Chapter 2: Mathematical Modelling of Translational Mechanical System

Admolljcg d` ]rbcslbtjdcbl Aonebcjnbl ^ystoa.

2. Japombcno bcm `drno-mjsplbnoaoct trbcslbtjdcbl rolbtjdcsejps `dr sprjcgs, vjsndus mbapor bcm abss

^yakdl ^prjcg



Japombcno



^tdros ocorgy bs

@drno-mjsplbnoaoct

    ( ) 1

( )

pdtoctjbl ocorgy

Pjsndus mbapor

 

 

^tdros ocorgy bs

      ( ) 1

( )

fjcotjn ocorgy

Abss



;

Mjssjpbtos ocorgy jctd eobt

    ( ) 1

( )

;.

Mrbw teo `roo kdmy mjbgrba

 ; 

( )

Motorajco oqubtjdc d` adtjdc

      ;    ( )+

( )+

( ) 1

( )

Motorajco teo trbcs`or `ucntjdc

      ;   ( )(  +

 +

)  1

( )

 1 2    ++ ( )

( )

>. Motorajco teo trbcs`or `ucntjdc

8  ( )

( )

Oqubtjdc d` adtjdc8

;        ( )+;

;

( ) + ? ( ) 1

( )

]d motorajco teo ]rbcs`or @ucntjdc

@bntdrjzo x(s)

  ;  ( )(;

 + ;  + ?)  1

8  ( )

( )



( )

Oljajcbto @(s) dc teo rjget ebcm sjmo

  ;+;+? 1       ;+;+? 1 2  ;     ;+;+?  1  2   ; +;+? ; +;+?  1 2   ; +;+? ( )(

)

( )

( )

( )(

( )

)

( )

Oljajcbto (;

 + ;  + ?) dc teo lo`t ebcm sjmo

( )(

)

( )(

)

(

)

( )

( )

(

)

=. Motorajco teo trbcs`or `ucntjdc8

]eo `rjntjdc, ? C-s/a ebs teo sbao prdporty bs vjsndus mbapor.

; ;                   ; 

(:

 + 29  + >  + ?  + 2;) ( ) 1 (:  + 25  + 2;) ( ) 1 ( )

( )

 1

( )

2

(:

 + 25  + 2;)

( )

?. @dr trbcscbtjdcbl aonebcjnbl systoa wjte ; abss,

Oqubtjdc d` adtjdc

       ℃     ∛   ℃      2 ;  2  2  ;  2        ℃ ∛ ℃            2 ;  2  ;  ;  ; ( )

( ) 1

( )+

( ) 1

Nrbaor‚s rulo nbc ko usom td sdlvo teo sjaultbcodus oqubtjdc. @dr oxbaplo, ndcsjmor b ljcobr systoa

Wejne jc abtrjx `drabt js

            -  ∛     ∛       ∛ ∛   +

 1

 +

 1

 1

]eoc x bcm y nbc ko `ducm wjte Nrbaor‚s rulo bs8

 1

 1

 1

 1

0. @dr teo trbcslbtjdcbl aonebcjnbl systoa wjte ; abss koldw,

Mrbw teo `roo kdmy mjbgrba

2 

0

 ;2   2   ; 

( )

?

=

( )

7

( )

=

 ;   ;;  2  =

( )

7

( )

5

( )

; 

( )

( )

Cdto `dr brrdw mjrontjdc8 2. Mjrontjdc `dr @(s) wjll stby ;. Mjrontjdc `dr bll dteor ndapdcoct ndccontom td x2 (`dr a2) bcm x; (`dr a;) wjll ko dppdsjto td @(s) >. Mjrontjdc `dr ndapdcoct sebrom kotwooc x2 bcm x; wjll ko `dlldwjcg mjrontjdc d` @(s).

Oqubtjdc d` adtjdc8

V?

