KINEMATICS - DESIGN OF MECHANISMS: INTRODUCTIONKinematics

The study of Kinematics of mechanisms and the machines, which are composed of one or more mechanisms, involves analysis of geometry of motion. Different components of any mechanism move relative to the each other following certain constraints to produce the desired motion. Kinematic analysis is of prime importance in design of mechanisms and machines.

For kinematic design of a mechanism analysis is done for positions of points on a solid body and the time derivatives of the position. The first derivative of position with respect to time is velocity, the second derivative is acceleration and further derivatives can be analyzed according to the design requirements. Similarly for angular position there is angular velocity and angular acceleration.

Mechanisms

The simplest example for a mechanism will be a liver hinged at a wedge. It transfers input motion at one end to the output motion on the other end. A scissors is a combination of two livers; the mechanical work from one end can be transformed to cutting motion on the output end. The two livers in scissors are connected together by a joint (revolute joint). A slightly more complex mechanism is a slider crank mechanism.

Thus mechanisms can be defined as assembly of rigid members connected to each other through joints. A mechanism transfers the input motion or work at the input point or point of actuation to one or more output points. Like in case of slider crank mechanism, the input rotational motion of the crank is transferred to the slider as a reciprocating motion.

simple Mechanisms

Kinematic Joints

The members in a mechanism are connected by kinematic joints. A kinematic joint is formed by direct contact between the surfaces of the members forming that joint. The contact between the surfaces of the members can be point contact, line contact

or area contact. The joints are classified according to the type of contact and relative motion of the members. The contact stresses developed will also depend on the contact type.

There are two types of joints according to the type of contact:

1. Lower pair joint

2. Higher pair joint

Lower pair joint has area contact between the two mating surfaces of the members forming joint, as in the case for slider, revolute and hinge.

Higher pair joint has the contact between the mating surfaces as point or line contact as in the case for cam pair and cam-follower.

Following sections will discuss basics of analysis and synthesis of mechanisms

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Kinematics – Design of Machines: Analysis and Synthesis:

The design of mechanisms has two aspects, analysis and synthesis of mechanisms.

Analysis of Mechanisms:

This is consisted of techniques of determining the positions, velocities and accelerations of certain points on the members of mechanisms. The angular positions, velocities and accelerations of the members of mechanisms are also determined during analysis of mechanisms. By analysis of mechanisms the trajectory of particular points and the orientation of the members at particular points of time are obtained.

Synthesis of Mechanisms:

If the desired set of positions/angular positions, velocities/angular velocities and acceleration/angular acceleration at definite points of time are stipulated. Then the synthesis of mechanisms comprises of mathematically determining the geometry of members of mechanisms such as to produce the desired results. When that mechanism is operated it will pass through the stipulated points with the required velocity and acceleration, and the members will have the desired orientation.

Synthesis of mechanisms as per the requirement can be achieved through two ways. First, Rational Synthesis, which consists of standard synthesis techniques developed by kinematicians. Being systematic these techniques can be automated using computer programs. Limitation of rational synthesis technique is that it is applicable only to some specific types of mechanisms.

Second technique commonly used by design engineers is Informal Synthesis. This design procedure involves first a guess of dimensions of members of mechanisms and then checking the resultant performance by analysis. The dimensions are modified based on previous performance and adjusted such that to obtain results close to desired. In this way the process of iterative synthesis and analysis is repeated to obtain acceptable design.

Forces in Kinematics:

Although kinematics does not have forces or their analysis in its purview, but velocity profile of mechanisms have symmetry with the force profile. Thus, the construction for analysis of geometry of motion (kinematic analysis) can be appropriately extended to static and dynamic force analysis (Kinetics) of mechanisms.

Slider Crank Analysis and Synthesis

To have a clear distinction between the analysis and synthesis procedure let us take a concise of slider crank. Slider crank mechanism has a rotating crank, a reciprocating slider and a coupler connecting crank to the slider.

Analysis of slider crank mechanism: it will involve finding the frequency of oscillation, speed at different positions and range of motion of the slider for given angular velocity of the crank and given lengths of crank and the coupler.

Synthesis of slider crank mechanism: One or more data about the frequency of oscillation, speed at different positions and range of motion of the slider may be provided and the requisite task will be finding the dimensions of the members of the mechanism and position of the slider such as to have the desired motion of the mechanism.

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KINEMATICS – DESIGN OF MACHINES: TYPES OF KINEMATIC JOINTS

Classification of Kinematic Joints

Basically the Kinematic Joints are classified into two categories based on the type of contact between the two members making a joint. It can be point, line or area contact.

1. Lower pair joint.

2. Higher pair joint.

But a third category of kinematic joint can also be created which is comprised of the joints formed by combination of two or more lower and/or higher pair joints. Such joints are termed as Compound Joints.

