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mechanism with decided input configuration and parameter settings. To supply a universal model for design purpose, a CHA

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Bulletin of the JSME

Vol.9, No.3, 2015

Journal of Advanced Mechanical Design, Systems, and Manufacturing

A characteristic triangle method on input vectors of scissor lift mechanism and its applications in modeling and analysis Wei ZHANG*,**,***,

Xuefei ZHANG****, Chao YAN****, Shujie XIANG** and Liwen WANG***

*Airport College, Civil Aviation University of China, 2898 Jinbei Road, Dongli District, Tianjin,300300, China E-mail: [email protected] **Sino-European Institute of Aviation Engineering, 2898 Jinbei Road, Dongli District, Tianjin,300300, China ***Aviation Ground Special Equipment Research Base, 2898 Jinbei Road, Dongli District, Tianjin,300300, China ****Aeronautical Automation College, Civil Aviation University of China, 2898 Jinbei Road, Dongli District, Tianjin,300300, China

Received 16 December 2014 Abstract Scissor mechanism, a basic element of structure and mechanism, can be used to make lots of deployable structures and functional motion devices. In such structures or devices, input configuration and parameter settings of actuator in scissor mechanism are variable and very important to structural forces and kinematics, however, has not been investigated systematically. Former researches on input vectors only consider a mechanism with decided input configuration and parameter settings. To supply a universal model for design purpose, a CHAracteristic Triangle (CHAT) is herein reported to calculate input vectors. The CHAT can be picked up from each of six input configurations of single group scissor lift mechanism. Based on former works and authors’ simulations, several value rules of input vectors in scissor mechanism are discovered, summarized, and approved via mathematics analysis. One of those rules shows that the maximum input force always occurs in the beginning position of lifting. Which indicates extreme importance of initial lift angle to design. Meanwhile, a further research on typology refines six input configurations into more detailed value cases according to possible parameter settings. Among them, three constant lift force cases, two singular cases and other cases are discussed individually. Typology and value rules are significant to power selection, mechanism configuration design and dimension optimization. Validity of presented equations is approved by four reported samples. Not limit to studied single group mechanisms, CHAT can also be found easily in every linear driven scissor lift mechanism such as multiple group mechanism, parallel mechanism, or nonstandard connected single group mechanism. Thus, presented universal equations are suitable for widespread applications with scissor mechanism. Key words : Ground support equipment, Scissor lift mechanism, Input vector, Mechanical modeling, Mathematic analysis

1. Introduction Scissor unit (SU) is two bars connected with a hinge joint. Earlier studies on SU can be traced back to scissor like element (SLE) in engineering and aerospace structures. SLE is basic unit in a sort of deployable structure belongs to kinetic structures (Akgün, et al., 2010, Babaei, M. and Sanaei, 2009, Chen, et al., 2002, Dai, 2004, Kolar, et al., 2010, Wei, et al., 2011). Among these researches, SLE constituted many complicated structures, such as: semi ellipse foldable structure (Babaei, M. and Sanaei, 2009), box SLE mast (Nagaraj, et al., 2009), flat and cylindrical deployable structure (Zhao, et al., 2009), and deployable ring type truss structure, etc (Chen, et al., 2002, Dai, 2004, Sun, 2009, Zhao, 2007). Most of papers about SLE considered mobility and kinematics of entire structure, or structural statics of system under folded state and unfolded state. However, basic element like SU or SLE is rarely appeared in the literature as target research object. Akgün, et al., (2010) investigated a complex structure consist of a modified SLE. In which the entire structure but basis unit, is mainly studied. SU is also frequently applied in mechanisms, such as orchestra or stage lift (Wei, 1997, Xiang and Wei, 2007), airport ground support equipment (Liu, et al., 2009, Guo, 2008, Xie and Jiang,

Paper No.14-00526 [DOI: 10.1299/jamdsm.2015jamdsm0042]

