mechines of mechinism chapter 1

Chapter 1: Introduction to Mechanisms

Yk

ME-309

Mechanics of Machines

I did not know it is aside-dumping car


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1.1. Course Objectives

The objective of the course ME309 is to introduce students to the concept of designing a machine that performs a desired useful function. In the basic design courses ME307 and ME308, the student was mainly concerned about design of mechanical components, provided that forces are known a priori. The focus in those courses was on component design in terms of shape, size, material, safety, etc. ME309, on the other hand introduces students to kinematic design; i.e. how to design a mechanism (an assembly of links and joints) to perform a desired motion. In this context, the student will be introduced to several types of special mechanisms, which are commonly encountered in real life. Analysis of motion of such mechanisms will be studied and accelerations will be calculated. The second part of the course focuses on the kinetics; i.e. how to determine dynamic forces acting on the mechanism. With this later task, it should be clear how ME309 complements its companions ME307 and ME308.

1.2. Mechanism or Machine?

A mechanism is a device which transforms motion to some desirable pattern and typically develops very low forces and transmits little power. A machine typically contains mechanisms which are designed to provide significant forces and transmit significant power. Some examples of common mechanisms are camera shutter, analog clock, folding chair, adjustable desk lamp, umbrella, wind-shield wiper, etc. Some examples of machines which possess motions similar to the mechanisms listed above are food blender, automobile transmission, IC engine, bulldozer, robot, etc. In fact, there is no clear-cut dividing line between mechanisms and machines, [1].

Schematic drawing of a valve-cam mechanism


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1.3. Kinematics & Kinetics for Mechanism Design

The role of kinematics is to ensure the functionality of the mechanism, while the role of dynamics is to ensure safety and acceptability of induced forces on different parts. The design process starts with meeting the functional requirements of the product. Let us consider the cam mechanism shown in Fig. 1. The basic functional requirements in this case is the proper opening, dwelling, and closing of the valve as a function of time. To achieve this objective, a corresponding cam profile producing the needed follower motion should be found. The rocker arm, being a lever, serves as a displacement amplifier/reducer. The timing of opening, dwelling, and closing is controlled by the speed of the camshaft. The function of the spring is to keep the roller always in contact with the cam. To meet this requirement the inertial forces developed during the followervalve system motion should be known, since the spring force must be larger than these forces at any time. Thus, it follows that the determination of component accelerations needed to find inertial forces is important for the choice of the proper spring stiffness.

In general, kinematic analysis allows one to satisfy the functional requirements for valve displacements. Dynamic analysis allows one to find forces in the system as a function of time. These forces are needed to continue the design process. The design process continues with meeting the constraints requirements, which in this case are [1]:

1. Dimensions of all parts; 2. Selection of materials; 3. Manufacturing; 4. Sealing between the valve and its seat; 5. Lubrication; 6. Assembly; 7. Cost; 8. Maintenance; 9. Safety, etc.;

The forces transmitted through the system during cam rotation allow one to determine the proper sizes of components, and thus to find the overall assembly dimension. The spring force affects the reliability of the valve sealing. If any of the requirements cannot be met with the given assembly design, then another set of parameters should be chosen, and the kinematic and dynamic analysis repeated for the new version. Thus, kinematic and dynamic analysis is an integral part of the machine design process, which means it uses input from this process and produces output for its continuation.


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1.4. Historical Overview

Figure 0-a: Ramses Cart

Figure 0-b: Stern-mounted steering oar of an Egyptian riverboat depicted in the Tomb of Menna (14221411 B.C.)


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Figure 0-c: Water extraction system with lever and counterweight (named as Shadoof)


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1- Archimedes:

Archimedes (287-212 BC) was born in Syracuse, in the Greek colony of Sicily. Archimedes went to Alexandria about 250-240 BC to study in the Museum under Conon of Samos, a mathematician and astronomer (the guardian of the Alexandrian library after Euclids death). In addition to formalizing the science of geometry and his many contributions in mathematics, Archimedes systematized the design of simple machines and the study of their functions and developed a rigorous theory of levers and the kinematics of the screw, the hydraulic clock, the catapult.

During his time in Egypt, he invented a hand-cranked manual pump, known as Archimedes screw (Figure 1), that is still used in many parts of the world. He used the screw principle to improve on the shaduf and other rudimentary pumping devices. Its open structure is capable of lifting fluids even if they contain large amounts of debris.