3. Compound Joints.

Lower Pair Joints

The two members forming a lower pair joint have area contact between the two mating surfaces. The contact stress is thus small for lower pair joint as compared to higher pair joints. Lower pair joints have long service life as the wear and stress is spread over larger surface area of contact and also allows for better lubrication. The degrees of freedom for a lower pair of joint is usually less as the requirement for area contact between the members constrains the geometry of the joint.

Examples of Lower Pair Joints

1. Revolute/Hinge Joint, 1 DoF

2. Prismatic/Slider Joint, 1 DoF

3. Srew/Helical Joint, 1 DoF

4. Cylindrical Joint, 2 DoF

5. Spherical/Ball Joint, 3 DoF

6. Planar Joint, 3 DoF

Higher Pair Joints

The contact between the two members of higher pair has point or line geometry. The contact stress for a higher pair joint is large because of very small contact area. If there is pure rolling contact between the members then at any point of time the contact point or line is at rest. There is no relative sliding between the contact surfaces and thus friction and wear will be negligible. The degrees of freedom for a higher pair of joint can be high as the point or line contact allows for less constrained motion of members.

Examples of Higher Pair Joints

1. Cylindrical roller, 1 DoF

2. Cam pair 2 DoF

Compound Joints

Lower pair and/or higher pair joints are combined as per the design requirement to obtain compound joints. Compound joints composed of higher pair joints can be kinematically equivalent to lower pair joints or vice verse. By such combinations desirable features from the combining joints are retained to obtain robust joints.

Examples of Compound Joints

Ball or Roller Bearings: The actual members in contact are balls or rollers with the inner and outer race. These are rolling contact which is a higher pair. But the overall joint has the motion geometry of revolute joint, a lower pair. A ball bearing has low friction properties of rolling contacts and higher load capacity of revolute joints. Ball or Roller Bearings are kinematically equivalent to simple revolute joint.

Universal of Hooke Joint: It is a combination of revolute joints and has two degrees of freedom.

KINEMATICS – DESIGN OF MECHANISMS: DEGREES OF FREEDOM

KINEMATIC DEFINITION

Degrees of freedom (DoF) is related to the motion possibilities of rigid bodies. Kinematic definition for DoF of any system or its components would be “the number of independent variables or coordinates required to ascertain the position of the system or its components”.

The concept of DoF in kinematics of machines is used in three ways. DoF of

1. A body relative to a reference frame.

2. A kinematic joint.

3. A mechanism.

DETERMINING DEGREES OF FREEDOM

Degrees of Freedom can be determined by analysis of motion of the concerned body or by determining the number of coordinates required to specify position of the body. In this article planar cases are considered which can be extended to spatial cases.

1. DEGREES OF FREEDOM OF A BODY RELATIVE TO A SPECIFIED REFERENCE FRAME

In a plane the position of a body relative to a reference frame can be specified by two position coordinates (say X and Y) and one coordinate (say theta)for specifying the orientation of the body. Total three coordinates are required to specify the position of the body if there are no constraints applied. The DoF will reduce as the motion of the body is restricted.

For example, a body is not allowed to move along one axis in the plane. As a result one DoF if lost thus leaving only two DoF.

2. DEGREES OF FREEDOM OF A KINEMATIC JOINT

Two bodies connect with each other to form a joint. One body can move in a number of ways relative to the other and may be constrained in other ways. DoF of a kinematic joint is number of ways in which one member of the joint can move relative to the other member.

For example, revolute joint has one DoF as one member can move only in one way relative to the other member. It can only rotate about the axis of the joint. Prismatic joint also has only one DoF as one of the two members can slide along the other in one direction only.

Cylindrical joint has two DoF as one of the two members can rotate about the axis of the joint and can also translate along it. Two motions possible so two DoF.

3. DEGREES OF FREEDOM OF A MECHANISM

The DoF for a mechanism is defined as the number of coordinates or variables required to be specified such that the position and orientation of all the members of the mechanism can be stated as a function of time.

For determining the DoF for a mechanism we will start with assuming all the members of the mechanism free in plane and thus having three DoF each. Then we will apply constraints and DoF will reduce as the members are joined together to form mechanism.

Take the mechanism to be composed of ‘n’ members or links. Initially each link is assumed to be free and thus the mechanism has 3n DoF. One of the members is to be a base or frame link thus have zero DoF or it lost its all three DoF. The DoF left in the mechanism at this stage is 3n-3 or 3(n-1).

When the pairs of links form joints they will loose DoF. If the formed joint have 'Fi' DoF each then reduction in DoF is (3-Fi) as they were initially free (having 3 DoF). If there are 'j' number of joints then total reduction in DoF will be summation of (3-Fi) over 'j' number of joints . The net DoF for a mechanism

can be given by

KINEMATICS – DESIGN OF MECHANISMS: KINEMATIC INVERSION

KINEMATIC INVERSION

Every mechanism has moving members which move relative to each other about the joints connecting them. These relative motions result in the trajectories of the points on members of the mechanism. In any mechanism one link or member is fixed and acts as the frame. The trajectories and motion characteristics of mechanism depend on the choice of the reference frame link.