;   2 ; ∛  ;    ∛  2     ;  ( ) + V =Q

 + 0  + =Q

V =Q

( ) + V7

( ) 1

 + 5  + =Q

( )

( ) 1 9

   teo oqubtjdc d` adtjdc js ndcvortom td abtrjx `dra. ()   ( )  ∛ ? ; + 0 + = = 2  1      ; ∛= 7  + 5 + =  ;() 9 ( )

]d nblnulbto teo trbcs`or `ucntjdc,

( )

Japdrtbct Cdto8

∀ ; ∛  ; ∛   ∀2    ; ∛   ∀;  ; ∛        ;   ;  ∀∀;  1

?

 + 0  + = =

( ) 1

( ) 9

( ) 1 ?

]eo trbcs`or `ucntjdc js

= 7  + 5  + =

7

=  + 5  + =

 + 0  + = =

( )

( )

( ) 9

, teoro`dro wo coom td nblnulbto

( ) 1

( )

( ).

; ∛   ;   ;  ∛  ;  ;  ∛ ∛; ?

( ) 1

?

9

( ) 1

(?

 ; ∛     ∛ ∛

 + 0  + = ( ) = 9  + 0  + = = = 7  + 5  + =

( =)( ( ))

 + 0  + =)(7

( =)

 + 5  + =)

;

 ;  = > ; > ;  ;  ∛  ;  = > ;  ( )

( )

1

=

>?

 + =9

 + ;9 ( )

( )

 + =;

1

 + =5

 + ;=  + ;5

 + >;  + 20

20

=

>?

 + 5;

 + :0

 + ?0

7. [o`or td @jguro koldw.

@jguro 2 Mrbw teo @roo kdmy mjbgrba `dr teo trbcslbtjdcbl aonebcjnbl systoa jc @jguro 2

 2 

 2   2  ; 2   2  2

( )

0

( )

; >

 ; 

( )

( )

 2  2   ;  ;  i

;

( )

>

;

( ) ( )

( )

( )

>

( )

  ;  ;   2   2  ;

 ;;  ;  ; 

( )

=

( )

?

( )

( )

;

( )

>

;

( )

>

( )

Wrjto teo Oqubtjdc d` Adtjdc

;   2  ; ∛  ∛  ;  ∛ ∛   2     ;      ∛; ∛ ;∛ ∛   2;      ;     ;  ;  ;  ∛ ∛  ∛ ; (>

( >

;)

Motorajco teo trbcs`or `ucntjdc,

>

 + ?  + :)

( ) + (=

;)

( ) 1 9

( ) 1

 + 29  + ;)

( )

( )

( )

 + ?  + : > ;

( ) 1

( )+( >

=

> ;  + 29  + ;

(>

(>

( )  1 ( )

9 ( )

 + ?  + :)( ( ))

 + ?  + :)(=

 + 29  + ;)

( >

;)

( )

 ;  = > ; >  ;  = ( )

( )

1

2;

 + >9

 + 0

 + ;9

( )

( )

1

2;

;  ; ; > >

 + ?9

 ;  ∛ ;  ; 

 + ?  + :

 + 29  + >0

>

 + ?9

 + :9  + 25

 + ?  + :

 + 5>

 + 55  + 2=

(:

 + 2;  + =)

5. [o`or td @jguro ;, motorajco teo trbcs`or `ucntjdc,

  

( )

( )

@jguro ;

;   2  ∛  ;  ∛;  2   ;     ∛  ∛   2;      ; ∛      ;   ;  ∛  ∛;  ;  ;  ∛   ;  ;;  ∛

(;9

(;9)

;9

( ) + ( ;9)

 + 29  + ;9)

( ) + ;9

 + 29  + ;9 ;9

( ) 1

;9 ;9

( )

( )  1 ( )

;9

 + 29 + ;9 ;9 ( ) 1 ;9  + 29  + ;9 ;9

( ) 1

(;9

=99

( )

( )

1

( ) 1 9

9 ( )

9 ( ) ;9 ;9

 + 29  + ;9) ( )

 + ;99  + ;99

;9

=99

=99

 + 29  + ;9

 + ;99

;99