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2007, Zhou, et al., 2009), assisted standing-up mechanism (Nango, 2010), foldable stairs (Zhao, 2011), deployable gate mechanism (Zhao, 2009), aerospace antenna or solar energy collection device (Miao, et al., 2004, Sun, 2007, Tang, et al., 2009, Zhao, 2007, Zhao, 2005), and so on. In these studies, geometry, mobility, kinematics, dynamics and statics are widely discussed (Nagaraj, 2009, 2010, Li, 1994, Li and Li, 2000, Liu, et al., 2009, Nango, et al., 2010, Song and Liu, 2004, Xie and Jiang, 2007, Zhao, et al., 2009, 2011, Zhou, et al., 2009, Zhao, et al., 2005). Statics is an important work of researches on structure and mechanism. Juan and Mirats (2008) systematically reviewed static analysis methods of tensegrity (composed of strut, bar, and cable). Based on graph theory, statics methods are surveyed via three points of view (motions, forces and energy approaches). Nagaraj, et al., (2010) analyzed a deployable pantograph mast with SLE. The stiffness matrix is obtained in symbolic form and the results agree with those obtained with the force and displacement methods. Zhao, et al., (2011, 2012) presented a SU based foldable stair and studied its geometry and statics. The studied stair is easily deployed for use and folded for storage with a four-bar linkage as actuator. Akgün, et al., (2011) introduced an adaptive deployable spatial mechanism with SLE. Structural response to different connecting locations of linear actuators is studied. Based on a modified SLE, a roof can be converted by means of linear actuators between a multitude of curvilinear arch-like shapes, where it can be stabilized and carry loads (Akgün, et al., 2010). The systematical influence of connecting position of actuators to structural forces and stability is out of discussion though. Scissor lift mechanism (SLM) is a device with SU to realizing a function of rising with loadings. There are many kinds of SLMs like parallel type (Xiang and Wei, 2007) and multiple group type (Hu, et al., 2006). It also has many variants (Juan and Mirats, 2008, Akgün, et al., 2011, Nango, 2010). For example, (Nango, 2010) proposed a unclassical SLM (arms are intersect instead of hinged) with gear train as a revolute input motion to achieve a tilting and lifting motion to help people standing up easily. Many airports ground support equipment are mounted with SLM, such as catering trucks (Liu, et al., 2009), platform trucks (Xie and Jiang, 2007, Guo, 2008), mobile boarding bridges (Zhao, et al., 2009) and mobile tower. These SLMs are always used to dock with aircraft doors at diversiform heights. Airport SLMs use mainly scissor lift unit (SLU). It is a scissor lift mechanism including only one group of scissor (Single group SLM). As a single degree of freedom mechanism (Zhang, et al., 2010), SLU is often driven by linear actuator like ball screw, chain drive and hydraulic actuator. Generally, connecting positions of actuator are very different and various. Studies on SLM consider not only deploying kinematics and supporting capacity of deployed state, but also process of deploying. For this purpose, many researches utilized quasi-static thought to investigate motions with instantaneous velocity center method and input forces with principle of virtual displacement (Wei, 1997, Li and Li, 2000, Spackman, 1989, 1994). A weakness of these works is that surveys of mechanical behavior, structural simulation and optimization (Hu, et al., 2006, Liu and Sun, 2009, Xie and Guo, 2007) were conducted to a determined connecting position of actuator with known structural parameters. For innovation, it is difficult to find out the optimum solution without variations. Selection and analysis should be executed among all possible input connecting positions and structural parameter values. To all SU based mechanisms or deployable structures, input power is needed to realize motion between different poses. Systematical investigation to input position of actuator, however, is not taken into account. From a geometric point of view, a different input position results in a different mechanism kinematics. It also affects following mechanical analysis. Without a universal description of different input positions, their performances are hard to compare. Former researches on SLM had to demand a pre-determined input position for its design, analysis and optimization (Liu, et al., 2009, Song and Liu, 2004, Feng, et al., Zhang, et al., 2012). Effective comparison to solutions with different inputs is therefore hardly made. Zhang, et al., (2010) classified all input positions of linear actuator of SLU into six configurations. Then, a SLU with one widely used input type is researched in (Zhang, et al., 2012). Using parameterized method, a unified mechanics model is constructed there. After that, Two SLUs with special input connections are unified into a common equation in (Zhang, et al., 2013). It is necessary to systematically investigate performances of all input position types and their influence to structural forces and stabilities. For that, a universal geometric and mechanical model of SLU needs to be built up. With quasi-static analysis, a geometric and mechanical modeling process is tried here in symbolic form to create a universal method that suitable for all SLMs. Through building up a series universal equations, influence of input position types and structural parameters to input vectors is discussed by observing and analyzing. Value rules of input vectors are also described in this paper. Value rules analysis makes it easier to compare performances of different input types of SLU, and clarifies which factors these differences are due to. System cost and selection of input power

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depends on this analysis. It is important to know that on which position we can decide model of input device. With reliable mathematic analysis, value rules may indicate optimization and design criteria of connection positions and structural parameters. To instruct design, typology of input type is discussed and summarized. An atlas of all possible input types would be helpful in practice. All analytical results are testified with examples of former literatures.