Inversions of a mechanism are the different configurations of the mechanism with change of the fixed reference link called frame. For different inversions of a mechanism although the motion characteristics are entirely different but the relative angular displacements of the members remain unchanged irrespective of the link chosen as frame.

The information obtained from one inversion of the linkage can be used to study other inversions of that linkage. Inversion technique is used extensively for analysis and synthesis of mechanisms.

DETERMINING THE INVERSIONS OF A MECHANISM

Before going into details of obtaining inversions of a mechanism I would like to make it very clear that Inverse Kinematics is different from Kinematic Inversion. Read more about Inverse Kinematics.

Every mechanism is formed of a kinematic chain. When one of the links in the kinematic chain is fixed it becomes a mechanism. To determine the inversions of a mechanism consider the kinematic chain forming the mechanism and obtain the desired inversions by fixing any one of the members as the frame link.

INVERSIONS OF A FOUR-BAR MECHANISM

A typical four bar mechanism, as the name denotes, is formed of a kinematic chain of four members connected by revolute joints. This mechanism can have four possible configurations with a different link fixed as frame each time.

Configuration 1

Link 1 is taken as the base link or frame. In this configuration the shortest link is jointed to the base link and this joint can fully rotate and hence called as crank. The other link jointed to the base link oscillates and called as a rocker. This configuration of the four-bar kinematic chain is called as Crank-Rocker mechanism.

Configuration 2

Link 2 is fixed as the base link. In this configuration shortest link is the base and both joints to the base can rotate completely. It is thus called as Double-Crank or a Drag-Link.

Configuration 3

Link 3 is fixed as the base link. It can be observed that this configuration is same as the Crank-Rocker mechanism.

Configuration 4

Link 4 is fixed as the base link. In this configuration shortest link is the coupler and both the links connected to the base link cannot rotate fully, both oscillate. In this configuration the four-bar kinematic chain is called as Double-Rocker mechanism.

KINEMATICS – ACTUATION OF MACHINES: PART I – ELECTRICAL ACTUATION

ACTUATION OF MACHINES

A machine is an integration of different mechanisms designed and assembled such that they can perform required work. The linkages and mechanisms inside the machine transfer the input mechanical work from the input point or generating site to the point where a useful effect is produced. Actuation means providing that required mechanical work for input. An actuator is a device, mechanical or electrical, which generates the required mechanical work in controlled manner.

Machines for printing and packaging, which perform the same task repeatedly at very high production rates, are generally powered by single motor called a prime mover and all the sub-functions are carried out by various linkages. This prime mover can be a large electric motor, an internal combustion engine, a steam engine or a turbine.

Advancements in actuator technology have resulted in a number of compact and precisely controllable actuating devices. The most commonly used actuators are:

Electrical Actuators Hydraulic Actuators Pneumatic Actuators

ELECTRICAL ACTUATORS

Today a wide variety of electrical actuators are available using new advanced power switching technologies which have resulted in greatly enhanced performance. Commonly used electrical actuators:

Electric motors: Electric motors have a rotating shaft which produces angular rotation and torque. Electric motors can operate at very high speeds which can be reduced to get proportional increase in torque by means of a mechanical gear assembly or transmission. There are basically three types of electric motors.

1. DC Motors or Commuted Motors.

They operate on Direct Current and have commutators to switch the current carrying coils so that the rotation of the shaft is in same direction.

2. AC Motors or Non-commuted Motors.

They operate on Alternating Current and do not need commutators for operation.

3. Stepper or Stepping Motors.

These motors have equally spaced discrete movements called as steps. Stepper motors are turned by feeding voltage pulses to it which can be controlled by stepper motor controller to determine the step size or amount of rotation for each pulse.

Electric motors when used as actuators can be assisted by a feedback controller to keep track of the position and make necessary corrections in input accordingly.

Solenoid: Solenoid has a cylindrical winding and a ferromagnetic core in and along the axis of this cylindrical winding. When this winding is supplied with direct current the ferromagnetic core is pulled inside the cylindrical winding. Thus, a solenoid is a two-state actuator.

ELECTRICAL ACTUATOR CONTROL

Electrical Actuators can be controlled to get optimum performance by using solid-state power-switching devices. Earlier methods for control of DC motors had inherent energy loss as they use potentiometers. Modern control devices such as pulse-width modulators and phase-controlled rectifiers can control DC motors without energy losses. The new control devices has made actuators more compact but at the same time the control unit may even cost more than the actuator itself.

KINEMATICS – ACTUATION OF MACHINES: PART II – HYDRAULIC ACTUATION

ACTUATION BY FLUID

Fluids can store energy in form of pressure, transfer energy in form of fluid flow and can transform the stored energy (in form of pressure) into motion. These properties of fluids are harnessed by designing the actuators based on fluid power. The actuators using liquid as the actuating fluid are termed as hydraulic actuators and actuators using gases (commonly compressed air) are called as pneumatic actuators.

