0

Fast Chain The study of dynamics of inclined linkage stick

system

Jinze,Shi Linchuan,Liu Xuanchen,Jia

Qingdao No.2 Middle School of China Date: 10. 20 , 2016

This paper utilizes Lagrange Equations to analyze parameters of a system consisting of several inclined

sticks. The solving process involves some tough problems, including a precise derivation of collision process

and unilateral constraint. We used Mathematica to obtain an Appropriate Approximation in 2 degrees of

freedom of system. During the theory process, we have simulated and verified the collision with the help of

ADAMS. In order to keep the precision of our experiments, we also utilized 3D printer, SCM tech and lathes to

manufacture objects. During the experiments, ABAQUS, professional software powered by Finite Element

Method was used to testify the precision of our experiments. Considerate conclusions were made at the end of

this paper, synthesizing all the theoretical analysis, modeling, and experiment data together.

Introduction

Free-falling model is the most common model in physics, it has been studies widely and deeply. However, a little modification on free-falling experiment conditions will absolutely change its attribute. A chain consisting of wooden blocks inclined relative to the vertical and connected by two ropes is suspended vertically and then released. Compared to free fall, the chain falls faster when it is dropped onto a horizontal surface. This paper will analyze this phenomenon and render theoretical formulas. There will also be ingenious experiments and modeling to testify our derivations. Ultimately, an appropriate conclusion will be drawn.

CONTENTS

Introduction Chapter 1 Theory Analysis 1.1 difficulties of ordinary analysis 1.2 Variables table 1.3 double degree-of-freedom system 1.4 Four degree-of-freedom system

1.5 Concrete Analysis on Collision 1.6 Unilateral Constraint Integrated in Lagrange

Equations 1.7 Eight Degrees of Freedom of the System

1.8Final Extra Velocity

1.9 ADAMS Simulation

Chapter 2 Experiment & ABAQUS Analysis

2.1 Experiment Facility And Objects

2.2 Measure Methods 2.3 Experiment 2.4 ABAQUS Model

2.5 Experiment on α ,β, θ, φ of six degrees of

freedom system

2.6 ABAQUS fittings

Chapter 3 3.1 Analyze about deviations 3.2 Conclusion 3.3 Acknowledgments 3.4 Reference Appendix Resume

0

1 1

7

6

10

13 14 15 16

23 23 23 24

1

10 11

19 19

11

1

CHAPTER1 THEORY&ADAMS ANALYSIS

The Chapter 1utilizes the Lagrange Equation of analytical mechanics to derive the theoretical analysis of the

system. Along with some appropriate approximations and the integration of unilateral constraint, the analytical

solutions of two degrees of the system are obtained, thus paving the way for the derivation of the optimum

maximum extra velocity of the whole system. After the analytical solutions, numerical solutions of four degrees

of system is drawn which are more consistent with the reality. Significant features such as collision of the

lowest stick and the unilateral constraint integrated into Lagrange Equation are discussed afterwards. Finally, a

conclusive model involving eight degrees of freedom is analyzed and the thought of calculating the final extra

velocity of the first stick of the system is given.

1.1 difficulties of ordinary analysis The whole falling procedure is made of two different processes:

accelerating process and free fall process. Accelerating processes are

separate. As this system is consisted of several sticks, the accelerating

process can be divided into discrete periods. As showing in the left figure,

the first period involves lever effect generated by gravity and the force along

the short string accelerates the whole system.

Difficulties emerged when analyzing the equations of motion of the first

accelerating period. First of all, the model of realistic string is difficult to

simulate; what's more, there are too many constraints in this system and the

force along the string will change its direction within two-dimension plane.

Due to the complicated priority of F and angular velocities and the huge

amount of constraints, utilizing analytic mechanics is simpler and more

appropriate.

1.2 Variables table Generalized coordinates of Lagrange Function:

The origin of this set of coordinates is a static point of the pivot of the first stick. The values on the horizontal and vertical axis will increase as moving to the right and up. θ:The angle of the lowest stick relative to the horizontal surface φ:The angle of the first string relative to the horizontal surface ψ:The angle of the second stick relative to the horizontal surface α:The angle of the second string(short/long) relative to the horizontal surface x: The displacement on x-axis of the pivot of the first stick y: The displacement on y-axis of the pivot of the first stick Other variants: x1, y1: The Cartesian coordinates of the barycenter of the lowest stick. (x2 and y2 represents for those of the second stick, etc) l:The length of the stick d: The length of short strings c: The length of long stringsμ: friction coefficient e: elastic coefficient m: the mass of one stick g: gravitational acceleration f: friction force F: force along the string N: normal force T: kinetic energy of the system U: potential energy of the system L: Lagrange Function of the system H: Hamiltonian Function of the system R: Routh Function of the system qi: a symbol represents for one generalized coordinate pi: a symbol represents for one generalized momentum q & p: symbols represent for all generalized coordinates or momentum n: the number of total sticks of the system I1: moment of inertia relative to the end point of one stick I2: moment of inertia relative to the barycenter of one stick

1.3 double degrees of freedom system

1.3.1 Lagrange Equations Every stick touches the horizontal surface can be treated as a lever in this falling system. Treating the

2

1

1sin

2y l

2 sin sinx l d

2 cos cosy l d

2 cos cosdx l 2 sin siny l d

2 2 2

1 2 2

1 1( 1) ( )

2 2T I n m x y

1 2( 1)U mgy n mgy

L T U

S

S S

dE

dt q q

second stick and sticks above as a whole, we give analysis of two degrees of freedom of the system, setting the

angle of the lowest stick and the lowest string as generalized coordinates. Here are the Cartesian coordinates

represented by theta and phi. Giving kinetic energy and potential energy, a set of Lagrange Equation can be

obtained by utilizing Euler Operator. Here are the concrete forms of the equation.

Lagrange Equations

1.3.2 Approximated Analytical Solutions Deriving accurate analytical solution of the system is impossible. A roughly analytical solution can be

drawn by omitting the second and forth terms in the first equation, because d is much smaller than constant g.

When theta0 is equal to 30 degrees, separating theta dot-dot and integrating the differential equation will render

the time t expressed by angle theta. Here F and k represent elliptic integral of the first form.

Elliptic Intergral:

,

，

1

1cos

2x l

0E L

0E L

2 ( 1)sin( ) 2 (3 2) 2 ( 1)cos( )cos 0

(2 1) 3(2 1) (2 1)

d n l n d n

n g n g n g

sin( ) cos( )cos 0

l l d

g g g

2 ( 1)sin( ) 2 (3 2) 2 ( 1)cos( )cos 0

(2 1) 3(2 1) (2 1)

d n l n d n

n g n g n g

sin( ) cos( )cos 0

l l d

g g g

2 (3 2) 2 (3 2)cos

3(2 1) 3(2 1)

l n l n d d

n g n g d dt

2

0

(3 2)sin sin

3(2 1)

l n

n g

00

0

3(2 1) 1 1( ) 2(2 ( 2 ) 4 )( )

(3 2) 4 4 6sin sin

n g dt i F k

l n

20 1 sin

dF m

m

2k m F m

00

0

3(2 1) 1 1( ) 2(2 ( 2 ) 4 )( )

(3 2) 4 4 6sin sin

n g dt i F k

l n

3

Here, inverse function of t is given. Am represents Jacobi Amplitude, and sigma is an expression extracted

from the function. Cn, sn, and dn represents for Jacobi Elliptic Function. Here is the comparison of the figure of

angle theta. The blue line now represents numerical solution of all the equations, and orange line represents

analytical solution. The margin of error is acceptable.

