computational method for the determination of forced...

Asian Research Journal of Mathematics

3(1): 1-12, 2017; Article no.ARJOM.31821 ISSN: 2456-477X

SCIENCEDOMAIN international www.sciencedomain.org

_____________________________________ *Corresponding author: E-mail: [email protected];

Computational Method for the Determination of Forced Motions in Mass-spring Systems

Y. Skwame1, A. I. Bakari2 and J. Sunday1*

1Department of Mathematics, Adamawa State University, Mubi, Nigeria.

2Department of Mathematics, Federal University, Dutse, Nigeria.

Authors’ contributions

This work was carried out in collaboration between all authors. Author YS derived the computational method. Author AIB analyzed the basic properties of the computational method while author JS managed the

literature searches and implemented the derived method on real world modeled problems. All authors read and approved the final manuscript.

Article Information

DOI: 10.9734/ARJOM/2017/31821

Editor(s): (1) Andrej V. Plotnikov, Department of Applied and Calculus Mathematics and CAD, Odessa State Academy of Civil Engineering and

Architecture, Ukraine. Reviewers:

(1) Yusuf Yesilce, Dokuz Eylul University, Turkey. (2) G. Y. Sheu, Chang-Jung Christian University, Tainan, Taiwan.

(3) Jorge F. Oliveira, Polytechnic Institute of Leiria, Portugal. (4) Abdullah Sonmezoglu, Bozok University, Turkey.

Complete Peer review History: http://www.sciencedomain.org/review-history/18136

Received: 26th January 2017 Accepted: 28th February 2017

Published: 9th March 2017 _______________________________________________________________________________

Abstract

In this paper, we propose a computational method for the determination of forced motions in mass-spring systems. The method of interpolation and collocation of Hermite polynomial (as basis function) was adopted to generate continuous computational hybrid linear multistep methods which were evaluated at grid points to form a discrete computational block method. The method was applied on two real life problems to determine the motions of weights in mass-spring systems. We also went further to analyze some properties of the method like zero-stability, consistence and convergence. From the results we obtained, it showed that the proposed method is computationally reliable.

Keywords: Computational method; forced motion; Hermite polynomial; mass-spring systems. AMS subject classification (2010): 65L05, 65L06, 65D30.

Original Research Article

Skwame et al.; ARJOM, 3(1): 1-12, 2017; Article no.ARJOM.31821

2

1 Introduction A lot of problems in science and technology can be formulated into differential equations. The analytical methods of solving differential equations are applicable to a limited class of equations. Quite often differential equations appearing in physical problems do not have exact solutions and one is obliged to resort to computational methods to solve such problems. In this paper, a computational method for the determination of motions on weights in a mass-spring systems in the form of second order differential equations of the form,

0 0 0 0'' ( , , '), ( ) , '( ) 'y f x y y y x y y x y= = = (1)

shall be proposed, where f is assumed to be continuous within the interval of integration. A lot of scholars have developed computational block methods for the solution of problems of the form (1). Computational block methods have the advantage of generating independent solutions at selected grid point without overlapping. They also possess the properties of Runge-Kutta method of being self-starting and do not require starting values. Some of the authors that proposed computational block methods using different approximate solutions are [1,2,3,4], among others.

2 The Differential Equation of the Vibrations of Mass-spring Systems Let l be the natural (unstretched) length of a coil spring. Suppose a mass m is attached to the lower end of

the spring so that it comes to rest in its equilibrium position O , this stretches the spring by an amount e, so

that the stretched length is l e+ . At the equilibrium position O , the mass m is acted upon by two forces

i.e. the weight mg acting vertically downwards and the spring force ke acting vertically upwards. Thus,

we have

mg ke= (2)

Supposing P is the position of the mass (below equilibrium position) at any time t so that the distance

from the equilibrium position O to the point P is given by OP x= . Then x may be positive, zero or negative according to whether the mass is below, at, or above its equilibrium position. When the mass is situated at P , it is acted upon by the following forces. The forces tending to pull the mass downward are positive, while those pulling it vertically upward are negative.