HYDRAULIC ACTUATION

Hydraulic Actuators uses flowing liquid to transmit the energy from generation point to actuation point. The required fluid power is generated in the form of high pressure liquid by a pump, which can be driven by electric motor as in the case of industrial robots and by engines directly for heavy machinery used in oil well drilling or construction operations.

The pressure generated by the hydraulic pump is distributed to different actuators through control valves according to their requirement, which is proportional to the amount of load needed to be supported by them. A different configuration for hydraulic actuator is having a separate variable displacement pump for each actuator. Some commonly used hydraulic actuators are

Hydraulic Cylinder

It is a linear actuator, that is, it can generate linear motion when powered by pressurized hydraulic fluid. Hydraulic cylinders are used when loads are large and strokes are long. They are generally used to drive the linkages in heavy engineering machines such as excavators, heavy duty cranes and JCB machines.

Hydraulic cylinder is basically a piston-cylinder arrangement with a piston closely fitting inserted in the cylinder. The piston can move in and out of the cylinder driven by the hydraulic fluid .

Hydraulic Motor

As the electric motors are similar in construction to those of electric generators, hydraulic motors are very similar in construction to hydraulic pumps. The functioning of an electric motor is just opposite to that of an electric generator. A generator converts rotation into electricity and a motor uses electricity to produce angular velocity and torque.

In the same way the working of a hydraulic motor is opposite of a hydraulic pump. A hydraulic pump produces fluid pressure and flow using rotation of shaft where as a hydraulic motor uses the fluid pressure and flow to produce torque and rotation of shaft.

A hydraulic actuation system is composed of hydraulic motors and pumps coupled in a circuit. Pumps generate the required fluid pressure and flow which is used by motors installed at different points for actuation.

Hydraulic Brake

For a brake it is generally convenient to have the point of application or actuation point away from the point of control. Hydraulic brake assembly has a cylinder filled with hydraulic fluid (called brake fluid), hydraulic pipe, brake pedal and the mechanical brake. Brake is applied by pressing the brake pedal which pressurizes the hydraulic fluid, the fluid is forced through the pipes and force is applied at the mechanical brake.

KINEMATICS – ACTUATION OF MACHINES: PART III – PNEUMATIC ACTUATION

PNEUMATIC ACTUATION

Pneumatic Actuation system transforms the energy stored in compressed gases to mechanical energy which is further used to move parts of mechanisms. The motion achieved can be either linear or rotary. The source of power in any pneumatic actuation system is the compressed gas. The energy is stored in the gases, generally air, by compression using compressors or pumps.

The advantage of pneumatic actuation system over hydraulic actuation system is the freely available working fluid for pneumatic actuators, that is air. Whereas the working fluid for hydraulic system, the hydraulic oil is very costly and requires major effort to check its leakage and need to replenished as and when required.

TYPES OF PNEUMATIC ACTUATORS

A typical Pneumatic Actuator has set up for supply of compressed gas, pipes for flow of compressed gas and valves to control the flow of gas. The requirement of any system and the convenience of use dictates which actuation system to be used. Based on application and design constraints pneumatic actuators are designed in different configurations.

Types of pneumatic actuators commonly used in industries are linear motion actuators, rotary motion actuators, grippers and special purpose pneumatic vacuum grippers for lifting smooth objects. A different type of pneumatic actuator whose functionality resembles the human muscle is also used for controlled motion of mechanisms or structures. It is called as Pneumatic Artificial Muscle (PAM).

LINEAR ACTUATOR

Linear pneumatic actuators are very similar to linear hydraulic actuators in construction but there is a basic difference in their operation. Like the linear hydraulic actuator its counterpart pneumatic actuator is an assembly of piston, cylinder and valves to control the actuator.

In the hydraulically operated actuator the hydraulic fluid is pumped in the cylinder to move the piston out and fluid is sucked or released in the hydraulic circuit itself to retract the piston, so as to keep the fluid in the circuit without any wastage of the hydraulic fluid. Whereas in the pneumatic actuator the moving out process is same but for moving the piston inside the pressure is released by letting the compressed air out of the cylinder by operating a release valve.

ROTARY PNEUMATIC ACTUATOR

Rotary pneumatic actuators converts the energy of pressurized flowing gases into the rotatory motion, similarly as a gas turbine. In this instead of compressed fluid being pressed against the piston walls, the gases flow past the actuator and generates the motion. Rotary Pneumatic actuators are generally fitted with rack and pinion to have high torque output.

PNEUMATIC GRIPPERS

Pneumatic gripper is an assembly of pneumatic actuators producing simultaneous motion to give a gripping action. They are used to ensure secure gripping while lifting heavy objects and are fitted with proximity switches to monitor open and closed position of the gripping jaws.

