ME306-Fall 2013- Chapter (2)- Mechanisms and machines.pdf

Chapter (2): Mechanisms and Machines ME306 - Fall 2013

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Chapter (2)

Mechanisms and Machines


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Chapter (2)

Mechanisms and Machines ( والماكينات األليات )

1. Introduction (مقدمة)

A mechanism is a combination of rigid parts or components arranged or connected in a

specific order to produce a required motion. Figure (1) shows a mechanism which is known

as slider-crank mechanics. The slider-crank mechanism converts the reciprocating motion of

the slider into a rotary motion of the crank or vice-versa.

Figure (1): Slider-crank Mechanism 1.1 Element or Kinematic Link ( الحركية الوصلة )

Each part of a machine that moves relative to another part is known as a kinematic link or

element (or a simple link). The link may consist of several parts which are rigidly joined

together so that they do not move relative to one another. For example, the slider-crank

mechanism shown in Figure (1) consists of four links: (1) frame, (2) crank shaft, (3)

connecting rod and (4) piston or slider.

Links can be classified into binary links ( الثنائية الوصالت ), ternary link ( الثالثية الوصالت ) and

quaternary links ( الرباعية الوصالت ), etc., depending upon their ends on which revolute or

turning pairs can be placed. The links illustrated in Figure (2) are rigid links and there is no

relative motion between the joints within the link.

Figure (2): Kinematic links

(a) Binary link (b) Ternary link (c) Quarternary link


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1.2 Rigid and Resistant Bodies ( والصلبة المقاومة األجسام )

A link or element needs not to be a rigid body, but it must be a resistant body. A body is said

to be a resistant body if it is capable of transmitting the required forces with negligible

deformation. Then the link should have the following two characteristics:

(i) The link should have relative motion, and

(ii) The link must be a resistant body.

1.3 Types of Links ( الوصالت أنواع )

(1) Rigid link ( الصلبة الوصلة ) does not undergo any deformation while transmitting

motion. For example: crank shaft in slider-crank mechanism (as shown in the figure).

(2) Flexible link ( المرنة الوصلة ) is partly deformed in a manner not to affect the

transmission of motion. For example: belts, chains and ropes are flexible links and

transmit tensile forces.

(3) Fluid link ( السائل وصلة ) is formed by having a fluid in a receptacle and the motion is

transmitted through the fluid by pressure or compression. For example: jacks and

brakes.

1.4 Kinematic Pair ( الحركية زوج )

The two links or elements of a machine (when in contact with each other) are said to form a

pair. If the relative motion between these two links is completely constrained, the pair is

known as a kinematic pair. For example, in the slider-crank mechanism shown in Figure (3),

link 2 (crank) rotates relative to link 1 (frame) and then links 1 and 2 is a kinematics pair.

Similarly link 2 is having relative motion to link 3 and then links 2 and 3 is also a kinematics

pair.

Figure (3): Slider-crank mechanism


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1.5 Classifications of Kinematics pairs (تصنيفات أزواج الحركيات)

Kinematics pairs can be classified according to:

1.5.1 Nature of contact between links

The kinematic pairs according to the nature of contact are classified as:

(1) Lower pair: A kinematic pair is known as lower pair if the two links has surface or

area contacts between them. Also the contact surfaces of the two links are similar.

Examples of lower pairs shown in Figure (4) are (i) shaft rotating in a bearing, (ii) nut

turning on a screw, and (iii) sliding pairs.

Figure (4): Lower kinematics pairs

(2) Higher Pair: If the two links (or a pair) has a point or line of contact between them,

then the kinematic pair is known as higher pair. Also the contact surfaces of the two

links are not similar. Examples for higher pairs are: (i) cam and follower, (ii) wheel

rolling on a surface, and (iii) meshing gear-teeth.

Figure (5): Higher pairs

(i) Shaft rotating in a bearing

(ii) Nut turning on a screw

(iii) Sliding pair

Cam-follower mechanism Meshing spur gears

Wheel rolling on a surface


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1.5.2 Relative motion between links

The kinematics pairs (Fig. 6) can be classified according to the following considerations:

1. Sliding pair: When one link slides relative to another link, it is known as sliding pair.

2. Turning pair: When one link turns or revolves relative to another link, it is known as

turning pair

3. Rolling pair: When one link rolls over the other pair, it is known as rolling pair.

4. Screw pair: If two pairs have turning as well as rolling motion between them, it is

known as screw pair.

5. Spherical pair: When a spherical link turns inside a fixed link, it is known as spherical

pair.