Archimedes was also known for his huge catapult; the largest stone-thrower on record, a 78-kg machine, Figure 2.

A good deal of Archimedes work survived only in Arabic translations of the Greek originals, and was not translated into Latin until 1543, [2].

Figure 1: Archimedes screw Figure 2: Archimedes catapult


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2- Heron of Alexandria: Heron of Alexandria was a mathematician, physicist and engineer who lived around 1085 AD. He taught at Alexandrias Museum and wrote many books on Mathematics, Geometry and Engineering, which were in use till the medieval times.

Heron wrote important three books on mechanics that describe simple mechanical machines and methods for lifting weights. Among his contributions to mechanisms: the pantograph, the pulley, wedges, sledges, cranes and the screw (incl. the worm gear), and their use in lifting heavy objects. Most of Herons work survived only in Arabic translations of the Greek originals, see Figure 3 for the worm gear train.

Figure 3: Endless screw (worm gear) and driven gear (worm), [2]. Inscriptions in Arabic.

The figure is used to calculate the velocity ratio of the gears.

Figure 4: Force multiplication Figure 5: Pulley system. Figure 6: Double-screw press


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3- Al-Jazari:

Ismail Al-Jazari (11361206) lived in the region in northern Syria and Iraq between the two rivers Tigre and Euphrates. He is an Arab inventor and one of the most famous mechanical engineer of the pre-medieval era, who is remembered for his design of water-raising machines and many unusual clocks and automata. His book entitled: Book of Knowledge of Ingenious Mechanical Devices presents a whole range of devices and machines, with a multiplicity of purposes. From all the presented machines, two are most remarkable: his famous elephant clock, which was by far the most sophisticated clock at that time and his invention of a water pump using a crank-slider-like system, which was the first known machine to use a crank.

Figure 7: Water raising machine Figure 8: Reciprocating Pump (fourth design) (Topkapi Palace Museum, Turkey)


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Figure 9: Candle Clock Figure 10: Elephant Clock (Smithsonian Institute) (NY Metropolitan Museum)

4- Leonardo da Vinci: Leonardo da Vinci (14521519) was an Italian polymath, displaying skills in numerous diverse areas of study. He is regarded as the genius of his time. Leonardo da Vinci is celebrated as a painter, sculptor, musician, writer, architect, engineer and anatomist. Most enduring to his legacy are his inventions and designs. Machines in Motion presents 40 full-scale machines that were built after in-depth study of Leonardo da Vincis designs by a group of scientists and skilled craftsmen in Florence, Italy.

Most enduring to his legacy are his inventions and designs. Machines in Motion Museum presents 40 full-scale machines that were built after in-depth study of Leonardo da Vincis designs by a group of scientists and skilled craftsmen in Florence, Italy. Its uniqueness lies in the fact that many of the mechanisms are life-sized and fully operational. Visitors may touch and set them in motion, combining a fascinating hands-on experience with an exploration of the principles he employed to create each machine.


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The machines are grouped into four sections based upon applications linked to the elements - air, water, land and fire - which held a strong fascination for Leonardo da Vinci. Exhibits include such visionary inventions as the helicopter and glider, the armored tank, the drive transmission, the printing press and the bicycle. Most of the materials used in the construction of these machines were available in Leonardo da Vincis era, and are the ones proposed by him in his codices, including wood, rope and glue. The materials were crafted by hand; using tools prescribed by the master himself.

Figure 11: Samples of assembled Leonardo da Vinci machines


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5- James Watt: James Watt was born on the 19th of January 1736 near the port of Greenock, west of the City of Glasgow, England. From a young age Watt showed signs of the chronic ill-health that was going to torment him through the greater part of his life.

Although Watt had no formal study of mechanisms he became a highly gifted designer of mechanisms. The windmill flyball governor for regulating the gap between millstones was adapted by Watt as an engine speed regulator giving the first closed-loop servomechanism.

Figure 12: Watts double acting steam engine

Watt considered several alternative devices for the conversion of reciprocating motion to rotating motion in the steam engine. See Figure 12 for the genius Watts parallel motion mechanism.

Figure 13: Watts parallel motion mechanism


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1.5. Basic Definitions

S LINK: A rigid body formed to represent a mechanical component.

Binary Link: one with two joint connections

Ternary Link: one with three joint connections

Quaternary Link: one with four joint connections

S JOINTS: (Kinematic Pair), connectors that join two different links and allow certain relative motions. These are classified in two types:

a) Lower-Pair Joints: (Nature of contact is surface contact) ; e.g. sleeves, journal bearings, hinges, screws & nuts.

b) Higher-Pair Joints: (Nature of contact is either line or point contact) ; e.g. spur gears (line contact), helical gears (point contact), ball bearings (point contact), cam-follower contact.