PNEUMATIC VACUUM GRIPPERS

Vacuum grippers are used lift objects with smooth surfaces which cannot be handled otherwise due to absence of any ends to pick the objects. Vacuum grippers work by creating a low pressure at the actuator end by venturi action generated by pressurized gases moving at fast speed past the nozzle. The actuator end is brought close to the object to be picked, the objects sticks to the end which can be released where required by stopping the gas flow. Advance vacuum grippers can pick multiple objects even with rough surfaces.

PNEUMATIC ARTIFICIAL MUSCLE (PAM)

PAMs work similar to the animal muscles, that is, they contract and expand when given the corresponding signal. Like other pneumatic actuators PAMs are operated by pressurized air. Between the two ends of the PAM there is a balloon type

construction, when inflated makes the two ends move closer thus resulting in compression of the PAM. As PAMs can only apply force while contracting, they are used in antagonistic arrangement where the two muscles in the pair can apply force in opposite directions. Spring loaded PAMs can work in single also, they will extend when air pressure is released.

KINEMATICS - ANALYSIS OF MECHANISMS: METHODS AND TECHNIQUES

ANALYSIS OF MECHANISMS

Analysis of mechanisms is the study of motion of different members constituting a mechanism and the mechanism as a whole entity while it is being operated or run. This study of motion involves linear as well as angular position, velocity and acceleration of different points on members of mechanisms. Analysis and synthesis are two different aspects of mechanisms and machine design .

Earlier design engineers used drafting equipments to graphically analyse the mechanisms. The continuous contribution by design engineers for years has lead to development of many methods and techniques for analysis of mechanisms. Recently, the development of computer techniques have offered a number of viable and attractive solutions.

METHODS AND TECHNIQUES OF MECHANISM ANALYSIS

Mechanism analysis methods are basically of two types, graphical and analytical. Each method has many techniques for analysis of mechanisms, where each technique is suitable for a particular category of mechanisms. With the development of sophisticated computer programs design engineers prefer to concentrate their effort on analytical approach. But still the graphical approach to mechanism analysis has not lost its utility, specially in some cases where graphical technique gives the most efficient solution and physical insight to visualize working of the mechanism.

Graphical Method of Mechanism Analysis

Graphical method starts with position analysis by simply drawing the linkage mechanism to scale. Then the velocity analysis is performed which requires the angular position of the links to be determined beforehand. Similarly it is necessary to know angular velocities of links for acceleration analysis. Thus, the sequence for

kinematic analysis of mechanisms is - position analysis, then velocity analysis and then acceleration analysis.

Different Techniques of Graphical Analysis

1. VELOCITY AND ACCELERATION POLYGON: Velocity and acceleration are vectors and thus their sum or difference will follow vector polygon laws. If velocity of one point on a link is known then the velocity of other points can be found using the vector polygons. This technique is based on vector polygon laws.

2. VELOCITY AND ACCELERATION IMAGE: This technique is used for graphical analysis of mechanisms with more than one loop. If the velocity and acceleration of two points on a link are known then the velocity and acceleration of third point on that link can be determined using velocity and acceleration image.

3. INVERSION TECHNIQUE: When it is not possible to analyse the linkage directly using vector polygon approach then Inversion Technique is used. In this technique the driven and driver cranks are interchanged to perform graphical analysis.

4. RELATIVE VELOCITY AND ACCELERATION: This technique is used to analyse mechanisms with large number of members. In this technique the relationships between relative linear/angular velocities and acceleration of points/members are used to analyse the mechanisms.

5. INSTANT CENTER OF VELOCITY: For a rigid body moving in a plane, at every instant there exists a point that is instantaneously at rest. This instant center of velocity for the given rigid body is found using standard methods. It is useful for finding input-output velocity relationships of complex mechanisms.

Analytical Method of Mechanism Analysis

Analytical method is used when repetitive and extensive analysis of mechanisms is required, as the analytical equations and solutions obtained can be conveniently programmed on a computer. In this approach vector position, velocity and acceleration equations are formulated based on the fact that there are two different paths connecting the points on a vector loop. The equations thus obtained are simplified and programmed using computers. Desirable solutions are obtained by varying the parameters

KINEMATICS - SYNTHESIS OF MECHANISMS: METHODS AND TECHNIQUES

SYNTHESIS OF MECHANISMS

The motion to be generated by machines are generally irregular, any motion except uniform rotation about a fixed axis and uniform translation. The machine designer's task is to design such mechanisms which can generate these required irregular motions. The simplest way to design such mechanisms is by clever combination and assembly of cams and/or linkages. Thus, basically there are two types of motion generators:

1. Cams2. Linkages

Each of the cam and linkage has their own advantages and disadvantages. Cams are easy to design but cams are difficult and expensive to manufacture. Cams

generally have continuous point or line contact and thus wear out early. On the other hand, linkages are difficult to design but linkages are less expensive and easy to manufacture and also linkage mechanisms are more reliable.

Manufacturing of machine parts is more expensive process than designing or synthesis, so to save on costs one can put more effort in synthesis of mechanisms. From this view point linkages have an advantage over cams to be prefered for machine design.

