:

will increase as c becomes smaller – As c approaches zero,



2

 

0



  k m

 

(11)

Euler-Cauchy Equation

– Try

– Substituting

– Auxiliary eqn.

– Three cases

2

     0

x y axy by



^m

y x

 

2 2 1

1

^ ^

0



^m



^m



^m



x m m x axmx bx

 

2

  1   0

m a m b

(12)

Case I. Distinct Real Roots

– Two real roots m₁, m₂ – General sol.

– Example

– Auxiliary eqn.

– Roots of the auxiliary eqn.

– General sol.

2

  2.5   2.0  0

x y xy y

1 2



^m



^m

y c x c x

2

 3.5  2.0  0

m m

1

  0.5,

2

 4

m m

1 4

 c 

2

y c x

x

(13)

Case II. Real Double Root (1)

– One double root m₁

– First basis

– Second solution --- by the reduction of order Set

– Substituting into the DE

– Rearranging

¹  ²

1



^a

y x

2



1

y uy

   

2

1

2

1 1 1 1 1

0

            x u y u y uy ax u y uy buy

 

1

1 1

 2 

m a

   

2 2

1

2

1 1 1 1 1

0

          

u x y u x xy ay u x y axy by

0

(14)

Case II. Real Double Root (2)

– 2^nd parenthesis

– Remaining

– Divide by y1 and separate

– General sol.

 u x 

²

 u x y  

¹

 0

1 , ln ln ,

1 , ln

     



  

u u x

u x

u u x

x

 

^¹ ^ ² ^¹ ^ ² ^¹ ^ ²

1 1 1

2 xy   ay   1 a x

^^a

 ax

^^a

 x

^^a

 y



1 2

ln 

^¹^ ^ ²

 

^a

y c c x x

(15)

Case III. Complex Conjugate Roots

– Two complex roots

– Using Euler formula and expressing

– Adding and subtracting

– General sol.

   

1

2

ln ln

cos ln sin ln cos ln sin ln

 

       

       

m i i x

x x x x e x x i x

    

 

1

  ,

2

 

m   i m   i

   

cos ln , sin ln x

^

 x x

^

 x

   

cos ln sin ln

     

y x

^

A  x B  x

(16)

Existence and Uniqueness of Sol.

• Homogeneous Linear 2

^nd

Order DE

– IC’s

– Existence and uniqueness of the general sol.

• Linear Independence of the sol.

– Linearly independent y₁, y₂on I

– Linearly dependent

    0

    

y p x y q x

1 1 2 2

 

y c y c y

 

0



0

,   

0



1

y x K y x K

   

1 1



2 2

 0

k y x k y x k

₁

 0, k

₂

 0

1



2

,

2



1

y ky y ly

(17)

Linear Dependence, Wronskian (1)

– Wronski determinant (Wronskian)

– y₁, y₂are linearly dependent for any x₀ – If y₁, y₂are linearly dependent, then

– If , consider the linear system



1 2



¹ ² 1 2 2 1

1 2

,     

  y y

W y y y y y y

y y



1

,

2

 0 W y y 

1 2

y  ky



1 2



² ² 2 2 2 2

2 2

, ky y 0

W y y ky y y ky

ky y  

   

 



1

,

2

 0 W y y 

   

1 1 2 2

0 0 k y x k y x

k y x k y x

 

   

Unknowns k₁, k₂

(18)

Linear Dependence, Wronskian (2)

– Coefficient matrix of the linear system = Wronskian – Nontrivial sol. of k₁, k₂must be obtained if W = 0

– Example – Wronskian

linearly independent if and only if

   

1 1 2 2

y  k y x  k y x

2

0

y    y  y

1

 cos  x y ,

2

 sin  x

 



² ²



cos sin

cos , sin

sin cos

cos sin

x x

W x x

x x

 

   

 

  

  0

(19)

Existence (1)

• Existence Theorem

--- If p(x) and q(x) continuous, general sol. exists.

– First sol. y₁ – Second sol. y₂ – Wronskian

linearly independent – General sol.

– Every sol. Y(x) can be expressed

   

1 0

1,

1 0

0

y x  y x  

   

2 0

0,

2 0

1

y x  y  x  1

W 

   

1 1 2 2

y  k y x  k y x

 

1 1

 

2 2

 

Y x  C y x  C y x

(20)

Existence (2)

– Does not have singular sol.

– Assume x0 which gives

– Matrix form

– Coefficient matrix = Wronskian unique sol.

– Everywhere on I

 

0

   

0

,

0

 

0

y x  Y x y x   Y x 

 0

 

1 1

 

2 2

 

y

^

x  C y x  C y x

     

1 1 0 2 2 0 0

c y x c y x Y x c y x c y x Y x

 

    

Unknowns c₁, c₂

1 1

,

2 2

c  C c  C

 

0

 

0

,  

0

 

0

y

^

x  Y x y

^

 x  Y x 

(21)

Non-homogeneous DE (1)

– Relation between sol. of corresponding homogeneous DE (1) Difference of two sol. of non-homogeneous DE:

sol. of homogeneous DE

(2) Sum of sol. To non-homogeneous DE and that to homogeneous DE: another sol. of non-homogeneous DE

    ,    

L y  r x L y  r x

          0

L y  y  L y  L y  r x  r x 

    0  

L y    y

^

   L y  L y    

^

 r x   r x

     

    

y p x y q x y r x

(22)

Non-homogeneous DE (2)

• General sol. to non-homogeneous DE

– y_h: general sol. of the corresponding homogeneous DE

– y_p: any sol. of the non-homogeneous DE with no arbitrary constants – Particular sol. can be obtained by assigning values to c₁and c₂

• Uniqueness and Existence of the General sol.