Jacobi Amplitude:

:

Jacobi Elliptic Function:

1 ( )( ) ( 4 4 )

2 4(3 2)

tt am

n l

( )

3 (3 2)(2 1) ( ) 4

2 (1 2 )

(3 2)t

n li g n dn t

g n

n l

2 2

2( )

3 ( 3 2)(2 1) ( ) 4 ( ) 4

(3 2) (1 2 )t

g n n cn t sn t

l n g n

,u F m am u m

(3 2) 1( ) 6 (2 1) 2(3 2)

(1 2 ) 4

n lt i g n t n lk

g n

coscn u m

21 sindn u m m

sinsn u m ( )am u m

4

1.3.3More Precise Analytical Solutions

In order to keep the term d in the function, we conserve the second and forth terms in the equation.

Utilizing the expression of t got before with only the first and third term, phi dot and phi dot-dot can be roughly

estimated by the expression over there. Set eta equals to a constant, we can separate the variables and integrate

the equation. F of theta now is also an expression extracted from the integral.

SET:

THEN:

,

Unfortunately, F of theta is a nonintegrable function; Taylor Expansion is utilized here to simplify the

function in the form of the summation of theta of different orders. The expression of theta can be got from the

inverse function of t. From the figure of the Cartesian coordinates x and y of the second stick, an appropriate

approximation can be drawn that the second stick and sticks above do not move along the x-axis. From the

equation over there and plugging theta, the expression of phi is put below. The figure now gives curves of phi.

The margin of error is also acceptable.

TAYLOR EXPANSION:

φ(t)=

2 ( 1)sin( ) 2 (3 2) 2 ( 1)cos( )cos 0

(2 1) 3(2 1) (2 1)

d n l n d n

n g n g n g

0

( )t

( )t

0

( )t

( )t

00

0

3(2 1) 1 1( ) 2(2 ( 2 ) 4 )( )

(3 2) 4 4 6sin sin

n g dt i F k

l n

2 2

0 0 0 02 ( 1)sin( ) 2 ( 1)cos( )

(2 1) (2 1)

d n d n

n g n g

0

0 0 0

( )

0

,( )

( ) ( ) (( ) )!

nn n

n

FF o

n

2 2 2

0( sin ) ( cos -lcos ) =dd l

2 (3 2)cos

3(2 1)

l n

n g

0 0

2(3 2)sin sin ( )

3(2 1)

l n

n g

0

0 0

,1

( )sin sin ( )

F

0

0,

3(2 1)*( ) ( )

(3 2)

n gt F d

l n

5

A comparison of theoretical figure and the situation in experiment is drawn here. It is easy to discover that

theory fits well the reality.

1.3.4 Unilateral Constraint Integrated in Analytical Solutions

Comparing the figures respectively shows theta and the acceleration of theta, one can discover that before

theta turns to 0, there is an interval that the acceleration of theta is smaller than 0, which is a unilateral situation,

because there will not be a push force along the string.

6

Because the strings are flexible, utilizing the function of absolute value and the variable x, it is able to

shield the part of the acceleration that is smaller than 0. Integrating the modified acceleration will render an

expression of theta containing unilateral constraint, because the part of the acceleration which is smaller than 0

is shielded and the integral will only contain the torque created by gravity, thus deleting the constraint of the

string. However, the function is nonintegrable and its derivative is discontinuous because the existence of

absolute value, so Taylor Expansion cannot be utilized here. Fourier Series can be applied here. The same as phi

here. Here is the integral.

THEN: Fourier Series

1.3.5 Max Extra Velocity Derived from Analytical Expressions Utilizing partial derivatives of analytical expressions of the vertical velocity of the center mass of the

second stick and the negative definite matrix consisted of second order derivatives, a set of variables of

optimum option can be set to determine the max extra velocity that sticks above can have in every rotating

process. Utilizing the concrete values of the variables got here will render the maximum velocity when two

devices are released at the same height.

1.4 Four degrees of freedom system

Setting more generalized coordinates can embody the rotation of the second short string. Taking the third

stick and sticks above as a whole, we render a more accurate set of equations involving four degrees of freedom.

1( ) ( ) 0, 0

2F x x x x 2 , 0x x

cos ( ) cos ( ) cos ( )1 3 3 3* *

2 2 2 2

g t g t g t

l l l

2 2 2 21cos cos sin sin cos cos sin sin

2l d l d g l d l d g g

( )t dt dt 2 2x dt x

2

0

0x

2 0x

l

2 0x

n

2 0x

d

2 0

x

t

2

2

2

2

2

2

2

2

2

2

2

2

0

2

2

2

x

n

x

l

x

d

x

x

t

( )n

ikt

k

k n

t C e

1

( )2

ikt

kC t e dt

7

0( , 0.1)6

l

*f N

*2

0

2

1sin( ) 1 ( )

21

12

l N t dt

ml

'

0

1sin

2Axv l

Lagrange Equations

Here is the comparison of the figure got from theory and experiment. The angle beta fits well with reality

when theta is equal to 0 degree.

1.5Concrete Analysis on Collision

Utilizing the theorem of momentum and the theorem of angular momentum, the critical condition of

sliding of coefficient mu is determined by judging the absolute velocity of the contact point A. When mu is

greater than critical value, it keeps static and remains that value responding to the need.

0E L

0E L

0E L

0E L

* ( )Cx

N t dtv

m

*

1arctan

8

( )N t dt ( )N t

When mu is less than the critical value, utilizing the conservation of energy, the theorem of momentum and

angular momentum, and the expression of energy decrement can render a set of equations. After appropriate

substitution of several variables, solving the equations will give the value of all the unknowns. The energy

decrement related to elastic coefficient e can also be calculated by the expression put below.

SET:

THEN:

When mu is greater than the value, similar equations can be drawn except the deletion of the decrement

expression of energy, due to the absent of sliding.

SET:

3 3

7

2 2 2 2

0

1 1 1 1( )

2 2 2 12my e E m x y ml

( )N t dtx

m

0

( )N t dty y

m

2 2

0

1 1sin( ) 1 ( )

2 12l N t dt ml

'

0

1sin ( )

2Ax Axv v x l x N t dt

2

0 0

1[3sin( ) 1 sin ]

m

( )[ ( ) ] ( )N t N t dt dt N t dt d

3 3

7

2 2 2 2 2 2

0

1 1 1 1( )

2 2 2 12my e m x y ml

' ( )N t dtx

m

0

( )N t dty y

m

'2 2

0

1 1sin( ) 1 ( )

2 12l N t dt ml

'

1arctan

' 3 3

7

( ) ( )[ ( ) ]AxE N t v dt N t N t dt dt

2 1 1 2

1 1 2 2 1 1 2 2

( )v v e u u

m u m u m v m v

2 21 21 2

1 2

1(1 ) ( )

2

m mE e u u

m m

2

2 2

1 1 20

1lim (1 ) ( )

2mE e m u u

9

Collision with horizontal velocity can be separated into two process. The first procedure turns the absolute

velocity of point A to 0 by both momentum and angular momentum. Solving the equation of final condition will

render the total impact of the first process.