LINKAGE MECHANISM SYNTHESIS

The most widely used mechanisms are four link mechanisms. The four link mechanisms are simplest mechanisms capable of performing most desired functions. The design techniques used for four link mechanisms can be extended to be used for design of five and six link mechanisms. In this article series emphasis will be on four link mechanism synthesis.

The members in linkage mechanisms are connected through joints having surface contacts. Surface contact in joints provide good lubrication and wear resistance. Revolute joints and prismatic joints are the only two kinematic joints available to be used in linkage mechanisms. A four link mechanism has four joints and only two types of joints can be used, it makes only four possibilities for types of four link mechanisms with joints having surface contact.

TYPES OF FOUR LINK MECHANISMS

1. The Four Bar Linkage Mechanism: This mechanism has all the four joints as revolute joints. The inversions of four bar linkage mechanisms are also four bar linkage mechanisms.

2. The Slider Crank Mechanism: This mechanism is used when either linear input is provided or a linear output is required. The slider crank mechanism is generally used to obtain linear oscillatory motion from rotary motion and vice versa. The inversions of slider crank mechanism also come under this same classification.

3. The Elliptical Trammel Linkage: This mechanism has two revolute joints and two prismatic joints on same links. The name Elliptical Trammel is given to this mechanism as the path of all the points on the coupler are ellipses.

4. The Rapson Slide Linkage: This mechanism also has two revolute joints and two prismatic joints but they are not on same links, each link has one revolute joint and one prismatic joint. Its inversions are also Rapson slide linkages.

KINEMATICS - FOUR BAR LINKAGE SYNTHESIS

FOUR BAR LINKAGES ARE MOST BASIC AND MOST WIDELY USED MECHANISMS FOR MACHINE DESIGN. FOUR BAR LINKAGES CAN PROVIDE SIMPLE SOLUTIONS TO THE COMPLEX MOTION GENERATION PROBLEMS WITH RELATIVELY LESS EFFORT. THESE MECHANISMS ARE RELIABLE AND AT THE SAME TIME EASY TO MANUFACTURE.

FOUR BAR LINKAGE SYNTHESIS

Four bar linkages are most preferred machine components as they have a large number of dimensions to be varied which allows for more flexibility in design and these dimensions can be varied to fit the design constraints of machines. But with flexibility in design comes the complexity, this results in complicated design techniques. The design techniques become simpler if one or more slider joints are included in the mechanisms.

The exact desired motion is very rare to be produced by using four bar linkages. By using four bar linkages synthesis techniques we can obtain approximate desired motions. With increase in the level of accuracy required for the desired motion, the complexity of computation increases greatly.

APPROACHES TO FOUR BAR LINKAGE SYNTHESIS

Mechanisms are required to follow the specified path and pass through the desired points as closely as possible. For some mechanisms it is more desirable that they should pass through the specified points and for some other mechanisms following the path is more important. There are two approaches to four bar linkage synthesis.

1. Precision Position Approach: In this approach the position through which the mechanism is desired to pass are selected and in the solution mechanism is compelled to exactly pass through these positions. In this approach it is difficult to control the path of mechanism between the specified points. The precision position approach generally employ graphical methods of synthesis. If the design positions are more than three than the solutions become complex and computer program is used for synthesis.

2. Path Optimization Approach: In this approach a large number of design positions are selected and the overall deviation of mechanism from these design points is minimized. For this approach numerical optimization techniques are employed using computers.

FOUR BAR LINKAGE SYNTHESIS PROBLEMS

There are infinite synthesis problems. The common classes of problems with practical importance are:

1. The Double Rocker Problem: It is desired to design a four bar linkage such that if the input link moves through certain angle the output link should move through a specified angle.

2. The Motion Generation Problem: For this problem the motion of coupler is specified and a linkage mechanism is to be synthesized such that it's coupler has the desired motion.

3. The Function Generation Problem: The mechanism is to be designed such that the two cranks follow a required functional relationship, that is, for a set of angles of one crank the other crank should move to the angles specified in the other set.

4. The Rocker Amplitude Problem: In this case a crank-rocker linkage is to be designed such that for the continuous rotation of the driving crank the output link oscillates through a specified angular amplitude.

5. The Point Path Problem: A four bar linkage is to be synthesized such that a point on the coupler follows a specified path.

KINEMATICS - SPECIAL MECHANISMS: STRAIGHT LINE MECHANISMS

THERE ARE SOME UNUSUAL MECHANISMS WHICH CAN MEET COMMON NEEDS OF MECHANICAL ENGINEERING PROBLEMS. THESE SPECIAL MECHANISMS CAN SERVE A RANGE OF PURPOSES LIKE GENERATING STRAIGHT LINES, TRANSFERRING TORQUE BETWEEN NON-COAXIAL SHAFTS, SELF CENTERING STEERING AND MECHANICAL PUNCHED CARD READERS

SPECIAL MECHANISMS

Some mechanisms have special motion characteristics different from those of generic mechanisms. These mechanisms are used for special purposes and few particular categories of motion. These mechanisms are unusual enough to be called as Special Mechanisms. Some common needs of mechanical engineering practice are:

Generation of a straight line motion by linkage mechanism. Reproduction of a path traced by one point at another tracing point with a change in scale. Transfer of torque and motion between non-coaxial shafts with changing relative alignment. Automotive steering mechanisms and suspension mechanisms. Indexing: Intermittent timed motion.