(i) 1F mg= , acting in the vertically downward direction.

(ii) Let 2F be the restoring force of the spring. When the mass is at P , 2F is acting in the upward

direction and so it is negative. By Hooke’s law, we have.

2 ( )F k x e= − + (3)

Using (2) in (3), we get

2F kx mg= − − (4)


3

(iii) Let 3F be the resisting force of the medium called damping force. It is known that for small

velocities, 3F is approximately proportional to the magnitude of the velocity. When the mass

moving downward (at P , say), 3F acts in the upward direction (opposite to that of the motion) and

so 3F is negative and is given by,

3 ( )F b dx dt= − (5)

(iv) External impressed force 4 4( )F F t= acting in downward direction.

By Newton’s second law, F ma= (6)

where 1 2 3 4F F F F F= + + + and 2 2a d x dt= . Thus,

( ) ( )2

2 4( )d x dxm mg kx mg b F tdtdt= − − − +

( ) ( )2

2 4( )d x dxm b kx F tdtdt+ + = (7)

which is a differential equation for the motion of the mass on a spring and is of the form (1). If 0b = , the motion is called undamped otherwise it is called damped. If there are no external impressed forces,

4( ) 0F t = for all t , the motion is called free, otherwise it is called forced, see [5,6].

3 Mathematical Formulation of the Method Hermite polynomial shall be used as a basis function in deriving the computational method. The derivation shall be carried out with the aid of Scientific Workplace 5.5 software. Thus,

( ) 2 26

2 3 4 5 6

0

( ) 1 109 110 676 152 464 35 64n

n x xn

n

dy x e e x x x x x x

dx−

=

= − = − + + − − + +∑ (8)

We interpolate equation (8) at 1 1

, ,4 2n sx s+ = and collocate its second derivative at

1, 0 1

4n rx r+ =

.

Note thats and r are the numbers of interpolation and collocation points respectively. Thus, we obtain a system of non linear equation of the form,

XA U= (9)

where

[ ]0 1 2 3 4 5 6 1 1 1 1 3 14 2 4 2 4

,

T

T

n nn n n n n

A a a a a a a a U y y f f f f f++ + + + +

= =


4

and

2 3 4 5 61 1 1 1 1 1

4 4 4 4 4 4

2 3 4 5 61 1 1 1 1 12 2 2 2 2 2

2 3 4

2 3 41 1 1 14 4 4 4

109 110 676 152 464 35 64

109 110 676 152 464 35 64

0 0 1352 912 5568 700 1920

0 0 1352 912 5568 700 1920

0 0 1352

n n n n n n

n n n n n n

n n n n

n n n n

x x x x x x

x x x x x x

x x x x

X x x x x

+ + + + + +

+ + + + + +

+ + + +

− − −

− − −

− −

= − −

2 3 41 1 1 12 2 2 2

2 3 43 3 3 34 4 4 4

2 3 41 1 1 1

912 5568 700 1920

0 0 1352 912 5568 700 1920

0 0 1352 912 5568 700 1920

n n n n

n n n n

n n n n

x x x x

x x x x

x x x x

+ + + +

+ + + +

+ + + +

− −

− − − −

Using Gaussian elimination method, we evaluate equation (9) for ' , 0(1)6ja s j = and substitute them into

(8) to obtain a continuous hybrid linear multistep method of the form,

12

1 1 1 104 4 2 2

1 1 3( ) ( ) ( ) , , ,

4 2 4j n j k n kn n

j

y x y y h x f x f kα α β β+ ++ + =

= + + + =

∑ (10)

where the coefficients of 1 1 1

, , , 0 14 2 4n j n jy j and f j+ +

= =

are obtained as follows;

( )

( )

( )

1

4

1

2

6 5 4 3 20

6 5 4 31

4

6 5 4 31

2

6 5 4 33

4

2 4

4 1

14096 15360 22400 16000 5760 962 57

115201

4096 13824 16640 7680 882 1532880

14096 12288 12160 3840 66 7

1920

14096 10752 8960 2560

2880

t

t

t t t t t t

t t t t t

t t t t t

t t t t

α

α

β

β

β

β

= −

= −

= − + − + − +

= − − + − + −

= − + − + +

= − − + −( )

( )6 5 4 31

70 3

14096 9216 7040 1920 54 3

11520

t

t t t t tβ

+ −= − + − + −

(11)


5

Note that, ( )nt x x h= − , ( )n j ny y x jh+ = + and ( )( ), ( ), '( )n j n n nf f x jh y x jh y x jh+ = + + + .