From to

SET:

GET:

SET:

THEN:

The other part of the collision is the same as the normal contact part because the absolute horizontal

velocity of point A is 0 initially at the second process. Combining two impulses of each process, the velocity

variable can be derived by the total impulse.

1

1

00

( )t

Cx x

N t dtv v

m

1arctan( )

1 2 2

0 0

1 11 ( ) sin( )

2 12

tN t l dt ml

1 1

'

0

1sin 0

2Ax Cx Cxv v l v

1 2

0 0 0 0

1( ) [ (3 1 sin( )sin )]

t

xN t dt vm

1

0 1( )t

N t dt

01 '

xv

' 2

0 0

1(3 1 sin( )sin )

m

0Axv

0Ax xv v 0Axv

1 1

'

Ax Cxv v

10

2

1

2 1

( )t

t

Cx Cx

N t dtv v

m

1.6Unilateral Constraint Integrated in Lagrange Equations

After the concrete theoretical analysis on collision, we utilize Lagrange multiplier to integrate unilateral

constraint into Lagrange Equations. By using a similar procedure of calculus of variation for deriving normal

Lagrange Function, a set of equations involving L Prime is given. In addition, KKT Condition of linear

complementarity is also valid in these Equations.

THEN:

KKT Condition：

1.7 Eight Degrees of Freedom of the System

After all the analysis above, a final model is set here, involving 8 degrees of freedom. Two degrees of

freedom have something to do with unilateral constraint, representing the extension of the string, and two degrees of freedom are related to collision. Utilizing KKT Condition to evaluate the value of lambda, 8

equations are given. Because x is a motion integral now, Routh Function can help to decrease the complexity of

the equations.

2 1

' '

0

1sin

2Ax Axv v l

2

1

2 2

' '

0

( )1sin

2

t

t

Ax Cx

N t dtv v l

m

'( , , )S L q q t dt ' '

( )L L

q q dtq q

' 'L Lqdt d q

q q

' ' '2

1

0

t

t

L L d Lqdt q q dt

q q dt q

' '

( ) 0L d L

qdtq dx q

'

1

( )a

L L f q

' 0sE L

1

a

s sE L f

( ) 0f q

0

1

( ) 0a

f q

11

0d R R

dt dq q

Routh Function：

1.8Final Extra Velocity Final velocity can be calculated by separating the system into rotating procedure and free fall process. The

total amount of extra velocity can be evaluated by adding all the effects circularly.

As mentioned before, the falling procedure is made of two process. Accelerating process and free fall

process are intersect with each other. After each accelerating process, Mathematica will plot a figure showing

the end state of the system. Utilizing them as initial conditions for free fall, one may calculate the next set of

initial values for the accelerating process. Final extra velocity can be calculated in this form of loop.

1.9 ADAMS Simulation After deriving the theory of critical value, the simulation in Adams has been drawn. We can discover from

the figure that when μ is small, the ratio of vertical and horizontal velocity is almost equal to μ; when μ increase

upper than a special value, the ratio of the vertical and horizontal velocity keeps the same, which is the critical

value calculated by us. Due to some assumptions made by us, the result got from the theory has the margin of

error of 4.3 percent.

We also compare the velocity calculated from theory to the result from Adams. Here is the figure of the

horizontal velocity of the stick, which is almost be 0; so we mainly consider the vertical velocity of the stick.

The margin of error is 4.01 percent.

1 1( ) 0f q q 2 2( ) 0f q q

1( ) 0q t 1( ) 0q t

2( ) 0q t 2( ) 0q t

1( ) 0q t 1 0

2( ) 0q t 2 0

0E L

0E L

0E L

0E L

0xE L

0yE L

1 1

1

a

q qE L f

2 2

1

a

q qE L f

( , , , , )x xR x p q q p x L x x

L LdR p dx xdp dq dq

q q

x

Rx

dp

x

Rp

dx

L R

dq dq

L R

q q

( , , )x

x

R p q qx dt

dp

0.2653

dVx/dVy=0.2639 dVx/dVy=0.6

12

0.8

dVx/dVy=0.71 dVx/dVy=0.71

Vx of last stick’s velocity

Vy of last stick’s velocity

13

CHAPTER2 EXPERIMENT &ABAQUS ANALYSIS

The Chapter 2 provides the manufacture process with facilities, tools for measurement and the experimenting

objects. As the phenomenon's nuance and stringent specification of precision, the high- accuracy of making

template and measure tools is indispensable, so the 3D printer has been used. In respect of influential factors,

we have measured the property and variables of different fast chains by methods introduced below, and studied

about the accelerated effect with CCD. ABAQUS simulation is also described in this section, and we have gave

analysis and made fittings for the results. Lastly, a final conclusion can be drawn considering experiment and

modeling.

2.1 Facilities And Objects

2.1.1voice operated switch Since the stable releasing state is mandatory to our experiment, we assembled a voice operated switch with

SCM techs. The switch can be divided into five parts: two monitors, one SCM main control panel, one sound sensor, one battery box and assembling parts. When the intensity of environmental sounds exceed the threshold value we set, the chain will be released at once.(SCM programming language in appendix 2)

2.1.2Major Facility

Facility is a prerequisite for experimenting process and measurement. We welded several 5mm×5mm steel tubes to install an experiment facility and stabilize it..The height of the releasing position can be measured by gradations(0-115cm) attached to the left side of this facility. And the fixator can change initial altitude by sliding up or down. The aim for employing EVA shock pad under the entire facility is to reduce the damage of rope’s structure after falling.

2.1.3 Sticks and Assembly Aim to accord with the theoretical requirements, sticks should satisfy the conditions

below : 1.All sticks are identical to each other; 2.Ropes must be constraint to be vertical to the ground ; 3.The points of acting force and the stick's centroid are in the same plane;

Considering the above conditions, we utilized 3D printer(the min precision is 0.4mm) to make sticks with PLA material.

14

d

With the help of lathe(The min precision is 0.001mm) and UG(3D painter), we manufactured a making template from abrasion-resistant composite coating（plan conception and objects are shown below）. Fast Chain parts can be manufactured with above conditions in a randomly assembled way.

2.2 Measure Methods

2.2.1Velocity Measurement On account of the quick falling process, the velocity of fast chain system’s last stick’s centroid can’t be

measured specifically with photogate, so we used computer-controlled display(CCD) whose frequency is 240

shooting times per second to experiment with different fast chains. We marked the centroid of last stick with

black logo for identification. After taking process, the photos are input into Matlab to get the final outcomes.

