Dynamic modelling of an intermittent slider–crank mechanism

Applied Mathematical Modelling 33 (2009) 2411–2420

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Dynamic modelling of an intermittent slider–crank mechanism Rong-Fong Fung a,*, Chin-Lung Chiang b, Shin-Jen Chen c a

Department of Mechanical and Automation Engineering, Graduate Institute of Electro-Optical Engineering, National Kaohsiung First University of Science and Technology, 1 University Road, Yenchau, Kaohsiung 824, Taiwan, ROC Institute of Engineering Science and Technology, National Kaohsiung First University of Science and Technology, 1 University Road, Yenchau, Kaohsiung 824, Taiwan, ROC c Department of Mechanical and Automation Engineering, National Kaohsiung First University of Science and Technology, 1 University Road, Yenchau, Kaohsiung 824, Taiwan, ROC b

a r t i c l e

i n f o

Article history: Received 14 November 2007 Received in revised form 17 June 2008 Accepted 14 July 2008 Available online 23 July 2008

Keywords: Intermittent slider–crank mechanism Permanent magnetic synchronous servomotor Hamilton’s principle

a b s t r a c t This paper investigates kinematic and dynamic analyses of a novel intermittent slider– crank mechanism, which consists of four parts: a crank, a connecting rod associated with a pneumatic cylinder, a slider and two stops at both ends of a stroke. When the crank rotates continuously, the slider will contact with the stops and the pneumatic cylinder is compressed or extended. One periodic motion of the intermittent slider–crank mechanism driven by a permanent magnetic (PM) synchronous servomotor could be divided into three stages. Three governing equations are formulated by Hamilton’s principle. A spring model instead of the pneumatic cylinder is also used for comparison. Finally, the transient amplitudes obtained by using Runge–Kutta method are compared with those of the conventional slider–crank mechanism. Ó 2008 Elsevier Inc. All rights reserved.

1. Introduction The slider–crank mechanism is widely applied in gasoline and diesel engines, where the gas force acts on the slider and the motion is transmitted through the links. Earlier research works in analysis of the slider–crank mechanism can be found in many publications. For examples, the steady-state solutions and the elastic stability of the mechanism were found in the references [1–3]. In addition, the responses of the system have been found to be dependent upon the five parameters: length, mass, damping, external piston force and frequency [4]. Recently, Fung [5] has investigated the transient responses on the basis of the length ratios and the rotating speeds. On the other hand, the slider–crank mechanism driven by a PM synchronous servomotor has applications in areas where the rotation motion transferring to the translation motion is needed. As the high precision was required, Lin et al. [6] introduced an adaptive controller to the slider position of the slider–crank mechanism for considering the existing of uncertainties. Moreover, Fung and Chen [7] developed the variable structure controllers with robust characteristics to control the crank rotating with a constant angular speed. The slider of a conventional slider–crank mechanism is designed to operate repetitively in a limited range, and is unable to stop at any desired position as the crank rotates continuously. In this paper, a novel intermittent slider–crank mechanism, which is combined with a manipulator and is driven by a PM synchronous servomotor, is dynamically analyzed. The system is shown in Fig. 1. At each end of the stroke, the slider contacts with the stop. The pneumatic cylinder is compressed or extended as the crank rotates continuously. Hence, the slider stays for a short duration at each stop. During the stop duration, the manipulator attached at the slider can operate upward and downward. The characteristics of the intermittent slider– crank mechanism are: (1) the locations of the two end stops are adjustable to change the stroke of the slider, therefore * Corresponding author. Tel.: +886 7 6011000x2238; fax: +886 7 6011066. E-mail address: [email protected] (R.-F. Fung). 0307-904X/$ - see front matter Ó 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2008.07.004

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Pressure-Regulating Valve

Pneumatic Cylinder

Slider

Crank

Manipulator

Fig. 1. The physical model of the intermittent slider–crank mechanism.