2. General Model of SLU 2.1 Model Simplification Because of symmetry, SLU is simplified into the 2-dimensional space. As shown in Fig.1, the input component I can be expressed as scalable linear bar without loss of generality. The length of scissor arm is l, the height of top

   ). Load is assumed as a vertical  2

platform is H, the length of input bar is I, lift angle is  (where    0, concentrated force.  W

 P



Fig. 1 Principal Schematic diagram of a scissor lift unit

Fig.2 Types and parameters of input configurations

Six types of possible linear input position were mentioned by Zhang, et al. (2010). As shown in Fig.2, SLU with all six input configurations is drawn in one mechanism. In which, each kind of input actuator is linked to 2 points selected from 4 SLU characteristic points B1, B2, B3, and B4. To simplify modeling process, one suppose that each point is 2 located on the line by center of two joints selected from A, B, C, and D. Therefore, it has C 4  6 kinds of input configurations. Input position is described by corresponding nonnegative parameter a, c, d, or f. Because SLU is a system of bars, load is much greater than its own weight, gravity of system is ignored.

2.2 Output Length Vector H Obviously, top platform can only move upward with respect to the lower platform, output length vector is expressed by parameter settings and lift angle  as following:

 T H  0 l sin  

(1)

2.3 Input Length Vectors I 2.3.1 A CHAracteristic Triangle In order to give a universal input calculation model, geometric properties of SLU are investigated. It is noticed that the scissor arm can be regarded as a hypotenuse (with thick dotted line shown in Fig. 3). Thus, this paper put forward a characteristic triangle to modeling and solving input length vectors. To each kind of input configuration, a right triangle can be found to make input side as a segment between hypotenuse and the right-angle side.

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c

c

 a



d

d



1

d

3

2 c

f



 

a

 f







a

4

 

f



6

5

Fig.3 Characteristic triangles of 6 different input configurations

As unified shown in Fig. 4, each characteristic triangle described in Fig. 3 has a right-angle side AB whose length is H. Hypotenuse BC is a SLU arm with length of l. The length of input vector I is expressed by segment B’B’’.

Fig.4 Characteristic triangle for calculating input length vector I

2.3.2 Length of Input Vector I According to the Law of cosines and Fig.4, input length vector is expressed by parameter settings and lift angle as following:



I  (l 2  2lu ) sin 2   v 2  u 2  2vu cos 

(2)

I  (l  u ) 2 sin 2   (v  u cos  ) 2

(3)

or,

Parameter u and v in equations (2) and (3) are provided from Table 1 according to different input configurations. Table 1.

Configuration u v

1 c a

Values of v and u under different input configurations

2 l-c lcos  -d

3 l-f d

4 f lcos  -a

5 0 d-a

6 l-f+c 2ccos 

Suppose

v  K cos   J

(4)

Then, equation (3) can be rewritten as

I  (l  u ) sin  2  (( K  u ) cos   J ) 2

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(5)

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Furthermore,

M  K  u   N  l u

(6)

I  ( N sin  ) 2  ( M cos   J ) 2

(7)

Hence,

Values of parameter are given in Table 2. Table 2. Values of K, J, M and N under different input configurations

Configuration K J M N

1 0 -a -c l-c

2 l d c c

3 0 -d f-l f

4 l a l-f l-f

5 0 a-d 0 l

6 2c 0 c+f-l f-c

2.3.3 Expression of Input Angle According to Fig.4, the input angle can be defined as:

  AB ' B '' With equation (7), trigonometric values of input angle are

N   sin   I sin   cos   M cos   J  I

(8)

2.3.4 Expression of Input Length Vector I According to Fig. 4, the expression of input length vector in reference system {x’y’} is:

 I   I cos 

or,

I sin  

T

 I   J  M cos 

(9)

N sin  

T

(10)

According to Fig. 3, expression of input length vector I in fixed reference system of Fig. 1 can be achieved through rotating or mirroring reference system of characteristic triangles in Fig. 4. Transformation relations are given in Table 3. Obviously, output length vector H in any characteristic triangle keeps the same form as equation (1). Table 3. Transformation of input length vector I in fixed reference system

coordinate transform Input configuration





I   I  xy



x' y'

I 

xy

1

  I



2

x' y'

I x  I x', I y  I y'

Ix  Ix', I y  I y'

3,6

4,5

2.4. Universal Expression of Input Force Vector Dynamics is unnecessary since acceleration of lift mechanism is always small. A quasi-static state of SLM is investigated as shown in Fig. 1, the load W is paralleled to the output length vector H. Meanwhile, input member is a two-force bar. The input length vector I is paralleled to input force vector P. Supposing SLU is an ideal restraint system, function of input force vector P can also be achieved in a characteristic triangle (See Fig. 5).