Variables：

Δh: distance from releasing position to ground;

f :frequency of CCD

d: distance of the displacement of stick’s centroid during the time of 1/240s;

g: gravity acceleration in Qingdao:9.7984m/s^2;

V1:vertical velocity of last stick’s centroid when Fast-Chain falls Δh to the ground;

V2:vertical velocity of last stick’s centroid when Fast-Chainfree fall Δh;

Solving equation:

V1=d1f

V2=d2f

15

2.2.2 Materials' properties

Variables:

μk：static friction coefficient of stick’s surface;

ψ：angle of slope；

H1: initial height of falling;

H2: final height of rising;

e:restitution coefficient of stick parts;

2.3.2.1 Static friction coefficient

Static friction factor of the stick’s surface measured by the angle between slope and the horizontal, we

elevated the slope until the stick just begin to slither .We recorded the value of angle and substituted in the

formula below:

μk=tanψ (3) 2.3.2.2 Restitution coefficient of stick parts

We hanged a homogenous PLA ball with rope and then cut the rope down. After the ball bounded against the

ground, we recorded the highest position of rising process. Formula as shown below:

e= H2/ H1 (4)

2.3 Experiment

2.3.1 Study on the influence of air resistance 2.3.1.1

Guess: If the phenomenon of “Fast Chain” is influenced by aerodynamic factors or stick’s structure?

2.3.1.2 Making: we used 3D printer to make two chains with material PLA in order to expand the influence of air.

Then we let them fall through the air instead of impact. Except the angle between the stick to the horizontal, the

other factors are kept the same. One of the angles is 30°，and the other is 0°.When the process going on, we

closed the windows, doors and air-condition to prevent the air flowing. The distance from releasing position to

measuring position is90cm.

2.3.1.3 Experiment and data forms:

V1(m/s): 4.175(30°) 4.186（parallel）

16

2.3.1.4

Conclusion：Through the final V of two chains, we can see that the values are almost same, so the

aerodynamic factor is not a main influential factor to this problem. 2.3.2 Actual experiment database

We used the method in 2.2 to experiment with Fast Chains and studied about different variables, then we

made comparisons to the free-fall velocity when under the same circumstance. The forms below show the

results.

(Remark: Each T average value below is the average of the 30-time measure results after expelling two

highest and lowest numbers)

According to the experimental forms,we calculated that the average of variances is 0.5636361. It shows the

stability of calculating outcomes. We used three sets of variables in experiments which have the lowest variance

values to construct models in ABAQUS for verification:2, 3, 6.

2.4ABAQUS Model

2.4.1 Cause of using ABAQUS Owning to the complex conditions of the actual experiment, the experimental results can not be defined as

absolutely precise. To dispel the influence of air resistance and other irrespective factors, we used software

ABAQUS to design a accurate model. The analysis method of ABAQUS is FEM(Finite Element Method ),a

mathematics technology which aims to get answers from partial differential equation. It assumes a proper,

simple and approximate solution to each unit element, then calculates the whole model’s satisfied conditions. So

the solution getting from this method is approximate. But the complex question can be solved through this way

fast and accurate. Because we tried to make a comparison of the Fast Chain's V1 andV2, and omit

thermomechanical effect and mechanics of material, so the calculated values of the two velocities is correct.

And they can be specified when we assign n,l,d and θto chain parts.

2.4.2Modeling process Considering the calculation efficiency and numerous experimental objects, simplifying query should not be

dismiss. After plenty of failed attempts about homogeneous solid element, we found that Truss model element

can accomplish this simulation, and refining the mesh of wire Truss part is the most effective method to imitate

the rope. Stick element can also set as wire instead of solid part to improve calculating efficiency.

To imitate the property of rope, the relation between the extreme points(A, B) of neighbouring sticks must be

restrained as following:

1.The line AB must be defined as a stretch-proof property, and the time of force

conduction must be limited to be exceedingly short ;

2.If there is compression on A or B, the force in line must disappear in an extremely

short time, too. And the restrain also loses effect.

3.The rope’s mass must set minimal compared to sticks.

After re-manufacture, the total CPU calculating time decreased to 719.4s, 0.0009515

times as short as homogeneous solid element:

θ=45° d=3cm，θ=30° d=3cm，θ=15° d=3cm， θ=30° d=3cm， θ=30° d=3cm， θ=30° d=3cm， θ=30° d=3cm

Variables n=8 l=11cm， n=12 l=8cm，n=10 l=11cm， n=8 l=8cm， n=10 l=11cm， n=10 l=8cm， n=9 l=14cm

Δh=80cm Δh=80cm Δh=54cm Δh=60cm Δh=70cm Δh=65cm Δh=80cm

Freefall (V2m/s)3.957752 4.19784421 3.2214 3.4275356 3.702 3.799 3.9578

Actual( V1m/s 4.22757 4.4878 3.451 3.62776 3.97532 3.6853 4.338

D—value(m/s) 0.2698 0.28996 0.2287 0.2 0.27311 0.231738 0.38

Variance 0.5639 0.4754 0.4593 0.64886 0.59788 0.3819 0.784

Step Increment Total CPU Step Stable Time Time Time Time inc

before 214476 0.0123208 480 0.0123208 5.72831e-08

after 1000063 0.40 719.4 0.40 4.01732e-07

17

Through the Time History and field output data images, the chain surely falls faster when it dropped onto a

horizontal surface.

2.4.3 Simulation of actual conditions In the ABAQUS modeling process, we constructed the Fast Chain with deformable wire planers, and

simulated impact surface with 3D deformable solid. We measured the property values with methods in 2.2 to

insure that the models are same to the reality. And we referred to the reference for other standardized values .

During the modeling process, we assigned Beam section to the stick parts, Truss section to rope parts, and

Homogeneous Solid to impact surface. Aiming to reduce the analysis time increment,we chose Dynamic

Explicit as analyzing step, the accuracy can also increase with this kind of step. And we set a Boundary

Condition which is used for locking impact surface's degrees of freedom in six. Considering the special feature

of ropes and the analysis efficiency, the global size of Truss mesh is set as 3. Which means the 120mm long

rope is divided into forty 3mm long elements connected by hinged joint. Here we used the three lowest VAR values of experiments to construct models in ABAQUS for verification.

Then we outputted the V2 and vertical displacement images of the last sticks' mass center.

The first initial conditions as follow: θ=30°，d=3cm，n=10，l=8.2cm，Δh=65cm. The V1in experiment is

3.6853 m/s，the result in ABAQUS is 3.6878m/s.

The second initial conditions as follow: θ=15°，d=3cm，n=10，l=11.4cm , Δh=54cm.The V1 in experiment

is 3.451 m/s，the result in model is 3.458m/s.

The third initial conditions as follow: θ=30°，d=3cm，n=12，l=8.2cm，Δh=80cm.The V1 in experiment is

4.4878 m/s，the result in model is 4.790m/s.

Contrasting to experiment, all the results in ABAQUS model are a bit larger. The average margin of error

between model and experiment is 2.33483%，and the min value is 0.067837%.