the stop duration is changeable, and (2) in order to avoid fluctuation of the driving torque of the motor when the pneumatic cylinder is compressed or extended, a pressure-regulating valve is used to maintain a constant air pressure in the pneumatic cylinder. The main objectives are focused on the dynamic formulation of the intermittent slider–crank mechanism. First, the kinematic analysis of the mechanism is performed. The motion of the intermittent slider–crank mechanism could be divided into three stages. The pneumatic cylinder is modelled by two collinear connecting rods. Next, the equations of motion of the motor-mechanism are formulated by Hamilton’s principle. In terms of the pneumatic cylinder, a spring model connecting two collinear connecting rods is also designed for comparisons. Finally, by using Runge–Kutta method, the numerical results validated the theoretical results of the motor-mechanism system are obtained. 2. Kinematic analysis The geometric illustration of the mechanism is shown in Fig. 2a. The m1, m2, m3, ms and m4 represent the masses of the crank, the second and third connecting rods, the pneumatic cylinder and the slider, respectively. The l1, l2 and l3 are the lengths of the crank OA, the second and third connecting rods (AB and CD), respectively. Moreover, x(t) and ls are the current and initial lengths of the pneumatic cylinder, respectively. The crank is driven by a PM synchronous servomotor, and the positive rotation direction is defined as counter-clockwise. In the kinematic analysis, the motion of the intermittent slider–crank mechanism is divided into three stages: (1) the firststage compression motion is shown in Fig. 2a, when the slider contacts with the right stop, (2) the second-stage extension motion is shown in Fig. 2b, when the slider contacts with the left stop, (3) the third-stage free-slide motion shown in Fig. 2c, when the slider is free to slide between the two stops. 2.1. The first-stage compression motion In Fig. 2a, when the slider contacts with the right stop, the crank angle is referred to as h = h4 and d is the distance from E to D. From the angle h = h4 to h = h1, the pneumatic cylinder is continuously compressed. This is the so-called compression motion of the first-stage. At the angle h = h1, the slider begins to leave the right stop. According to the geometric relationship in Fig. 2a, we obtain the critical angles, h1 and h4, and the holonomic constraint equation as follows:

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a

X

l2 l1

θ4

l3

A θ1

x (t) B

ms

m2

m1

φ

m3 Y

E O

D m4

d

l1

b

C

X

θ3

D

θ2

Y E

l1

c

e

X

ls

θ

Y

Fig. 2. The geometric illustration of the three stages of the intermittent slider–crank mechanism (a) the first-stage compression motion, (b) the secondstage extension motion, (c) the third-stage free-slide motion.

" h1 ¼ cos1

# 2 l1 þ ðl1 þ dÞ2  ðl2 þ l3 þ ls Þ2 ; 2l1 ðl1 þ dÞ

h 4 ¼ 2p  h 1 ;

ð1Þ

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 U1 ðQ Þ ¼

 l1 sin h  ðl2 þ l3 þ xÞ sin / ¼ 0; l1 cos h þ ðl2 þ l3 þ xÞ cos /  l1  d

h4 6 h 6 2p;

0 6 h 6 h1 ;

ð2Þ

where Q ¼ ½ / x h T is the vector of generalized coordinates. It is seen that three variables are used to describe the system and two constraint equations are contained in U1(Q) = 0, thus, the system has one degree-of-freedom. From the constraint equation U1(Q) = 0, we obtain the current length of the pneumatic cylinder and the angle of the connecting rod: 1

2

x ¼ ½l1 þ ðl1 þ dÞ2  2l1 ðl1 þ dÞ cos h2  ðl2 þ l3 Þ;   l1 sin h 1 / ¼ sin ; h4 6 h 6 2p; 0 6 h 6 h1 : l2 þ l3 þ x