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S KINEMATIC CHAIN: An assembly of interconnected links; and is classifies as either closed or open kinematic chain:

a) Closed Kinematic Chain: Each link is connected to at least two other links.

b) Open Kinematic Chain: Any kinetic chain that does not satisfy condition (a).

S MECHANISIM: A constrained kinematic chain.


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S LINKAGE: A class of mechanisms consisting only of lower-pair joints.

S KINEMATIC DIAGRAM: A representation of the mechanical components (links) in skeleton form so that only dimensions that affect their motions are considered.


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Links &Joints presented in skeleton form:


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Mechanisms presented in skeleton form:


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The skeleton is composed of 206 bones. It is divided into two parts: the axial skeleton and appendicular skeleton. The numbers in parentheses indicate the number of bones of a certain type (or in a certain subgroup). The names of the major bones of the skeleton are identified in the figure. Approximately 700 muscles pull on different parts of the skeleton.


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Kinematical structure of the human body proposed by Morecki [3].


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Robots & Robotic Manipulators


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INDUSTRIAL ROBOTS

PUMA 560


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1.6. Types of 2-dimensional Joints:

Let us first define Degrees of Freedom (DOF).

DOF: is the number of independent motions the body can execute. Alternatively, it is equal to the number of independent coordinates required to uniquely describe the motion of the body.

We know that the general planar motion of a rigid body can be described by three coordinates; these are the x and y coordinates of the center of mass, and the body rotation , as shown in Figure 14.

Figure 14: DOF of a rigid body in general planar motion

a) Revolute Joint: (RJ): also called hinge, pin joint

Figure 15: Revolute Joint

Now, consider link j as a reference. If we fix link j, then link i can only execute rotational motion relative to link j due to the presence of the revolute joint. But any unconstrained link has three DOF. In other words, we can say that the RJ eliminates two DOF (i.e. imposes two constraints); thus leaving only ONE relative DOF. Accordingly, RJ is classified as a one-DOF joint.

j i

x

y


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b) Prismatic Joint: (PJ): also called sliding or translating joint

Figure 16: Prismatic Joint

Consider link j as a reference. If we fix link j, then link i can only execute sliding motion relative to link j due to the presence of the prismatic joint. Similarly, we can say that the PJ eliminates two DOF (i.e. imposes two constraints); thus leaving only ONE relative DOF. Accordingly, PJ is classified as a one-DOF joint.

c) Rolling-Contact Joint: (RCJ):

A rolling contact joint preserves the rolling without slipping contact relationship. That is. As shown in the Figure below

Consider link j as a reference. If we fix link j, then link i can execute both translation and rotational motions relative to link j , however due to the condition of rolling without slipping, the two motions are related (note that s R = ). Accordingly, the rolling contact joint allow only one independent degrees of freedom and therefore RCJ is classified as a one-DOF joint.

s

j

i

j

i


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d) Sliding Contact Joint: (SCJ): also called cam joint, Pin-in-Slot, sliding contact joint or half-joint

Figure 16: (a) Pin-in-Slot Joint (b) Cam Joint

Consider link j as a reference. If we fix link j, then link i can execute both rotational and translational motion relative to link j due to the presence of the SCJ joint. Now, we can say that the RC eliminates one DOF (i.e. imposes one constraint); thus leaving TWO relative DOF. Accordingly, SCJ is classified as a two-DOF joint.

REMARK:

Kinematic equivalence between sliding-contact joint and rolling-contact joint arrangements. Not that the following two configurations are kinematically equivalent.

This kinematic equivalence will be clarified after introducing mobility of mechanisms in Section 1.7.

2

1 1

2

3

RCJ SCJ

j

i


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EXAMPLE 1.1

To demonstrate how joints eliminate DOF, let us consider the following example:


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1.7. Mobility of a Mechanism: M The mobility of a mechanism is its number of degrees of freedom . This translates into a number of independent input motions the mechanism can execute. This very crucial to know prior to the selection or design of a mechanism to perform a certain function, as the mobility implies the number of motors required to drive the mechanism.