According to these sets of the comparative trial, the results are similar to an overwhelming extent. However, the dissimilarity still exists due to the deviation in the process of property measurement. The air

resistance is also a decisive factor of the differences. But the order of error's magnitude is about two decimal

Stick’s density(t/mm^3) 1.25E-009 Rope’s density(t/mm^3) 2.03E-011

μ 0.2653 PLA’s Young modulus(Mpa) 3500

L(m) 0.14,0.11,0.08 S(mm^2) 1×1

Stick's Poisson Ratio 0.4 Rope’s Poisson Ratio 0.05

Rope’s Young modulus（Mpa）10000

18

1( , , )nP a a1( , , )nQ b b

1 1 1( , , )n n nM Q P c b a c b a

places (0.01),we supposed that stick's deformation and material's property are inconsequential in terms of these

dissimilarities. 2.4.4 Additional analyze on model

We used the model with the initial conditions：θ=30°，d=3cm，n=10，l=8.2cm，Δh=65cm to analyze.

Convenient for describing, here we lead into three number assemblages:

The value of n equals to the declining sticks' order. Here is 1-10. The physics meaning of the elements in P

shows the time nodes of each stick when it just impact the surface, and which in Q represent the time of each

stick just finish rotation process, M shows the time period of each stick's accelerated process. Each two

adjoining images in the first picture above displays the displacement of each stick's extreme points. We can

attain the values in P, Q and M through the image's turning point. We calculated and outputted the center's stress tendency chart of the few topper short ropes and long ropes

(the images as above),here are the conclusions: 1.The tendency of stresses in ropes is in the shape of wave curve, and the start times of each wave have the

same value in assemblage P，the cycles are almost equal to the element in assemblage M； 2.The lower the rope's initial position is (104,363,371), the bigger the amplitude of stress fluctuation is ,

and the shorter time period last. If the initial position of rope is higher(43-1,367),then the situation become

opposite, and the waving process keep lasting to the end； 3.As time goes on, the amplitude of stress fluctuation (43-1,367) keep dying down. This phenomenon

shows that the longer the falling displacement is, the less effect of acceleration sticks provide. This situation

accord with the theory in Chapter 1； 4.The output pictures display the vertical displacement and V1 of the last stick's mass center compared to

free-fall. We discovered that the gradient of the V1 became cliffier and cliffier as time going by, then got placid

when near the end time. It might be caused by the effect of acceleration reduction and impact interreaction

between sticks.

19

2.5 experiment on α, β, θ, φ of six degrees of freedom system

At theoretical analysis section, we have discussed about α, β, θ, φ of six degrees of freedom system:

To verify these results(d=0.022m,n=11,l=0.114m,θ0=30°),we set the same conditions of fast chains and

used CCD to track the sticks’ path, getting angular displacement. In the section of theory, when θ=0°,α=146°，β=96°, φ=117°.With the measure of tracker: α=149.1°，β=95.4°, φ=117.2°,which can prove that the

theory is better accord with experiment.

2.6 ABAQUS modeling on different variable sets and data fitting

In this section, we have studied about the influences on different variables of fast chain and made fittings

about n,l,d,θand μ.We changed these variables one by one when the other variables keep constant.The picture

below shows all chains manufactured by ABAQUS part function.

20

B30-10-10-1, 0.481

A30-10-10-2, 0.75

C30-10-10-3, 0.99

V3010-10-4, 0.83

T30-10-10-5, 0.65

Y30-10-10-6, 0.41

0

0.2

0.4

0.6

0.8

1

1.2

B30-10-10-1 A30-10-10-2 C30-10-10-3 V3010-10-4 T30-10-10-5 Y30-10-10-6

D-value

（m/s）

d(cm)

Convenient for describing, the objects in experiment are named like : A(numbers of part)-θ- l-n-d.All

outcomes areshown in the forms below:

d B30-10-10-1 A30-10-10-2 C30-10-10-3 V30-10-10-4 T30-10-10-5 Y30-10-10-6

V1(m/s) 2.92 3.26 3.626 3.66 3.91 3.9

V2(m/s) 3.401 4.01 4.625 4.49 4.56 4.31

D-values(m/s) 0.481 0.75 0.99 0.83 0.65 0.41

Θ S5-10-10-2 J10-10-10-2 H15-10-10-2 W20-10-10-2 X25-10-10-2 A30-10-10-2 I45-10-10-2

V1(m/s) 2.049 2.39 2.54 2.94 2.958 3.26 3.97

V2(m/s) 2.43 3.15 3.07 3.57 3.728 4.01 4.91

D-values(m/s) 0.381 0.76 0.53 0.63 0.77 0.75 0.94

L K30-1-10-1 D30-5-10-1 B30-10-10-1 E30-15-10-1 Z30-20-10-1

V1(m/s) 2 2.218 2.92 3.61 4.146

V2(m/s) 2.04 2. 409 3.401 4.91 5.768

D-values(m/s) 0.04 0.191 0.481 1.3 1.622

n M15-10-5-2 H15-10-10-2 N15-10-15-2 O15-10-20-2 Aa15-10-25-2

V1(m/s) 1.81 2.54 3.12 3.8 4.1

V2(m/s) 2.24 3.07 3.81 5.15 5.284

D-values(m/s) 0.43 0.481 0.69 1.35 1.184

μ H-0 H-0.2653 H-0.5 H-0.8 H-1.0 H-1.2

V1(m/s) 2.54 2.54 2.54 2.54 2.54 2.54

V2(m/s) 3.55 3.07 3.1 3.2 3.196 3.15

D-values(m/s) 1.01 0.481 0.56 0.66 0.656 0.61

We made image analysis which consists of these results and forecast the trend of each variable’s influence，as shown in black line.

A few conclusions can be drawn from these fittings:

1. According to the fitting trend of the first variable, n, the D-value has positive correlation with sticks’

number (n). As n increases, the incremental gradient of fast chain’s final velocity becomes larger and larger.

Which means that the more sticks the system consists of, the more obvious the accelerated effect is. And the

tendency of variable l is the same as n.

2. When the friction conditions between sticks are set as frictionless, the chain moves faster than which

under friction force. The moment of friction force will reduce the stick’s angular velocity and the speed of

extreme points when they get strained. We also supposed that if the surface can not provide the friction as large

as the demand of impact, a part of total energy will lose due to the micro slip of sticks. Once the friction

satisfies the demands, the accelerated effect will rise and keep in a stable level.

3. Generally considering all the results from fitting of Θ, the D-values also positivly change as Θchanges.

4. With the increase of d, the D-value first increases and then decreases.

21

S5-10-10-2, 0.381

J10-10-10-2, 0.76

H15-10-10-2, 0.53

W20-10-10-2, 0.63

X25-10-10-2, 0.77

A30-10-10-2, 0.75

I45-10-10-2, 0.94

0

0.2

0.4

0.6

0.8

1

S5-10-10-2 J10-10-10-2 H15-10-10-2 W20-10-10-2 X25-10-10-2 A30-10-10-2 I45-10-10-2

D-value

（m/s）

Θ(°)

K30-1-10-1, 0.04

D30-5-10-1,

0.191B30-10-10-1,

0.481

E30-15-10-1, 1.3 Z30-20-10-1,

1.622

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

K30-1-10-1 D30-5-10-1 B30-10-10-1 E30-15-10-1 Z30-20-10-1

D-value

（m/s）

l(cm)

H-0, 1.01

H-0.2653, 0.481H-0.5, 0.56

H-0.8, 0.66

H-1.0, 0.656

H-1.2, 0.61

0

0.2

0.4

0.6

0.8

1

1.2

H-0 H-0.2653 H-0.5 H-0.8 H-1.0 H-1.2

D-value

（m/s）

μ

M15-10-5-2 , 0.43 H15-10-10-2, 0.481N15-10-15-2, 0.69

O15-10-20-2, 1.35

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

M15-10-5-2 H15-10-10-2 N15-10-15-2 O15-10-20-2

D-value

（m/s）

n

22

2 2 2

1 2 ( )ln 6.8V V V kl n d d

Without considering μ,we created a approximate equation to describe all the data. After testing, theaverage

margin of error of this description is 6%. (k≈2.1391905e-6):

As the lack of experiment apparatus, some variables such as stress or angular velocity can't be measured.