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 W

 P 



Fig.5 Characteristic triangle of input force vector P

As shown in Fig.5:

 T  W  0 W    T  P   P cos  P sin  

(11)

According to the principle of virtual displacement:





i

i

 F  r  0

(12)

In an ideal restraint system, only active forces work,

    W  H  P  I  0

(13)

Considering equation (1),



 H  0 l cos  

T

(14)

Using equation (10),



 I    M sin 

N cos  

T

(15)

Takes equation (11), (14) and (15) into equation (13), one can sort out

lW cos   P cos  M sin   P sin  N cos   0

(16)

That is

P ( N sin  cos   M cos  sin  )  lW cos 

(17)

If one supposes

N sin  cos   M cos  sin   0

(18)

There is

P

lW cos  N sin  cos   M cos  sin 

(19)

P

lW N sin   M cos  tan 

(20)

or,

Utilizing equation (7) and (8), value of input force is a function of lift angle,

P

lWI  N  M  sin   MJ tan  2

2

(21)

or,

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P

lW ( N sin  )2  ( M cos   J )2

N

2

(22)

 M 2  sin   MJ tan 

According to equation (11), input force is

 P

lW   J  M cos   N  M  sin   MJ tan 

 P

 lW I  N  M  sin   MJ tan 

2

2

N sin  

T

(23)

or, 2

2

(24)

In fact, the reported characteristic triangle can be found in any linear driven scissor mechanism, not just in presented SLUs here. Two kinds of special complicated SLMs were discussed with triangles (Zhang, et al., 2013). It is also suitable for multiple group SLM, parallel SLM, multi-input SLM, or nonstandard connected single group SLM.

3. Distribution rules of input functions in SLM According to equation (7) and (22), one can investigate magnitude of input length and input force for design purpose. Some rules of length of input vectors are observed from former articles. Using presented equations, many computation tests of different parameters and all 6 input configurations were conducted. Based on these tests and former reported data in articles, some rules of length variation of input vectors are discovered. There are mainly 3 types of magnitude variations of input vectors. These variation possibilities are described in Fig. 6.1 to Fig. 6.3. To obtain these figures, we set l = 1200 mm, W = 2000 N. Considering parameter definitions shown in Fig. 2 and their applicable positions, we set c, f  [0, l ] and a, d  [0, 2l ] . With randomized parameter settings and input configurations, many curves are achieved and can be summarized into curves shown in Fig. 6. Although parameter a and d can be a bigger value than l, there is no new curve type found. According to Fig 6.1, configuration 1 have 3 possible variations of input length and input force if one sets parameter a and c to different values. So does configuration 3. In Fig 6.2, configuration 6 has 2 possible variations if its parameter c and f change. While input configuration 2, 4 and 5 obey with one certain magnitude variation no matter how much values of parameters are (See Fig 6.3).

Fig. 6.1 Input vector magnitude possibility of configuration 1 and 3. These two configurations are found to obey same rule. The left graph shows 3 possibilities of input length I. The right graph shows 3 possibilities of input force P. In two graphs, curves with same mark have same parameter settings.

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Fig 6.2 Input vector magnitude possibility of configuration 6. This configuration has 2 possibilities of input length I as shown in the left. The right graph shows 2 possibilities of input force P. In two graphs, curves with same mark have same parameter settings.

Fig. 6.3 Input vector magnitude possibility of configuration 2, 4 and 5. This kind of configurations has only one possibility of input length I and input force P. Fig. 6. Input vector magnitude possibilities of all kinds of input configurations.

According to Fig. 6, some meaningful results are observed. The following statements are effective to all input configurations in any parameter settings except one situation which is drawn in Fig. 6.1 with mark diamond. These statements are: Statement 1: Monotonicity of input length I relate to sign of input force P. Statement 2: Signs of input length I is constantly positive. Statement 3: Signs of input force P reversely relates to its monotonicity. Statement 4: Maximum amplitude of P occurs at initial lift angle. To the discontinuous case of Fig. 6.1, there exists a turning point as SLU lifts up. In Fig. 7, it shows a parameter settings of input configuration 3 and the corresponding lift angle to achieve the discontinuous point of diamond curve that shown in Fig. 6.1. To input force P, its value has a sudden change at this turning point. This situation results from a change of signs of P and thus produces a discontinuity. To input length I, its monotonicity changes when curve crosses the turning point. This means I get an extremum at the turning point. As one can see, both sides of turning point in curves also obey above statements. Mathematically, turning point is proved to appear only if equation (18) is not satisfied. This is a situation of singularity (dead point) in mechanism.