Evaluating equation (10) for the independent solution at the grid points gives the continuous block method,

( )( )1 1( ) 2

0 0

1 1 3( ) ( ) , , ,

! 4 2 4

m

mn j n j k n k

j j

jhy x y h x f f k

mσ σ+ +

= =

= + + =

∑ ∑ (12)

where the ' 'j ks and sα α are given by,

( )

( )

( )

( )

( )

5 4 3 20

5 4 3 21

4

5 4 3 21

2

5 4 3 23

4

5 4 3 21

1192 600 700 375 90

901

384 1080 1040 36045

1192 480 380 90

15

1384 840 560 120

45

1192 360 220 45

90

t t t t t

t t t t

t t t t

t t t t

t t t t

σ

σ

σ

σ

σ

= − + − +

= − − + − = − + −

= − − + − = − + −

(13)

Evaluating equation (12) at 1 1

14 4

t =

gives a discrete block method as,

( ) ( )1

(0) ( ) ( ) 2 2

0

, 0,1i i im i n i n i m

i

A Y h e y h d f y h b f Y i=

= + + =∑ (14)

where

1 1 3 1 1 1 3 1

4 2 4 4 2 4

, ( )

T T

m n m nn n n n n n

y y y y f f f f f+ ++ + + + + +

= =

Y Y

[ ]( ) ( ) ( ) ( ) ( )1 2 3 1 2 3, ( )

T Ti i i i in n n n n n n n n ny y y y f f f f f− − − − − − = = y y

and (0) 4 4A = × identity matrix.

When 0 :i =


6

0 1 0 0

734 6 94 58 141 0 0 0

0 0 0 69120 384 11520 11280 2304040 0 0 1 106 2 2 2 21 0 0 0

0 0 0 1 0 0 0 4320 30 144 270 1440, , ,20 0 0 1 294 234 54

3 0 0 00 0 0 7680 1920 3840 0 0 1 4

140 0 0 1 0 0 0

270

e e d b

− − − − = = = =

6 27

0 384 76808 2 8

045 30 135

−

When 1:i =

1 1 1

502 646 22 106 380 0 0

8640 4320 360 4320 86400 0 0 1 58 62 2 2 2

0 0 00 0 0 1 1080 270 45 270 1080, ,0 0 0 1 54 102 18 42 6

0 0 0960 480 120 480 9600 0 0 114 48 6 48 14

0 0 0270 135 45 135 270

e d b

− − − = = = −

4 Analysis of the Method 4.1 Order of accuracy of the method

The linear operator { }( );L y x h of the method given by equation (14) is defined by,

{ } ( )1

(0) ( ) ( ) 20 0

0

( ); ( ) ( )i i im i n n m

i

L y x h A h e y h d f y b F=

= − − +∑Y Y (15)

If we expand equation (15) in Taylor series and compare the coefficients of h , we get,

{ } 2 1 1 2 20 1 2 1( ); ( ) '( ) ''( ) ... ( ) ( ) ( )...p p p p p p

p p pL y x h c y x c hy x c h y x c h y x c h y x c h y x+ + + ++= + + + + + +

(16)

Definition 1 [7]: The linear operator L and the associated block formula (14) are said to be of order p if

0 1 2 1 2... 0 0.p p pc c c c c and c+ += = = = = = ≠ 2pc + is called the error constant and implies that the

local truncation error is given by,

( 2) ( 2) 32 ( ) ( )p p p

pn kt c h y x O h+ + +++ = + (17)

Taylor series is applied to expand the method derived and then compare with the coefficients of h .