In 2-dimensional analysis, the mobility of a given mechanism is equal the sum of DOF of all links comprising the mechanism, minus sum of all the constraints imposed by different joints including the fixed frame. This can be written as M = SUM (all DOF) SUM (all Constraints), or alternatively in a more detailed form as

1 23( 1) 2n J J= M (1) where n total number of links of the mechanism

1J Number of single-DOF joints 2J Number of two-DOF joints

EXAMPLE 1.2 Determine the mobility of the following mechanisms: SOLUTION

(a) 3(3 1) 2 3 0 0= =M (b) 3(4 1) 2 4 0 1= =M (c) 3(4 1) 2 4 0 1= =M (d) 3(5 1) 2 5 0 2= =M


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Before we move on calculating mobility, we need to define the order of a joint. This characteristic is commonly associated with the revolute joint (RJ). Recall the definition of a joint that we stated earlier, which defines joint as a connection between two different links. There are situations where more than two links are connected by a RJ; a case in which the stated definition of a joint is not applicable. Consider the assembly of three links in Figure 17-a. If one applies the definition of a joint each pair of the assembly, then one can arrive at two RJs; one connecting links 1 & 2, and another connecting links 2 & 3. This in turns guarantee connection between 1 & 3.

Figure 17-a: Order of RJ

As a RULE; we can state the following:

The order of a revolute joint connecting ( n ) links is equal to ( 1n ). For the sake of mobility calculations, let us substitute the phrase number of revolute joints for order of revolute joint. That is, if we have ( n ) links hinged by one pin, then they are effectively connected by ( 1n ) revolute joints.

To explain further, consider the mechanism shown in Figure 17-b, in which the four links AD, CD, ED and FD are hinged by one pin at D, then the number of revolute joints connecting these four links at D is given by (4-1) = 3 RJ.

Figure 17-b: Number of RJ

Let us demonstrate this procedure through the following examples.


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EXAMPLE 1.3 Determine the mobility of the following mechanisms:

SOLUTION

(a) 3(4 1) 2 4 0 1= =M

(b) 3(5 1) 2 6 0 0= =M

(c) 3(6 1) 2 8 0 1= = M

When we added one more link to the four-bar linkage in Part (b), the mobility becomes equal to zero, which means that the mechanism cannot move. According, this is not a mechanism anymore, and instead it is called structure.

In Part (c), we further added on more link, and the mobility became negative. What is the meaning of a negative mobility? Recall static analysis, when you studied over-constrained structures, you encountered what is called statically-indeterminate structures. In fact, this is it. The structure in Part (c) is statically-indeterminate with one redundant constraint; i.e. the number next to the minus sign indicates the degree of redundancy in the statically-indeterminate structure.

Accordingly, it is important to note that the mobility equation is applicable to mechanisms and structures; although the result is used to indicate different intrinsic features in each case.


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EXAMPLE 1.4 Determine the mobility of the following mechanisms:

(a) (b)

SOLUTION

(b) 3(8 1) 2 10 0 1= =M (c) 3(6 1) 2 7 1 0= =M

EXAMPLE 1.5 Determine the mobility of the following mechanism:

2


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SOLUTION

Note that the mechanism is driven by the hydraulic cylinder (HC), which is labeled as links 2 & 3 in the schematic diagram on the right. In this kinematic diagram, the HC is modeled as two links connected from the two ends by two revolute joints and the other two ends by one prismatic joint.

Let us first calculate the mobility of the HC alone, as a free body. This gives 3(2 0) 2 3 0 0HC = =M

This may suggest that the HC has no effect on the mobility calculation of the whole mechanism. To verify this, let us do the mobility calculations twice; once with HC and another without including the HC. HC included:

3(6 1) 2 7 0 1= =M

HC ignored: 3(4 1) 2 4 0 1= =M

As a RULE; we can state the following:

Force elements; e.g. hydraulic cylinders, springs, etc. do not contribute to the mobility calculations in mechanisms, and can be ignored in mobility calculations.

1.8. The Planar Four-Bar Linkage:

Why the four-bar linkage is of particular interest?

Figure 18: Basic Configuration of four-bar linkage

RJ

RJ

PJ


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The 4-bar linkage is considered as the basic building block of many complex and higher-order mechanisms. In fact, it exists everywhere to the extent that almost any machine may have at least one 4-bar mechanism. Even in daily life objects, 4-bar linkages are there in the folding chair, folding table, door-closer, etc.