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Fig. 7. The singular parameter settings of configuration 3 and its sudden change position to produce a discontinuous curve in Fig. 6.1. At which, input length has a minimum value in the whole lift process and input force changes suddenly from a pulling force to an equal pushing force.

As shown in Fig. 7, if we set d  800 and f  200 , it has a turning point while lift angle  where, input length has minimum and input force changes its direction.

 33.5573o . At

4 Mathematic Explanations In this section, mathematic principles are discussed to explain rules from statements in section 3, including the exceptional case. This will tell us how to select a specific configuration and what parameter settings are for an application.

4.1 Derivation of Input Length Function According to equation (10), input length is a nonnegative variable. It is helpful to investigate its derivative function.

4.1.1 Derivative function of I Because I and

 are functions of time t ,

So,

dI (sin 2 ) d  dt 2I dt

(25)

  MJ sec    N 2  M 2 

(26)

Where

If one considers a lift process, monotonicity of length function I only depends on signs of discriminate equation (26). According to Table 2 and equation (20), values of ∑ under different input configurations can be obtained (See Table 4). Table 4. Values of



under different input configurations

Configuration MJ

1 ac

2 Cd

3 d(l-f)

4 a(l-f)

5 0

6 0

N2 M 2

l 2  2lc

0

l 2  2lf

0

l2

(l  2 f )(2c  l )

As Table 4 shows, the first part of ∑, MJ sec   0 . However, the second part of ∑ has different signs. It will lead a different monotonicity of I. A detail discussion about signs of ∑ is omitted here.

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4.2 Derivation of Input Force Function and its utilization 4.2.1 Derivative function of input force P According to equation (21) and (26), we have

lWI sin  Where I and  are functions of time t , P

(27)

Take the first order derivative

1 dP I sin   I ( cos   sin  )  2 lW dt  sin   Where,  

(28)

d d  dI  d  d ,I  ,  .   N 2  M 2   MJ sec    MJ tan  sec  dt dt dt dt dt

Defuse the equation

cos 2   sin   I dP d  /  A cos3   B cos 2   C cos   D lW dt dt 2

(29)

Where

A  M 4  M 2N 2  J 2N 2 B  3MJN 2  3M 3 J C  3M 2 J 2 D   MJ ( N 2  J 2 ) Assume

F  A cos 3   B cos 2   C cos   D

(30)

It can be easily concluded that monotonicity of function P is determined by signs of equation (30).

4.2.2 Boundary values of equation F Because of   (0, When  

 2

 2

) , cos3  , cos 2  , cos   (0,1)

, right boundary value of equation (30) is

F  D   MJ ( N 2  J 2 )  0

(31)

2

When 

 0 , left boundary value of equation (30) is

F0  A  B  C  D

(32)

One expands equation (32)

F0  M 4  3 JM 3  (3 J 2  N 2 ) M 2  J (2 N 2  J 2 ) M  J 2 N 2

(33)

Simplifies,

F0    M  J   ( N 2  M 2  MJ )    M  J    inf   2

2

(34)

Where inf  is the infimum of ∑. Based on analysis of equation (31) and (34), right boundary of F is non-positive, and left boundary is decided by inf  . According to section 4.1.2, inf  is changeable with different parameter settings (see Table.4). To analyze signs of F, one discusses whether a cubic equation has roots between 0 and 1.

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4.2.3 Roots of a cubic equation and related signs of F One sets equation (30) equals zero

F  Ax 3  Bx 2  Cx  D  0

(35)

While

x  cos   (0,1) The nature of roots of equation (35) can be achieved by Cardano formula. Discriminant of equation (35) is as follow,



J 2 M 2 N 4 ( J 2  M 2 )2 ( J 2  M 2  N 2 )2 4( M 4  M 2 N 2  J 2 N 2 )4

(36)

One finds that equation (36) is nonnegative. This means that the cubic equation (35) has two real roots at most. In addition, a degeneration of equation (35) is taken into account. Since only real root is meaningful to our analysis, three cases need to be considered: I. A0. II. A  0,  0 III. A  0,  0 Under these cases, root of equation (35) and signs of equation (30) are investigated to determine monotonicity of P. a detailed analysis about root of F among 0 to 1 is also done.

4.3 Proof to Statements of Input Functions 4.3.1 Proof of Properties of Input Length I With equations in section 4.1 and 4.2, one proves statement 1 and statement 2 about input function I. For statement 1, one divides input force P by first derivative of input length I. Using equation (25) and (27), one has

P 2lWI 2  I sin  sin 2 2

(37)

According to equation (37) P and I have same signs. This means that input length I rises by lifting when input length P get a positive value (Pushing). Or, input length I decreases by lifting when input length P get a negative value (Pulling). For statements 2, one proves it directly with equation (7). Or one knows its validity because of its meaning of length of an input device.