This gives, 0 1 2 3 4 5 6 0c c c c c c c= = = = = = = and with the error constant

7

7 6 6 66.4790 10 1.5501 10 2.4523 10 3.1002 10T

c − − − − = × × × × .


7

Therefore, the one-step computational method is of uniform fifth order.

4.2 Zero stability analysis of the method Definition 2 [8]: The block method (14) is said to be zero-stable, if the roots , 1,2,...,sz s k= of the first

characteristic polynomial ( )zρ defined by (0)0( ) det( )z zA eρ = − satisfies 1sz ≤ and every root

satisfying 1sz = have multiplicity not exceeding the order of the differential equation. Moreover, as

0,h → ( ) ( 1)rz z zµ µρ −= − where µ is the order of the differential equation, r is the order of the

matrices (0)0A and e , see [9] for details.

Applying this analysis on the method derived, we obtain

1 0 0 0 0 0 0 1

0 1 0 0 0 0 0 1( ) 0

0 0 1 0 0 0 0 1

0 0 0 1 0 0 0 1

z zρ

= − =

(18)

Thus, 31 2 3 4( ) ( 1) 0, 0, 1z z z z z z zρ = − = ⇒ = = = = . We therefore conclude that the method is zero-

stable.

4.3 Consistency and convergence analysis of the method Since the method (14) has order 5 1p = ≥ , it implies that it is consistent. The method is also convergent by

virtue of the Dahlquist theorem stated below. Theorem 1 [10]: The necessary and sufficient conditions that a continuous LMM be convergent are that it be consistent and zero-stable. 4.4 Region of absolute stability analysis of the method Definition 3 [11]: Region of absolute stability is a region in the complex z plane, where z hλ= . It is

defined as those values of z such that the numerical solutions of ''y yλ= − satisfy

0jy as j→ → ∞ for any initial condition.

In determining the region of absolute stability of the method, we employ the boundary locus method. This gives the polynomial of stability as,

8 3 4 6 4 3

4 4 3 2 4 3 4 3

7 521 7019 1272109( )

3686400 1848115200 554434560 249495520

5309 349109 5 592

34652160 10395648 12 96

h w h w w h w w

h w w h w w w w

= − − − +

− + − + − −

(19)


8

Thus, the region of absolute stability of the method is presented in the figure below.

Fig. 1. Region of absolute stability of the method

5 Results We shall apply the computational method developed on two problems of mass-spring systems that has to do with the determination of the motion of mass. The following notations shall be used in the Tables 1 and 2.

ERR - |Exact Solution - Computed Solution| t - Time

x - Motion of the mass at time t

EvlTime - Evaluation time per seconds

The results for the problems to be considered below were programmed using MATLAB software version R2010a.

Problem 5.1

A 128lb weight is attached to a spring having a spring constant of 64lb ft . The weight is started in

motion with no initial velocity by displacing it 6inches above the equilibrium position and by

simultaneously applying to the weight an external force 4( ) 8sin 4F t t= . Assuming no air resistance,

compute the subsequent motion of the weight at : 0.01 0.10t t≤ ≤ . Source: [12]

-6 -5 -4 -3 -2 -1 0-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Re(z)

Im(z

)


9

Firstly, we model this problem into a mathematical equation of the form (1) using (7) and then apply our method to compute the motion on the weight attached to the spring. Here,

44, 64, 0 ( ) 8sin4m k b and F t t= = = =

Thus, problem 5.1 boils down to

2

2

116 2sin 4 , (0) , '(0) 0

2

d xx t x x

dt+ = = − = (20)

with the exact solution of (20) is given by,

1 1 1( ) cos4 sin 4 cos4

2 16 4x t t t t t= − + − (21)

Problem 5.2

A 10 kg mass is attached to a spring having a spring constant of 140N m. The mass is started in motion

with an initial velocity of 1 secm in the upward direction and with an applied external force

4( ) 5sinF t t= . Find the subsequent motion of the mass ( :0.10 1.00)t t≤ ≤ if the force due to air

resistance is 90dx

Ndt

.