In this section, we will show how to perform kinematic design a four-bar linkage. In this context, the following questions may need to be answered:

1. How can we choose dimensions of different links? 2. Which link can be chosen as the fixed frame? 3. Which link(s) will be able of making full revolution (continuous rotation)? 4. How the motion changes as we alternate the fixed link? 5. If the mechanism is to be cranked by a continuous rotation motor, which link is the best choice

for a crank? The answers to all of the posed questions will be answered by Grashofs Theory. GRASHOFs Theory: For a planar four-bar linkage, the sum of the shortest and longest links

cannot be greater than the sum of the other two links, if there is to be continuous relative rotation between two links.

Figure 19: The four-bar linkage

Referring to the 4-bar linkage in Figure 19, one can write Grashofs theory in symbolic form as

l s p q+ + (2) Classification of the Grashofs four-bar linkages:

Now, assume we assembled a four-bar linkage that obeys Grashofs theory. That is, we guarantee that at least one link will execute continuous rotation. Let us alternate the fixed link, and see how this can affect the relative motion between different links. Accordingly, we expect four possibilities, as we fix each of the four links, one at a time.

l

s

p

q


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Fix the link adjacent to the shortest: This results in the two possibilities (a) and (b), as shown in Figure 20.

Figure 20: Crank-Rocker Mechanism

In this case, the shortest link will execute continuous rotation, while the other two links will execute rocking motion. The resulting mechanism is called Crank-Rocker Mechanism.

Fix the shortest: This results in the configuration shown in Figure 21.

Figure 21: Drag-Link Mechanism

In this case, all moving links will execute continuous rotation. The resulting mechanism is called Drag-Link Mechanism.


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Fix the link opposite to the shortest: This results in the configuration shown in Figure 22.

Figure 22: Double-Rocker Mechanism

In this case, the shortest link will execute continuous rotation, while the other two links will execute rocking motion. The resulting mechanism is called Double-Rocker Mechanism. Note that a four-bar linkage that do not satisfy Grashofs criterion is called double-rocker of the second kind or triple-rocker. Change-Point or Crossover mechanism (folding mechanism: Now, consider the critical

case of Grashofs theory; i.e.

l s p q+ = + (3)

Figure 23: (a) Parallelogram, (b)Anti-Parallelogram; (c) Deltoid or kite Mechanism

Note that if l s p q+ > + , the mechanism is non-Grashof and non of the links will execute continuous motion.

See animations in my WebCT

(a)

(b)

(c)


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The Transmission Angle:

Figure 24: Transmission angle of a four-bar linkage

Another useful test that can be very quickly applied to a linkage design to judge its quality is the measurement of its transmission angle . This can be done analytically or graphically. The transmission angle is defined as the angle between the output link and the coupler. It is usually taken as the absolute value and varies continuously from some minimum to some maximum value as the linkage goes through its range of motion. It is a measure of the quality of force and velocity transmission at the joint.

As shown in the figure, the useful component of the force 34F transmitted from the coupler CD to the output link 4DO is simply 34 sinF . That is, the larger the the larger the transmitted force. But since takes a range of values, it may reach very small values.

Figure 25: Transmission Angles

Notice that is max when dL is max, and visa versa.

oL

1L

2L

3LdL


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Figure 26: Max & Min Transmission Angles

As a design guideline for a good design; the range of is recommended to be within the following range:

45 135 (4) Referring to Fig. 25, and using the cosine-rule, an expression of can be derived as follows:

2 2 22 3 2 32 cosdL L L L L = + (5)

Or,

2 2 22 3

2 3cos

2dL L L

L L + = (6)

where 2 2 2

0 1 12 cosd oL L L L L = + (7) Discuss the following:

Methods of design refinements to improve the transmission angle.

Problems associated with driving the change-point mechanism.

B

A

B

A

B

A max min


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1.9. Some Special Mechanisms:

(a) The Slider-Crank Mechanism

Figure 27: The slider-crank mechanism


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Figure 28: The slider-crank mechanism & its inversions

(b) Quick-Return Mechanisms These are mechanisms, which take shorter time in the return stroke within one cycle of the crank rotation. In this context, one needs to define the forward and return strokes of a mechanism. Let us consider the slider-crank mechanism shown in Fig. 28.a, which is called in-line slider-crank mechanism.