4.3.2 Proof of Properties of Input Force P Statement 3 and statement 4 are analyzed via roots of discriminant of equation (35). Based on which, it is proved that there has one real root at most in the request range. Thus, F must take different signs at two boundary of range to take a zero value within the range. According to equation (31) and (34), one has

F0    M  J    inf    0 2

(38)

Namely,

inf   N 2  M 2  MJ  0

(39)

Since    inf ,   , there has one position that makes  zero, namely

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( x* ) 

MJ N2  M 2  0 x*

(40)

On the opposite way, it is easy to prove that a root of equation (40) within (0, 1) must results in satisfaction of equation (38). One concludes that either F and ∑ both have a real root within (0, 1) or they have no root within the range. At position of x* , one deduces expression (30) as following:

( x* )  x* 

2

F ( x* ) 

 N 2  M 2  N 2  J 2   M 2  N 2  x*3  MJ  N2 M 2

(41)

In simplification,

F ( x* )  N 2  M 2  N 2  J 2  x*2  1 x*

(42)

With equation (39), one has

J 2

M 

2

 N2

2

(43)

M2

And M N J M N 2

2

2

2

2

M 

2

 N2

M

2

2



M 2  N2 2 N 0 M2

(44)

So equation (42) is negative, namely

F ( x* )  0

(45)

According to equation (27) and (29), F and ∑ imply signs and monotonicity of input force P. Because F and ∑are both monotonic in range (0, 1), a sketch is drawn as below.

Fig 8. General distribution of discriminants within lift range. In zone A, input force P is negative while mechanism rises. In zone C, input force P is positive while mechanism drops. In zone B, input force P is negative while mechanism drops. The situation in zone B should not occurs in a practical mechanism because of its singularity. A rising of parameter J makes function of ∑ move up and function of F move down. By which, zone B and zone C move toward left while zone A diminishes till to disappear. Three zones exist simultaneously only in some settings of configuration 1 or 3. To configuration 2, 4 and 5, curves are always in zone C. To configuration 6, curves are either in zone A or in zone C under a certain parameter settings.

In Fig 8, parts of curves in zone A and zone C obey with rules of statement3 and statement 4. According to Table 2 and Table 4, configuration 2, 4, and 5 always have zone C in the entire range with any parameter settings. Configuration 6 has two possibilities of curves distribution: zone A in the range or zone C in the range. Configuration 1 or 3 has a possibility to appear zone B in the range. This violates rules of statements 3 and statement 4. However, discriminant ∑ equals 0 makes equation (27) meaningless. It also indicates appearance of a dead point (position of

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singularity). One therefore concludes that every applicable configuration obeys statement 3 and statement 4. One finds that a change of parameter settings leads a different coverage of three zones. To sum up, input force P and input length I are both proved to be monotonic in the cosine value range (0, 1) of lift angle. This conclusion covers all nonsingular parameter settings of all configurations. This conclusion and above four rules are significant to design. In type selection, the presented universal equations may be used to determine and optimize parameter settings, and choose an input configuration. Even more, input force at initial lift angle becomes a key index to select power of input device since it is the maximum force in the entire lift process.

5 Typology of Input Functions A rigorous proving to statement 3 and 4 are done according to different configurations and parameter settings. As a byproduct, distribution types of input configurations with different parameter setting are achieved. It is a direct application of proposed method and its mathematic analysis. It shows input performance of SLM with different input parameters and configurations. From simplicity to complexity, one supply a series of tables here to introduce typology of SLU’s input functions. Through these tables, all types of parameter settings to input function distribution are given out by graph samples.

5.1 Vertical Input: An Exception

Fig. 9. Exceptional input configurations with boundary parameter settings. If M=J=0, Configuration 1 and 3 delegate a same input setting between point A and B. This is applied into platform truck to lift luggage platform. Only push force is switched into pulling force with a chain drive. Also, configuration 5 and 6 have vertical input settings. They can also be set at AB.

An exceptional applicable parameter setting is founded in configuration 1, 3, 5 and 6: if parameter M and J both equal zero, input linkage will be vertical in entire lift process. This makes input force P a proportional constant of W, and input length I a sine function. This is a boundary case of parameter settings. It still obeys rules only but statement 3. This kind of parameter settings has a revised application in platform trucks in airport (Xie and Jiang, 2007, Guo, 2008).