ð3Þ ð4Þ

2.2. The second-stage extension motion When the slider reaches the left stop as shown in Fig. 2b, the crank angle is h = h2 and the distance between E and D is e. The pneumatic cylinder will be extended until the crank rotates to the angle h = h3. This is the extension motion of the second stage. Similarly, from the geometry of Fig. 2b, we obtain the critical angles, h2 and h3, and the holonomic geometric constraint equation as follows: 1

h2 ¼ cos

 U2 ðQ Þ ¼

" # 2 l1 þ ðl1 þ eÞ2  ðl2 þ l3 þ ls Þ2 ; 2l1 ðl1 þ eÞ

h3 ¼ 2p  h 2 ;

l1 sin h  ðl2 þ l3 þ xÞ sin /



l1 cos h þ ðl2 þ l3 þ xÞ cos /  l1  e

¼ 0:

ð5Þ ð6Þ

From constraint equation U2(Q) = 0, we have 1

2

x ¼ ½l1 þ ðl1 þ eÞ2  2l1 ðl1 þ eÞ cos h2  ðl2 þ l3 Þ;   l1 sin h 1 / ¼ sin ; h2 6 h 6 h3 : l2 þ l3 þ x

ð7Þ ð8Þ

In the paper, it is assumed that the current length x of the pneumatic cylinder is merely dependent on the crank angle h in the first- and second-stage motions. 2.3. The third-stage free-slide motion As the slider freely moves between the two stops, i.e., the crank rotates during the angles, h1 < h < h2 and h3 < h < h4, the system could be regarded as a conventional slider–crank mechanism. It is shown in Fig. 2c with the assumption that the length of the pneumatic cylinder is always x = ls. This is called the free-slide motion of the third stage. From the geometric relationship, we obtain the holonomic geometric constraint equation and the angle / as follows:

U3 ðQ Þ ¼ l1 sin h  ðl2 þ l3 þ ls Þ sin / ¼ 0;   l1 sin h 1 / ¼ sin ; h1 < h < h2 ; h3 < h < h4 ; l2 þ l3 þ ls

ð9Þ ð10Þ

where the vector of generalized coordinates becomes Q ¼ ½ / h T . In the third-stage motion, two variables are used to describe the system with only one constraint equation U3(Q) = 0, and the system still has one degree-of-freedom. 2.4. The velocity and acceleration The constraint velocity and acceleration equations could be obtained by taking the first and second time derivatives of Eqs. (2), (6) and (9) for all the three stages:

UQ Q_ ¼ 0; € ¼ ðUQ Q_ Þ Q_  c: UQ Q Q

ð11Þ ð12Þ

For the first- and second-stage motions, i.e., h4 6 h 6 2p, 0 6 h 6 h1, and h2 6 h 6 h3, we have

"

UQ Q_ ¼ " € ¼ UQ Q

# l1 cos hh_  ðl2 þ l3 þ xÞ cos //_  sin /x_ ; l1 sin hh_  ðl2 þ l3 þ xÞ sin //_ þ cos /x_

# l1 sin hh_ 2  ðl2 þ l3 þ xÞ sin //_ 2 þ 2 cos //_ x_ : l1 cos hh_ 2 þ ðl2 þ l3 þ xÞ cos //_ 2 þ 2 sin //_ x_

For the third-stage motion, i.e., h1 < h < h2 and h3 < h < h4, we have

ð13Þ ð14Þ

R.-F. Fung et al. / Applied Mathematical Modelling 33 (2009) 2411–2420

_ UQ Q_ ¼ ½l1 cos hh_  ðl2 þ l3 þ ls Þ cos //; € ¼ ½l1 sin hh_ 2  ðl2 þ l3 þ ls Þ sin //_ 2 : UQ Q

2415

ð15Þ ð16Þ

Eqs. (13)–(16) can be solved for the angular velocity of the connecting rod and the deformed speed of the pneumatic cylinder as

/_ ¼

8 <

l1 cosðhþ/Þ _ h; l2 þl3 þx

the first and second stages

_ : l1 cos h h; ðl2 þl3 þls Þ cos /

( x_ ¼

ð17Þ

the third stage

_ l1 sinðh þ /Þh;

the first and second stages

0;

the third stage

ð18Þ

:

The angular acceleration of the connecting rod and the deformed acceleration of the pneumatic cylinder can be solved by taking the first time derivatives of Eqs. (17) and (18) as:

€¼ /

€x ¼

8 <

/_ € h h_

_ 2 sinðhþ/Þ2/_ x_

þ l1 h

l2 þl3 þx

;

: /_ €h þ l1 h_ 2 sin hþðl2 þl3 þls Þ/_ 2 sin / ; ðl2 þl3 þls Þ cos / h_ ( x_ € _ _ 2 cosðh þ /Þ þ ðl2 þ l3 þ xÞ/; h þ l h 1 h_ 0;

the first and second stages

ð19Þ

the third stage the first and second stages the third stage

:

ð20Þ

The position of slider D can be expressed as*

xD ¼

8 > < > :

l1 þ d l1 þ e l1 cos h þ ðl2 þ l3 þ ls Þ cos /

the first stage the second stage :

ð21Þ

the third stage

Only in the free-slide motion of the third stage, h1 < h < h2 and h3 < h < h4, the slider D has the velocity and acceleration as follows:

_ x_ D ¼ l1 sin hh_  ðl2 þ l3 þ ls Þ sin //; €  ðl2 þ l3 þ ls Þ cos //_ 2 : €xD ¼ l1 sin hh€  l1 cos hh_ 2  ðl2 þ l3 þ ls Þ sin //

ð22Þ ð23Þ

3. Dynamic formulations In this section, dynamic formulations of the three stages of the intermittent slider–crank mechanism driven by a PM synchronous motor will be performed in conjunction with the pneumatic cylinder and spring models. The Hamilton’s principle, Lagrange multiplier and geometric constraints are employed to derive the differential–algebraic equation for the motormechanism system. Here, only the first stage compression motion is derived explicitly. The dynamic formulations of the second and third-stage motions are almost the same as the first stage. The complete derivations of the equations of motion can be seen in [8]. 3.1. Governing equations with the pneumatic model In order to formulate the governing equation by Hamilton’s principle, the kinetic energy and potential energy of each component must be found first. The kinetic energy of the crank with mass m1 is

T1 ¼

1 2 m1 l1 h_ 2 : 6

ð24Þ

The kinetic energy of the second connecting rod with mass m2 is

T2 ¼

 2 1 1 1 1 1 1 2 2 2 2 2 m2 l2 /_ 2 þ m2 l1 sin hh_ 2 þ m2 l2 sin //_ 2 þ m2 l1 l2 sin h sin /h_ /_ þ m2 l2 þ l3 þ x cos2 //_ 2 24 2 8 2 2 2  2 1 1 2 _ þ m2 sin /x_ 2 þ m2 l2 þ l3 þ x cos / sin //_ x: 2 2

The kinetic energies of the third connecting rod with mass m3 and the spring with mass ms are respectively

ð25Þ

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1 2 m3 l3 /_ 2 ; 24  2 1 1 1 1 1 2 2 2 Ts ¼ ms x2 /_ 2 þ ms l1 sin hh_ 2 þ ms l2 þ x sin //_ 2 þ ms cos2 /x_ 2 24 2 2 2 8   1 1 1 1  ms l1 sin h cos /h_ x_  ms l2 þ x sin / cos //_ x_ þ ms l1 ðl2 þ xÞ sin h sin /h_ /_ 2 2 2 2  2   1 1 1 1 1 2 _ þ ms l3 þ x cos2 //_ 2 þ ms sin /x_ 2  ms l3 þ x sin / cos //_ x: 2 2 8 2 2 T3 ¼

ð26Þ

ð27Þ

The kinetic energy of the motor shaft is

Tr ¼

1 _2 J h : 2 m

ð28Þ

The potential energies of the crank, the second and third connecting rods and the spring are respectively given as