STRAIGHT LINE MECHANISMS

Generation of straight line motion using linkage mechanisms has always been a common requirement in machine design practice. Although exact straight line cannot be generated using simple mechanisms though some simple mechanisms are designed such that they can produce approximate straight lines for short range of motion. These approximate straight line mechanisms has important applications in machine design. These mechanisms were used extensively in classical machines such as steam engines. Perfect straight lines can also be generated using linkage mechanisms but those are relatively complex mechanisms.

There are two classes of straight line mechanisms:

1. Approximate Straight Line Mechanisms2. Exact Straight Line Mechanisms

The straight line mechanisms were mostly developed in industrial revolution days when many machines required straight line paths in their operations, whether it was guiding the piston of engines or for operating valves. Straight line mechanisms were developed by continuous effort in trail and error process with making intelligent variations in linkage mechanisms.

APPROXIMATE STRAIGHT LINE MECHANISMS

Watt's Straight Line Mechanism

Approximate straight line mechanisms can generate straight line motion to a good deal of accuracy for short range. Such mechanisms are generally four bar linkage

mechanisms. The straight line mechanism developed by James Watt, to guide the piston of steam engines through a straight line path, is considered to be as the best and simplest mechanism able to generate close to straight line motion for considerable distance. This mechanism is called as Watt's straight line mechanism or simply Watt's Linkage.

Watt's linkage is a simple four bar mechanism of double-rocker type with the two rockers connected through a coupler. When the two rockers move the mid-point of the coupler moves in an almost straight line path for the motion close to coupler's mean position. If something is hinged to the middle point of the coupler of Watt's linkage it will be constrained to move in straight line path close to the coupler's mean position.

This property of Watt's linkage is utilized in construction of rear axle suspension system of car to prevent sideways motion of car body relative to the rear axle.

KINEMATICS - SPECIAL MECHANISMS: STRAIGHT LINE MECHANISMS - IIIN CONTINUANCE TO THE LAST ARTICLE, WHICH GAVE INTRODUCTION ABOUT SOME SPECIAL MECHANISMS AND BRIEFLY EXPLAINED WATT'S STRAIGHT LINE MECHANISM, IN THIS ARTICLE FEW MORE APPROXIMATE STRAIGHT LINE MECHANISMS WILL BE DISCUSSED. THESE ARE DIFFERENT CONFIGURATIONS OF FOUR BAR LINKAGES.

CHEBYSHEV'S STRAIGHT LINE MECHANISM

The Chebyshev approximate straight line mechanism is also a four bar linkage mechanism that is both historically important and also of practical importance. After the invention of steam engine and straight line mechanism by Watt a range of straight line mechanisms were designed. Chebyshev's mechanism is the first mechanism to be designed after Watt's linkage by a Russian Mathematician Pafnuty

Chebyshev. This mechanism was invariable used for linear guidance of the piston and valves.

Like the Watt's Linkage Chebyshev's straight line mechanism is simple in construction. It is a double rocker and the mid point of the coupler is the point tracing the approximate linear path. Chebyshev's mechanism has two critical advantages over Watt's linkage, viz, a very long segment of the path of the coupler midpoint is approximately linear and both fixed points of the linkage are on the same side of the linear path, which in case of Watt's linkage are on opposite sides. The required proportions of the length of members of the linkage are a = 1, b = 2.5 and c = 2.

ROBERT'S STRAIGHT LINE MECHANISM

Like the Chebyshev's mechanism Robert's approximate straight line mechanism is a symmetrical four bar linkage. The construction of Robert's mechanism is different from the approximate straight line mechanisms discussed so far, in the sense that, this mechanism has an extension to the coupler at the coupler mid point. this extension is perpendicular to the line joining the two adjacent joints. The end point of the coupler extension generates an approximate straight line for the motion between the fixed pivots.

This mechanism designed by Richard Robert has the proportions of lengths of members of the linkage as, a = 1, b = 1.2, c = 2 and d = 1.09. This approximate straight line mechanism is generally used for linear guidance of the tracing point. The point required to traverse on straight line is constrained to the end point of the coupler extension. Robert's straight line mechanism is normally used in the coupler driven mode, that is, the mechanism is not driven by either of the cranks or rockers instead the coupler extension is used to just guide the requisite point along an approximate straight line.