5.2 Input Function Distribution of Configuration 2 and 4 To configuration 2, and 4, one can easily see that  is always positive. This makes I a monotonic increasing function in a lift process. Table 5.

Input function variation of configuration 2 and 4

Configuration

Parameter Settings

2

c   0, l ,d  0

4

f   0, l ,a  0

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P

I





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Notice: in Table 5 and other series tables,   is used to express input force P decrease on area of positive numbers, while  is used to express input length I increase on area of positive numbers.

5.3 Input Function Distribution of Configuration 5 and 6 To configuration 5 and 6, one can easily see that  is always positive. This makes I a monotonic increasing function in a lift process. Except the boundary settings, configuration 5 has only one distribution (See Table 6), and configuration 6 have four kinds of distributions (See Table 7). Table 6. Input function variation of configuration 5

Parameter Settings

P

ad

Positive Constant

ad



I



Table 7. Input function variation of configuration 6

Parameter Settings

cl f, f 

l 2

 l l  c   0,  , f   , l   2 2  l   l c   , l  , f   0,  2   2

 l  l c  0,  , f  0,   2  2 l  l  c  ,l , f  ,l 2  2 

P

I

Positive Constant

 





Notice: in Table 7 and other series tables,   is used to express input force P increase on area of negative numbers, while  is used to express input length I decrease on area of positive numbers.

5.4 Input Function Distribution of Configuration 1 and 3 To configuration 1 and 3, there have 11 types of parameter settings individually. Actually, if one substitutes f with l-c, and a with d, configuration 3 can be turned into configuration 1 without any difference in value distribution. Therefore, one gives one typology table to describe these two configurations. According to Table 8, two kinds of parameter settings bring changeable monotonicity of input scalar functions. They actually result from occurrence of zone B in Fig. 8. These two settings need a sudden change of signs of input force P, and therefore are hardly to realize with single drive device. During lifting, one can utilize any side of turning point except position that makes  =0 .

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Table 8. Input function variation of configuration 1

Parameter Settings

P

I

ac0

Positive Constant



 l a  0, c   0,   2





















l  a  0, c   , l  2 

 l a  0, c  0,   2

  l  l  a   0,  2   l  , c   , l  c  2   

l  l  a   2   l, c   , l  c  2  a

c l  2lc  c 2 , c   , l  l c 2 

l  a  2lc  c 2 , c   , l  2  a  c   0, l 

a   0, l  , c  l

a  l, c  l

Notice: There have two kinds of parameter settings with turning points. For them, sample curves are drawn to express distribution of input functions’. They both have zone B defined in Fig. 8. Their occurrences are due to discriminant  =0 at a position within range (0, 1). In configuration 1, one gives examples of these two parameter settings (See Fig. 10 and Fig. 11). There have similar cases in configuration 3 as mentioned. The tenth parameter setting in table 8 describes a special case. In which, there exist a position to make discriminant  and F equal zero simultaneously during lifting (See Fig. 11). It also need a sudden change of signs of input force P. Currently, it is hard to construct a solution to entirely realize this kind of lift process. One can utilize a part of this lift process though.

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6. Validation of Input Functions All parameter settings are supposed to be positive numbers, which can fit all known practical cases. In which, ends of input linkage are located at joint connections. This kind of mechanism is prototype of most SLMs. However, there exist some more complex cases. Besides multi-group SLM (Zhao, et al., 2005) or parallel SLM (Wei, 1997), endpoints of input linkage are sometimes not in the joints connections of scissor arm (Nagaraj, et al., 2009, Raskin, 1998). In practice, two, four or even more hydraulic drives are mounted into SLU for demand of reliability, stability, symmetry or other design needs. Actually, they can also be used to compute multiple group of SLM or multi-input SLM. With adjusting, suitable input equations can be achieved by characteristic triangle for every SLM that is not included in this paper.

Fig. 10. A singular position of configuration 1 with parameter settings. The 5th parameter settings of Table 8 include a singular position (dead point). As described, input drive linkage has a minimum length in a special position after it coincides with arm BD in one line and moves wheel D to point A. This makes input drive cannot extend anymore no matter how much P is.

Fig. 11. A special horizontal setting of configuration 1. The 10-th parameter settings of Table 8 include a singular position. This is not like a dead point as the 5-th parameter settings. But, one hardly selects a realistic transmission solution to force wheel D pass point F without a break.