Source: [12]

Applying the same procedure, where 410, 140, 90 ( ) 5sinm k b and F t t= = = =

Problem 5.2 reduces to

2

2

19 14 sin , (0) 0, '(0) 1

2

d x dxx t x x

dt dt+ + = = = − (22)

with the exact solution of (22) is given by,

( )2 71( ) 90 99 13sin 9cos

500t tx t e e t t− −= − + + − (23)

6 Discussion of Results From the results obtained in Tables 1 and 2, it is clear that the computational method derived is convergent because the computed solutions agree with the exact solutions. Thus, at a particular time t , one is able to

know the motion of weights x in a mass-spring systems. The evaluation time per seconds (EvlTime) in Tables 1 and 2 are also seen to be very small; implying that the computational method generates results very fast. Therefore, this method has a greater advantage over manual computations where one has to spend hours before computing the results. The graphical plots obtained in Figs. 2 and 3 also show the natures of the two problems considered.


10

Table 1. Result for Problem 5.1 showing motion of the mass at time t

t Exact solution ( )x Computed solution ( )x ERR EvlTime

0.0100 -0.4995987202104767 -0.4995987185483694 1.662107e-009 0.0579 0.0200 -0.4983901933097495 -0.4983901817239240 1.158583e-008 0.0587 0.0300 -0.4963683697402797 -0.4963683399977432 2.974254e-008 0.0592 0.0400 -0.4935285266081794 -0.4935284705319387 5.607624e-008 0.0599 0.0500 -0.4898672879689450 -0.4898671974653856 9.050356e-008 0.0604 0.0600 -0.4853826428970993 -0.4853825099832920 1.329138e-007 0.0609 0.0700 -0.4800739612905668 -0.4800737781214341 1.831691e-007 0.0615 0.0800 -0.4739420073643619 -0.4739417662596349 2.411047e-007 0.0620 0.0900 -0.4669889507920278 -0.4669886442629255 3.065291e-007 0.0624 0.1000 -0.4592183754572240 -0.4592179962327859 3.792244e-007 0.0629

Fig. 2. Graphical result showing the nature of Problem 5.1

Fig. 3. Graphical result showing the nature of Problem 5.2

-5

0

5

-5

0

5-2

-1

0

1

2

tx

-5

0

5

-5

0

5-2

0

2

4

6

8

10

x 1012

tx


11

Table 2. Result for Problem 5.2 showing motion of the mass at time t

t Exact solution ( )x Computed solution ( )x ERR EvlTime

0.1000 -0.0643620515455246 -0.0643619450778007 1.064677e-007 0.0238 0.2000 -0.0843072052264477 -0.0843083921765293 1.186950e-006 0.0247 0.3000 -0.0840522531339004 -0.0840545166131738 2.263479e-006 0.0252 0.4000 -0.0752930421333338 -0.0752958639884895 2.821855e-006 0.0257 0.5000 -0.0635706396035580 -0.0635735934752395 2.953872e-006 0.0262 0.6000 -0.0514211706938451 -0.0514239893808955 2.818687e-006 0.0267 0.7000 -0.0399305295643870 -0.0399330761412450 2.546577e-006 0.0272 0.8000 -0.0294986586280357 -0.0295008820820101 2.223454e-006 0.0277 0.9000 -0.0202126913125913 -0.0202145903719575 1.899059e-006 0.0284 1.0000 -0.0120269942540317 -0.0120285930298131 1.598776e-006 0.0289

7 Conclusion We have been able to develop computational method for determining the motion of weights in a mass-spring system. The computational method developed is convergent, consistent and stable. This method is therefore recommended for the solution of problems of the form (1).

Competing Interests Authors have declared that no competing interests exist.

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Peer-review history: The peer review history for this paper can be accessed here (Please copy paste the total link in your browser address bar) http://sciencedomain.org/review-history/18136

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