Figure (6)

1.5.3 Mechanical constraint between links

Based on the nature of mechanical constrained, the pairs are classified as:

Sliding pair: Piston and cylinder

Turning pair: Cycle wheels turning over their axles Rolling pair: Ball bearing

Screw pair: Bolt with nut

Spherical pair: Car mirror


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(1) Closed pairs: When the links (or elements) of the pair are held together

mechanically. All lower kinematic pairs are closed pairs, for example; sliding

pairs, spherical pairs, turning pairs and screw pairs.

(2) Unclosed pairs: If the two links of the pair are not held mechanically but are

held in contact by the action of external forces. A cam-follower mechanism which

held in contact due to spring force is an example for unclosed pair.

1.6 Kinematic Chain (السلسلة الحركية)

The combination of links and pairs without a fixed link is not a mechanism but a kinematic

chain. Figure (7) shows the kinematic chains in car engine and bike.

Figure (7): Kinematic chains in car engine and bike.

In order to check whether the mechanism of links having lower pairs forms a kinematic

chain or not, the following two equations are used:

2 4L p , and

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2J L

in which L = number of links, p = Number of pairs, and 2 3 42 3 4

2

n n nJ

is the total

number of binary joints, where, 2n = number of binary links, 3n = number of ternary links,

and 4n = number of quaternary links.

The above equations are applied only for kinematics chains having lower pairs but if these

equations are applied for the kinematic chains having higher pairs, then each higher pair

must be taken equivalent to two lower pairs plus one additional link.


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In the above two equations,

(i) If LHS > RHS then the chain is locked (مؤمنة).

(ii) If LHS = RHS then the chain is constrained (مقيدة).

(iii) If LHS < RHS then the chain is unconstrained (غير مقيدة).

Example: Determine whether each of the following mechanisms is a kinematic chain or not.

Check that the mechanism is locked, constrained or unconstrained.

(a) For three-bar links: 3L , 3p , 3j , then:

Equation 1: 3LHS , 2 4 2(3) 4 2RHS p , The left hand side (LHS) > right hand side

(RHS).

Equation 2: 3LHS , 3 3

2 3 2 2.52 2

RHS L , LHS > RHS

Since the three links mechanism does not satisfy the two equations, then, the mechanism is

not a kinematic chain, and as LHS > RHS, the mechanism is locked chain.

(b) For four-bar links: 4L , 4p , 4j :

Equation 1: 4LHS , 2 4 2(4) 4 4RHS p , LHS=RHS.

Equation 2: 4LHS , 3 3

2 4 2 42 2

RHS L , LHS = RHS

Then, both equations are satisfied, thus the four-links mechanism is a constrained kinematic

chain.

(a) Three-bar links (b) Four-bar chain

(c) Five-bar chain


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1.7 Degrees of Freedom (Mobility) of a Mechanism

The number of degrees of freedom (or mobility) of the mechanisms is the first important

consideration in the design or analysis of the mechanisms. The degrees of freedom (DOF) is

defined as the number of independent translational and rotational motions the body can

have. The degrees of freedom for some mechanisms are shown (Fig. 8) and given as:

(1) A rigid body in space has six degrees of freedom,

(2) A rectangular bar sliding in a rectangular hole has one degree of freedom, and

(3) A ball-socket joint has three degrees of freedom.

Figure (8): Degrees of freedom for some mechanisms

2.8 Degrees of Freedom for Planar Mechanism

The degrees of freedom for planar mechanism can be determined using the Grubler’s

criterion, i.e.,

1 23 1 2FD L p p

where FD is the degrees of freedom (DOF) of the mechanism,

L is the total number of links in the mechanism,

1p is the number of lower pairs, and

2p is the number of higher pairs.

Note 1: For a kinematic chain consists of different types of links, then the number of lower

pairs is determined as follows:

1 2 3 4

12 3 4 .....

2p L L L

where 2L is the number of binary links, 3L is the number of ternary links, and so on.

Note 2: when the joint has k links at a single joint, it must be counted as 1k joints.

S

3 translations + 3 rotationsthen DOF = 6

TranslationS

Rotation

θ

DOF = 1 (S or θ)

S

Translation

DOF = 1 (Translation S)


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Note 3:

(1) If 0FD the device is a structure (i.e., there is no movability of the device).

(2) If 1FD the device is a mechanism with DF degrees of freedom.

The number degrees of freedom (DF) of a mechanism in terms of number of links (L)

and number of joints (J) can be obtained by:

3 1 2FD L J

where L is the number of links in the mechanism, J is the number of simple joints

(having two links).

Example: Find the number of degrees of freedom for the mechanisms shown in Figure

(9).