1 m1 l1 g sin h; 2 1 V 3 ¼ m3 gl3 sin /; 2

V1 ¼

  1 V 2 ¼ m2 g l2 þ l3 þ x sin /; 2   1 V s ¼ ms g l3 þ x sin / 2

ð29a—eÞ

where g is the gravitational constant. Note that the kinetic and potential energies of the slider D in contact with the stop are T4 = 0 and V4 = 0. In Fig. 3a and b, two collinear connecting rods are in conjunction with the pneumatic cylinder at both ends B and C. The distance between the two ends is x(t). By using the pressure-regulating valve, the air pressures inside the pneumatic cylinder will remain the same, i.e., there is no pressure difference around the points B and C. Thus, the virtual work done by the air forces is zero. The total kinetic energy and potential energy of the system in the first-stage motion are respectively

T ¼ T1 þ T2 þ T3 þ Ts þ T4 þ Ta;

U ¼ V 1 þ V 2 þ V 3 þ V s þ V 4;

ð30Þ

where T1, V1, T2, V2, T3, V3, TS, Vs, T4, V4 are the kinetic and potential energies of the crank, the second and third connecting rods, the pneumatic cylinder and the slider D, respectively. Tr is the kinetic energy of the motor shaft.

a B

X

C

Y O

Pressure-Regulating Valve

b

X

B C

Y

Pressure-Regulating Valve

Fig. 3. The geometric illustration of the pneumatic model (a) the first-stage compression motion, (b) the second-stage extension motion.

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A PM synchronous servo motor system including a gear reduction set and an output torque is applied to the intermittent slider–crank mechanism. The virtual work done by the driving torque sa applied on the crank [9] is 

_ dW A ¼ sa dh ¼ g r ðK t iq  g r Bm hÞdh;

ð31Þ



where iq is the control input, Bm is the damping coefficient in the rotation direction, and gr and Kt are the gear ratio and the motor torque constant, respectively. The virtual work can be expressed in the general coordinate as

dW A ¼ dQ T Q A1 ;

ð32Þ

where

2 6 Q A1 ¼ 4

3

0  g r K t iq

7 0 5:  g 2r Bm h_

ð33Þ

The virtual work done by all constraint forces can be expressed in terms of Lagrange multiplier k [10] as

dW C ¼ dQ T UTQ k ¼ dQ T Q C1 ;

ð34Þ

where

2 6 Q C1 ¼ UTQ k ¼ 4

ðl2 þ l3 þ xÞ cos / ðl2 þ l3 þ xÞ sin /  sin / l1 cos h

cos / l1 sin h

3 7 5k:

ð35Þ

Applying Hamilton’s principle, we have

0¼

Z

t2

½dL þ dW A þ dW C  ¼

t1

Z

t2

" dQ T

t1

oL d oL  oQ dt oQ_

!

# þ Q A1 þ Q C1 dt þ dQ T

t2 oL   ; oQ_ 

ð36Þ

t1

where L is Lagrangian function defined as the total kinetic energy minus the potential energy:

L  T  U:

ð37Þ

From Eq. (36), we obtain the Euler–Lagrange equation for the first-stage compression motion as

€ þ N1 ðQ ; Q_ Þ þ Q C ¼ B1 U þ D1 ðQ Þ; M1 ðQ ÞQ 1

ð38Þ



where U ¼ ½iq  and M1,N1, B1 and D1 are detailed in [8]. 3.2. Governing equations with the spring model In terms of the pneumatic cylinder, a spring model connecting with two collinear rods is shown in Fig. 4 and is designed for comparisons. The spring model decreases the production cost and provides the same intermittent motion for the slider– crank mechanism. By using Hamilton’s principle to formulate the equations of motion, the total kinetic energy remains the same, but the total potential energy of the spring model is modified as

U ¼ U þ V x;

ð39Þ 2 1 kðls 2

2

1 kðx2 2

2 ls Þ

where Vx is the potential energy of the spring and is equal to  x Þ in the first-stage motion,  in the secondstage motion and zero in the third-stage motion. Finally, the formulation processes for the equations of motion have the same form as Eq. (38). The detail derivations are given in [8]. 4. Decouple the differential equations For the first-stage motion, Eqs. (12) and (38) can be combined in the matrix form as

"

M

UTQ

UQ

0

#"

€ Q k

#

" ¼

BU  NðQ ; Q_ Þ þ DðQ Þ c

# :

ð40Þ

This is a system of differential–algebraic equation (DAE). Implicit function method could be employed to solve equation (40) by reordering and partitioning. In order to solve the DAE, we adopt the coordinate partitioning method [11] to partition the coordinate vector as

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X

l2 l1

l3

A θ1

θ4

x (t)

m2

m1

B

ms C

m3 Y

E D m4

O

l1

d

Fig. 4. The geometric illustration of the spring model.