Figure 29: In-line slider-crank mechanism

B B B A A

A

R

F S


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where the symbols in Figure 29 are defined as:

F angle of forward stroke R angle of return stroke

S stroke; S = 2r where r is the length of the crank Now, let us introduce a parameter called Time Ratio (TM) :

TMTime of Forward stroke

Time of Backward stroke= (8)

For a mechanism driven by a constant velocity motor; as given by the crank rotation , and knowing that for constant velocity we have t = , then eq. (8) becomes

TM FR

= (9)

If TM > 1 , then the mechanism is called a Quick-Return Mechanism. For the in-line slider crank mechanism shown in Fig. 29; TM = 1. Can we modify the kinematic design of the slider-crank mechanism to achieve a quick-return motion.

Figure 30: Offset slider-crank mechanism

In this case, the time ratio is given by TM 1FR

= > , which shows that the offset slider-crank can be

operated as a quick return mechanism

Exercise: Show that the stroke in the case of the offset slider-crank is S > 2r . (see proof in page ----)

S

e

B B B A

A

A R

F

O


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The Crank-Shaper Mechanism:

Figure 31: Shaper Machine and its Crank-shaper Quick-return mechanism

Figure 32: Crank-shaper Quick-return mechanism

S

R

F B B B A

A A

2O

1O

C


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In this case, the time ratio is obviously greater than one, and is given by TM ( / ) 1F R = > . Referring to Fig. 32, it is noted that link 2O A lies at the same angle at the two extreme positions

2O Aand 2O A ; i.e. the distance A A is equal to B B . Now, the stroke can be expressed as 2 22 sin(90 / 2) 2 cos( / 2)R RS B B A A L L = = = = (10) Design considerations: What are the proper link dimensions to design a slider-crank mechanism as a quick-return mechanism to satisfy a desired stroke while maintaining smooth accelerations?

In this case we need to choose the dimensions 2L and 3L , in addition to the value of the offset 1L , as shown in Fig. 33. Apparently, for smooth motion one needs 3 2L L>> . Figure 34 shows a simulation of the links acceleration for different length ratios different length ratios

Figure 33: Offset slider-crank mechanism

Figure 34: Acceleration Profile

As a RULE of thumb, choose dimensions such that 3 24L L :

S


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EXAMPLE 1.6

An adjustable slider drive mechanism consists of a crank-slider with an adjustable pivot, which can be moved up and down, as shown.

c) How many bodies (links) can be identified in this mechanism?

d) Identify the type (and corresponding number) of all kinematic joints.

e) What is the function of this mechanism and how will it be affected by moving the pivot point up and down?

SOLUTION

Refer to classroom discussions.

EXAMPLE 1.7

Determine the number of fully rotating links in the planar mechanism shown.

SOLUTION

Check Grashofs condition: l s p q+ < + , which is satisfied. Now choose l as the frame, results in Double-Rocker with the shortest making full revolution.

Choose p or q as the frame, results in Crank-Rocker with the shortest as the crank which makes full revolution.

Now choose s as the frame, results in Drag link with all links making full revolution.


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EXAMPLE 1.8

Determine 3L for 0 120L = , 1 220L = and 2 80L = , such that the mechanism performs as:

a) Crank-rocker b) Drag-link c) Double-rocker d) Change-point mechanism

SOLUTION

a) Crank-rocker (i.e. 3L is shortest) Check Grashofs formula: 3220 120 80L+ +; ; i.e. impossible

b) Drag-link (i.e. 0L is shortest; impossible )

c) Double-rocker (i.e. 2L is shortest ) Now, we have two possibilities; (i). 3L is longest

3 380 120 220 260L L+ + (ii). 3L is intermediate

3 3220 80 120 180L L+ + i.e. 3180 260L

d) Change-Point (i.e. l s p q+ = + ) Two possibilities; (i). 3L is longest

3 380 120 220 260L L+ = + = (ii). 3L is intermediate

3 3220 80 120 180L L+ = + =

2L

0L 3L 1

L


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REFERENCES

[1] Oleg Vinogradov, FUNDAMENTALS of KINEMATICS & DYNAMICS of MACHINES and MECHANISMS, CRC Press 2000.

[2] Marco Ceccarelli, HISTORY OFMECHANISM ANDMACHINE SCIENCE, Volume 1, Springer 2007.

[3] Marco Ceccarelli, HISTORY OFMECHANISM ANDMACHINE SCIENCE, Volume 2 1, Springer 2010.

[4] R.L. Norton, DESIGN OF MACHINERY AN INTRODUCTION TO THE SYNTHESIS AND ANALYSIS OF MECHANISMS AND MACHINES, 2nd Edition, McGraw-Hill 1999.

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mechines of mechinism chapter 1

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