With the help of Dynamic Explicit analyze step in ABAQUS, we can know more details about the chains.

Neglecting air resistance and considering the above images and results at an overall scale, our ABAQUS model

is overwhelmingly similar to the experiment. And we can change initial conditions to get more different chains

instead of manual building method whose efficiency is comparatively low. The toughest problem of this

construct process is the rope feature's simulation. We have tried many times and search plenty of references, but

there are nothing about the simulation of flexible ropes. Finally, we completed the imitation of ropes with Truss

element by ourselves, and the experiment results accord with it. This method can also apply to other projects

about rope. With the comparison of CCD measure results and ABAQUS modeling, we finally get a descriptive

equation by and large.

(when μ=0.2653)

0s 0.175s 0.245s 0.315s 0.35s 0.385s

23

CHAPTER3 Analyze about deviations

According to the slow motion payback in the actual experiment and Time History in ABAQUS mode，we

found that some unmeasured and unspecified factors have caused disregard influence to the final results. The

quantitative analysis of these factors are impossible, so we make qualitative analysis：

1.After A stick upspring, it will reduce the speed and start

rotating. But B stick which over A still have A’s velocity that before

upspringing, the D-value between their speed may cause A and B

impact if the d is short. The up impulse will reduce the higher stick's

acceleration effect. One after another ,the situation must cause the

system’s energy dissipation.

2.No mater in ABAQUS model or in reality experiment, the ropes and

sticks are not idealized. When the ropes or sticks get winding and

twining, the system may out of plane X-Y because of the interferential

force. Due to the randomness of sticks and ropes’ winding, the precise position of where they impact can not be

specified.

3.The positions of the sticks finish falling are uncertain ,so the moving sticks might graze

static sticks before colliding. Which also cause additional energy losing and acceleration

reducing.

4.The experiment was going under the static-air environment. However, the system probably

influenced by the spontaneously air-turbulence when it falls. And we can't experimentalize in

vacuum owing to the size of facility, so the air-resistance is not able to omit.

5.The perfect states such as smooth edge, massless rope and completely homogeneous

material are proved unpractical, so the energy may dissipate more on account of the intrinsic

energy, extra function and tiny vibration.

After analyzing about these unmeasured and uncertain factors，we discovered that all the

factors can make a difference which reduce the system's general energy on the final velocity.

So the theoretical results are larger than which in reality is normal and absolutely definite.

The margin of error is 5.7%

Conclusion The special inclined sticks system discussed in this paper can be solved by analytic mechanics. Lagrange

Equations are able to conceal the complicated forces along the strings, and reduce the degrees of freedom of the

system according to the constraints. SCM techs and the utilization of lathe ensure the stability and precision of

our experiments, while constructing model made byABAQUS further verify our derivations and experiments.

One convincing thing is that our theoretical analysis, experiments, and modeling are finely fitted to each

other, as well as consistent with real situation.

Though approximation like Taylor Expansion and precise abbreviations of secondary factors are made, the

model we constructed still has some deviations compared to software simulation or reality. A precise solution of

collision and unilateral constraint still need to be further researched. Acknowledgments

We would like to show our appreciation for Qingdao No.2 Middle School for the laboratory and facilities

they provided. Some measure tools of experiment can’t be used without their help. The working conditions they

offered is comfortable and sufficient for resources. We will also never forget the indication from Mr. Liyong Jia,

which is working at the First Aircraft Design And Research Institute in China now. He has given us a few clues about ABAQUS modeling.

θ=30°，d=3cm， θ=30°，d=3cm，

Variables n=10 l=11.4cm， n=10 l=8.2cm，

Δh=70cm Δh=65cm

V2 in theory(m/s) 4.1651 3.942

V2 in experiment(m/s) 3.97532 3.6853

24

REFERENCE:

[1]ABAQUS6.11 中文版有限元分析从入门到精通/马晓峰编著.-北京：清华大学出版社，2013.2（CAX 工

程应用丛书）ISBN 978-7-302-31077-8

[2]朗道《力学》解读/鞠国兴编著.——北京：高等教育出版社，2014.7（2016.4 重印）ISBN 978-7-04-039945-5

[3]力学：第 5 版/（俄罗斯）朗道，（俄罗斯）粟弗席兹著：李俊峰，鞠国兴译校.——北京：高等教育出

版社，2007.4（2016.1 重印）ISBN 978-7-04-020849-8

[5]ABAQUS 动力学有限元分析指南作者：张文元费红姿连慰安丁玉坤中国图书出版社 ISBN

988-98508-1-8

[6]有限云分析中的材料性能单位.pdf

[7]ABAQUS-实例分析，球的下落谈起问题.pdf

[8]ABAQUS 软件版本 windows 64.2016

[10]Mathematica 软件版本 10.3

[11]http://wenku.baidu.com/link?url=Z14kQRrsnEAcaDt6nf8pa7oMN3-ViaP4JQ8yxhQ64IjLArjzrQwcAfVM6F5L

zzC1jTvvsDHCVp06hfV0F7JR085RUp3rBI6dtaJK3w_QjvO

[12]http://wenku.baidu.com/link?url=a2pHz2ye-0b7q3RHEVARbI_7fm8HyBGksaNHx0SfnYFMR8Q9N3Yax_Jq4

_8M-13vIQcRyfBAg0OFQxIK7WbcYX6TAD59WbGEW1FtSKqEjvW

[13]http://tieba.baidu.com/p/3484186817

[14]http://wenku.baidu.com/link?url=gXcKx94X9Hd1-PpuzAD4ZW0e5Z6o-yZ_YqV7kpT5np40o11mMGo1m_tw

-vnTnF829N08ERXMDEUXycjrtConHMezCkJQhiAx6zmx8N5Tqs7

[15]http://v.youku.com/v_show/id_XNDIzODAwMTY=.html?f=3744431&from=y1.2-3.4.7

[16]http://reference.wolfram.com/language/tutorial/DSolveSecondOrderPDEs.zh.html

[17]http://www.docin.com/p-459594352.html

25

Appendix

Appendix 1

The ABAQUS programming language of the chain whose initial condition is θ=30°，d=3cm，n=10，l=8.2cm，Δh=65cm as below:

*Heading ** Job name: new Model name: Model-1 ** Generated by: Abaqus/CAE 2016 *Preprint, echo=NO, model=NO, history=NO, contact=NO ** ** PARTS ** *Part, name=Part-4 *Node 1, -100., -100., 20. 2, -100., -80., 20. 3, -100., -60., 20. ...... 240, 100., 60., 0. 241, 100., 80., 0. 242, 100., 100., 0. *Element, type=C3D8R 1, 23, 24, 35, 34, 1, 2, 13, 12 2, 24, 25, 36, 35, 2, 3, 14, 13 3, 25, 26, 37, 36, 3, 4, 15, 14 ......... 98, 228, 229, 240, 239, 206, 207, 218, 217 99, 229, 230, 241, 240, 207, 208, 219, 218 100, 230, 231, 242, 241, 208, 209, 220, 219 *Nset, nset=Set-1, generate 1, 242, 1 *Elset, elset=Set-1, generate 1, 100, 1 ** Section: Section-3 *Solid Section, elset=Set-1, material=earth , *End Part ** *Part, name=new-3 *Node 1, 195.333633, 710., 0. 2, 266.347809, 710., 0. 3, 266.347809, 638.999939, 0. ...... 395, 195.333633, 124., 0. 396, 195.333633, 130., 0. 397, 195.333633, 136., 0. *Element, type=B31 1, 1, 23 2, 23, 24 3, 24, 25 ...... 354, 355, 356 355, 356, 357 356, 357, 22 *Element, type=T3D2

26

14, 3, 35 15, 35, 36 16, 36, 37 ...... 404, 395, 396 405, 396, 397 406, 397, 17 *Nset, nset=Set-1 1, 2, 3, ......355, 356, 357 *Elset, elset=Set-1 1, 2, 3, ......354, 355, 356 *Nset, nset=Set-2 1, 2, 3, ...... 395, 396, 397 *Elset, elset=Set-2 14, 15, 16, ......404, 405, 406 *Nset, nset=Set-3 1, 2, 3, ......355, 356, 357 *Elset, elset=Set-3 1, 2, 3, ......354, 355, 356 ** Section: Section-1 Profile: Profile-1 *Beam Section, elset=Set-1, material=timber, temperature=GRADIENTS, section=RECT 6.3, 6.9 0.,0.,-1. ** Section: Section-2 *Solid Section, elset=Set-2, material=chain 0.01, *End Part ** ** ** ASSEMBLY ** *Assembly, name=Assembly ** *Instance, name=Part-4-1, part=Part-4 50.000000133619, 0., 0. 50.000000133619, 0., 0., 51.000000133619, 0., 0.,

90 *End Instance ** *Instance, name=new-3-1, part=new-3 -195.333636, 0., 0. *End Instance ** *Instance, name=new-3-1-lin-2-1, part=new-3 75.680564, 0., 0. *End Instance ** *Nset, nset=Set-2, instance=Part-4-1, generate 1, 242, 1 *Elset, elset=Set-2, instance=Part-4-1, generate 1, 100, 1 *Nset, nset=Set-5, instance=new-3-1, generate 1, 397, 1 *Nset, nset=Set-5, instance=new-3-1-lin-2-1, generate 1, 397, 1 *Elset, elset=Set-5, instance=new-3-1, generate 1, 406, 1 *Elset, elset=Set-5, instance=new-3-1-lin-2-1, generate

27

1, 406, 1 *End Assembly ** ** MATERIALS ** *Material, name=chain *Density 2e-11, *Elastic 15000., 0.1 *Material, name=earth *Density 7.9e-09, *Elastic 21000., 0.1 *Material, name=timber *Density 6.6959e-10, *Elastic 10000., 0.33 ** ** INTERACTION PROPERTIES ** *Surface Interaction, name=IntProp-1 *Friction 0.2653, *Surface Behavior, pressure-overclosure=HARD ** ---------------------------------------------------------------- ** ** STEP: Step-1 ** *Step, name=Step-1, nlgeom=YES *Dynamic, Explicit , 0.5 *Bulk Viscosity 0.06, 1.2 ** ** BOUNDARY CONDITIONS ** ** Name: BC-1 Type: Symmetry/Antisymmetry/Encastre *Boundary Set-2, ENCASTRE ** ** LOADS ** ** Name: Load-1 Type: Gravity *Dload Set-5, GRAV, 9798.4, 0., -1., 0. ** ** INTERACTIONS ** ** Interaction: Int-1 *Contact, op=NEW *Contact Inclusions, ALL EXTERIOR *Contact Property Assignment , , IntProp-1 ** ** OUTPUT REQUESTS

28

** *Restart, write, number interval=1, time marks=NO ** ** FIELD OUTPUT: F-Output-1 ** *Output, field *Node Output A, U, V *Element Output, directions=YES EVF, S, SVAVG ** ** HISTORY OUTPUT: H-Output-1 ** *Output, history, variable=PRESELECT *End Step

Appendix 2 SCM language of electromagenetic sound-control switch switch: <?xml version="1.0" encoding="utf-8"?> <FutureStarProgram ProgramName="electromagenetic-sound control switch "> <FutureStarProgress Name="electromagenetic-sound control switch " Info="" Type="Main" Start="cbc285a9-1f72-4aac-b839-aa26e7c72642" End="fc4b6396-0cee-4a42-aac4-d699bf605a28" Show="yes"> <Instruction Type="程序开始" Id="cbc285a9-1f72-4aac-b839-aa26e7c72642" /> <Instruction Type="程序结束" Id="fc4b6396-0cee-4a42-aac4-d699bf605a28" /> <Instruction Type="电动机正转" Id="9fbee193-46c8-4ad1-b764-ac9397941261"> <Argument Format="10">2</Argument> </Instruction> <Instruction Type="延时" Id="f2095316-1f25-44cb-b4d4-e585b8386482"> <Argument Format="10">0</Argument> <Argument Format="10">5</Argument> </Instruction> <Instruction Type="电动机停转" Id="044ff879-a430-475f-9927-9469b6ced911"> <Argument Format="10">2</Argument> </Instruction> <Instruction Type="传感器" Id="37929148-2b81-4f7e-8977-f119dac9571e"> <Argument Format="10">0</Argument> <Argument Format="10">0</Argument> </Instruction> <Instruction Type="无条件转移" Id="ef639ca8-1c44-4857-a687-2aec0099721e" /> <Instruction Type="传感器" Id="ae483f20-1087-4072-b546-4c5fe865ffb7"> <Argument Format="10">0</Argument> <Argument Format="10">0</Argument> </Instruction> <Instruction Type="电动机反转" Id="06464c04-366a-45b9-9f57-a37d0c75fdac"> <Argument Format="10">2</Argument> </Instruction> <Instruction Type="延时" Id="211c87f0-80b9-46c6-858e-94bde3e8c07c"> <Argument Format="10">0</Argument> <Argument Format="10">5</Argument> </Instruction> <Instruction Type="电动机停转" Id="2e7bcdb4-6003-4fcf-9dab-e6b84709f392"> <Argument Format="10">2</Argument> </Instruction> <Instruction Type="电动机正转" Id="897a3a99-729c-4918-9425-7844926bf3d1"> <Argument Format="10">1</Argument> </Instruction> <Instruction Type="电动机停转" Id="d3228d99-18a3-4c5b-8287-ca31b160d7a6">