Q ¼ ½ / x h T ¼ ½pT qT T ; T

ð41Þ

T

where p ¼ ½ / x  and q = [h] are the dependent and independent coordinates, respectively. According to the decomposition of Q into p and q, we have

€ þ Mpq q € þ UTp ¼ Bp U  Np þ Dp ; Mpp p € þ Mqq q € þ UTq c ¼ Bq U  Nq þ Dq ; Mqp p

ð42Þ

€ þ Uq q € ¼ c: Up p € yields Eliminating k and p

b U þ D; c b b € þ Nðq; _ ¼Q MðqÞq qÞ

ð43Þ

where 1 pq T c M ¼ Mqq  Mqp U1  Mpp U1 p Uq  Uq ðUp Þ½M p Uq 

b ¼ ½Nq  UT ðU1 ÞNp  þ ½Mqp U1  UT ðU1 ÞMpp U1 k; N q p p q p p b ¼ Bq  UT ðU1 ÞT Bp ; Q q p

b ¼ Dq  UT ðU1 ÞT Dp : D q p

Eq. (43) is a set of differential equation with only one independent generalized coordinate h in the vector q. Similarly, the dynamic equations of the second- and third-stage motions can be reduced as the same form as Eq. (43). The system becomes an initial value problem and can be integrated by using fourth order Runge–Kutta method. 5. Numerical simulations In the numerical results, the parameters of the intermittent slider–crank mechanism are chosen as follows: m1 = 1.311 kg, m2 = 0.32 kg, m3 = 0.32 kg, m4 = 3 kg, ms = 0.5 kg, l1 = 0.065 m, l2 = 0.052 m, l3 = 0.053 m, ls = 0.075 m, d = 0.16 m, e = 0.07 m, g = 9.81 m/s2, l = 0.1, k = 100N/m, gr = 1. Since the system has one degree-of-freedom, a crank angle h will correspond to a position of slider D. From the above geometric parameters, we have the critical angles h1 = 39.9°, h2 = 124.5°,h3 = 235.5° and h4 = 320.1°. By using the coordinate partitioning method and the Runge–Kutta fourth order numerical integration method, Eq. (43) is solved for the motor-mechanism coupling system. 5.1. Constant angular velocity Fig. 5a–c shows the comparisons of the displacement, velocity and acceleration of slider between the intermittent (solid _ line) and conventional slider–crank mechanism (dash line). The crank rotates with a constant angular velocity hð0Þ ¼ 4 rad=s for two cycles. The initial crank angle is h(0) = 0. In Fig. 5a, each stage motion for the first cycle is described as follows: the end of the first-stage compression motion is point a, the third-stage free-slide motion is from a to b and from c to d, and the second-stage extension motion is from b to c. It takes 0.346 s to finish the first-stage compression motion, while it takes

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R.-F. Fung et al. / Applied Mathematical Modelling 33 (2009) 2411–2420

a

b

0.26 0.24

1

0.22

a

0.3 0.2

d

0.1

0.20

. x D 0.0

3

3

x D 0.18 0.16

-0.1 2

0.14

b

0.12

-0.2

c

-0.3

0.10 0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0

0.5

1.0

Time (sec)

c

1.5

2.0

2.5

3.0

Time (sec)

d

0.8 0.4

0.10

2 0.09

0.0

.. x D -0.4

0.08

x (m) 0.07

-0.8

0.06

-1.2 -1.6 0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.05 0.0

Time (sec)

a

d 3 b

c 3

1

0.5

1.0

1.5

2.0

2.5

3.0

Time (sec)

Fig. 5. Comparisons between the intermittent and conventional slider–crank mechanisms, (a) the slider position; (b) the slider speed; (c) the slider acceleration; (d) the current length of pneumatic cylinder. ‘‘—”: intermittent slider–crank, ‘‘. . .”: conventional slider–crank, one: the first-stage motion, two: the second-stage motion, three: the third-stage motion.