Figure (9)

Solution:

(a) Total number of links L=5, number of lower pairs p1=6, number of higher pairs p2=0

1 23 1 2 3 5 1 2(6) 0FD L p p

(b) Total number of links L=4, number of lower pairs p1=4, number of higher pairs p2=0

1 23 1 2 3 4 1 2(4) 1FD L p p

Or (another solution based on number of links and joints):

Number of links L=4, number of simple joints j=4

3 1 2 3 4 1 2(4) 1FD L j

(c) Total number of links L=4, number of lower pairs p1=3, number of higher pairs p2=1

1 23 1 2 3 4 1 2(3) 1 2FD L p p

(e)

(b)

2

3

4 5

1

2

3

4

2

1

1

3

4

(a) (b)


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Examples 1– Kinematic Chains

Find the type of chain for the mechanisms shown in Figure (1).

(a)

(a) L=5, p=5, J=5

Equation: L= 2p-4

LHS=5, RHS=2p-4=2(5)-4=6, LHS < RHS.

Then, the mechanism is not a kinematic chain but it is an

unconstrained chain.

(b)

(b) L=4, p=4, J=3, (four lower pairs having surface contacts

between links 1-2 , 2-3, 3-4 and 1-4 ).

Eqn. 1: LHS=4, RHS=2p-4=2(4)-4=4, LHS=RHS.

Eqn. 2: LHS=3, RHS=(3L/2)-2= (34/2)-2=4 , LHS=RHS

Then, the mechanism is a constrained kinematic chain.

(c)

(c) L=4, p=4, (two lower pairs (turning pairs) having

surface contacts between links 1-2 and 1-3 and one higher

pair (rolling pair) having a line contact between links 2-3).

Eqn. 1: LHS=4, RHS=2p-4=2(4)-4=4, LHS=RHS.

Then, the mechanism is a constrained kinematic chain.

(d)

(d) L=5, p=5 (three lower pairs having surface contacts

between links 1-2, 2-3 and 3-4 and one higher pair (rolling

pair) having a line contact between links 1-4).

Eqn. 1: LHS=5, RHS=2p-4=2(5)-4=6, LHS<RHS.

Then, the mechanism is an unconstrained chain.

(e) L=6, (four binary links 2, 3, 5 and 6 and two ternary

links 1 and 4). The number of joints is obtained as:

2 3 42 3 4 02 3 4

72 2

n n nJ

Eqn. 2: LHS=7, RHS=(3L/2)-2= (36/2)-2=7 , LHS=RHS

Then, the mechanism is locked kinematic chain. (e)


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Examples 2– Degrees of Freedom for Mechanisms

(a)

(a) Total number of links (N) =number of binary links (n2)+

number of ternary links (n3)=6+1=7

Number of joints (J) =

2 3 42 3 4 02 3 4

82 2

n n nJ

DF= 3(N-1)-2J = 3(7-1)- 2(8)= 2

Then, the mechanism shown has two degrees of freedom.

(b)

(b) The mechanism has a higher pair (rolling pair).

Total number of links (N) =4

Number of lower pairs (p1) = 3

Number of higher pairs (p2) = 1

DF= 3(N-1)-2 p1 –p2= 3(4-1)- 2(3)-1= 2

DF= 3(N-1)-2J = 3(7-1)- 2(8)= 2


1

2

3 4

5

(c)

(d) Total number of links (N) =6

Number of binary joints (n2) = 3 (at B, C and E)

Number of ternary joints (n3) = 2 (at A and D)

Total number of joints (J)= 3+ 2(2)=7

DF= 3(N-1)-2 J= 3(6-1)- 2(7)= 1

Then, the mechanism shown has one degree of freedom.

1

2

3 4

5

(d)

6

A

B

C

D

E

(c) Total number of links (N) =5

Number of lower pairs (p1) = 5

Number of higher pairs (p2) = 0

DF= 3(N-1)-2 p1 –p2= 3(5-1)- 2(5)= 2

DF= 3(N-1)-2J = 3(7-1)- 2(8)= 2


(e) Number of binary links (n2) = 4

Number of ternary links (n3) = 2

Total number of links (N)= n2 +n3 = 4+ 2=6

Number of joints 2 3

2 32 37

2 2

n nJ

DF= 3(N-1)-2 J= 3(6-1)- 2(7)= 1

Then, the mechanism has one degree of freedom.

(e)


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Assignment (1)

Q1: Find the degrees of freedom for the shown mechanisms.

Q2: For the given linkages, determine the mobility (degrees of freedom) for each linkage.

Ans. of Q2: (a) N=5, n2=3, n3=2, J= 6, DF=0.

(b) DF=1, (c) DF=1, (d) DF=1, (e) DF=1, (f) DF=1.

ME306-Fall 2013- Chapter (2)- Mechanisms and machines.pdf

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