EXACT STRAIGHT LINE MECHANISMS

DRAWING A STRAIGHT LINE LOOKS TO BE A VERY EASY TASK BY HANDS BUT TO MAKE A MACHINE SUCH THAT IT CAN GENERATE AN EXACT STRAIGHT LINE REQUIRES EXPERTISE IN ANALYSIS AND SYNTHESIS OF MECHANISMS. EXACT STRAIGHT LINE GENERATING MECHANISMS ARE MUCH MORE COMPLEX THAN APPROXIMATE STRAIGHT LINE GENERATORS.EXACT STRAIGHT LINE MECHANISMS

As detailed in the previous articles there are many four bar linkage based mechanisms which can generate straight lines. These mechanisms are simple linkage mechanisms with revolute joints, but they can only generate approximate straight lines and that too only for short lengths. In certain design requirements such as design of production machinery it is desirable to have more accurate straight line paths or sometimes it becomes inevitable to have exact straight line trajectories of mechanisms.

Perfect straight lines can also be generated using a linkage mechanism. When linkage mechanisms are designed to generate exact straight lines the level of complexity increases as compared to the mechanisms designed to generate approximate straight line paths. The first exact straight line generating mechanism was invented by a French army officer Charles Nicolas Peaucellier in 1864. This mechanism is called as Peaucellier Exact Straight Line Mechanism and commonly more as Peaucellier's linkage. There are many mechanisms based on slider crank linkage which can generate exact straight lines for limited intervals.

PEAUCELLIER EXACT STRAIGHT LINE MECHANISM

Peaucellier linkage can convert an input circular motion to the exact straight line motion. The construction of this mechanism is such that the point which is connected to the crank moves in a circular path and the point traversing the straight line is selected as the output point. The linkage has a rhombic loop formed of the equal lenght members, 5, 6, 7 and 8. Two equal length length links are connected to the opposite corners of the rhombus at one end and to a common fixed point at the other ends. The point A of the rhombus is connect to fixed point O2 through the link 2. The length of the link 2 is equal to the distance between points O2 and O4. By the constraints of the geometry point A moves in a circular path and as the point A moves in a circle point P traverses an exact straight line path normal to the line joining O2 and O4.

From the construction of the Peaucellier linkage it is clear that this is a much more complex mechanism than the mechanisms generating approximate straight lines, which were simple four bar linkages. This mechanism has eight members and six joints.

SCOTT-RUSSELL EXACT STRAIGHT LINE MECHANISM

The complexity of the mechanisms to generate exact straight lines can be reduced by introduction of one or more slider crank linkages. It is possible to generate an exact straight line using the slider crank mechanism but the range of motion is limited. One such example is Scott-Russell Mechanism as shown in the figure. Based on the geometry of the linkage the output motion is a simple sine function of the drive link or a simple harmonic motion. It is evident from the figure that this mechanism is made up of isosceles triangles, AB, AC and AO2 are of equal lengths.

KINEMATIC INVERSION

Every mechanism has moving members which move relative to each other about the joints connecting them. These relative motions result in the trajectories of the points on members of the mechanism. In any mechanism one link or member is fixed and acts as the frame. The trajectories and motion characteristics of mechanism depend on the choice of the reference frame link.

Inversions of a mechanism are the different configurations of the mechanism with change of the fixed reference link called frame. For different inversions of a mechanism although the motion characteristics are entirely different but the relative angular displacements of the members remain unchanged irrespective of the link chosen as frame.

The information obtained from one inversion of the linkage can be used to study other inversions of that linkage. Inversion technique is used extensively for analysis and synthesis of mechanisms.

DETERMINING THE INVERSIONS OF A MECHANISM

Before going into details of obtaining inversions of a mechanism I would like to make it very clear that Inverse Kinematics is different from Kinematic Inversion. Read more about Inverse Kinematics.

Every mechanism is formed of a kinematic chain. When one of the links in the kinematic chain is fixed it becomes a mechanism. To determine the inversions of a mechanism consider the kinematic chain forming the mechanism and obtain the desired inversions by fixing any one of the members as the frame link.

INVERSIONS OF A FOUR-BAR MECHANISM

A typical four bar mechanism, as the name denotes, is formed of a kinematic chain of four members connected by revolute joints. This mechanism can have four possible configurations with a different link fixed as frame each time.

Configuration 1

Link 1 is taken as the base link or frame. In this configuration the shortest link is jointed to the base link and this joint can fully rotate and hence called as crank. The other link jointed to the base link oscillates and called as a rocker. This configuration of the four-bar kinematic chain is called as Crank-Rocker mechanism.

Configuration 2

Link 2 is fixed as the base link. In this configuration shortest link is the base and both joints to the base can rotate completely. It is thus called as Double-Crank or a Drag-Link.

Configuration 3

Link 3 is fixed as the base link. It can be observed that this configuration is same as the Crank-Rocker mechanism.

Configuration 4

Link 4 is fixed as the base link. In this configuration shortest link is the coupler and both the links connected to the base link cannot rotate fully, both oscillate. In this configuration the four-bar kinematic chain is called as Double-Rocker mechanism.