 

h

 

p

 

y x  y x  y x

 

1 1

 

2 2

 

y

h

x  c y x  c y x

 

0



0

,   

0



1

y x K y x K

 

0 0 _p

   

0

,

0 0 _p

 

0

y x  K  y x y x   K  y  x

(23)

Non-homogeneous DE (3)

– Every sol. can be obtained by assigning values to c₁and c₂in y_h : any sol. of the non-homogeneous DE

: any general sol. of the same DE : sol. of the homogeneous DE

– Sol. of the non-homogeneous DE

--- find the sol. to corresponding homogeneous DE, y_h and y_p

y   y y

p

y

Unique sol. to non-homogeneous DE

 

h

 

p

 

y x  y x  y x

   

p

 

Y x  y x  y x

   

p

 

y x  Y x  y x

(24)

Example of Non-homogeneous DE

• Initial Value Problem

– General sol. to the corresponding homogeneous DE – General sol. to the non-homogeneous DE

Try

– Substituting

– General sol. of the non-homogeneous DE

– From IC

   

2 101 10.4 ,

^x

0 1.1, 0 0.9 y   y   y  e y  y   

 cos10 sin10 

x

y

h

 e

^

A x  B x

x

y

p

 Ce

 1 2 101    Ce

^x

 10.4 e

^x

C  0.1

 cos10 sin10  0.1

x x

h p

y  y  y  e

^

A x  B x  e cos10 0.1

x x

y  e

^

x  e

(25)

Example of Non-homogeneous DE



0.1

 e

^x

 e

^x

^x

 0.1

^x

e e

(26)

Sol. by Undetermined Coefficients

• Finding y

_p

in the non-homogeneous DE

• Method of Undetermined Coefficients

(A) Basic rule

(B) Modification rule --- If your choice for y_p happens to be a sol. of the corresponding DE, then multiply by x or x² (if it is a double root).

(C) Sum rule --- If r(x) is a sum of several functions in Table, then choose for y_p the sum of the corresponding trial functions.

 

    

y ay by r x

(27)

Term in r(x) Choice for y_p

 0,1, 

cos sin



x n

x x

ke

kx n

k x

ke x







 

1

1 1 0

cos sin







  



x

n n

x x

Ce

K x K x K x K

K x M x

e K x M x





 

Basic Rule

(28)

Example of Undetermined Coefficients (1)

4 8

2

   

y y x

2

2 1 0

,  2

2

   

p p

y K x K x K y K



²



²

2 2 1 0

2 K  4 K x  K x  K  8 x

2

1

 

y

p

x

cos 2 sin 2 2

2

1



_h



_p

   

y y y A x B x x

– Choice of y_p

– Substituting

– Equating coefficients of x², x, and x⁰

– General sol.

(29)

– Sol. of the corresponding homogeneous DE y_h

– Choice of y_p

– Substituting

– General sol.

2

1 2



^x



^x

y

h

c e c e

   

,  ,  2



^x



^x



^x



^x



^x

p p p

y Cxe y C e xe y C e xe

2

1 2



_h



_p



^x



^x



^x

y y y c e c e xe

3 2

    

^x

y y y e

 2  

^x

 3 1   

^x

 2

^x



^x

,   1

C x e x e Ce e C

Same as y_h

(30)

Example of Undetermined Coefficients (3)

– Choice of y_p

– Substituting

– General sol.



1 2



 

^x

y

h

c c x e

   

2 2 2

, 2 , 2 4













^x

 

^x

  

^x

p p p

y Cx e y C x x e y C x x e



1 2



²

1 2

 



_h



_p

 

^x



^x

y y y c c x e x e

 

²

   

2 1

^

, 0 1, 0 1

      

^x

   

y y y D y e y y

 2 4

²

 2  2

²



²

, 1

2

   

 

^x

 

^x



^x



^x



C x x e C x x e Cx e e C

Same as y_h,, double root

(31)

– From IC

1

2

2 1

 



     

y x e

x

(32)

– Choice of y_p

 cos 2 sin 2 



^x



y

h

e A x B x

0.5 0.5

0.5

cos 4 sin 4 ,

0.5 4 sin 4 4 cos 4 , 0.25 16 cos 4 16 sin 4

  

   

   

x p

y Ce K x M x

   

2 5 1.25

0.5

40 cos 4 55sin 4 , 0 0.2, 0 60.1

      

  

y y y e

x

x x

y y

(33)

Example of Undetermined Coefficients (6)

– Substituting and equating

– General sol.

– Particular sol.

 cos 2 sin 2  0.2

^0.5

5sin 4



^x

 

^x



y e A x B x e x

0.2, 0, 5

  

C K M

20

^

sin 2 0.2

0.5

5sin 4



^x



^x



y e x e x

(34)