0.482 s for the second-stage extension motion. In Fig. 5b and c, it is obviously seen that the transient slider speed and acceleration of the intermittent slider–crank mechanism have jump phenomena and are zero when the slider contacts with the stops. From Fig. 5d, it can be found that the deformed range of the pneumatic cylinder is 0.055 m 6 x 6 0.095 m. 5.2. Startup behavior with a separately excited DC motor The separately excited DC (SEDC) motor [12] has the advantages of low cost and easily available, and is widely used in commercial applications. The significant feature of the SEDC motor configuration is its ability to produce high starting torque at low operating speeds. It is usually chosen for traction applications where high torque at lower operation speeds is typically required. In this section, an SEDC motor is applied to investigate the startup behavior of the intermittent slider–crank mechanism. The total potential energy of the motor-mechanism system is the same as Eq. (30), while the total kinetic energy can be modified as

T ¼ T þ Ta;

ð44Þ

2 1 L i 2 a a

where T a ¼ is the magnetic co-energy function of inductance, the current is ia ¼ q_ a and qa is the charge of the armature. The virtual work is obtained as

_ dW A ¼ ðkt ia  Bm hÞdh þ ðV T  K b h_  Ra ia Þdq;

ð45Þ

where Kt and Bm are defined as the same as Eq. (31), VT is the electrical potential, kb h_ is the back emf, Raia is the circuit resistance, dh and dqa are the virtual angular displacement and virtual charge, respectively. It is noted that the charge qa is chosen as the generalized coordinate for the circuit. The governing equation of the SEDC motor-mechanism system can also be derived by using Hamilton’s principle, and can be seen in [8]. The physical properties of an SEDC motor [13] are selected as follows:

kb ¼ 0:678 V s=rad; Ra ¼ 0:4 X; K t ¼ 0:678 N m s=A; La ¼ 0:05 H; Jm ¼ 0:0565 kg m2 ; Bm ¼ 0:226 N m s=rad; ia ð0Þ ¼ 0 A; V T ¼ 60 V: The transient angular speed responses of the crank and the corresponding armature currents at startup are shown in Fig. 6a and b, respectively. Startup means that the system is at a complete standstill with zero armature currents at time t = 0 when an initial input displacement is chosen and an armature voltage is applied. In Fig. 6a, the crank angular velocity reaches its averaged speed 70 rad/s after 0.8 s. Fig. 6b illustrates the armature currents for startup. It is obviously seen that the speed curve has a significant variation and the currents lead to high initial currents corresponding to a very short startup time.

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a

100

b

80

.

70 60 50

60

θ ( rad s ) 40

I a 40 ( A) 30 20

20

10

0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1`.4 Time (sec)

0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 Time (sec)

Fig. 6. The transient responses of the intermittent slider–crank mechanism driven by the SEDC at startup. (a) The crank angular speed; (b) the corresponding armature currents.

6. Conclusions This paper presents the kinematic and dynamic analyses of the intermittent slider–crank mechanism. The connecting rod is connected with a pneumatic cylinder and a spring model. Due to the intermittent motions of the slider–crank mechanism, the analyses are divided into three stages. The governing equations of motion are formulated by Hamilton’s principle successfully. From the numerical results, some conclusions are drawn as follows: (1) The pneumatic model solutions are closer to the kinematic solutions and have faster responses than those of the spring model. (2) The spring model can be designed to decrease the production cost and to achieve the same intermittent motion. Acknowledgement Support of this work by the National Science Council of the Republic of China under Contract NSC95-2212-E327-005 is gratefully acknowledged. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]

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