29

<Argument Format="10">1</Argument> </Instruction> <Instruction Type="电动机反转" Id="97e5153f-6f61-4cf9-9056-1f6d3df36165"> <Argument Format="10">1</Argument> </Instruction> <Instruction Type="电动机停转" Id="d7bb34b0-fa6c-477b-96c3-e7788c3f78b4"> <Argument Format="10">1</Argument> </Instruction> </FutureStarProgress> </FutureStarProgram>

Appendix3 Mathematica programming language of Six degrees of freedom system:: a1 = y[t]

a2 = y[t]+1/2*l*Sin[θ[t]]+d*Sin[ϕ[t]]+1/2*l*Sin[α[t]]

a3 = y[t]+1/2*l*Sin[θ[t]]+d*Sin[ϕ[t]]+l*Sin[α[t]]+d*Sin[β[t]]

x1 = x'[t]

y1 = y'[t]

x2 = x'[t]-1/2*l*θ'[t]*Sin[θ[t]]-d*ϕ'[t]*Sin[ϕ[t]]-1/2*l*α'[t]*Sin[α[t]]

y2 = y'[t]+1/2*l*θ'[t]*Cos[θ[t]]+d*ϕ'[t]*Cos[ϕ[t]]+1/2*l*α'[t]*Cos[α[t]]

x3 = x'[t]-1/2*l*θ'[t]*Sin[θ[t]]-d*ϕ'[t]*Sin[ϕ[t]]-l*α'[t]*Sin[α[t]]-d*β'[t]*Sin[β[t]]

y3 = y'[t]+1/2*l*θ'[t]*Cos[θ[t]]+d*ϕ'[t]*Cos[ϕ[t]]+l*α'[t]*Cos[α[t]]+d*β'[t]*Cos[β[t]]

T = 1/2*m*(x1^2 + y1^2) + 1/24*m*l^2*(θ'[t])^2 + 1/2*m*(x2^2 + y2^2) + 1/24

*m*l^2*(α'[t])^2 + m*(n-2)/2*(x3^2 + y3^2)

U = m*g*a1 +m* g*a2+m*(n-2)*g*a3

L = T-U

R =1/24 m (12 g (l (-3+2 n) Sin[α[t]]+2 d (-2+n) Sin[β[t]]+(-1+n) (l Sin[θ[t]]+2 d Sin[ϕ[t]]))+24 g

n y[t]+12 n ((PX/m*2-l (3-2 n) Sin[α[t]] (α^′)[t]+4 d Sin[β[t]] (β^′)[t]-2 d n Sin[β[t]] (β^′)[t]+l

Sin[θ[t]] (θ^′)[t]-l n Sin[θ[t]] (θ^′)[t]-2 d (-1+n) Sin[ϕ[t]] (ϕ^′)[t])/(2*n))2-12 n (y^′)[t]2+36 l

Cos[α[t]] y′[t] α′[t]-24 l n Cos[α[t]] y′[t] α′[t]+20 l2 (α^′)[t]2-12 l2 n (α^′)[t]2+48 d Cos[β[t]] y′[t] β′[t]-24 d n Cos[β[t]] y′[t]

β′[t]+48 d l Cos[α[t]-β[t]] α′[t] β′[t]-24 d l n Cos[α[t]-β[t]] α′[t] β′[t]+24 d2 (β^′)[t]2-12 d2 n (β^′)[t]2+12 l Cos[θ[t]] y′[t] θ′[t]-12

l n Cos[θ[t]] y′[t] θ′[t]+18 l2 Cos[α[t]-θ[t]] α′[t] θ′[t]-12 l2 n Cos[α[t]-θ[t]] α′[t] θ′[t]+24 d l Cos[β[t]-θ[t]] β′[t] θ′[t]-12 d l n

Cos[β[t]-θ[t]] β′[t] θ′[t]+2 l2 (θ^′)[t]2-3 l2 n (θ^′)[t]2-12 d (2 (-1+n) Cos[ϕ[t]] y′[t]+l (-3+2 n) Cos[α[t]-ϕ[t]] α′[t]+2 d (-2+n)

Cos[β[t]-ϕ[t]] β′[t]+l (-1+n) Cos[θ[t]-ϕ[t]] θ′[t]) ϕ′[t]-12 d2 (-1+n) (ϕ^′)[t]2)

RY = D[D[R,y'[t]],t]==D[R,y[t]]

RTheta = D[D[R,θ'[t]],t]==D[R,θ[t]]

RPhi = D[D[R,ϕ'[t]],t]==D[R,ϕ[t]]

RAlpha = D[D[R,α'[t]],t]==D[R,α[t]]

RBeta = D[D[R,β'[t]],t]==D[R,β[t]]

d = 0.022

n = 11

l = 0.114

g = 9.7984

c = 0.144

μ = 0.2653

θ0 = 30°

30

m= 0.003318207

a = Cos[θ0]-μ*Sin[θ0]

b = 1+μ^2

v0 = -2

PX = m*(-μ*2*v0/(3*a^2+b))

12*v0*a/((3*a^2+b)*l)

{ysol,θsol,ϕsol,αsol,βsol}=NDSolveValue[{RY,RTheta,RPhi,RAlpha,RBeta,y[0]==0,θ[0]==θ0,ϕ[

0]==90°,α[0]==180°-θ0,β[0]==90°,y'[0]+1/2*l*θ'[0]*Cos[θ[0]]==-2*v0/(3*a^2+b)+v0,θ'[0]==12

*v0*a/((3*a^2+b)*l),α'[0]==0,β'[0]==0,-μ*2*v0/(3*a^2+b)-1/2*l*θ'[0]*Sin[θ[0]]-d*ϕ'[0]*Sin[ϕ[0

]]==0},{y,θ,ϕ,α,β},{t,0,0.02}]

Export["D:\4.0B1.jpg",Plot[{θsol[t]*180/Pi,ϕsol[t]*180/Pi,αsol[t]*180/Pi,βsol[t]*180/Pi},{t,0,0.0

2},GridLines->Automatic,AxesLabel

->{"time/s","degree/°"},PlotLegends->{"θ","φ","α","β"},ImageSize->Large]]

31

Resume

1.

姓名：石金泽

性别：男

民族：汉

所在单位：山东省青岛第二中学

政治面貌：团员

出生日期：2000.02.02

身份证号：370213200002025213

联系方式：18563925816

个人事迹：

1)太空城市设计大赛“中国赛区二等奖”

2)青岛二中阿卡贝拉社长，B-box 爱好者

3)山东省青少年航空航天锦标赛“山东省一等奖”

2.

姓名：贾宣辰

性别：男

民族：汉

所在单位：山东省青岛市第二中学

政治面貌：团员

出生日期：2000.03.03

身份证号：370202200003030058

联系方式：15964268752

个人事迹：

1)英国伦敦国际青少年科学论坛 LIYSF

2)AAPT 美国物理碗竞赛全国前３％

3.

姓名：刘林川

性别：男

民族：汉族

所在单位：山东省青岛市第二中学

政治面貌：团员

出生日期：2000.09.08

身份证号：370205200009086510

联系方式：15692382187

个人事迹：

1)参加全国中学生 botball 机器人挑战赛获得二等奖、最佳策略奖。

2)参加“市长杯”足球比赛，进入十六强

3)参加“FLL”机器人工程挑战赛，获得省一等奖第二名