The presented input vector functions can be verified by any former reported mechanisms in articles. For consistency, one takes four articles (Li, et al., 2000, 2011, Song and Liu, 2004, Wei, 1997) to do verification. These four samples are drawn in Fig.12. Sample 1: WEI Fakong (1997) investigated input force of orchestra SLU with horizontal screw drive (Fig. 12.1). Screw drive and hydraulic transmission are exchangeable in geometry, so it matches input configuration 1.In addition, this paper drew a conclusion that the input force decreases as the lifting angle increases, which matches the proposed analysis here in situation N 2  M 2  0 . Sample 2: (Li, et al., 2000) formulated equation of input force value under configuration 6 (Fig.12.2). A function of input force value by output height is drawn into curve. In this example, it had drawn a conclusion that the input force is monotonic decreasing. According to its settings N  k  a , M  k  a . Thus N 2  M 2  4ak  0 . This proves monotonicity research of situation N 2  M 2  0 . Sample 3: (Song and Liu, 2004) also analyzed configuration 6 SLU and its maximum value position (Fig.12.3). Through a set of practical parameters, results of 18 point-positions are achieved. The maximum of input force and its position is found. In this example, it concluded that the input force is monotonic decreasing. Here, N 

M

l  l  b , 2

l  l   b , that is N 2  M 2 =2l   l  2b   0 . This also belongs to situation N 2  M 2  0 . 2

Sample 4: Differences of Input forces and valid work ranges were discussed between configuration 1 and configuration 4 under the same load and symmetric parameters by (Li, et al., 2011) in (Fig.12.4).

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 Q

 W  P   F 

 F



 P



 M

 W

 P1



 P2

'



'

 P Fig.12 Four reported Samples. Notations keep original as it was reported in corresponding articles. It is easy to see that all results are matched with presented equation, if one substitute original notations into corresponding parameter settings and configurations used here (See Table 9).

Samples 1

Table 9. Results come from different examples Configuration Notation substitution 1 a  0,c  l

l l  a, f   k 2 2

2

6

c

3

6

l c   l ',f  l  b 2

1

a  770mm, c  1465mm

4

a  3210mm,f  1465mm

4

7. Conclusion Input length vector and input force vector are analysis basis of internal forces, stresses, movement performance and optimal design. These input vectors of widely used scissor lift mechanisms are studied in this paper. To all six input configurations of single group scissor lift mechanism, a characteristic triangle is discovered. By which, a series universal equations suitable for all input configurations are achieved to compute input vectors. In fact, the reported characteristic triangle can be found in every linear driven scissor mechanism, not just in presented SLMs here. It is also suitable for multiple group SLM, parallel SLM, or nonstandard connected single group SLM. Therefore, it is a universal method to compute input vectors of SLM. Based on observations to existed works, some value features of input vectors are found. Utilizing presented universal equations, a large amount of parameter settings and input configurations are simulated, and value distribution types are approximately achieved. Four rules of input scalar functions are therefore summarized: 1.Monotonicity of input length I relates to sign of input force P; 2. Signs of input length I is constantly positive; 3. Signs of input force P

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reversely relates to its monotonicity; and 4. Maximum amplitude of P occurs at initial lift angle. Furthermore, mathematical analysis of universal equations is proceeded including monotonicity, signs of value, and real roots distribution of discriminants, etc. By discussing different configurations with every kind of parameter setting, four presented rules are proved. Meanwhile, input value distribution typology of every input configuration is revealed. To configuration 1, 3, 5 and 6, there have three vertical input vector cases to supply a constant lift force. In configuration 1 and 3, two cases with changeable monotonicity are found during lifting. In these cases, input force P has an instant change point and input length I has an extremum point simultaneously. It results from a zero value of discriminant, and result in a singularity of mechanism. These two cases are unrealistic and should be avoided in design. Otherwise, a part of entire lift process should be applied to exclude turning point. To all nonsingular parameter settings of all configurations, input force P and input length I are proved to be monotonic in the range. This conclusion and above four rules are significant to design. In type selection, the presented universal equations may be used to determine and optimize parameter settings, and choose an input configuration. Even more, input force at initial lift angle becomes a key index to select power of input device since it is the maximum force in the entire lift process. Finally, validity of presented equations is verified by four reported samples in former articles. It is easy to see, characteristic triangle widely exists in scissor mechanisms. For each not mentioned SLM herein, if a linear input is applied, its parameter settings always can be adjusted and replaced for using presented universal equations. Thus, methodology and corresponding equations are also applicable to analyze and design other scissor mechanisms.

Acknowledgements This work is supported by National Natural Science Foundation of China and Civil Aviation Administration of China jointly funded project (Grant # U1233106), the Fundamental Research Funds for the Central Universities funded project (Grant # ZXH2012H007) and the university scientific research project of Civil Aviation University of China (Grant # 2012KYE05). Corresponding author is grateful to all who provided helps for this research.

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